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Tensor product

In mathematics, the tensor product of two vector spaces V and W over a field K is defined as a vector space V \otimes_K W together with a bilinear map \iota: V \times W \to V \otimes_K W that satisfies the following universal property: for any vector space Z over K and any bilinear map \phi: V \times W \to Z, there exists a unique linear map \tilde{\phi}: V \otimes_K W \to Z such that the diagram commutes, i.e., \tilde{\phi} \circ \iota = \phi.^[1]^[2] This property characterizes the tensor product up to unique isomorphism and makes it the "universal" object for linearizing bilinear forms.^[3] One standard construction of V \otimes_K W proceeds by taking the free vector space on the set V \times W and quotienting by the subspace generated by the relations enforcing bilinearity, such as (v_1 + v_2, w) - (v_1, w) - (v_2, w), (v, w_1 + w_2) - (v, w_1) - (v, w_2), and (\lambda v, w) - \lambda (v, w), (v, \lambda w) - \lambda (v, w) for \lambda \in K.^[3] If \{e_i\} and \{f_j\} are bases for V and W, respectively, then \{e_i \otimes f_j\} forms a basis for V \otimes_K W, so \dim_K(V \otimes_K W) = (\dim_K V) \cdot (\dim_K W).^[1] Elements of V \otimes_K W are finite sums of simple tensors v \otimes w, though not all elements are simple in general.^[2] The tensor product extends naturally to modules over a commutative ring R, where for R-modules M and N, M \otimes_R N is an R-module generated by symbols m \otimes n subject to analogous distributive and scalar relations, again characterized by the universal property for R-bilinear maps.^[4] It is associative up to isomorphism, (M \otimes_R N) \otimes_R P \cong M \otimes_R (N \otimes_R P), and commutative, M \otimes_R N \cong N \otimes_R M.^[3] Examples include the tensor product of scalars (yielding the field itself) and of a vector space with its dual, which recovers the space of linear endomorphisms.^[1] Historically, the concept arose in the late 19th century through work on multilinear forms in physics and geometry; J. Willard Gibbs introduced an "indeterminate product" for vectors in 1884 to analyze strain, while Gregorio Ricci and Tullio Levi-Civita developed absolute differential calculus in the 1890s–1900s, applying tensors to curvature and influencing general relativity.^[4] In modern applications, tensor products underpin multilinear algebra, where tensors of type (p,q) are elements of (V^{\otimes p} \otimes (V^*)^{\otimes q}), and play crucial roles in physics for describing stress, electromagnetic fields, and spacetime metrics that transform multilinearly under coordinate changes.^[4]^[1]

Definitions and Constructions for Vector Spaces

Universal Property

The tensor product of two vector spaces V and W over a field K is characterized by a universal property that makes it the representing object for the functor from the category of vector spaces to sets, which sends a vector space U to the set of K-bilinear maps \mathrm{Bil}_K(V \times W, U). This means that the tensor product V \otimes_K W comes equipped with a canonical bilinear map \otimes: V \times W \to V \otimes_K W, and the induced map \mathrm{Bil}_K(V \times W, U) \to \mathrm{Hom}_K(V \otimes_K W, U) sending a bilinear map f to the unique linear map g such that f = g \circ \otimes is a natural isomorphism.^[5]^[6] Formally, for any vector spaces V, W, U over K and any bilinear map f: V \times W \to U, there exists a unique linear map g: V \otimes_K W \to U such that the following diagram commutes:

\begin{CD} V \times W @>{\otimes}>> V \otimes_K W \\ @V{f}VV @VV{g}V \\ U @= U \end{CD}

That is, f(v, w) = g(v \otimes w) for all v \in V, w \in W. This property uniquely determines V \otimes_K W up to unique isomorphism satisfying the universal mapping condition.^[5]^[6] The existence of such a tensor product can be established by constructing V \otimes_K W as the quotient of the free vector space on the set V \times W by the subspace generated by the relations enforcing bilinearity: specifically, elements of the form (v_1 + v_2, w) - (v_1, w) - (v_2, w), (v, w_1 + w_2) - (v, w_1) - (v, w_2), and (\lambda v, w) - \lambda (v, w) (and similarly for the second factor) for all v, v_1, v_2 \in V, w, w_1, w_2 \in W, and \lambda \in K. Any bilinear map f: V \times W \to U then descends uniquely to a linear map on this quotient, as the relations are preserved by f.^[5] A simple example illustrates this universal property: the tensor product \mathbb{R}^2 \otimes \mathbb{R}^3 is the universal vector space for bilinear maps from \mathbb{R}^2 \times \mathbb{R}^3 to another space U. For instance, the standard bilinear form that sends (x_1, x_2) \in \mathbb{R}^2 and (y_1, y_2, y_3) \in \mathbb{R}^3 to x_1 y_1 + x_2 y_2 (extended trivially in the third coordinate) factors uniquely through the linear map induced on \mathbb{R}^2 \otimes \mathbb{R}^3, which has dimension 6.^[5]

Construction from Bases

One explicit construction of the tensor product of two vector spaces V and W over a field k proceeds by choosing bases for each space, assuming for simplicity that V and W are finite-dimensional. Let \{v_i \mid 1 \leq i \leq n\} be a basis for V and \{w_j \mid 1 \leq j \leq m\} be a basis for W. The tensor product V \otimes_k W is then defined as the k-vector space having \{v_i \otimes w_j \mid 1 \leq i \leq n, 1 \leq j \leq m\} as a basis.^[7]^[5] A simple tensor is an element of the form v \otimes w, where v \in V and w \in W. To define this, express v = \sum_{i=1}^n a_i v_i and w = \sum_{j=1}^m b_j w_j with respect to the bases, and set

v \otimes w = \sum_{i=1}^n \sum_{j=1}^m a_i b_j (v_i \otimes w_j).

This assignment extends by linearity to a bilinear map \otimes: V \times W \to V \otimes_k W, meaning it is linear in each argument separately: for scalars \alpha, \beta \in k and vectors v, v' \in V, w, w' \in W,

(\alpha v + \beta v') \otimes w = \alpha (v \otimes w) + \beta (v' \otimes w), \quad v \otimes (\alpha w + \beta w') = \alpha (v \otimes w) + \beta (v \otimes w').

By construction, every element of V \otimes_k W is a finite linear combination of basis elements v_i \otimes w_j, so the image of the bilinear map spans V \otimes_k W.^[7]^[5] This basis construction yields a vector space satisfying the universal property of the tensor product, as any bilinear map from V \times W to another space factors uniquely through it.^[5] For an example, consider the tensor product of polynomial spaces over \mathbb{R}. Let P_n = \mathbb{R} be the space of polynomials in one variable (infinite-dimensional with basis \{1, x, x^2, \dots \}) and similarly Q_m = \mathbb{R} with basis \{1, y, y^2, \dots \}. Then P_n \otimes_{\mathbb{R}} Q_m has basis \{x^i \otimes y^j \mid i, j \geq 0\}, and the bilinear map sends f(x) \otimes g(y) to the bivariate polynomial \sum_{i,j} a_i b_j x^i y^j, yielding an isomorphism P_n \otimes_{\mathbb{R}} Q_m \cong \mathbb{R}[x, y].^[5]

Quotient Space Approach

The quotient space approach constructs the tensor product of two vector spaces V and W over a field K as an algebraic quotient that enforces bilinearity without relying on explicit bases. Let F(V \times W) denote the free vector space on the set V \times W, whose elements are formal finite linear combinations \sum \lambda_i (v_i, w_i) with \lambda_i \in K, v_i \in V, and w_i \in W. This space serves as the ambient space before imposing relations.^[8]^[2] To obtain the tensor product, define the subspace N \subseteq F(V \times W) generated by the bilinearity relations: for all \lambda \in K, v, v_1, v_2 \in V, and w, w_1, w_2 \in W,

(\lambda v, w) - \lambda (v, w), \quad (v, \lambda w) - \lambda (v, w),

(v_1 + v_2, w) - (v_1, w) - (v_2, w), \quad (v, w_1 + w_2) - (v, w_1) - (v, w_2).

The tensor product is then the quotient space V \otimes W = F(V \times W) / N, where the image of (v, w) under the quotient map is denoted v \otimes w. This ensures that the map V \times W \to V \otimes W given by (v, w) \mapsto v \otimes w is bilinear, and elements of V \otimes W are equivalence classes of these formal combinations modulo the relations.^[8]^[9] This construction satisfies the universal property of the tensor product: for any vector space E and bilinear map \phi: V \times W \to E, there exists a unique linear map \tilde{\phi}: V \otimes W \to E such that \phi(v, w) = \tilde{\phi}(v \otimes w) for all v \in V, w \in W. The map \tilde{\phi} is induced by the universal bilinear map to the quotient, as the relations in N are precisely those required for bilinearity, ensuring the diagram commutes. Conversely, any linear map from V \otimes W to another space composes to yield a bilinear map on V \times W. This equivalence holds because the quotient precisely captures the relations needed for the universal bilinear object.^[8]^[2] The quotient approach is particularly advantageous for abstract vector spaces, where bases may not be specified or finite-dimensionality is not assumed, as it relies solely on the algebraic structure of free spaces and subspaces without enumerating spanning sets.^[9]^[10] For finite-dimensional spaces, this construction yields a natural isomorphism V^* \otimes W \cong \mathrm{Hom}(V, W), where V^* is the dual space and \mathrm{Hom}(V, W) is the space of linear maps, which can be identified with the space of matrices when bases are chosen for V and W. Specifically, if \{\mathbf{e}_i\} and \{\mathbf{f}_j\} are bases for V and W, the elements \mathbf{e}_i^* \otimes \mathbf{f}_j correspond to matrix units that span the space of m \times n matrices, where \dim V = m and \dim W = n.^[8]^[2]

Linear Disjointness

Two subspaces U and V of a vector space W over a field K are said to be linearly disjoint over K if every linearly independent set in U remains linearly independent in W after tensoring its elements with a basis of V. More precisely, if \{u_1, \dots, u_m\} is linearly independent in U and \{v_1, \dots, v_n\} is a basis for V, then the set \{u_i \otimes v_j \mid 1 \leq i \leq m, 1 \leq j \leq n\} (viewed in W via the natural bilinear map) is linearly independent in W. This condition ensures that the canonical map U \otimes_K V \to W is injective, preserving the structure without introducing additional linear dependencies.^[11] An equivalent characterization is that the dimension of the image of U \otimes_K V in W equals (\dim_K U) \cdot (\dim_K V), assuming finite dimensions; since \dim_K (U \otimes_K V) = (\dim_K U) \cdot (\dim_K V) always holds for vector spaces, linear disjointness holds precisely when there is no kernel in the map to W. This equivalence ties directly to the universal property of the tensor product, where bases of U and V generate a basis for the tensor product, and injectivity confirms no collapse in the ambient space.^[12] In the context of field extensions, linear disjointness plays a key role in analyzing tensor products V \otimes_K L, where V and L are extensions of the base field K. Here, the tensor product measures the extent to which the extensions "split" or interact minimally in a common extension field; if V and L are linearly disjoint over K, the natural map V \otimes_K L \to compositum is injective, and for finite extensions, the degree of the compositum equals the product of the individual degrees. This property is crucial for understanding independence in Galois theory and separability, as inseparable extensions often fail linear disjointness due to shared minimal polynomials leading to nilpotent elements in the tensor product.^[11] A concrete example illustrates this: viewing the rational numbers \mathbb{[Q](/page/Q)} and the real numbers \mathbb{[R](/page/R)} as vector spaces over \mathbb{[Q](/page/Q)}, they are linearly disjoint, since \mathbb{[Q](/page/Q)} \otimes_{\mathbb{[Q](/page/Q)}} \mathbb{[R](/page/R)} \cong \mathbb{[R](/page/R)} and \dim_{\mathbb{[Q](/page/Q)}} \mathbb{[R](/page/R)} = \mathfrak{c} = 1 \cdot \mathfrak{c}, with the isomorphism being injective and preserving the infinite-dimensional structure without relations. This reflects the transcendental nature of \mathbb{[R](/page/R)} over \mathbb{[Q](/page/Q)}, ensuring full dimensionality in the tensor product.^[12]

Properties of Vector Space Tensor Products

Dimension Formula

For finite-dimensional vector spaces V and W over a field K, with \dim V = m and \dim W = n, the dimension of the tensor product V \otimes_K W is mn.^[4] This follows from the basis construction of the tensor product, where if \{v_1, \dots, v_m\} is a basis for V and \{w_1, \dots, w_n\} is a basis for W, then the set \{v_i \otimes w_j \mid 1 \leq i \leq m, 1 \leq j \leq n\} forms a basis for V \otimes_K W, consisting of exactly mn elements.^[4] The linear independence and spanning properties of this set ensure that the dimension is precisely the product of the individual dimensions.^[4] In the infinite-dimensional case, the tensor product construction extends naturally, and if V and W admit Hamel bases of cardinalities \kappa and \lambda respectively, then V \otimes_K W has a basis of cardinality \kappa \cdot \lambda, where \cdot denotes cardinal multiplication.^[7] For example, consider the polynomial rings \mathbb{R} and \mathbb{R} over \mathbb{R}, each of which has a countable infinite basis \{1, x, x^2, \dots\} and \{1, y, y^2, \dots\} respectively. Their tensor product \mathbb{R} \otimes_{\mathbb{R}} \mathbb{R} is isomorphic to \mathbb{R}[x, y] as \mathbb{R}-modules, with basis \{x^i \otimes y^j \mid i, j \geq 0\}, which is also countably infinite.^[4]

Associativity

The tensor product of vector spaces exhibits associativity, meaning that for finite-dimensional vector spaces V, W, and U over a field k, there is a natural isomorphism of vector spaces

(V \otimes_k W) \otimes_k U \cong V \otimes_k (W \otimes_k U).

This isomorphism is induced by the bilinear map ((V \otimes_k W) \times U \to V \otimes_k (W \otimes_k U)) defined by ((v \otimes w) \otimes u) \mapsto v \otimes (w \otimes u) for v \in V, w \in W, u \in U, which extends uniquely from the universal property of the tensor product.^[5]^[13] The isomorphism is natural in each argument, meaning that for linear maps f: V \to V', g: W \to W', and h: U \to U', the following diagram commutes:

\begin{CD} (V \otimes W) \otimes U @>{\cong}>> V \otimes (W \otimes U) \\ @V{(f \otimes g) \otimes h}VV @VV{f \otimes (g \otimes h)}V \\ (V' \otimes W') \otimes U' @>{\cong}>> V' \otimes (W' \otimes U') \end{CD}

This compatibility ensures the isomorphism respects the category of vector spaces and linear maps.^[5]^[13] One proof of the isomorphism relies on the universal property: both (V \otimes W) \otimes U and V \otimes (W \otimes U) represent the same quotient space of the free vector space on V \times W \times U by the relations for multilinearity, yielding inverse maps f: (V \otimes W) \otimes U \to V \otimes (W \otimes U) via (v \otimes w) \otimes u \mapsto v \otimes (w \otimes u) and g: V \otimes (W \otimes U) \to (V \otimes W) \otimes U via v \otimes (w \otimes u) \mapsto (v \otimes w) \otimes u, which are mutual inverses by uniqueness of bilinear extensions.^[5]^[13] Alternatively, for finite-dimensional spaces with bases \{e_i\} for V, \{f_j\} for W, and \{g_k\} for U, the set \{(e_i \otimes f_j) \otimes g_k\} spans (V \otimes W) \otimes U and is mapped bijectively to \{e_i \otimes (f_j \otimes g_k)\}, which spans V \otimes (W \otimes U), preserving dimension \dim(V) \cdot \dim(W) \cdot \dim(U) and establishing the isomorphism.^[14] This associativity extends multilinearly to n-fold tensor products, where V_1 \otimes_k \cdots \otimes_k V_n is well-defined up to canonical isomorphism regardless of parenthesization, via iterative application of the three-factor case, with basis elements e_{1,i_1} \otimes \cdots \otimes e_{n,i_n} forming a basis of cardinality \prod_{j=1}^n \dim(V_j).^[5]^[14]

Symmetry and Commutativity

The tensor product of two vector spaces V and W over a field K exhibits a form of symmetry through the flip map \tau: V \otimes_K W \to W \otimes_K V, defined by \tau(v \otimes w) = w \otimes v for all v \in V and w \in W. This map is K-linear because the tensor product is constructed to respect bilinearity, and it is invertible with inverse \tau^{-1} = \tau, as applying \tau twice yields the identity map.^[5] The flip map induces a natural isomorphism V \otimes_K W \cong W \otimes_K V, meaning the tensor products are isomorphic as K-vector spaces via this canonical correspondence, which preserves the bilinear structure.^[5] However, the tensor product operation itself is not commutative in the sense that v \otimes w \neq w \otimes v in general within V \otimes_K W, as the elementary tensors are ordered and the space lacks an intrinsic mechanism to equate them without the isomorphism.^[5] This symmetry underlies constructions like the symmetric and exterior products, which are quotients of tensor powers that enforce commutativity or anticommutativity. The symmetric product \mathrm{Sym}^2(V) is the quotient of V \otimes_K V by the subspace generated by elements of the form v \otimes w - w \otimes v, imposing the relation v \otimes w = w \otimes v.^[15] In contrast, the exterior product \Lambda^2(V) is the quotient of V \otimes_K V by the subspace generated by v \otimes w + w \otimes v, enforcing antisymmetry where v \otimes w = -w \otimes v.^[15] These quotients illustrate how the flip map's action can be used to define invariant subspaces under symmetry.

Tensor Products of Linear Maps

Definition and Action

Given linear maps f: V \to V' and g: W \to W' between vector spaces over a field k, their tensor product is the linear map f \otimes g: V \otimes_k W \to V' \otimes_k W' defined on simple tensors by

(f \otimes g)(v \otimes w) = f(v) \otimes g(w)

for all v \in V and w \in W, and extended by linearity to the entire domain.^[16]^[17] This definition preserves the structure of simple tensors, ensuring that the image of a pure tensor remains a pure tensor in the codomain.^[16] The map f \otimes g arises uniquely from the k-bilinear map V \times W \to V' \otimes_k W' given by (v, w) \mapsto f(v) \otimes g(w), via the universal property of the tensor product, which guarantees the existence and uniqueness of such a linear extension.^[16]^[17] Moreover, the assignment (f, g) \mapsto f \otimes g is bilinear as a map from \Hom_k(V, V') \times \Hom_k(W, W') to \Hom_k(V \otimes_k W, V' \otimes_k W').^[17] A concrete example occurs when V = W = k^n and V' = W' = k^m, so that f and g are represented by n \times n and m \times m matrices A and B, respectively. In this case, f \otimes g corresponds to the Kronecker product A \otimes B, an nm \times nm block matrix whose (i,j)-block is a_{ij} B.^[16]^[17] This matrix acts on the standard basis of k^n \otimes_k k^m \cong k^{nm} by scaling blocks according to the entries of A.^[16]

Induced Maps and Functoriality

The tensor product operation on vector spaces extends to linear maps, inducing a structure-preserving map between tensor products. For fixed vector spaces V, V' and W, and a linear map f: V \to V', the induced map f \otimes \mathrm{id}_W: V \otimes W \to V' \otimes W is defined by linearity on simple tensors as (f \otimes \mathrm{id}_W)(v \otimes w) = f(v) \otimes w. Similarly, for fixed W, W' and g: W \to W', the map \mathrm{id}_V \otimes g: V \otimes W \to V \otimes W' is defined by v \otimes g(w). More generally, for linear maps f: V \to V' and g: W \to W', the induced map f \otimes g: V \otimes W \to V' \otimes W' satisfies (f \otimes g)(v \otimes w) = f(v) \otimes g(w), and this construction respects composition and identities, making the tensor product functorial.^[16] Fixing a vector space W, the assignment V \mapsto V \otimes W defines a covariant functor -\otimes W: \mathbf{Vect}_K \to \mathbf{Vect}_K, where K is the base field, sending linear maps to their induced tensor maps as above. This functor is exact, preserving both kernels and cokernels of short exact sequences of vector spaces, since every vector space over a field is flat. For instance, if $0 \to A \to B \to C \to 0 is a short exact sequence, then $0 \to A \otimes W \to B \otimes W \to C \otimes W \to 0 is also exact. In particular, for a linear map f: V \to U, the kernel of f \otimes \mathrm{id}_W equals (\ker f) \otimes W.^[18] The tensor product functor admits an adjunction with the Hom functor. Specifically, there is a natural isomorphism \mathrm{Hom}_K(V \otimes W, U) \cong \mathrm{Bilin}_K(V \times W, U), where \mathrm{Bilin}_K(V \times W, U) denotes the vector space of K-bilinear maps from V \times W to U. This isomorphism arises from the universal property of the tensor product: every bilinear map \phi: V \times W \to U factors uniquely through the canonical bilinear map \alpha: V \times W \to V \otimes W via a linear map \tilde{\phi}: V \otimes W \to U such that \phi = \tilde{\phi} \circ \alpha, and conversely, every linear map T: V \otimes W \to U yields a bilinear map by precomposition with \alpha. This adjunction is natural in all variables, providing a categorical characterization of the tensor product.^[5] The tensor product category \mathbf{Vect}_K is symmetric monoidal, with the symmetry given by natural transformations such as the flip map \sigma_{V,W}: V \otimes W \to W \otimes V, defined by \sigma_{V,W}(v \otimes w) = w \otimes v on simple tensors and extended linearly. This flip is a natural isomorphism satisfying \sigma_{W,V} \circ \sigma_{V,W} = \mathrm{id}_{V \otimes W}, and it interchanges the roles of the factors, enabling the treatment of tensors without regard to order in many applications. For higher tensor powers, permutations induce natural transformations via iterated flips.^[16]

General Tensors and Operations

Multilinear Forms and Evaluation

A tensor of type (k, l) generalizes the construction of the tensor product to multiple factors by taking an element of the space V^{\otimes k} \otimes (V^*)^{\otimes l}, where V is a vector space over a field K and V^* denotes its dual space. This space is the iterative tensor product, generated as the linear span of elementary tensors of the form v_1 \otimes \cdots \otimes v_k \otimes \phi_1 \otimes \cdots \otimes \phi_l with v_i \in V and \phi_j \in V^*. Such tensors provide a coordinate-free way to encode multilinear relationships between vectors and covectors.^[16] The evaluation of these tensors is defined through their correspondence to multilinear maps. Specifically, a tensor T \in V^{\otimes k} \otimes (V^*)^{\otimes l} induces a (k, l)-linear map, which is multilinear in l arguments from V and k arguments from V^*, mapping to the base field K. For an elementary tensor, the map is given by

(v_1 \otimes \cdots \otimes v_k \otimes \phi_1 \otimes \cdots \otimes \phi_l)(\psi_1, \dots, \psi_k, w_1, \dots, w_l) = \left( \prod_{i=1}^k \psi_i(v_i) \right) \left( \prod_{j=1}^l \phi_j(w_j) \right),

extended linearly to the full space; this realizes the universal property of the tensor product for mixed multilinear functionals.^[19] A key aspect of evaluation involves contraction operations that pair factors from V^* and V to produce scalars. For instance, consider a tensor in the space V^{\otimes k} \otimes (V^*)^{\otimes k}; the evaluation map is the linear contraction

T: V^{\otimes k} \otimes (V^*)^{\otimes k} \to K, \quad T(v_1 \otimes \cdots \otimes v_k \otimes \phi_1 \otimes \cdots \otimes \phi_k) = \prod_{i=1}^k \phi_i(v_i),

again extended linearly, which fully contracts the tensor to a scalar by pairing each vector with a dual element. This map is natural and functorial, preserving the structure under linear transformations.^[16] An important example is the metric tensor, which serves as a (0,2)-tensor in this framework. In the context of an inner product space, the metric g is an element of (V^*)^{\otimes 2}, a symmetric, nondegenerate element that defines evaluation g: V \times V \to K via g(v, w) = \sum g_{ij} v^i w^j in coordinates, encapsulating distances and angles in applications like differential geometry.^[20]

Contraction and Trace

In multilinear algebra, the contraction of a tensor is an operation that pairs a contravariant index with a covariant index, effectively reducing the tensor's rank by two through summation over the paired indices. This arises from the natural pairing between a finite-dimensional vector space and its dual space, where a tensor T \in V^{\otimes r} \otimes (V^*)^{\otimes s} can be contracted by composing with the evaluation map V^* \otimes V \to \mathbb{R}, or \langle \cdot, \cdot \rangle, on specific factors.^[21] In component notation, for a tensor T^{i_1 \dots i_r}_{j_1 \dots j_s}, contracting the p-th upper index i_p with the q-th lower index j_q yields a new tensor C^{i_1 \dots \hat{i_p} \dots i_r}_{j_1 \dots \hat{j_q} \dots j_s} = \sum_k T^{i_1 \dots k \dots i_r}_{j_1 \dots k \dots j_s}, equivalent to inserting the Kronecker delta \delta^k_k and summing over the repeated index k.^[21] More generally, contractions can involve multiple pairs of indices across higher-rank tensors, allowing for partial or full reductions in tensor order while preserving multilinearity. For instance, in a rank-4 tensor, one might contract two non-adjacent index pairs sequentially, resulting in a scalar if all indices are paired, or a lower-rank tensor otherwise; this operation is associative and corresponds to composing multilinear maps with dual pairings.^[21] The contraction is basis-independent, as it relies on the intrinsic duality structure rather than coordinates, ensuring invariance under change of basis.^[13] A special case of contraction is the trace, which fully contracts a (1,1)-tensor, or endomorphism, T \in \mathrm{End}(V) \cong V \otimes V^*, to a scalar via \mathrm{tr}(T) = \sum_i T^i_i, or abstractly as T composed with the identity endomorphism under the pairing \langle T, \mathrm{id}_V \rangle.^[13] This generalizes the matrix trace and is independent of the choice of basis, reflecting the canonical isomorphism between V \otimes V^* and the space of linear functionals on endomorphisms.^[13] An important application appears in differential geometry, where the Ricci tensor R_{\mu\nu} is obtained by contracting the Riemann curvature tensor R^\rho_{\ \mu\sigma\nu} over the first and third indices: R_{\mu\nu} = R^\rho_{\ \mu\rho\nu} = \sum_\rho R^\rho_{\ \mu\rho\nu}.^[22] This contraction captures the trace of the Riemann tensor's action on bivectors, encoding volumetric aspects of spacetime curvature essential in general relativity.^[23]

Adjoint Representation

In the adjoint representation of a Lie algebra \mathfrak{g}, the tensor product \mathfrak{g} \otimes \mathfrak{g} arises naturally as a representation space under the extended adjoint action. For x \in \mathfrak{g}, the adjoint map \ad_x: \mathfrak{g} \to \mathfrak{g} is given by \ad_x(y) = [x, y] for all y \in \mathfrak{g}, where [ \cdot, \cdot ] denotes the Lie bracket. This action extends to \mathfrak{g} \otimes \mathfrak{g} via the derivation property: (\ad_x \otimes \id + \id \otimes \ad_x)(y_1 \otimes y_2) = [x, y_1] \otimes y_2 + y_1 \otimes [x, y_2]. This construction ensures that \mathfrak{g} \otimes \mathfrak{g} becomes a \mathfrak{g}-module, preserving the Lie algebra structure in higher tensor powers used in representation theory.^[24] The structure constants of \mathfrak{g} encode the Lie bracket as components of a multilinear map, interpretable as a tensor in (\mathfrak{g} \otimes \mathfrak{g})^* \otimes \mathfrak{g}. Given a basis \{e_i\}_{i=1}^{\dim \mathfrak{g}} for \mathfrak{g}, the bracket expands as [e_i, e_j] = \sum_k c^k_{ij} e_k, where the coefficients c^k_{ij} are the structure constants. These constants transform under change of basis as the components of a (1,2)-tensor, reflecting the alternation c^k_{ij} = -c^k_{ji} from antisymmetry of the bracket and satisfying the Jacobi identity \sum_l (c^l_{im} c^j_{lk} + c^l_{jm} c^k_{li} + c^l_{km} c^i_{lj}) = 0. This tensorial perspective facilitates computations in the universal enveloping algebra and decomposition of tensor representations.^[25] Tensors invariant under the adjoint action play a central role in the geometry and classification of Lie algebras. An element T \in \mathfrak{g}^{\otimes p} \otimes (\mathfrak{g}^*)^{\otimes q} is ad-invariant if (\ad_x \otimes \id^{\otimes p-1} + \sum \id^{\otimes i-1} \otimes \ad_x \otimes \id^{\otimes p-i})(T) = 0 for all x \in \mathfrak{g}, extended appropriately to the dual factors. Such invariants determine Casimir operators and symmetry properties. A prominent example is the Killing form, a symmetric bilinear form B: \mathfrak{g} \times \mathfrak{g} \to \mathbb{R} (or the base field) defined by B(x, y) = \tr(\ad_x \circ \ad_y), which corresponds to an element of \mathfrak{g}^* \otimes \mathfrak{g}^*. For semisimple \mathfrak{g}, B is nondegenerate and ad-invariant, satisfying B([x, y], z) + B(y, [x, z]) = 0 for all x, y, z \in \mathfrak{g}, enabling identification of \mathfrak{g} \cong \mathfrak{g}^* and facilitating root space decompositions.^[26]

Tensor Products of Modules over Rings

Definition for Modules

In the context of modules over a commutative ring R, the tensor product M \otimes_R N of two R-modules M and N generalizes the construction for vector spaces, where R is a field.^[5] The tensor product is constructed as the quotient of the free R-module generated by the set M \times N by the submodule generated by the relations enforcing R-bilinearity: for all m, m' \in M, n, n' \in N, and r \in R,

(m + m') \otimes n - m \otimes n - m' \otimes n = 0, \quad m \otimes (n + n') - m \otimes n - m \otimes n' = 0, \quad (r m) \otimes n - m \otimes (r n) = 0.

Elements of M \otimes_R N are thus formal finite sums \sum_i m_i \otimes n_i, subject to these relations, and the module structure is given by r (\sum_i m_i \otimes n_i) = \sum_i (r m_i) \otimes n_i.^[5]^[27] This construction satisfies the universal property for R-bilinear maps: the canonical bilinear map \otimes: M \times N \to M \otimes_R N given by (m, n) \mapsto m \otimes n is universal, meaning that for any R-module P and any R-bilinear map f: M \times N \to P, there exists a unique R-linear map \phi: M \otimes_R N \to P such that f = \phi \circ \otimes.^[5]^[27] A concrete example is the tensor product of cyclic groups viewed as \mathbb{Z}-modules: \mathbb{Z}/m\mathbb{Z} \otimes_{\mathbb{Z}} \mathbb{Z}/n\mathbb{Z} \cong \mathbb{Z}/\gcd(m,n)\mathbb{Z}.^[5]^[27] For free modules, if M has R-basis \{e_i\}_{i \in I} and N has R-basis \{f_j\}_{j \in J}, then M \otimes_R N is free with basis \{e_i \otimes f_j\}_{(i,j) \in I \times J}.^[5]

Bilinear Maps and Universal Property

The tensor product of two modules M and N over a commutative ring R, denoted M \otimes_R N, is characterized by a universal property with respect to R-bilinear maps. Specifically, there exists a canonical R-bilinear map \phi: M \times N \to M \otimes_R N given by (m, n) \mapsto m \otimes n, such that for any R-bilinear map f: M \times N \to P into another R-module P, there is a unique R-linear map \overline{f}: M \otimes_R N \to P satisfying f = \overline{f} \circ \phi.^[4] An [R](/page/R)-bilinear map f: M \times N \to P is additive in each argument separately, so f(m + m', n) = f(m, n) + f(m', n) and f(m, n + n') = f(m, n) + f(m, n'), and it respects the [R](/page/R)-module scalar multiplication in a balanced way: f(rm, n) = r f(m, n) = f(m, rn) for all r \in [R](/page/R).^[4] This balance condition ensures compatibility between the actions on M and N, reflecting the commutative nature of [R](/page/R).^[4] The uniqueness of \overline{f} follows from the fact that the elements m \otimes n generate M \otimes_R N as an R-module, so any linear map is determined by its values on these generators.^[4] Moreover, any two R-modules satisfying this property are canonically isomorphic.^[4] To establish the universal property, one constructs M \otimes_R N as the quotient of the free R-module on the set M \times N by the submodule generated by the relations enforcing bilinearity: (m + m', n) - (m, n) - (m', n), (m, n + n') - (m, n) - (m, n'), and (rm, n) - r(m, n) (or equivalently (m, rn) - r(m, n)) for all m, m' \in M, n, n' \in N, and r \in R.^[4] The induced map on the quotient then satisfies the universal property because any bilinear f factors uniquely through this quotient, as the relations are precisely those needed to make f well-defined on the generators.^[4] This construction via generators and relations also shows how the tensor product can be defined abstractly without explicit computation, though it aligns with the explicit quotient presentation.^[4] The universal property highlights the tensor product's role in linearizing bilinear maps, but the functor -\otimes_R N is only right exact, measuring its failure to preserve all short exact sequences through the derived functors \Tor^R_i(M, N) for i \geq 1, where \Tor^R_0(M, N) \cong M \otimes_R N.^[28]

Tensor Product over Non-Commutative Rings

When the ring R is non-commutative, the tensor product of modules requires careful specification of module sidedness to ensure compatibility with the ring action. Specifically, for a right R-module M and a left R-module N, the tensor product M \otimes_R N is defined as an abelian group equipped with a map \mu: M \times N \to M \otimes_R N that is additive in each variable and R-balanced, meaning \mu(mr, n) = \mu(m, rn) for all m \in M, n \in N, and r \in R.^[5]^[29]^[30] This R-balanced bilinearity distinguishes the non-commutative case from the commutative one, where the tensor product is simply R-bilinear without needing to balance left and right actions explicitly, as scalars commute. In the non-commutative setting, M \otimes_R N is generally only an abelian group and does not inherit a natural R-module structure unless additional bimodule structures are imposed, such as when N is also a right S-module for some ring S, making M \otimes_R N a right S-module via (m \otimes n)s = m \otimes (ns).^[5]^[30] The tensor product satisfies a universal property: for any abelian group P and any R-balanced bilinear map f: M \times N \to P, there exists a unique group homomorphism \tilde{f}: M \otimes_R N \to P such that \tilde{f} \circ \mu = f, with \tilde{f}(m \otimes n) = f(m, n). This property characterizes M \otimes_R N up to unique isomorphism and justifies its existence, which can be constructed explicitly as the free abelian group on the set M \times N modulo the subgroup generated by the relations enforcing additivity and R-balancing.^[5]^[29]^[30] A concrete realization arises in representation theory over group rings. For a finite group G and field k, let R = kG (non-commutative if G is non-abelian) and let M be a left kH-module corresponding to a representation of a subgroup H \leq G, with R viewed as a right kH-module. The induced module \mathrm{Ind}_H^G M = R \otimes_{kH} M is then a left R-module whose dimension over k equals |G|/|H| times \dim_k M, illustrating how the tensor product extends representations from subgroups to the full group.^[5]

Computation Methods

One effective method for computing the tensor product M \otimes_R N of two R-modules involves using projective resolutions, particularly when direct computation is challenging. To compute M \otimes_R N, take a projective resolution P_\bullet \to N \to 0 of N, then tensor the resolution with M to obtain the complex M \otimes_R P_\bullet, and the tensor product is the zeroth homology of this complex, H_0(M \otimes_R P_\bullet). This approach leverages the fact that tensoring with M preserves projectivity under certain conditions, allowing explicit chain complexes to be formed and homology computed via kernel-image calculations. For instance, if N is finitely presented, a finite free resolution suffices for practical computation.^[31] For cyclic modules over the integers, the tensor product \mathbb{Z}/m\mathbb{Z} \otimes_\mathbb{Z} \mathbb{Z}/n\mathbb{Z} simplifies explicitly to \mathbb{Z}/d\mathbb{Z}, where d = \gcd(m, n). This isomorphism arises because the generators $1 \otimes 1 satisfy the relation d(1 \otimes 1) = 0, and the module is cyclic with no smaller annihilator. The proof proceeds by noting that the bilinear map \mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}/d\mathbb{Z} given by (a, b) \mapsto ab \mod d factors through the quotients, satisfying the universal property. This result extends to cyclic modules over principal ideal domains via analogous arguments.^[32] Under flatness conditions, change of rings provides another computational tool: if S is a flat R-algebra, then for an R-module M, flatness ensures that tensoring exact sequences with S remains exact, facilitating base change computations. Specifically, if $0 \to [K](/page/K) \to F \to M \to 0 is a presentation of M with F free, then $0 \to [K](/page/K) \otimes_R S \to F \otimes_R S \to M \otimes_R S \to 0 is exact, allowing iterative construction. This is particularly useful when S is a localization or polynomial extension.^[33] A concrete example occurs with quotient modules of polynomial rings: for a field k and monic polynomials f \in k, g \in k, the tensor product k/(f) \otimes_k k/(g) is isomorphic to k[x,y]/(f(x), g(y)). This follows from the universal property, as the map sending p(x) \otimes q(y) to p(x)q(y) modulo the ideals (f(x), g(y)) is bilinear over k and surjective, with the kernel generated by the relations from f and g. Such computations are essential in algebraic geometry for fiber products of varieties.^[33]

Tensor Products of Algebras

Definition and Structure

The tensor product of two R-algebras A and B, where R is a commutative ring, is constructed as an R-algebra whose underlying R-module is the tensor product of A and B as R-modules.^[34] The multiplication in A \otimes_R B is defined by extending the rule (a \otimes b)(a' \otimes b') = aa' \otimes bb' for elementary tensors a \otimes b \in A \otimes_R B and a' \otimes b' \in A \otimes_R B, where a, a' \in A and b, b' \in B, via R-linearity to the entire module.^[34]^[35] This structure equips A \otimes_R B with a ring multiplication that is associative and distributive over addition, making it an R-algebra with unit $1_A \otimes 1_B.^[34] The tensor product A \otimes_R B satisfies a universal property characterizing it as the representing object for R-bilinear maps that respect the algebra structures. Specifically, for any R-algebra C, there is a canonical R-bilinear map \iota: A \times B \to A \otimes_R B given by (a, b) \mapsto a \otimes b, such that any R-bilinear map f: A \times B \to C that is multiplicative in each argument separately (i.e., f(aa', b) = f(a, b)f(a', b) and f(a, bb') = f(a, b)f(a, bb')) factors uniquely through an R-algebra homomorphism \tilde{f}: A \otimes_R B \to C with f = \tilde{f} \circ \iota.^[34]^[35] This property positions A \otimes_R B as the coproduct in the category of R-algebras.^[34] When A and B are commutative R-algebras, the tensor product A \otimes_R B is also commutative, as the multiplication satisfies (a \otimes b)(a' \otimes b') = (a' \otimes b')(a \otimes b) for all elementary tensors.^[34] A representative example is the tensor product of polynomial rings: for a commutative ring R, R \otimes_R R \cong R[x, y] as R-algebras, where the isomorphism sends x \otimes 1 \mapsto x and $1 \otimes y \mapsto y.^[34] Similarly, over the complex numbers \mathbb{C}, \mathbb{C} \otimes_{\mathbb{C}} \mathbb{C} \cong \mathbb{C}[x, y].^[34]

Properties and Examples

For finite-dimensional algebras A and B over a field k, the dimension of their tensor product satisfies \dim_k (A \otimes_k B) = (\dim_k A) \cdot (\dim_k B). This property arises because the tensor product of finite-dimensional vector spaces over k has dimension equal to the product of the individual dimensions, and algebras are vector spaces equipped with additional multiplicative structure.^[36] The center of the tensor product algebra Z(A \otimes_k B) equals the tensor product of the centers Z(A) \otimes_k Z(B). This inclusion holds more generally, with equality in the case of algebras over a field, reflecting how central elements commute with all factors in the product.^[37] Idempotents and nilpotents multiply across factors in a compatible way: if e \in A and f \in B are idempotents (satisfying e^2 = e and f^2 = f), then e \otimes f is idempotent in A \otimes_k B, as (e \otimes f)^2 = e^2 \otimes f^2 = e \otimes f. Similarly, if x \in A is nilpotent with x^n = 0 for some n, then x \otimes 1_B is nilpotent in the tensor product, and elements of the form x \otimes y where y \in B is also nilpotent satisfy (x \otimes y)^n = x^n \otimes y^n = 0. These behaviors extend the radical structure, where the nilradical of A \otimes_k B contains the sum of the images of the nilradicals under the natural embeddings.^[38]^[37] The tensor product of algebras inherits associativity from the underlying module tensor product over the base ring. A key example is the tensor product of group algebras: for finite groups G and H and a field k, there is a canonical k-algebra isomorphism kG \otimes_k kH \cong k(G \times H), where G \times H is the direct product group. This isomorphism maps basis elements via (g \otimes h) \mapsto (g, h), preserving the algebra structure induced by group multiplication.^[39] Another significant example is complexification: given a real algebra A (over \mathbb{R}), the tensor product A \otimes_\mathbb{R} \mathbb{C} provides the complexification, turning A into a complex algebra where scalar multiplication extends naturally by \mathbb{C}-linearity on the second factor. This construction allows real algebraic structures to be analyzed over the complexes, facilitating techniques like spectral decomposition.^[40]

Advanced and Specialized Tensor Products

Topological Tensor Products

In the context of topological vector spaces, the tensor product is endowed with a topology to preserve continuity of bilinear maps and facilitate analysis in infinite-dimensional settings. The algebraic tensor product of two topological vector spaces V and W provides the underlying vector space structure, upon which various compatible topologies are defined.^[41] The projective tensor product V \otimes_\pi W is equipped with the finest locally convex topology such that the canonical bilinear map V \times W \to V \otimes_\pi W is continuous. This topology is generated by seminorms of the form

p_\pi(z) = \inf\left\{ \sum_i p(v_i) q(w_i) : z = \sum_i v_i \otimes w_i \right\},

where p and q are continuous seminorms on V and W, respectively, and the infimum is taken over all finite representations of z. Introduced by Grothendieck, this construction ensures the projective tensor product is functorial and complete when V and W are Banach spaces, making it suitable for studying bounded operators and approximations in normed spaces.^[42] The injective tensor product V \otimes_\varepsilon W carries the coarsest locally convex topology making the bilinear map continuous, defined by seminorms

p_{\varepsilon}(z) = \sup\left\{ \left| \sum_i \langle \hat{v}, v_i \rangle \langle \hat{w}, w_i \rangle \right| : \|\hat{v}\|_{V^*} \leq 1, \|\hat{w}\|_{W^*} \leq 1 \right\},

where the supremum is over the unit balls in the dual spaces V^* and W^*. This topology corresponds to uniform convergence on equicontinuous subsets of the dual and is particularly useful for embeddings and extensions of operators. The inductive tensor product topology, in contrast, is the finest locally convex topology compatible with the bilinear map, often arising as an inductive limit in the context of strict inductive limits of spaces, such as in distribution theory.^[41]^[42] A key development is the notion of nuclear spaces, where the projective and injective topologies on V \otimes W coincide for every topological vector space W. Such spaces, pioneered by Grothendieck, are stable under subspaces, quotients, products, and projective limits, and their completions under these topologies yield well-behaved structures ideal for applications in partial differential equations and quantum field theory. For instance, the Schwartz space of rapidly decreasing functions satisfies \mathcal{S}(\mathbb{R}^2) \cong \mathcal{S}(\mathbb{R}) \hat{\otimes}_\pi \mathcal{S}(\mathbb{R}), forming a nuclear Fréchet space.^[41]^[42] In functional analysis, topological tensor products of L^p spaces play a central role; the projective tensor product L^p(\mu) \hat{\otimes}_\pi L^q(\nu) (for \sigma-finite measures \mu, \nu and $1 \leq p, q < \infty) identifies with the Banach space of Bochner-integrable functions equipped with the projective norm, facilitating the study of integral operators and Schatten classes. When $1/p + 1/q = 1/r with r \geq 1, this construction embeds densely into L^r(\mu \times \nu), providing a bridge between convolution and operator theory.^[41]^[43]

Graded Vector Spaces

In a graded vector space V over a field k, the underlying vector space decomposes as a direct sum V = \bigoplus_{n \in \mathbb{Z}} V_n, where each V_n is a subspace corresponding to homogeneous elements of degree n. Similarly, a graded vector space W decomposes as W = \bigoplus_{m \in \mathbb{Z}} W_m. The tensor product V \otimes W inherits a natural grading defined by (V \otimes W)_k = \bigoplus_{n+m=k} V_n \otimes_k W_m, where the components V_n \otimes_k W_m are the ordinary tensor products of vector spaces placed in degree k = n + m. This construction ensures that the tensor product preserves the grading and satisfies the universal property for bilinear maps that respect the degrees. For \mathbb{Z}/2-graded vector spaces, often called super vector spaces with even (degree 0) and odd (degree 1) parts, the graded tensor product incorporates a sign convention known as the Koszul sign rule to maintain compatibility with algebraic structures like supercommutativity. Specifically, when interchanging homogeneous elements v \in V_p and w \in W_q, the relation (v \otimes w) = (-1)^{p q} (w \otimes v) holds, introducing a sign for products involving odd-degree elements. This convention, originating in Koszul's work on homology, ensures that morphisms and operations in graded categories behave consistently under composition and tensoring. A key example arises in the construction of the exterior algebra \Lambda V on a vector space V, which can be viewed through the lens of graded tensor products with antisymmetry enforced by the sign rule. Placing V in odd degree (as a purely odd super vector space), the graded-symmetric algebra on V—generated by symmetric products under the Koszul convention—coincides with the exterior algebra, where the antisymmetry v \wedge w = - w \wedge v for v, w \in V emerges naturally from the signs in the graded tensor product. This perspective unifies the exterior algebra with symmetric algebras in super geometry. In cohomology theory, graded tensor products play a central role via the Künneth theorem, which computes the cohomology ring of a product space. For topological spaces X and Y with cohomology coefficients in a field k, the theorem asserts that H^*(X \times Y; k) \cong H^*(X; k) \otimes H^*(Y; k) as graded commutative rings, where the isomorphism uses the graded tensor product to match degrees: the class in degree k on the right arises from sums over p + q = k of tensor products of classes from degrees p and q. This equips cohomology rings with a product structure preserved under tensoring, facilitating computations in algebraic topology. The graded tensor product extends associativity to multi-graded settings, allowing iterated products to be well-defined up to canonical isomorphisms.

Representations of Groups

In representation theory, the tensor product provides a fundamental construction for combining two representations of a group into a new one. Given a group G and two representations \rho: G \to \mathrm{GL}(V) and \sigma: G \to \mathrm{GL}(W), where V and W are vector spaces over \mathbb{C}, the tensor product representation \rho \otimes \sigma acts on the tensor product space V \otimes W by the diagonal action:

(\rho \otimes \sigma)(g)(v \otimes w) = \rho(g)v \otimes \sigma(g)w

for all g \in G and v \in V, w \in W.^[44] This defines a representation of G on V \otimes W, preserving the bilinear structure inherent to the tensor product.^[44] The character of the tensor product representation, which encodes key information about its trace under group elements, is the pointwise product of the individual characters: \chi_{\rho \otimes \sigma}(g) = \chi_\rho(g) \cdot \chi_\sigma(g) for all g \in G.^[44] This multiplicative property simplifies computations, such as orthogonality relations and decomposition multiplicities, in the character ring of representations. For finite groups or compact Lie groups, it facilitates the projection onto irreducible components using the inner product of characters.^[44] In general, the tensor product of two irreducible representations is reducible, decomposing into a direct sum of irreducible representations. For the special unitary group \mathrm{SU}(2), whose finite-dimensional irreducible representations are labeled by non-negative integers n (with dimension n+1), the decomposition of V_n \otimes V_m (where V_k denotes the irrep of dimension k+1) is given by the Clebsch-Gordan series:

V_n \otimes V_m = \bigoplus_{j = |n - m|}^{n + m} V_j,

with each summand appearing exactly once, assuming n \geq m.^[45] These coefficients, known as Clebsch-Gordan coefficients, determine the explicit intertwining maps between the tensor product and the summands, playing a central role in applications like angular momentum coupling in quantum mechanics.^[45] As an example, consider the tensor product of two fundamental representations of \mathrm{SU}(2), each of dimension 2 (corresponding to n = m = 1). This yields V_1 \otimes V_1 = V_0 \oplus V_2, decomposing into the trivial representation (dimension 1) and the 3-dimensional irrep. This illustrates how tensoring finite-dimensional complex representations over \mathbb{C} generates higher-weight irreps while isolating invariant subspaces.^[45]

Quadratic Forms and Multilinear Forms

A quadratic form q on a vector space V over a field K of characteristic not 2 is associated to a unique symmetric bilinear form B_q via the polarization identity:

B_q(x, y) = \frac{1}{4} \left( q(x + y) - q(x - y) \right).

This bijection allows the tensor product of quadratic forms to be defined through their bilinear forms. For quadratic forms q on V and r on W, the tensor product bilinear form B_{q \otimes r} on V \otimes W is given by B_{q \otimes r}(v_1 \otimes w_1, v_2 \otimes w_2) = B_q(v_1, v_2) B_r(w_1, w_2), extended by bilinearity to the entire space. The corresponding quadratic form q \otimes r on V \otimes W is then recovered by polarization: q \otimes r (z) = B_{q \otimes r}(z, z).^[46] Multilinear forms on vector spaces can be constructed using tensor products. A k-linear form on V^k is an element of the tensor product (V^*)^{\otimes k}, where V^* is the dual space. For alternating multilinear forms, which vanish upon swapping any two arguments, the relevant subspace is the exterior power \bigwedge^k V^*, obtained by antisymmetrizing the tensor product: for a multilinear form T \in (V^*)^{\otimes k}, the alternating projection is

\text{Alt}(T)(v_1, \dots, v_k) = \frac{1}{k!} \sum_{\sigma \in S_k} \operatorname{sgn}(\sigma) T(v_{\sigma(1)}, \dots, v_{\sigma(k)}),

where S_k is the symmetric group. This construction yields the space of alternating k-forms A^k(V^*). A key example is the determinant on \mathbb{R}^n, which is the unique alternating multilinear form \det: (\mathbb{R}^n)^n \to \mathbb{R} that equals 1 on the standard basis, up to scalar multiple.^[47] Over the real numbers, Sylvester's law of inertia classifies nondegenerate quadratic forms up to orthogonal equivalence. For a quadratic form of dimension n, there exists an orthogonal basis in which it is diagonal with p entries of +1 and s entries of -1, where p + s = n, and the pair (p, s) is invariant under the action of the orthogonal group O(V). This signature (p, s) is preserved for the tensor product q \otimes r: if q has signature (p_1, s_1) and r has (p_2, s_2), then q \otimes r has signature (p_1 p_2 + s_1 s_2, p_1 s_2 + s_1 p_2), as the eigenvalues multiply pairwise in the diagonal representation, and the orthogonal group O(V \otimes W) acts to preserve this inertia.^[46] An illustrative example arises with inner products, which are positive definite quadratic forms. The tensor product of inner products \langle \cdot, \cdot \rangle_V on V and \langle \cdot, \cdot \rangle_W on W defines an inner product on V \otimes W by \langle v_1 \otimes w_1, v_2 \otimes w_2 \rangle = \langle v_1, v_2 \rangle_V \langle w_1, w_2 \rangle_W, extended by linearity. This induces a metric on the tensor product space equivalent to the product structure in finite dimensions, where orthonormal bases yield an orthonormal basis for the tensor product.^[46]

Sheaves and Line Bundles

In algebraic geometry, given a ringed space (X, \mathcal{O}_X) and sheaves of \mathcal{O}_X-modules \mathcal{F} and \mathcal{G}, their tensor product \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{G} is the sheaf associated to the presheaf U \mapsto \Gamma(U, \mathcal{F}) \otimes_{\Gamma(U, \mathcal{O}_X)} \Gamma(U, \mathcal{G}), which satisfies the stalkwise property (\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{G})_x = \mathcal{F}_x \otimes_{\mathcal{O}_{X,x}} \mathcal{G}_x for every point x \in X. This local computation via module tensor products ensures that the tensor product sheaf inherits the universal property of the tensor product in the category of modules. Line bundles on X, also known as invertible sheaves, are locally free \mathcal{O}_X-modules of rank 1. The tensor product of two line bundles \mathcal{L} and \mathcal{M} is defined using local trivializations: if \{U_i\} is an open cover with transition functions g_{ij}: U_{ij} \to \mathbb{C}^\times for \mathcal{L} and h_{ij}: U_{ij} \to \mathbb{C}^\times for \mathcal{M}, then \mathcal{L} \otimes \mathcal{M} has transition functions g_{ij} h_{ij}.^[48] This multiplication of transition functions makes the tensor product operation compatible with the cocycle condition, yielding another line bundle. For instance, on the projective line \mathbb{P}^1, the bundles \mathcal{O}(m) and \mathcal{O}(n) satisfy \mathcal{O}(m) \otimes \mathcal{O}(n) \cong \mathcal{O}(m+n).^[48] The isomorphism classes of line bundles on X form the Picard group \operatorname{Pic}(X), an abelian group under tensor product, where the identity is the trivial bundle \mathcal{O}_X and the inverse of \mathcal{L} is its dual \mathcal{L}^\vee \cong \mathcal{L}^{-1}, with transition functions g_{ij}^{-1}.^[48] This group structure captures the extent to which line bundles twist sections over X, and \operatorname{Pic}(X) \cong H^1(X, \mathcal{O}_X^\times) via the exponential sequence. A representative example is the tensor product involving the canonical bundle \omega_X, the determinant of the cotangent sheaf on a smooth variety X. On projective space \mathbb{P}^n, \omega_{\mathbb{P}^n} \cong \mathcal{O}_{\mathbb{P}^n}(-n-1), so the tensor product with itself yields \omega_{\mathbb{P}^n} \otimes \omega_{\mathbb{P}^n} \cong \mathcal{O}_{\mathbb{P}^n}(-2n-2), corresponding to the bundle of holomorphic (n,n)-forms twisted by the anticanonical divisor.

Fields and Graphs

In the context of field extensions, consider a base field K and two extensions L and M containing K. The tensor product L \otimes_K M, regarded as an M-algebra (or L-algebra), is a ring whose structure reflects the compatibility of the embeddings of L into extensions of M. If both L/K and M/K are finite separable extensions, then L \otimes_K M is a finite étale K-algebra, meaning it decomposes as a direct product of finite separable field extensions of K.^[49] This decomposition arises because separability ensures that the minimal polynomials of primitive elements split into distinct linear factors over the algebraic closure, allowing the Chinese Remainder Theorem to apply after base change. Linear disjointness, which holds for separable extensions, guarantees that the tensor product has no nilpotent elements and is reduced.^[49] A concrete illustration occurs when tensoring a number field with the reals. For instance, let K = \mathbb{Q}(\alpha) where \alpha is a root of the irreducible cubic polynomial x^3 - x - 1 \in \mathbb{Q}, which has one real root and two complex conjugate roots. This extension K/\mathbb{Q} has degree 3 and is separable. Then K \otimes_{\mathbb{Q}} \mathbb{R} \cong \mathbb{R}/(x^3 - x - 1). Over \mathbb{R}, the polynomial factors as a product of a linear factor and an irreducible quadratic, yielding

K \otimes_{\mathbb{Q}} \mathbb{R} \cong \mathbb{R} \times \mathbb{C}

by the Chinese Remainder Theorem, where the \mathbb{R} component corresponds to the real embedding and the \mathbb{C} to the complex pair.^[16] The normal basis theorem plays a key role in understanding such tensor products, particularly for Galois extensions. If L/K is a finite Galois extension with Galois group G = \mathrm{Gal}(L/K), then L \otimes_K L \cong \prod_{\sigma \in G} L as L-algebras, where the isomorphism identifies the left L with the diagonal embedding and the right L acts via the Galois action. The normal basis theorem asserts the existence of an element \beta \in L such that \{\sigma(\beta) \mid \sigma \in G\} forms a basis for L over K, enabling an explicit construction of this isomorphism by mapping the regular representation L \cong K[G] (as K-vector spaces) compatibly with the group action.^[50] This basis simplifies computations of traces and norms in the tensor product, which are essential in algebraic number theory for deducing properties like the decomposition of primes.^[50] In graph theory, the tensor product (also known as the direct product or categorical product) of two graphs G and H is defined on the Cartesian product of their vertex sets V(G \times H) = V(G) \times V(H), with an edge between distinct vertices (u, v) and (u', v') if and only if u is adjacent to u' in G and v is adjacent to v' in H.^[51] This operation corresponds to the Kronecker product of the adjacency matrices of G and H, preserving structural properties such as girth and chromatic number in certain cases. For example, the tensor product of two cycles C_m \times C_n is a 4-regular graph used to model lattice structures or interconnection networks in computer science. Applications include analyzing graph homomorphisms, where the tensor product encodes simultaneous colorings, and studying spectral properties for eigenvalue multiplicities in combinatorial optimization.^[51]

Categorical Perspectives

Monoidal Categories

In category theory, the tensor product generalizes to the structure of a monoidal category, providing a framework that unifies various instances of tensor products across different mathematical contexts. A monoidal category is a quintuple (\mathcal{C}, \otimes, I, \alpha, \lambda, \rho), where \mathcal{C} is a category, \otimes: \mathcal{C} \times \mathcal{C} \to \mathcal{C} is a bifunctor called the tensor product, I \in \mathrm{Ob}(\mathcal{C}) is the unit object, \alpha_{A,B,C}: (A \otimes B) \otimes C \to A \otimes (B \otimes C) is a natural isomorphism known as the associator for all objects A, B, C \in \mathrm{Ob}(\mathcal{C}), \lambda_A: I \otimes A \to A is the left unitor, and \rho_A: A \otimes I \to A is the right unitor, both natural in A. These structure morphisms satisfy two coherence conditions: the pentagon identity, which ensures associativity up to isomorphism in a diagram involving four applications of the tensor product,

\begin{CD} ((A \otimes B) \otimes C) \otimes D @>{\alpha_{A,B,C \otimes D}}>> (A \otimes (B \otimes C)) \otimes D \\ @V{\alpha_{A,B,C} \otimes D}VV @VV{\alpha_{A,B \otimes C,D}}V \\ (A \otimes B) \otimes (C \otimes D) @>>{\alpha_{A,B,C \otimes D}}> A \otimes ((B \otimes C) \otimes D) \\ @V{A \otimes \alpha_{B,C,D}}VV \\ A \otimes (B \otimes (C \otimes D)) \end{CD}

and the triangle identity, which relates the unitors and associator in a diagram for three objects.^[52] Mac Lane's coherence theorem asserts that in any monoidal category, every diagram composed solely of instances of the associator \alpha, the unitors \lambda and \rho, and identity morphisms commutes. This result implies that the monoidal structure behaves "as if" it were strict (where \alpha, \lambda, and \rho are identities), up to canonical isomorphism, simplifying computations and proofs by allowing one to ignore the isomorphisms in many cases. The theorem is proven by showing that the free strict monoidal category on a set of generators is the classifying category for monoidal structures, and all such diagrams reduce to the same morphism via normal forms.^[52]^[53] A prototypical example of a symmetric monoidal category is \mathbf{Vect}_K, the category of vector spaces over a field K (with linear maps as morphisms), equipped with the usual tensor product of vector spaces \otimes_K, the one-dimensional space K as unit, and the standard associator, unitors, and braiding \sigma_{V,W}: V \otimes_K W \to W \otimes_K V given by swapping basis elements. The symmetry satisfies \sigma_{W,V} \circ \sigma_{V,W} = \mathrm{id} and coheres with the associator via two hexagon identities.^[52] Another example is the category \mathbf{Rel} of sets (as objects) and binary relations (as morphisms, composed via relational composition), which forms a symmetric monoidal category with tensor product given by the cartesian product of sets A \times B, unit the singleton set \{*\}, and the induced structure on relations (where a relation R \subseteq A \times B and S \subseteq C \times D tensor to R \times S \subseteq (A \times C) \times (B \times D)). The associator and unitors are the canonical isomorphisms from the associativity of products and the identification of A \times \{*\} \cong A.^[54]

Quotient Constructions

In algebra, many important constructions involving tensor products arise as quotients of the tensor algebra T(V) = \bigoplus_{n=0}^\infty V^{\otimes n} of a vector space V by suitable two-sided ideals generated by relations. For instance, the symmetric algebra S(V), which encodes symmetric multilinear forms, is obtained as the quotient T(V) / I, where I is the ideal generated by elements of the form v \otimes w - w \otimes v for all v, w \in V. This quotient enforces commutativity in the product, making S(V) the free commutative algebra generated by V. Similarly, the exterior algebra \Lambda(V), used for alternating multilinear forms and determinants, is the quotient T(V) / J, where J is the ideal generated by v \otimes v for all v \in V, imposing antisymmetry and nilpotency on repeated factors. These constructions highlight how quotienting the tensor algebra by homogeneous ideals of degree 2 yields graded algebras that capture specific symmetry properties essential in multilinear algebra.^[55]^[56]^[57]^[58] A prominent example of such a quotient is the Clifford algebra Cl(V, Q) associated to a vector space V equipped with a quadratic form Q. It is defined as the quotient T(V) / K, where K is the two-sided ideal generated by elements v \otimes v - Q(v) \cdot 1 for v \in V, with $1 the unit in degree 0. This relation generalizes both the symmetric and exterior cases: when Q = 0, it recovers the exterior algebra, while positive definite Q yields structures used in spinor representations and quadratic form theory. The Clifford algebra thus provides a unified framework for studying quadratic relations in tensor products, with applications in geometry and physics.^[59]^[60] In the category of modules over a commutative ring R, which is an abelian category equipped with a tensor product \otimes_R, the functor - \otimes_R N preserves all colimits for any R-module N, as it is left adjoint to the internal Hom functor \operatorname{Hom}_R(N, -). This property holds symmetrically for the second variable and ensures that tensor products commute with direct sums, coproducts, and filtered colimits, facilitating computations in homological algebra. However, the tensor product does not always preserve finite limits or exact sequences, leading to the need for derived constructions. The left derived functors \operatorname{Tor}_i^R(M, N) measure the failure of exactness, providing "derived quotients" that capture higher-order obstructions in tensoring with quotients of modules; for example, in a short exact sequence $0 \to K \to F \to M \to 0, tensoring with N yields a long exact sequence involving \operatorname{Tor}_i^R(K, N) and \operatorname{Tor}_i^R(M, N), allowing resolution of the derived tensor product in the derived category. These Tor groups thus enable precise handling of quotients in non-exact tensor scenarios.^[61]

References

[1]
[PDF] Introduction to the Tensor Product - UCSB Math
In mathematics, a tensor refers to objects that have multiple indices. Roughly speaking this can be thought of as a multidimensional array. A good starting ...
[2]
[PDF] Lecture 16: Tensor Products
We can do the same for the tensor product by specifying its universal property. Definition. Let U and V be vector spaces. Then the tensor product is vector.
[3]
[PDF] Notes on Tensor Products - Brown Math
May 3, 2014 · Existence of the Universal Property: The tensor product has what is called a universal property. the name comes from the fact that the ...
[4]
[PDF] TENSOR PRODUCTS 1. Introduction Let R be a commutative ring ...
(Mathematician) Tensors belongs to a tensor space, which is a module defined by a multilinear universal mapping property. • (Physicist) “Tensors are systems ...
[5]
[PDF] TENSOR PRODUCTS 1. Introduction Let R be a commutative ring ...
For instance, in linear algebra it is useful to enlarge Rn to Cn, creating in this way a complex vector space by letting the real coordinates be extended to ...
[6]
[PDF] notes on tensor products and the exterior algebra - UMD MATH
Feb 4, 2019 · (This is in most linear algebra books used in 405. For example ... Theorem 17 (Universal property of tensor product). Suppose V1,...,Vn ...
[7]
[PDF] Tensor Product of vector spaces - Harvard Mathematics Department
A basis for this vector space is given by the “delta functions” ... A basis for the tensor product. At the end of class, I claimed that ...
[8]
[PDF] SOME MULTILINEAR ALGEBRA 1. Tensor products. Let V,W be ...
The quotient space F(V,W)/R(V,W) is called the tensor product of V and W (denoted V ⊗ W). The projection of (v, w) is denoted v ⊗ w, and in the quotient ...
[9]
[PDF] multilinear algebra notes - Eugene Lerman
The existence of the map ¯b satisfying the above conditions is called the universal property of the tensor product. This definition is quite abstract. It is not ...
[10]
[PDF] Notes on Tensor Products and the Exterior Algebra
Jul 16, 2012 · The tensor product can also be defined for more than two vector spaces. ... tensor product, we need the notion of a free vector space.
[11]
[PDF] Graduate Texts in Mathematics - Serge Lang - Algebra
Library of Congress Cataloging-in-Publication Data. Algebra / Serge Lang. ... Linearly disjoint extensions. 357. 360. 363. 4. Separable and regular extensions.
[12]
Section 9.27 (09IC): Linearly disjoint extensions—The Stacks project
9.27 Linearly disjoint extensions. Let k be a field, K and L field extensions of k. Suppose also that K and L are embedded in some larger field \Omega .Missing: textbook | Show results with:textbook
[13]
[PDF] 27. Tensor products
A tensor product of R-modules M, N is an R-module M ⊗R N with an R-bilinear map τ : M × N −→ M ⊗R N. The notation m ⊗ n denotes the image of m × n.
[14]
[PDF] Tensor spaces - the basics
Here is our basic definition of the tensor product of two vector spaces: Definition 1.5 (The tensor product of two vector spaces). Let V1 and V2 be vector ...
[15]
[PDF] extra material on tensor, symmetric and exterior algebras
Tensor algebra of a vector space. Let V be a vector space over F and V ∗ be its dual space. There is a natural (canonical) isomorphism of graded vector spaces.
[16]
[PDF] tensor products ii - keith conrad
Proof. Use induction and associativity of the tensor product of linear maps. Remark 2.10. Equation (2.1) in the setting of vector spaces and ...
[17]
[PDF] Math 55a: Honors Abstract Algebra Tensor products Slogan. Tensor ...
Tensor products of vector spaces are to Cartesian products of sets as direct ... A universal property. Suppose now that we have a linear map. F : U ⊗ V ...
[18]
2. Tensor products and the Tor functors
Since vector spaces are free (Proposition VI.1.7), tensoring is exact in k-Vect: if. 0. // V1. // V2. // V3. // 0 is an exact sequence of k-vector spaces and W ...
[19]
[PDF] Multilinear Mappings and Tensors
A multilinear mapping is linear in each variable. A tensor is a scalar-valued multilinear function with variables in both V and V*.<|control11|><|separator|>
[20]
[PDF] What is a tensor? (Part II)
Apr 29, 2021 · contravariant 1-tensor: a0 = X−1a a0 = Xa covariant 1-tensor: a0 = XTa a0 = X−Ta covariant 2-tensor: A0 = XTAX. A0 = X−TAX−1.
[21]
[PDF] Introduction to representation theory - MIT Mathematics
Jan 10, 2011 · 1.1 What is representation theory? In technical terms, representation theory studies representations of associative algebras. Its general.
[22]
[PDF] Introduction to Lie Algebras and Representation Theory
This book is designed to introduce the reader to the theory of semisimple. Lie algebras over an algebraically closed field of characteristic 0, with emphasis on ...Missing: action Harris
[23]
[PDF] The Killing Form, Reflections and Classification of Root Systems 1 ...
This can be done using the Killing form, and since this provides an adjoint-invariant inner product, it will turn out that the Weyl group acts by isometries ...
[24]
[PDF] 1. The Tensor Product - UC Berkeley math
A bilinear map T : M ×N → P is a map such that for any m ∈ M, the induced map T(m, •) : N → P (that is, f : N → P sending f(n) = T(m, n)) is linear and for any ...
[25]
[PDF] Notes on Tor and Ext - UChicago Math
These notes cover basic homological algebra, Torsion products, and Ext groups, including chain complexes, tensor products, and short/long exact sequences.
[26]
[PDF] NONCOMMUTATIVE RINGS Michael Artin class notes, Math 251 ...
Tensor products. If MA and AP are right and left A-modules, we can form the tensor product M ⊗A P. It is defined as the abelian group generated by elements ...
[27]
[PDF] Notes on tensor products Robert Harron - Department of Mathematics
We may now define the tensor product M ⊗R N by a universal property as we did in the case of commutative R. For simplicity, we will eschew the language of ...<|control11|><|separator|>
[28]
[PDF] with a View Toward Algebraic Geometry
David Eisenbud. Commutative Algebra with a View Toward Algebraic Geometry. With 90 Illustrations. $. Springer-Verlag. New York Berlin Heidelberg London Paris.
[29]
[PDF] 1 Chapter 2 - 1.1 Tensor product
... nZ are cyclic, generated by 1, (Z/mZ)⊗Z. (Z/nZ) is cyclic and ... Atiyah-MacDonald. 3. Exercise 3. Prove that if A is a local ring, M and N are ...
[30]
[PDF] Hideyuki Matsumura - Commutative Algebra
• Hideyuki Matsumura, Commutative Algebra. – https://aareyanmanzoor.github ... Bourbaki. Commutative Algebra: Chapters 1-7. Elements de mathema- tique ...
[31]
[PDF] 5. Tensor Products ∑ ∑
universal property we conclude that ψ ◦ϕ = idT1 . In the same way we see ... Construction 5.19 (Tensor product of algebras). Let R be a ring, and let ...
[32]
[PDF] 18.783 Elliptic Curves Lecture 12
Mar 9, 2022 · Every ring is a Z-algebra. Definition. The tensor product of two R-algebras A and B is the R-algebra A ⊗R B generated by the formal symbols ...
[33]
[PDF] MULTIVARIABLE ANALYSIS What follows are lecture notes from an ...
Feb 2, 2021 · . (21.22) Tensor product of algebras. Let A1,A2 be algebras. In particular, they are vector spaces and so we can form the tensor product ...
[34]
[PDF] Tensor Factorizations of Group Algebras and Modules
To show this, we use the following. 5.1. LEMMA. The centre of a tensor product of algebras is the tensor product of the centres of the tensor factors. Page ...Missing: center | Show results with:center
[35]
[PDF] Tensor Product and Local Interior G-Algebras
Let i, j be idempotents in F-algebras A and B, respectively. Then i ⊗ j is an idempotent in A ⊗ B; furthermore, i ⊗ j is primitive if and only if i and j ...
[36]
[PDF] Representation theory of finite-dimensional algebras
Nov 18, 2014 · Show that there is a canonical k-algebra isomorphism k(G × H) ∼= kG ⊗k kH. (6) Let n be a positive integer. Show that there is a canonical k- ...
[37]
[PDF] complexification - keith conrad
Introduction. We want to describe a procedure, called complexification, that enlarges real vector spaces to complex vector spaces in a natural way.
[38]
[PDF] Essays in analysis Nuclear spaces - UBC Math
Oct 12, 2012 · This essay is intended to be an introduction to the theory of nuclear spaces adapted to the needs of non-experts.
[39]
https://www.ltcc.ac.uk/media/qmul-images/Representation-Theory-Notes-.pdf
[40]
[1407.0457] Projective tensor products and Apq spaces - arXiv
Jul 2, 2014 · In particular, the generalised form of the classical result Lp*Lq is a subset of Lr; where 1/r = 1/p + 1/q - 1; is shown to be true in this ...
[41]
[PDF] Linear Representations of Finite Groups
1 Generalities on linear representations. 3. 1.1 Definitions. 3. 1.2 Basic examples. 4. 1.3 Subrepresentations. 5. 1.4 Irreducible representations.
[42]
[PDF] SU(2) Representations and Their Applications
This decomposition of the tensor product goes under the name “Clebsch-. Gordan” decomposition. The general issue of how the tensor product of two irreducibles ...
[43]
[PDF] Quadratic forms Chapter I: Witt's theory
In these notes we give a detailed treatment of the foundations of the algebraic theory of quadratic forms, starting from scratch and ending with Witt Cancella-.
[44]
[PDF] Note 12. Alternating tensors Differential Geometry, 2005 Let V be a ...
2. Let V = Rn. The n × n determinant is multilinear and alternating in its columns, hence it can be viewed as an alternating map (Rn)n → R.
[45]
[PDF] introduction to algebraic geometry, class 23
These are all the line bundles on Pm. Hence. PicPn ∼= Z, with O(1) the generator. ii) Tensor product of two invertible sheaves. Take 1: Suppose you have two.
[46]
[PDF] 4Étale algebras, norm and trace - 4.1 Separability - MIT Mathematics
Sep 18, 2017 · The decomposition of an étale algebra into field extensions is unique up to permutation and isomorphisms of factors. 1There are many ...
[47]
[PDF] Galois theory and the normal basis theorem - UC Berkeley math
Dec 3, 2010 · The inverse of β takes an element t of T to the function g 7→ gt. The key to our proof is the so-called “Normal basis theorem.” It gives an ...
[48]
Graph Tensor Product -- from Wolfram MathWorld
The graph tensor product of simple graphs G and H is given by A(G×H)=A(G) tensor A(H), where tensor denotes the Kronecker product.
[49]
[PDF] maclane-categories.pdf - MIT Mathematics
... GREUB. Linear Algebra. 4th ed. Theory I: Elements of Functional. 24 ... multilinear algebra which were important to French Mathe- matical traditions ...
[50]
[PDF] A Coherent Proof of Mac Lane's Coherence Theorem
Both categories can be regarded monoidal: (LieAlg,⊕,{•}) is the monoidal category where we apply the cartesian product between Lie algebras, and. (k-Alg,⊗,k) is ...
[51]
[PDF] Chapter 5: Monoidal Categories
Monoidal categories combine objects and morphisms, like monoids. A strict monoidal category has a tensor product and a unit, such as (N,+,0) with addition and ...
[52]
[PDF] FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 26
Jan 25, 2008 · The symmetric algebra Sym∗ M is a symmetric algebra, graded by Z≥0, defined as the quotient of T∗(M) by the (two-sided) ideal generated ...
[53]
[PDF] arXiv:1812.02452v5 [math.CT] 26 Feb 2021
Feb 26, 2021 · coproducts, the symmetric algebra. Sym•X = M i∈N. SymiX ∈ algC is the maximal commutative quotient of the tensor algebra of X. It is ...
[54]
[PDF] Clifford Algebras as Filtered Algebras
May 11, 2005 · Definition. The Clifford algebra of V C(V ) is the quotient T(V )/D(V ). The Clifford algebra is no longer graded. This is because now when one ...
[55]
[PDF] The exterior algebra and central notions in mathematics
The exterior algebra is the quotient algebra of T(V ) by the relations R. More formally, let hRi be the two-sided ideal in T(V ) generated by R. The ...<|control11|><|separator|>
[56]
[PDF] Math 210C. Clifford algebras and spin groups
The description of this Clifford algebra as an n-fold Z/2Z-graded tensor product (in the b⊗-sense) of copies of C(R,−x2) = C should not be confused with the ...
[57]
[PDF] On the bundle of Clifford algebras over the space of quadratic forms
Dec 13, 2023 · Abstract. For each quadratic form Q ∈ Quad(V ) on a vector space over a field K, we can define the Clifford algebra Cl(V,Q) as the quotient.
[58]
[PDF] AN INTRODUCTION TO HOMOLOGICAL ALGEBRA
... HOMOLOGICAL ALGEBRA terms of use, available at https://www.cambridge ... Cartan-Eilenberg resolution P** of a chain complex A* in A is an upper half ...