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Tensor algebra

The tensor algebra of a vector space V over a field k (or more generally, of a module M over a commutative ring) is the free associative algebra generated by V, constructed as the direct sum T(V) = \bigoplus_{n=0}^\infty V^{\otimes n}, where V^{\otimes n} is the n-fold tensor power of V, and multiplication is given by the tensor product. This structure generalizes multilinear algebra to higher-order multilinear objects known as tensors, which can be viewed as elements of these tensor powers and defined as multilinear maps from Cartesian products of V and its dual V* to k, enabling coordinate-independent descriptions of multi-dimensional linear relationships.^[1]^[2] Tensors are classified by type (p, q), with p contravariant indices and q covariant indices; examples include scalars as (0,0)-tensors, vectors as (1,0)-tensors, covectors as (0,1)-tensors, and linear transformations as (1,1)-tensors.^[1]^[3] The algebraic structure includes the tensor product, combining tensors into higher-order ones via a universal bilinear property, and contraction, reducing order by summing over paired indices.^[4]^[2] Tensors support addition, scalar multiplication, and, in inner product spaces, inner products, forming vector spaces themselves. These structures transform under basis changes, with contravariant components scaling by the inverse Jacobian and covariant by the direct Jacobian, preserving physical and geometric meanings.^[1] Historically, tensor concepts arose in the 19th century with August Cauchy’s 1822 stress tensor in continuum mechanics and Bernhard Riemann’s 1854 curvature tensor in differential geometry, followed by Josiah Willard Gibbs’s 1884 "indeterminate product" for vector tensor products and Woldemar Voigt’s 1898 coining of "tensor" in crystal physics.^[5] Gregorio Ricci-Curbastro and Tullio Levi-Civita formalized tensor calculus in their 1900 work on absolute differential calculus, influencing Albert Einstein’s general relativity by enabling covariant physical laws.^[5]^[1] In the 20th century, tensor products extended to Hilbert spaces by F.J. Murray and John von Neumann in 1936 with ⊗ notation, to abelian groups by Hassler Whitney in 1938, and to modules over commutative rings by the Bourbaki group in 1948, establishing tensor algebra in abstract algebra.^[5] Tensor algebra applies in physics for stress-strain and spacetime curvature, engineering in continuum mechanics, and computer science for multidimensional data and machine learning.^[3]^[2] It supports exterior algebra for differential forms, symmetric algebras for polynomials, and tensor decompositions in computational problems like matrix multiplication.^[4]^[5]

Fundamentals

Definition

In tensor algebra, the tensor algebra of a vector space V over a field k, denoted T(V), is defined abstractly as the free associative unital k-algebra generated by V. This means that T(V) is an associative unital k-algebra equipped with a k-linear map \iota: V \to T(V) satisfying the following universal property: for any associative unital k-algebra A and any k-linear map f: V \to A, there exists a unique k-algebra homomorphism \tilde{f}: T(V) \to A such that \tilde{f} \circ \iota = f.^[4]^[6]^[7] More generally, the construction applies when V is a module over a commutative ring k with unity, though the focus here is on the case where k is a field for simplicity, ensuring that T(V) is a free associative unital algebra without additional relations imposed on the generators from V.^[4] The universal property characterizes T(V) up to unique isomorphism, guaranteeing its existence and uniqueness in the category of associative unital k-algebras.^[6]^[7] As a graded algebra, T(V) includes a zero-degree component isomorphic to k \cdot 1, where $1 denotes the multiplicative unit of T(V), corresponding to the empty tensor product.^[4]^[6] This explicit realization as the direct sum of tensor powers is detailed in the construction of tensor algebra.^[7]

Motivations and examples

Tensor algebra arises as a fundamental structure in multilinear algebra, where it provides a framework for handling multilinear maps and their compositions without imposing additional relations, allowing the encoding of all possible multilinear expressions on a vector space.^[8] This generalization extends the concept of polynomial rings to the case of non-commuting variables, serving as the free associative algebra generated by a vector space V, which captures the universal property for extending linear maps to algebra homomorphisms.^[9] In representation theory, tensor algebra facilitates the decomposition of tensor powers into irreducible representations under group actions, such as those of \mathrm{GL}(V), highlighting its role in organizing invariant subspaces.^[8] Moreover, it acts as a starting point for constructing quotient algebras, including the symmetric algebra (encoding commuting variables) and the exterior algebra (incorporating antisymmetry), which are obtained by imposing specific relations on the tensor product structure.^[8] A basic example illustrates this in the one-dimensional case: if V = k is a one-dimensional vector space over a field k, then the tensor algebra T(V) is isomorphic to the polynomial ring k, where x generates V, and elements are formal sums \sum a_i x^i with the usual commutative multiplication.^[9] For a higher-dimensional space V = k^n with basis \{e_1, \dots, e_n\}, T(V) consists of non-commutative polynomials, which can be expressed as formal sums \sum_{m=0}^\infty \sum_{i_1, \dots, i_m} a_{i_1 \dots i_m} e_{i_1} \cdots e_{i_m}, where the multiplication is associative but not necessarily commutative, reflecting the tensor product concatenation.^[9] This structure emphasizes the infinite-dimensional nature of T(V) even when V is finite-dimensional, as the grading allows arbitrarily high tensor powers unless artificially truncated. In physics, tensor algebra provides the foundational structure for applications in areas like crystal physics and electromagnetism, from which specialized algebras such as those for multivectors can be derived. It also underpins operator algebras by providing a non-commutative setting for composing linear operators, as seen in quantum mechanics where tensor products model composite systems.^[10]

Construction

Tensor powers and grading

The tensor algebra T(V) of a vector space V over a field k is constructed as the graded vector space given by the direct sum of the tensor powers of V. Specifically, for each nonnegative integer n, the n-th tensor power V^{\otimes n} is defined inductively: V^{\otimes 0} = k, V^{\otimes 1} = V, and for n \geq 2, V^{\otimes n} = V^{\otimes (n-1)} \otimes V, where \otimes denotes the tensor product of vector spaces over k.^[11] The tensor product operation is multilinear, meaning that if f: V_1 \times \cdots \times V_n \to W is a map that is linear in each argument separately (while fixing the others), then there exists a unique linear map \tilde{f}: V_1 \otimes \cdots \otimes V_n \to W such that \tilde{f}(v_1, \dots, v_n) = f(v_1, \dots, v_n).^[11] If \{v_i\}_{i \in I} is a basis for V, then the set \{ v_{i_1} \otimes \cdots \otimes v_{i_n} \mid i_1, \dots, i_n \in I \} forms a basis for V^{\otimes n}.^[12] Thus, the dimension of V^{\otimes n} is (\dim V)^n when V is finite-dimensional, and infinite otherwise. The full tensor algebra is then the graded direct sum

T(V) = \bigoplus_{n=0}^\infty V^{\otimes n},

where addition and scalar multiplication are defined componentwise: for homogeneous elements x \in V^{\otimes m} and y \in V^{\otimes n}, their sum is zero in degrees other than m and n, and scalar multiplication \lambda \cdot x lies in the same component as x.^[11] The grading is made explicit by the projection maps \pi_n: T(V) \to V^{\otimes n}, which send an element to its degree-n component and zero elsewhere; these projections are linear and satisfy \pi_n \circ \iota_m = \delta_{n m} \mathrm{id}_{V^{\otimes m}}, where \iota_m: V^{\otimes m} \hookrightarrow T(V) is the inclusion.^[11] If \dim V = d \geq 1 is finite, then \dim T(V) = \infty, as the dimensions of the components grow without bound. The Hilbert series of T(V), which encodes the dimensions of the graded pieces, is the formal power series

\sum_{n=0}^\infty (\dim V^{\otimes n}) t^n = \sum_{n=0}^\infty d^n t^n = \frac{1}{1 - d t}.

This geometric series reflects the exponential growth in the size of the tensor powers.^[12]

Associative multiplication

The tensor algebra T(V) over a vector space V is equipped with an associative multiplication derived from the tensor product structure. For homogeneous elements x \in V^{\otimes m} and y \in V^{\otimes n}, the concatenation map defines the product as x \cdot y = x \otimes y \in V^{\otimes (m+n)}, where the tensor product on the right is the standard one between tensor powers. This map extends bilinearly to the full graded direct sum T(V) = \bigoplus_{k \geq 0} V^{\otimes k}, yielding a well-defined multiplication on all of T(V).^[4]^[13] Formally, the multiplication is given by a linear map \mu: T(V) \otimes T(V) \to T(V) that restricts on graded components to the identity maps \mu|_{V^{\otimes m} \otimes V^{\otimes n}} = \mathrm{id}_{V^{\otimes m}} \otimes \mathrm{id}_{V^{\otimes n}}: V^{\otimes m} \otimes V^{\otimes n} \to V^{\otimes (m+n)}, extended by linearity across the direct sum decomposition. This construction ensures the multiplication is bilinear over the base field and compatible with the grading.^[9]^[13] Associativity follows directly from the associativity of the tensor product: for homogeneous elements x \in V^{\otimes m}, y \in V^{\otimes n}, and z \in V^{\otimes p}, one has (x \cdot y) \cdot z = (x \otimes y) \otimes z = x \otimes (y \otimes z) = x \cdot (y \cdot z) \in V^{\otimes (m+n+p)}, with the equality extending bilinearly to general elements. This property makes T(V) an associative algebra.^[4]^[9] The algebra is unital, with the degree-0 component T^0(V) \cong [k](/page/K) (the base field) providing the unit element $1. For any x \in T(V), the relations $1 \cdot x = x \cdot 1 = x hold, as multiplication by scalars in degree 0 acts as the identity on higher-degree components via the bilinear extension.^[13]^[4]

Universal Properties

Adjunction to forgetful functor

The forgetful functor U: \mathbf{AssAlg}_k \to \mathbf{Vec}_k from the category of associative unital algebras over a field k to the category of vector spaces over k sends an algebra A to its underlying vector space U(A), while forgetting the multiplication and unit maps. This functor has a left adjoint F: \mathbf{Vec}_k \to \mathbf{AssAlg}_k, which assigns to each vector space V the tensor algebra T(V), also denoted F(V). The adjunction F \dashv U is natural in both variables, establishing an isomorphism of sets

\mathbf{Hom}_{\mathbf{AssAlg}_k}(T(V), A) \cong \mathbf{Hom}_{\mathbf{Vec}_k}(V, U(A))

for any vector space V and algebra A. The unit of the adjunction is the canonical inclusion \eta_V: V \to U(T(V)), which embeds V into the tensor algebra as the degree-1 component. Explicitly, the bijection sends a linear map f: V \to A to the unique algebra homomorphism \tilde{f}: T(V) \to A such that \tilde{f} \circ \eta_V = f, obtained by extending f multilinearly to all tensor powers and preserving the concatenation product. Conversely, any algebra homomorphism g: T(V) \to A restricts to a linear map g \circ \eta_V: V \to A, yielding the inverse correspondence. This construction relies on the explicit formation of T(V) as the direct sum of tensor powers with associative multiplication. Categorically, this adjunction characterizes T(V) as the free associative unital algebra on the vector space V, meaning it is initial among algebras equipped with a linear map from V. In the broader context of universal algebra, such free constructions arise as left adjoints to forgetful functors for varieties of algebras defined by operations and identities, here the binary multiplication and unitality for associative algebras. The tensor algebra thus exemplifies the general principle that free algebras encode generators without relations, facilitating universal extensions in homological and operadic settings.

Universal mapping property

The tensor algebra T(V) of a vector space V over a field k possesses a universal mapping property that characterizes it as the free associative unital algebra generated by V. Specifically, for any associative unital k-algebra A and any k-linear map f: V \to A, there exists a unique algebra homomorphism \tilde{f}: T(V) \to A such that \tilde{f} restricts to f on V, and thus \tilde{f}(v_1 \cdots v_n) = f(v_1) \cdots f(v_n) for all v_i \in V and n \geq 1.^[14]^[15] This property ensures that T(V) universalizes the construction of algebra homomorphisms from linear maps into V, making it the "freest" such algebra. This universality arises from the graded structure of T(V) = \bigoplus_{n=0}^\infty V^{\otimes n}, where each homogeneous component V^{\otimes n} itself satisfies a universal property for multilinear maps. For an n-multilinear map \phi: V \times \cdots \times V \to A (n factors), the universal property of the tensor power yields a unique k-linear map \psi_n: V^{\otimes n} \to A such that \psi_n(v_1 \otimes \cdots \otimes v_n) = \phi(v_1, \dots, v_n). When n=1, this reduces to the linear extension case, directly composing with the inclusion V \hookrightarrow T(V) to yield the full algebra homomorphism via the overall property of T(V). In general, the multilinear maps approximate the algebra structure by specifying behavior on pure tensors, which the universal property of T(V) then extends consistently across degrees using the associative multiplication.^[14]^[15] To see this, the existence of \tilde{f} is constructed recursively by degree: on degree 0, \tilde{f}(1) = 1_A; on degree 1, \tilde{f}|_V = f; and for higher degrees, \tilde{f} on V^{\otimes n} is defined using the universal property of the tensor product iteratively, combined with the bilinear multiplication map of T(V), ensuring associativity and unitality. Uniqueness follows from the freeness of T(V), as every element is a finite sum of products of elements from V, so \tilde{f} is determined by its values on the generators V.^[14] This perspective interprets T(V) analogously to the algebra of non-commutative polynomials over k in non-commuting variables from a basis of V, where the universal property ensures that evaluation of such a "polynomial" at elements of A via f yields the product f(v_1) \cdots f(v_n) for monomials v_1 \cdots v_n. This connection underscores the role of T(V) in linearizing multilinear operations within algebraic structures.^[15]

Algebraic Structure

Graded components

The tensor algebra T(V) over a vector space V is \mathbb{Z}_{\geq 0}-graded, with the graded components given by the tensor powers T_n(V) = V^{\otimes n} for n \geq 0, where T_0(V) \cong k is the base field.^[15] Elements of T_n(V) are called homogeneous of degree n, such as pure tensors v_1 \otimes \cdots \otimes v_n for v_i \in V.^[15] Any element x \in T(V) decomposes uniquely as a finite sum x = \sum_{n \geq 0} x_n, where each x_n is homogeneous of degree n, reflecting the direct sum structure T(V) = \bigoplus_{n \geq 0} T_n(V).^[15] This decomposition ensures that algebraic operations respect the grading in a controlled manner. The subspace generated by V (identified with T_1(V)) forms a graded ideal I = \bigoplus_{n \geq 1} T_n(V), known as the augmentation ideal, which is the kernel of the projection onto the degree-0 component.^[16] Multiplication in T(V) is graded, meaning the product of a homogeneous element of degree m and one of degree n lies in degree m+n, thereby preserving or increasing the total degree of elements.^[15] Thus, T(V) itself is a graded algebra, and subalgebras generated by homogeneous elements inherit this structure, with the ideal I serving as a canonical graded subalgebra excluding scalars.^[15] The grading induces a descending filtration on T(V) defined by F^m T(V) = \bigoplus_{n \geq m} T_n(V) for m \geq 0, where each F^m T(V) is a subalgebra and F^{m+1} T(V) \subseteq F^m T(V).^[16] The associated graded algebra is then \mathrm{gr} T(V) = \bigoplus_{m \geq 0} \mathrm{gr}_m T(V), with \mathrm{gr}_m T(V) = F^m T(V) / F^{m+1} T(V) \cong T_m(V), yielding \mathrm{gr} T(V) \cong T(V) as graded algebras.^[16] This isomorphism highlights the compatibility between the filtration and the inherent grading of the tensor algebra. The augmentation map \varepsilon: T(V) \to k is the graded algebra homomorphism projecting onto the degree-0 component, defined by \varepsilon\left( \sum x_n \right) = x_0 and satisfying \varepsilon(1) = 1.^[15] Its kernel is precisely the augmentation ideal I, which consists of all elements with vanishing degree-0 part and generates the positive-degree components.^[15] This map provides a canonical way to extract scalar components while emphasizing the role of the grading in separating homogeneous parts.

Unit and scalar inclusion

The degree-0 component of the tensor algebra T(V) over a field k is isomorphic to k via the map \lambda \mapsto \lambda \cdot [1](/page/1), where [1](/page/1) denotes the multiplicative identity element in T(V).^[5] This unit element satisfies [1](/page/1) \cdot x = x \cdot [1](/page/1) = x for all x \in T(V), ensuring the algebra is unital.^[5] Scalar multiplication in T(V) is central, meaning that for any \lambda \in k and x \in T(V), the relation (\lambda \cdot 1) \cdot x = \lambda x = x \cdot (\lambda \cdot 1) holds, which aligns with the underlying vector space structure on V.^[5] The degree-0 component, consisting of scalar multiples of the unit, lies in the center of T(V).^[17] The inclusion map k \to T(V) given by \lambda \mapsto \lambda \cdot 1 endows T(V) with the structure of a unital k-algebra.^[5] This map is the unique algebra homomorphism from k to T(V), as k is the initial object in the category of commutative unital algebras.^[5] The image of this inclusion forms the center of the degree-0 component, which coincides with k itself since the degree-0 part is commutative; and in fact comprises the entire center of T(V).

Interpretation as non-commutative polynomials

The tensor algebra T(V) over a vector space V with basis \{ e_i \mid i \in I \} can be interpreted as the algebra of non-commutative polynomials in the indeterminates \{ e_i \}. Specifically, T(V) is the k-vector space with basis consisting of all finite words e_{i_1} \cdots e_{i_n} for n \geq 0 and i_j \in I, where the empty word (for n=0) corresponds to the unit element $1. Elements of T(V) are then formal linear combinations \sum c_{i_1 \dots i_n} e_{i_1} \cdots e_{i_n}, with coefficients c_{i_1 \dots i_n} \in k.^[18]^[19] Multiplication in T(V) is defined by concatenation of words, extended linearly to all elements, which preserves the non-commutativity inherent in the basis elements. For instance, e_1 e_2 \neq e_2 e_1 in general, distinguishing this structure from the commutative polynomial ring k[e_i \mid i \in I], where such relations would hold. This concatenation operation makes T(V) associative but not necessarily commutative, unless additional relations are imposed via quotients.^[18]^[19] As an algebraic structure, T(V) is isomorphic to the free associative algebra k \langle e_i \mid i \in I \rangle generated by the set \{ e_i \}, where the generators satisfy no relations beyond associativity. If \dim V < \infty, this is the free algebra on finitely many generators; otherwise, it involves infinitely many indeterminates. This isomorphism underscores the "freest" nature of T(V) among associative algebras containing V as a subspace.^[18] Algebra homomorphisms from T(V) to any associative algebra A over k are in one-to-one correspondence with linear maps from V to A, achieved by assigning images to the basis elements e_i and extending via the universal property, akin to substituting values for non-commuting variables in polynomials.^[19]^[18]

Quotients

General quotient algebras

In tensor algebra, the construction of more specific associative algebras often proceeds by forming quotients of the free tensor algebra T(V) by suitable two-sided ideals. For any two-sided ideal I \subset T(V), the quotient algebra T(V)/I inherits a natural structure as an associative unital algebra over the base field, where the multiplication is defined by (\alpha + I)(\beta + I) = \alpha \beta + I for homogeneous components \alpha, \beta \in T(V), and the unit is the image of the scalar inclusion in T^0(V). This quotient preserves the grading of T(V) when I is homogeneous, meaning I = \bigoplus_n (I \cap T^n(V)), ensuring that the induced algebra remains graded by tensor degree.^[20] Two-sided ideals in T(V) are typically generated by relations imposed on elements of V or higher tensor powers, extended multiplicatively across the algebra. For a subset R \subset V (or more generally, homogeneous elements in some T^k(V)), the ideal \langle R \rangle is the two-sided ideal generated by R, consisting of all finite sums of elements of the form \alpha r \beta where \alpha, \in T(V), r \in R, and \beta \in T(V). Such ideals allow the imposition of algebraic relations, like commutativity or anticommutativity, while maintaining the associative product in the quotient. The generation process leverages the universal property of T(V) as the free associative algebra, ensuring that the relations propagate consistently through tensor products.^[20]^[21] The quotient construction preserves a form of the universal mapping property: given a linear map \rho: V \to A to another associative unital algebra A such that the relations defining I hold in the image (i.e., the extension \tilde{\rho}: T(V) \to A vanishes on I), there exists a unique algebra homomorphism \overline{\rho}: T(V)/I \to A extending \rho. This makes T(V)/I the "freest" associative algebra satisfying the imposed relations via \rho. Furthermore, this framework is complete in the sense that every associative unital algebra A equipped with a linear map \iota: V \to A arises as such a quotient: specifically, A \cong T(V)/\ker(\tilde{\iota}), where \tilde{\iota}: T(V) \to A is the unique extension of \iota to an algebra homomorphism, and \ker(\tilde{\iota}) is a two-sided ideal. This representation underscores the tensor algebra's role as a universal generator for associative structures.^[20]^[22]

Key examples: symmetric, exterior, and universal enveloping

The symmetric algebra S(V) of a vector space V over a field k is constructed as the quotient T(V) / I, where I is the two-sided ideal generated by all commutators v \otimes w - w \otimes v for v, w \in V.^[23] This ideal, known as the commutator ideal, is explicitly [T(V), T(V)] = \span\{ ab - ba \mid a, b \in T(V) \}, ensuring that multiplication in S(V) is commutative.^[23] As a graded algebra, S(V) has a basis consisting of symmetric tensors, which are equivalence classes of pure tensors invariant under permutation of factors.^[24] In algebraic geometry, S(V) serves as the coordinate ring for affine space, providing a polynomial ring structure that encodes geometric properties of varieties.^[24] The exterior algebra \wedge(V), also called the Grassmann algebra, arises as the quotient T(V) / J, where J is the two-sided ideal generated by squares v \otimes v for all v \in V.^[21] This imposition yields a graded-commutative algebra, where the n-th graded component \wedge^n(V) consists of alternating multilinear forms, with the wedge product \alpha \wedge \beta = (-1)^{pq} \beta \wedge \alpha for \alpha \in \wedge^p(V) and \beta \in \wedge^q(V).^[21] The algebra is finite-dimensional if V is, with dimension $2^{\dim V}, and its basis elements are wedge products of distinct basis vectors from V.^[25] In differential geometry, \wedge(V) underpins the algebra of differential forms on manifolds, enabling the formulation of integration, Stokes' theorem, and de Rham cohomology.^[26] For a Lie algebra \mathfrak{g} over k, the universal enveloping algebra U(\mathfrak{g}) is the quotient T(\mathfrak{g}) / K, where K is the two-sided ideal generated by elements xy - yx - [x, y] for all x, y \in \mathfrak{g}.^[27] This relation embeds the Lie bracket into the associative multiplication of U(\mathfrak{g}), making it a non-commutative algebra that "envelops" \mathfrak{g} via the inclusion i: \mathfrak{g} \hookrightarrow U(\mathfrak{g}) in degree 1.^[27] As a Hopf algebra, U(\mathfrak{g}) admits a coproduct extending the primitive structure of \mathfrak{g}.^[28] It plays a central role in representation theory, where modules over U(\mathfrak{g}) correspond to representations of \mathfrak{g}, facilitating the study of weights, characters, and highest weight modules for semisimple Lie algebras.^[28]

Coalgebra Structure

Deconcatenation coproduct

The deconcatenation coproduct equips the tensor algebra T(V) over a vector space V with a natural coalgebra structure, defined as the unique unital algebra homomorphism \Delta: T(V) \to T(V) \otimes T(V) such that \Delta|_V = \mathrm{id}_V \otimes 1 + 1 \otimes \mathrm{id}_V.^[29]^[30] Explicitly, this yields \Delta(1) = 1 \otimes 1, \Delta(v) = v \otimes 1 + 1 \otimes v for v \in V, and for a pure tensor of degree n \geq 1,

\Delta(v_1 \otimes \cdots \otimes v_n) = \sum_{k=0}^n (v_1 \otimes \cdots \otimes v_k) \otimes (v_{k+1} \otimes \cdots \otimes v_n),

where empty tensors are the unit element.^[29]^[30]^[31] The coproduct \Delta is coassociative, satisfying (\Delta \otimes \mathrm{id}) \circ \Delta = (\mathrm{id} \otimes \Delta) \circ \Delta, a property arising from the iterative nature of the tensor product construction underlying T(V).^[29]^[30] Given the graded structure of T(V) = \bigoplus_{n \geq 0} T_n(V), the map \Delta preserves grading by sending T_n(V) into \bigoplus_{i+j=n} T_i(V) \otimes T_j(V).^[29]^[30] If \{e_i\} is a basis for V, then the monomials e_{i_1} \cdots e_{i_n} form a basis for T_n(V), and \Delta acts by summing over all deconcatenations of the index sequence: \Delta(e_{i_1} \cdots e_{i_n}) = \sum_{k=0}^n (e_{i_1} \cdots e_{i_k}) \otimes (e_{i_{k+1}} \cdots e_{i_n}).^[29]^[30]

Counit map

In tensor algebra, the counit map \varepsilon: T(V) \to k equips T(V) with a counital coalgebra structure by projecting onto the degree-zero component, where k is the base field and T(V) = \bigoplus_{n=0}^\infty V^{\otimes n}. For a general element \sum_n x_n with x_n \in V^{\otimes n}, \varepsilon(\sum_n x_n) extracts the scalar coefficient of the unit $1inx_0. Explicitly, \varepsilon(1) = 1, \varepsilon(v) = 0for allv \in V, and \varepsilon(v_1 \otimes \cdots \otimes v_n) = 0forn \geq 1$. This definition ensures the counital property (\varepsilon \otimes \mathrm{id}) \Delta = \mathrm{id} = (\mathrm{id} \otimes \varepsilon) \Delta, where \Delta is the deconcatenation coproduct. The property holds on the unit by direct computation: (\varepsilon \otimes \mathrm{id})\Delta(1) = (\varepsilon \otimes \mathrm{id})(1 \otimes 1) = 1 \otimes 1 = 1, and similarly for the other side. On generators v \in V, \Delta(v) = v \otimes 1 + 1 \otimes v, so

(\varepsilon \otimes \mathrm{id})\Delta(v) = \varepsilon(v) \otimes 1 + \varepsilon(1) \otimes v = 0 \otimes 1 + 1 \otimes v = v,

with an analogous verification for (\mathrm{id} \otimes \varepsilon)\Delta(v) = v; it extends by multiplicativity and linearity to all of T(V). The kernel \ker \varepsilon = \bigoplus_{n \geq 1} V^{\otimes n} forms the augmentation ideal, which is the unique maximal coideal of T(V) as a coalgebra. As an algebra homomorphism, \varepsilon is dual to the unit map \eta: k \to T(V) sending $1 \mapsto 1, embodying the natural duality between the algebra and coalgebra structures on T(V).

Hopf Algebra Structure

Antipode definition

In the context of the bialgebra structure on the tensor algebra T(V), the antipode is the unique algebra anti-endomorphism S: T(V) \to T(V) that inverts the coproduct in the convolution algebra, thereby endowing T(V) with a Hopf algebra structure. Specifically, since T(V) is a graded connected bialgebra with \varepsilon(V) = 0, there exists a unique linear map S: T(V) \to T(V) satisfying

m \circ (S \otimes \mathrm{id}) \circ \Delta = u \circ \varepsilon = m \circ (\mathrm{id} \otimes S) \circ \Delta,

where m: T(V) \otimes T(V) \to T(V) is the multiplication and u: k \to T(V) is the unit map embedding the base field k. This antipode is an anti-algebra homomorphism, meaning S(xy) = S(y)S(x) for all x, y \in T(V) and S(\lambda \cdot 1) = \lambda \cdot 1 for scalars \lambda \in k. The explicit form of the antipode on T(V) is determined degree by degree: S(1) = 1 on the degree-zero component, S(v) = -v for v \in V (the degree-one component), and for a decomposable tensor v_1 \cdots v_n of degree n \geq 2, S(v_1 \cdots v_n) = (-1)^n v_n \cdots v_1, extended linearly and via the anti-multiplicativity to all of T(V). This reversal with sign ensures compatibility with the coproduct \Delta. To verify, consider v \in V: the coproduct is \Delta(v) = v \otimes 1 + 1 \otimes v, so

m \circ (S \otimes \mathrm{id}) \circ \Delta(v) = S(v) \cdot 1 + 1 \cdot v = -v + v = 0 = u \circ \varepsilon(v),

since \varepsilon(v) = 0. The relation m \circ (\mathrm{id} \otimes S) \circ \Delta(v) = 0 holds symmetrically. For higher-degree elements, the formula satisfies the defining equation by induction on degree, leveraging the graded connectedness. For example, in degree 2 with x = v w, \Delta(x) = 1 \otimes x + v \otimes w + w \otimes v + x \otimes 1, and the computation yields x - v w - w v + w v = 0.

Convolution properties

The presence of the antipode S equips the tensor algebra T(V), together with its multiplication \cdot, coproduct \Delta, counit \varepsilon, and antipode S, with the structure of a Hopf algebra (T(V), \cdot, \Delta, \varepsilon, S). In this structure, the antipode S is invertible, with its inverse given explicitly by S^{-1}(v_1 \cdots v_n) = (-1)^n v_n \cdots v_1 for v_i \in V. In the convolution algebra \mathrm{Hom}(T(V), A) over any algebra A, the antipode provides the convolution inverse of the identity map, satisfying \mathrm{id} * S = u \circ \varepsilon = S * \mathrm{id}, where u denotes the unit map of A and * is the convolution product defined by f * g = m_A \circ (f \otimes g) \circ \Delta. This invertibility follows from the general properties of Hopf algebras, where the antipode serves as the unique two-sided inverse to the identity under convolution.^[32] Left Hopf modules over T(V) consist of vector spaces M equipped with a compatible left T(V)-module action and right T(V)-comodule coaction, satisfying the intertwining condition \rho(m \cdot h) = \rho(m_{(0)}) \cdot h_{(2)} \otimes m_{(1)} h_{(1)} in Sweedler notation, where \rho: M \to M \otimes T(V) is the coaction.^[32] By the fundamental theorem of Hopf modules, every such module M is isomorphic to T(V) \otimes M^{\mathrm{coinv}}, where M^{\mathrm{coinv}} = \{ m \in M \mid \rho(m) = m \otimes 1 \} is the subspace of coinvariants, reflecting the free nature of T(V) as an algebra.^[32] Unlike finite-dimensional Hopf algebras, T(V) lacks a nonzero integral element, as the existence of a unique (up to scalar) left integral requires finite dimensionality.^[32] Quotient Hopf algebras of T(V), such as the universal enveloping algebra U(\mathfrak{g}) of a Lie algebra \mathfrak{g} with underlying vector space V, inherit the full Hopf structure provided the defining ideal is a Hopf ideal (i.e., a biideal that is also a coideal).

Categorical Aspects

Cofree coalgebra properties

In the category of coalgebras over a field k, the tensor algebra T(V) equipped with the deconcatenation coproduct serves as the cofree coalgebra cogenerated by a vector space V.^[33] This structure endows T(V) with a dual role to its algebraic free property, where the underlying vector space is \bigoplus_{n \geq 0} V^{\otimes n}, and the coproduct \Delta: T(V) \to T(V) \otimes T(V) is defined by deconcatenation on tensor powers, extended by linearity.^[34] The key universal property characterizing this cofreeness is as follows: for any k-coalgebra C and any linear map \gamma: V \to C, there exists a unique coalgebra morphism \hat{\gamma}: T(V) \to C such that \hat{\gamma} \circ i = \gamma, where i: V \to T(V) is the inclusion of V as the degree-1 component.^[33] This property arises because coalgebra morphisms from T(V) are uniquely determined by their action on the generators in V, with the deconcatenation coproduct ensuring coassociativity and compatibility through recursive extension across tensor degrees.^[34] Dually to the free algebra universal property, this makes T(V) the universal object for coextensions of linear maps from V to arbitrary coalgebras. The functor V \mapsto T(V) from vector spaces to coalgebras preserves colimits, rendering it cocontinuous; this follows from the colimit-preserving nature of finite tensor products and direct sums in the construction of T(V).^[34] In the broader context of the adjunction between the forgetful functor U: \mathbf{CoAlg}_k \to \mathbf{Vec}_k and its right adjoint (the cofree coalgebra functor), the object T(V) aligns with the cofree construction via duality: the cofree coalgebra on V is T(V^*)^*, the dual of the tensor algebra on the dual space V^*.^[34] When V is finite-dimensional, T(V) exhibits self-duality in the category of coalgebras, as the double dual aligns the algebraic and coalgebraic structures: T(V)^{**} \cong T(V) as vector spaces, and the deconcatenation coproduct on T(V) corresponds to the dual of the concatenation product on T(V^*).^[34] This finite-type self-duality underscores the symmetric roles of tensor algebras and coalgebras in finite settings, though in infinite dimensions, the full cofree object requires the completed dual construction.^[35]

Adjunctions in monoidal categories

In a monoidal category (\mathcal{C}, \otimes, I) equipped with colimits such that the tensor product \otimes preserves these colimits, the tensor algebra functor T: \mathcal{C} \to \mathrm{AssAlg}(\mathcal{C}), which assigns to each object V \in \mathcal{C} its free associative algebra T(V), is left adjoint to the forgetful functor U: \mathrm{AssAlg}(\mathcal{C}) \to \mathcal{C}.^[36] This construction generalizes the classical tensor algebra from vector spaces to arbitrary monoidal settings, where T(V) serves as the free monoid generated by V.^[36] The adjunction is characterized by a natural isomorphism

\mathrm{Hom}_{\mathrm{AssAlg}(\mathcal{C})}(T(V), A) \cong \mathrm{Hom}_{\mathcal{C}}(V, U(A))

for any V \in \mathcal{C} and associative algebra A \in \mathrm{AssAlg}(\mathcal{C}), with the unit of the adjunction providing the canonical inclusion V \to T(V) and the counit inducing the structure map T(A) \to A.^[36] This extends naturally to enriched monoidal categories or symmetric monoidal categories, where the free algebra inherits coherence from the underlying structure, preserving associativity via the monoidal constraints.^[37] Assuming \mathcal{C} is cocomplete, the tensor algebra admits an explicit realization as a colimit:

T(V) = \mathrm{colim} \left( I \leftarrow V \leftarrow V \otimes V \leftarrow V \otimes V \otimes V \leftarrow \cdots \right),

known as the bar construction, which provides a free resolution of V in the category of associative algebras.^[36] This bar construction ensures that T preserves colimits, reinforcing its left adjoint nature.^[36] In algebraic topology, the tensor algebra functor relates to models of free loop spaces: the SO(2)-equivariant homology of the free loop space \Lambda X of a path-connected space X can be computed as the cyclic hyperhomology of the singular chains on the Moore loop space MX via a cyclic bar construction.^[38] This connection underscores the categorical generalization's importance, linking monoidal adjunctions to homotopy-theoretic invariants like cyclic homology.^[38]

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