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Symmetric algebra

In mathematics, particularly in abstract algebra, the symmetric algebra of an R-module M, where R is a commutative ring, is the free commutative R-algebra generated by M. It is constructed as the quotient of the tensor algebra T(M) by the two-sided ideal generated by all commutators of the form x \otimes y - y \otimes x for x, y \in M.^[1] The symmetric algebra \Sym(M) is naturally graded as \Sym(M) = \bigoplus_{n=0}^\infty \Sym^n(M), where \Sym^0(M) = R and \Sym^n(M) for n \geq 1 is the nth symmetric power of M, obtained by quotienting the nth tensor power T^n(M) by the relations imposing commutativity.^[1] This grading reflects its role as a universal object for symmetric multilinear maps: any symmetric R-multilinear map from M^n to an R-module N factors uniquely through \Sym^n(M).^[2] A key feature is its universal property: given any commutative R-algebra A and any R-linear map \phi: M \to A, there exists a unique R-algebra homomorphism \tilde{\phi}: \Sym(M) \to A such that \tilde{\phi}|_M = \phi, making \Sym(M) the "freest" such algebra.^[1]^[2] When M is a free R-module of finite rank r, \Sym(M) is isomorphic to the polynomial ring R[x_1, \dots, x_r], highlighting its connection to classical polynomial algebras.^[1] In general, \Sym(M) serves as a foundational structure in multilinear algebra, paralleling the tensor algebra (for non-commutative extensions) and the exterior algebra (for anti-commutative ones).^[2] It finds applications in representation theory, where symmetric powers classify polynomial representations of groups, and in commutative algebra, such as analyzing ideals via \Sym(I) for an ideal I.^[3]^[4]

Construction

Quotient of the tensor algebra

The tensor algebra T(M) of an R-module M over a commutative ring R is the free associative R-algebra generated by M, constructed as the direct sum T(M) = \bigoplus_{n=0}^\infty T^n(M), where T^0(M) = R and T^n(M) = M^{\otimes n} for n \geq 1, with multiplication given by concatenation of tensors extended linearly.^[1]^[5] This makes T(M) the universal R-algebra containing M as an R-submodule with no relations imposed on the generators other than associativity.^[1] To obtain a commutative structure, consider the two-sided ideal J in T(M) generated by all elements of the form m \otimes n - n \otimes m for m, n \in M.^[1]^[5] This ideal J consists of all finite sums of terms involving these commutators multiplied on the left and right by arbitrary elements of T(M), and it is homogeneous with components in degrees n \geq 2.^[5] The symmetric algebra \Sym(M) is defined as the quotient T(M) / J, equipped with the canonical projection \pi: T(M) \to \Sym(M).^[1]^[5] In this quotient, the relations m \otimes n \equiv n \otimes m \pmod{J} force the images of elements from M to commute. The multiplication on T(M) descends to an associative, unital, and commutative multiplication on \Sym(M), making \Sym(M) a commutative R-algebra.^[1]^[5] Specifically, for homogeneous elements \pi(x_1 \otimes \cdots \otimes x_n) and \pi(y_1 \otimes \cdots \otimes y_m) with x_i, y_j \in M, their product is \pi(x_1 \otimes \cdots \otimes x_n \otimes y_1 \otimes \cdots \otimes y_m), and this operation is bilinear in each factor.^[1] The image \pi(M) generates \Sym(M) as an R-algebra, since every element is an R-linear combination of products of images of elements from M.^[1]^[5] Moreover, for all m, n \in M, the commutativity relation holds in \Sym(M):

\pi(m \otimes n) = \pi(n \otimes m),

ensuring that the generators from M satisfy no further relations beyond commutativity.^[1]^[5] In general, for non-free modules, \Sym(M) does not simplify to a polynomial ring but retains the quotient structure.^[1]

Presentation as a polynomial ring

For a free R-module M of finite rank n over a commutative ring R with basis \{e_1, \dots, e_n\}, the symmetric algebra \Sym(M) is isomorphic to the polynomial ring R[x_1, \dots, x_n] in n indeterminates.^[6] This isomorphism arises from the universal property of the symmetric algebra as the free commutative R-algebra generated by M: there exists a unique R-algebra homomorphism \phi: \Sym(M) \to R[x_1, \dots, x_n] such that \phi(e_i) = x_i for each i, and this map is an isomorphism because both sides are freely generated by the images of the basis elements under the relations of commutativity.^[6] Consequently, \Sym(M) serves as the free commutative R-algebra on n generators, where the generators correspond to the basis elements of M.^[7] The multiplication in \Sym(M) under this identification mirrors ordinary polynomial multiplication, enforcing commutativity. For instance, the image of the elementary tensor e_i \otimes e_j in the quotient construction maps to x_i x_j = x_j x_i, reflecting the symmetrization that identifies e_i \otimes e_j with e_j \otimes e_i.^[6] This structure highlights how \Sym(M) encodes multilinear symmetric forms in a commutative algebraic framework. A concrete example occurs when the rank of M is 1, so M \cong R with basis \{e\}; then \Sym(M) \cong R, the univariate polynomial ring, where powers of x correspond to iterated symmetric products of e.^[6] In the infinite rank case, if M is free of rank \kappa, then \Sym(M) is isomorphic to the polynomial ring R[\{x_i\}_{i \in I}] in \kappa indeterminates, but this algebra is not Noetherian and lacks finite generation.^[8] However, applications often require completed versions, such as formal power series rings, to handle convergence or topological structures, though the finite rank setting remains the primary focus for explicit computations.^[8]

Fundamental Properties

Grading

The tensor algebra T(M) of an R-module M over a commutative ring R is the free associative R-algebra generated by M, graded as T(M) = \bigoplus_{n=0}^\infty T^n(M), where T^0(M) = R and T^n(M) = M^{\otimes_R n} for n \geq 1 is the space of n-fold tensor products.^[1]^[5] In the special case where M = V is a vector space over a field k (so R = k), the symmetric algebra \Sym(V) is constructed as the quotient T(V) / J, where J is the two-sided ideal generated by elements of the form x \otimes y - y \otimes x for x, y \in V. Since these generators are homogeneous of degree 2, the ideal J is homogeneous, meaning J = \bigoplus_{n=0}^\infty (J \cap T^n(V)). Consequently, the quotient inherits a natural grading: \Sym(V) = \bigoplus_{n=0}^\infty \Sym^n(V), with each homogeneous component given by \Sym^n(V) = T^n(V) / (J \cap T^n(V)). This grading extends to the general module case, where \Sym(M) = \bigoplus_{n=0}^\infty \Sym^n(M).^[1]^[5] The zeroth component is \Sym^0(V) = k (or R generally), while the first is \Sym^1(V) \cong V (or M), as J \cap T^1(V) = 0. For n \geq 2, \Sym^n(V) is the quotient of T^n(V) by the subspace J \cap T^n(V), which is the span of all elements arising from the relations imposing commutativity, effectively identifying tensors up to symmetrization. The canonical projection \pi: T(V) \to \Sym(V) restricts to a surjective map \pi: T^n(V) \to \Sym^n(V) on each graded piece, with kernel precisely J \cap T^n(V), the subspace spanned by commutators embedded in degree n.^[1]^[5] This grading is compatible with the algebra multiplication: the product map induces a bilinear map \Sym^m(V) \otimes \Sym^n(V) \to \Sym^{m+n}(V) that is well-defined and respects the direct sum decomposition (similarly over rings). When V is finite-dimensional with \dim(V) = d < \infty, each component has dimension \dim(\Sym^n(V)) = \binom{n + d - 1}{d - 1}, corresponding to the number of monomials of degree n in d commuting variables.^[1]^[9]

Relation to symmetric tensors

A symmetric n-tensor over a vector space V over a field k is an element of the n-th tensor power \otimes^n V that remains invariant under the action of the symmetric group S_n on the factors, meaning it is unchanged by any permutation of the tensor indices. This invariance distinguishes symmetric tensors from general tensors, forming a subspace S^n(V) \subseteq \otimes^n V. In the general setting over a commutative ring R, symmetric tensors are defined analogously in M^{\otimes_R n} as elements fixed by permutations, though tensor powers over rings lack a group action in the same way unless R is a field.^[10] Over fields, the symmetrization map s: \otimes^n V \to S^n(V) projects arbitrary tensors onto this subspace by averaging over all permutations in S_n, explicitly given by

s(t) = \frac{1}{n!} \sum_{\sigma \in S_n} \sigma \cdot t

for t \in \otimes^n V, where \sigma \cdot t denotes the permuted tensor—provided that n! is invertible in k (i.e., \mathrm{char}(k) = 0 or \mathrm{char}(k) > n). This map is a projection onto the invariants, yielding the isomorphism S^n(V) \cong (\otimes^n V)^{S_n}, the subspace of S_n-invariants in \otimes^n V. In general, even without inverses, S^n(V) aligns with the n-th graded piece of \Sym(V) via the quotient construction. In the symmetric algebra \mathrm{Sym}(V), the n-th graded component aligns with S^n(V), interpreting symmetric products as symmetrized tensors.^[10] Symmetric multilinear forms on V connect naturally to this structure: a symmetric n-linear map \phi: V^n \to k factors uniquely through the projection \otimes^n V \to S^n(V), inducing a linear functional on S^n(V). This factorization arises because symmetric tensors represent the domain for such forms, ensuring compatibility with the S_n-action. Over rings, symmetric R-multilinear maps factor through \Sym^n(M).^[10] The polarization identity recovers symmetric multilinear forms from elements of the symmetric algebra, particularly for homogeneous polynomials. For a quadratic form q: V \to k corresponding to an element of S^2(V), over fields of characteristic not 2, the associated symmetric bilinear form is

\phi(u,v) = \frac{1}{4} \left( q(u+v) - q(u-v) \right),

extending to higher degrees via multilinearity and differences. This identity demonstrates how algebra elements encode multilinear data.^[10] In the symmetric algebra, the symmetric product of vectors v_1, \dots, v_n \in V is defined as v_1 \cdot \dots \cdot v_n = s(v_1 \otimes \dots \otimes v_n) \in S^n(V) when the symmetrizer is available, providing a concrete realization of symmetrized tensors. For instance, in coordinates with respect to a basis of V, elements of S^n(V) correspond to homogeneous polynomials of degree n in the dual basis variables, where monomials like x_1^{a_1} \dots x_d^{a_d} with \sum a_i = n span the space.^[10]

Universal and Categorical Aspects

Universal property

The symmetric algebra \Sym(M) of an R-module M over a commutative ring R satisfies the following universal property: for any commutative R-algebra A and any R-linear map f: M \to A, there exists a unique R-algebra homomorphism F: \Sym(M) \to A such that F \circ \iota = f, where \iota: M \to \Sym(M) is the canonical inclusion map. This characterizes \Sym(M) up to unique isomorphism as the free commutative R-algebra generated by M. To see this, recall that \Sym(M) can be constructed as the quotient of the tensor algebra T(M) by the two-sided ideal generated by elements of the form m \otimes n - n \otimes m for m, n \in M, enforcing commutativity. Given f: M \to A, the universal property of the tensor algebra T(M) induces a unique R-algebra homomorphism \tilde{F}: T(M) \to A such that \tilde{F} \circ j = f, where j: M \to T(M) is the inclusion; since A is commutative, \tilde{F} vanishes on the ideal, hence factors uniquely through the quotient as F: \Sym(M) \to A with the desired property. Uniqueness follows from the freeness of \Sym(M), as it is generated as an R-algebra by the image \iota(M) subject only to the relations that elements of \iota(M) commute. If A is \mathbb{N}-graded and f is a degree-1 map (i.e., lands in the degree-1 component), then the induced F is a graded homomorphism preserving the grading on \Sym(M), where \Sym(M) = \bigoplus_{n \geq 0} \Sym^n(M) with \Sym^0(M) = R and \Sym^n(M) the nth symmetric power. Explicitly, F sends \iota(m) to f(m) for m \in M, and extends by multiplicativity: for homogeneous elements x \in \Sym^m(M) and y \in \Sym^n(M), F(xy) = F(x) F(y) with \deg F(x) = m and \deg F(y) = n. As a corollary, \Sym(M) realizes the coproduct in the category of commutative R-algebras obtained by freely adjoining the elements of M to R with commutativity relations.

Categorical characterization

The symmetric algebra construction defines a functor \Sym: \RMod \to \CommAlg_R from the category of R-modules to the category of commutative R-algebras, which is left adjoint to the forgetful functor U: \CommAlg_R \to \RMod that sends a commutative algebra to its underlying R-module.^[6] This adjunction is characterized by a natural isomorphism of hom-sets \Hom_{\CommAlg_R}(\Sym(M), A) \cong \Hom_{\RMod}(M, U(A)) for any R-module M and commutative R-algebra A, where the isomorphism is natural in both M and A.^[6] As the left adjoint to the forgetful functor, \Sym serves as the free functor generating commutative algebras, freely adjoining a commutative multiplication to the generators in M.^[6] The adjunction comes equipped with a unit \eta_M: M \to U(\Sym(M)) and a counit \epsilon_A: \Sym(U(A)) \to A, satisfying the usual triangular identities that encode the universal property functorially.^[11] This structure extends beyond \RMod to more general settings, such as abelian categories where \Sym can be defined via colimit-preserving constructions, or to the category of sheaves of modules over a scheme, where the symmetric algebra sheaf \mathcal{S}(E) over a sheaf of modules E on a space X is given by \mathcal{S}(E) = \bigoplus_{n \geq 0} E^{\otimes n} / (x \otimes y - y \otimes x) on each open set, preserving the adjointness to the forgetful functor on sheaves of commutative algebras.^[12] In the context of algebraic geometry, the spectrum functor \Spec: \CommAlg_R^{\op} \to \AffSch_R, which assigns to each commutative R-algebra its relative affine scheme over \Spec R, reverses the direction of the adjunction. For example, over a field k and finite-dimensional vector space V, \Sym(V) is the coordinate ring of the affine space whose underlying vector space is dual to V, so that \Spec_k(\Sym(V^*)) represents the affine space \mathbb{A}^n_k where n = \dim V.^[13] The functor \Sym preserves colimits, including direct sums; for instance, \Sym(M \oplus N) \cong \Sym(M) \otimes_R \Sym(N) as commutative algebras, reflecting its free nature.^[6]

Geometric Interpretations

Symmetric algebra of an affine space

In algebraic geometry, the symmetric algebra plays a central role in describing the polynomial functions on an affine space. Let k be a field and E an affine space over k modeled on a finite-dimensional vector space V, meaning E is a principal homogeneous space under the action of the additive group of V. The ring of polynomial functions \mathcal{O}(E) on E is isomorphic to the symmetric algebra \mathrm{Sym}_k(V^*) on the dual vector space V^*.^[14]^[15] This identification arises because polynomial functions on E are precisely those that, in any choice of coordinates translating E to V, become polynomial expressions in the coordinates of V, and such functions are independent of the choice of origin due to the affine structure.^[16] The symmetric algebra \mathrm{Sym}_k(V^*) is graded as

\mathrm{Sym}_k(V^*) = \bigoplus_{n=0}^\infty \mathrm{Sym}^n_k(V^*),

where \mathrm{Sym}^n_k(V^*) is the space of homogeneous polynomials of degree n on V, which extends naturally to homogeneous polynomial functions of degree n on E.^[14] For each point x \in E, there is an evaluation map \mathrm{ev}_x: \mathrm{Sym}_k(V^*) \to k defined by \mathrm{ev}_x(p) = p(x), where p(x) is computed using coordinates on E relative to a fixed origin in V; this map is a k-algebra homomorphism and is independent of the coordinate choice.^[15] If \dim V > 0, then \mathrm{Sym}_k(V^*) is infinite-dimensional as a k-vector space, reflecting the unbounded nature of polynomial functions on non-trivial affine spaces.^[16] Morphisms between affine spaces correspond contravariantly to homomorphisms of their symmetric algebras. Specifically, k-algebra homomorphisms \mathrm{Sym}_k(V^*) \to \mathrm{Sym}_k(W^*) are in bijection with affine maps E \to F, where F is an affine space modeled on another vector space W; such a homomorphism is induced by the dual of the linear part of the affine map.^[17] For a concrete example, consider the affine line E = \mathbb{A}^1_k modeled on V = k, so V^* \cong k with basis corresponding to the coordinate function t. Then \mathrm{Sym}_k(V^*) \cong k, the polynomial ring in one indeterminate, whose elements are precisely the polynomial functions on \mathbb{A}^1_k.^[15]

Coordinate ring of affine varieties

In algebraic geometry, the coordinate ring of an affine variety provides a fundamental algebraic structure encoding the geometry of the variety. Consider an affine variety X \subset \mathbb{A}^n_k, where \mathbb{A}^n_k is the affine n-space over an algebraically closed field k, defined as the zero locus V(I) of an ideal I \subset k[x_1, \dots, x_n]. The coordinate ring k[X] is the quotient ring k[x_1, \dots, x_n]/I(X), where I(X) is the vanishing ideal consisting of all polynomials in k[x_1, \dots, x_n] that vanish on X.^[18] This ring k[X] consists of the regular functions on X, and it is finitely generated as a k-algebra.^[18] The polynomial ring k[x_1, \dots, x_n] is isomorphic to the symmetric algebra \operatorname{Sym}((k^n)^*), where (k^n)^* is the dual vector space to k^n. Thus, the coordinate ring k[X] can be realized as a quotient \operatorname{Sym}((k^n)^*)/J, where J is an ideal containing the vanishing ideal I(X). More precisely, there is a surjective quotient map \operatorname{Sym}(V^*) \to k[X] with kernel exactly the vanishing ideal I(X), for V = k^n.^[19] This construction links the symmetric algebra of the ambient affine space to the subvariety X via ideal quotients. Hilbert's Nullstellensatz establishes a bijection between radical ideals in k[x_1, \dots, x_n] and affine varieties in \mathbb{A}^n_k, ensuring that the radical of I(X) defines X precisely and that maximal ideals correspond to points of X.^[20] For an affine variety X, the coordinate ring k[X] coincides with the ring of global sections \Gamma(X, \mathcal{O}_X) of the structure sheaf \mathcal{O}_X, which assigns to each open set the ring of regular functions on that set. Since X is affine, \Gamma(X, \mathcal{O}_X) = k[X] is finitely generated over k. In a more general perspective, the coordinate ring arises as the symmetric algebra on sections of the structure sheaf modulo relations imposed by the geometry of X.^[18] A concrete example is a hypersurface X defined by the equation f(x_1, \dots, x_n) = 0, where f is irreducible. Here, k[X] = k[x_1, \dots, x_n]/(f), which is an integral domain reflecting the irreducibility of X.^[18] Morphisms between affine varieties also translate algebraically: a morphism \phi: X \to Y induces a k-algebra homomorphism k[Y] \to k[X] by precomposition, sending a function on Y to its pullback along \phi. This contravariant correspondence underlies the functorial relationship between affine varieties and their coordinate rings.^[21]

Analogies and Extensions

Analogy with exterior algebra

The symmetric algebra \Sym(V) and the exterior algebra \wedge V share a common origin as quotients of the tensor algebra T(V), but they impose contrasting relations that yield distinct algebraic structures.^[22] Specifically, \Sym(V) is obtained by quotienting T(V) by the two-sided ideal generated by all commutators v \otimes w - w \otimes v for v, w \in V, enforcing commutativity in the product.^[23] In contrast, the exterior algebra \wedge V is the quotient T(V) / \langle v \otimes w + w \otimes v \mid v, w \in V \rangle, where the ideal is generated by anticommutators, introducing antisymmetry; additionally, this forces v \wedge v = 0 for all v \in V since v \otimes v + v \otimes v = 2v \otimes v lies in the ideal (and equals zero when the base field has characteristic not 2).^[22] Thus, both algebras arise from homogeneous ideals concentrated in degree 2, but one symmetrizes via commutators while the other alternates via anticommutators.^[23] Their gradings highlight further differences: \Sym(V) = \bigoplus_{n \geq 0} \Sym^n(V) is \mathbb{N}-graded with each \Sym^n(V) infinite-dimensional if V is, allowing arbitrary powers like v^n for v \in V; whereas \wedge V = \bigoplus_{n \geq 0} \wedge^n(V) has \dim \wedge^n(V) = \binom{\dim V}{n}, finite-dimensional in each degree and vanishing for n > \dim V, reflecting the nilpotency v \wedge v = 0 that caps higher terms.^[24] This contrasts sharply with \Sym(V), where v \cdot v = v^2 \neq 0 in general, enabling polynomial-like growth.^[23] The universal properties underscore their complementary roles: \Sym(V) is universal among commutative algebras A equipped with a linear map V \to A, representing symmetric multilinear maps from V^n to the base field; analogously, \wedge V is universal for alternating multilinear maps, capturing antisymmetric forms.^[22] Geometrically, \wedge V^* models the algebra of differential forms on a manifold, where the wedge product \wedge encodes antisymmetric integration over oriented simplices, while \Sym(V^*) serves as the coordinate ring of polynomials on the affine space underlying V, facilitating algebraic geometry via commutative multiplication.^[24] A deeper analogy emerges in superalgebra, where both fit into graded-commutative frameworks: the exterior algebra \wedge V coincides with the symmetric algebra on a purely odd supervector space, whose parity grading introduces the sign flip (-1)^{|u||w|} in the product u \otimes w, recovering anticommutativity for odd elements; this unifies them under the umbrella of Clifford algebras, which generalize both by deforming the relations with a quadratic form.^[25]^[26]

Relation to universal enveloping algebras

The universal enveloping algebra U(\mathfrak{g}) of a Lie algebra \mathfrak{g} over a field k is defined as the quotient of the tensor algebra T(\mathfrak{g}) by the two-sided ideal generated by elements of the form x \otimes y - y \otimes x - [x, y] for all x, y \in \mathfrak{g}.^[27] This construction ensures that U(\mathfrak{g}) is an associative algebra containing \mathfrak{g} as a Lie subalgebra via the canonical inclusion, satisfying a universal property for Lie algebra homomorphisms into associative algebras.^[27] When \mathfrak{g} is abelian, meaning the Lie bracket [\cdot, \cdot] vanishes identically, the defining relations simplify to x \otimes y - y \otimes x for x, y \in \mathfrak{g}, which is precisely the ideal used to construct the symmetric algebra \Sym(\mathfrak{g}).^[28] Thus, U(\mathfrak{g}) \cong \Sym(\mathfrak{g}) as associative algebras in this case.^[27] More generally, viewing a vector space V as an abelian Lie algebra with zero bracket yields \Sym(V) = U(V).^[28] The algebra U(\mathfrak{g}) is always associative by construction, but its multiplication is commutative if and only if \mathfrak{g} is abelian, as the relations enforce commutativity precisely when the bracket is zero.^[27] In the abelian case, the multiplication in U(\mathfrak{g}) = \Sym(\mathfrak{g}) is therefore commutative, aligning with the symmetric product on tensors.^[28] For nilpotent Lie algebras, U(\mathfrak{g}) is "close" to \Sym(\mathfrak{g}) in the sense that the associated graded algebra \gr U(\mathfrak{g}) \cong \Sym(\mathfrak{g}) by the Poincaré–Birkhoff–Witt theorem, differing only by the additional relations imposed by the non-zero bracket.^[28] A concrete example is the three-dimensional Heisenberg Lie algebra \mathfrak{h} over k with basis \{x, y, z\} and relations [x, y] = z, [x, z] = [y, z] = 0. Here, U(\mathfrak{h}) is generated by x, y, z with the relation xy - yx = z (and z central), forming a non-commutative algebra akin to the Weyl algebra in one dimension, whereas \Sym(\mathfrak{h}) would impose full commutativity without the z-relation.^[28]

Algebraic Structures

Multiplicative structure

This construction endows \Sym(M) with the structure of a commutative associative R-algebra with multiplicative unit $1, the canonical image of the identity in T^0(M) \cong R. The multiplication is induced from that in T(M), ensuring commutativity: for any m, n \in M, the product m \cdot n = n \cdot m in \Sym(M).^[29] The scalar multiplication in \Sym(M) distributes over the ring operations in the standard way for an R-algebra. Specifically, for \lambda \in R and m_1, \dots, m_n \in M, one has \lambda \cdot (m_1 \cdot \dots \cdot m_n) = (\lambda m_1) \cdot m_2 \cdot \dots \cdot m_n = m_1 \cdot \dots \cdot m_{n-1} \cdot (\lambda m_n), with linearity extending to sums in each factor. As a commutative ring, the center of \Sym(M) coincides with \Sym(M) itself. Moreover, \Sym(M) is free as a module over itself, with rank 1 and generator $1.^[29] \Sym(M) admits a natural \mathbb{N}-grading \Sym(M) = \bigoplus_{r \geq 0} \Sym^r(M), where \Sym^r(M) is the r-th symmetric power, and this grading interacts with the multiplicative structure by preserving degrees in products. Ideals in \Sym(M) can thus be considered in the graded sense, with homogeneous components. When M is a free R-module of finite rank n < \infty, \Sym(M) \cong R[x_1, \dots, x_n] as R-algebras via a choice of basis for M, and in this polynomial ring case, the ideals are the polynomial ideals; notably, for n=1, all ideals are principal.^[29] Derivations of \Sym(M) are the R-linear endomorphisms D: \Sym(M) \to \Sym(M) satisfying the Leibniz rule D(ab) = D(a)b + aD(b) for all a, b \in \Sym(M). The module of derivations \Der_R(\Sym(M), \Sym(M)) is free as a \Sym(M)-module, isomorphic to \Sym(M) \otimes_R M^*, where M^* is the R-dual of M; each \phi \in M^* determines a derivation via contraction on degree-1 elements, extended by the Leibniz rule.^[30] A concrete example arises when M has finite rank n and basis \{e_1, \dots, e_n\}, yielding \Sym(M) \cong R[x_1, \dots, x_n] with e_i mapping to x_i. Here, multiplication is the usual polynomial multiplication, and the group of units consists precisely of the nonzero constant polynomials, i.e., R^\times. The derivations are then spanned over the ring by the partial derivatives \partial/\partial x_i, corresponding to the dual basis of M^*.^[29]^[30]

Hopf algebra structure

Let k be a field and V a k-vector space. The symmetric algebra \Sym(V) admits a natural bialgebra structure. The multiplication m: \Sym(V) \otimes \Sym(V) \to \Sym(V) is the algebra multiplication inherited from the tensor algebra quotient, and the unit \eta: k \to \Sym(V) maps the scalar $1 to the identity element of \Sym(V).^[31]^[32] To define the coalgebra structure, the coproduct \Delta: \Sym(V) \to \Sym(V) \otimes \Sym(V) is specified on the generators by \Delta(v) = v \otimes 1 + 1 \otimes v for v \in V, and extended as an algebra homomorphism to the entire symmetric algebra, leveraging its commutative grading. The counit \varepsilon: \Sym(V) \to k satisfies \varepsilon(1) = 1 and \varepsilon(v) = 0 for all v \in V, extended by multiplicativity to higher degrees. This ensures compatibility, as \Delta(v w) = \Delta(v) \Delta(w) for v, w \in V, establishing \Sym(V) as a bialgebra.^[31]^[32] The bialgebra structure extends to a Hopf algebra via the antipode S: \Sym(V) \to \Sym(V), defined by S(v) = -v on generators and extended as an algebra anti-homomorphism. For monomials, this yields S(v^n) = (-v)^n, preserving the convolution inverse property required for the Hopf axiom. This construction parallels the Hopf algebra structure on the group algebra of an abelian group, but here it arises from the additive structure of the vector space V treated as an abelian Lie algebra with trivial bracket.^[31]^[32] A key application of this Hopf structure lies in studying invariants under group actions on V. If a Hopf algebra \Lambda (such as the group algebra of a finite group acting linearly on V) coacts on V, it induces a coaction on \Sym(V), and the invariants \Sym(V)^\Lambda form a subalgebra whose properties, such as Cohen-Macaulayness, depend on the reductivity of \Lambda.^[33] Additionally, the Peter-Weyl theorem in the context of Hopf algebra representations decomposes the dual structure into matrix coefficients of irreducibles, facilitating analysis of \Sym(V)-modules like symmetric powers. Finally, V itself acquires a right \Sym(V)-comodule structure via the restriction of \Delta to V, mapping v \mapsto v \otimes 1 + 1 \otimes v, which encodes the primitive nature of generators and enables extensions to tensor products in representation theory. For explicit computations, the polynomial presentation of \Sym(V) as k[x_1, \dots, x_n] for \dim V = n simplifies evaluation of these maps.^[31]^[32]

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Free delivery 14-day returnsThis is the softcover reprint of the English translation of 1974 (available from Springer since 1989) of the first 3 chapters of Bourbaki's 'Algèbre'.
[8]
[PDF] Infinite-dimensional methods in commutative algebra
Jul 8, 2022 · In the polynomial ring Rn, the only elements of infinite strength are linear. ... symmetric algebra Sym(W) (which is a polynomial ring in the ...
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[PDF] Math 395. Bases of symmetric and exterior powers Let V be a finite ...
To see the spanning aspect, one can use the universal properties of symmetric and exterior powers much as in our analogous argument for why tensors of basis ...
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Multilinear Algebra | SpringerLink
Symmetric tensor algebra. W. H. Greub. Pages 190-204. PDF · Multilinear functions ... Chapter I contains standard material on multilinear mappings and the tensor ...
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symmetric algebra in nLab
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[PDF] 1 Chapter 2: Categories of Sheaves
Oct 28, 2014 · Let S(E) = Ln≥0 E⊗n/(x ⊗ y − y ⊗ x) be the sheaf of OX-algebras, called symmetric algebra over E. For every open U ⊂ X we get an ...
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[PDF] ELEMENTARY INTRODUCTION TO REPRESENTABLE ...
There is an analogous notation for affine spaces: V (V ) = Spec Sym(V ) is the affine space whose underlying vector space is the dual of V . For Grassmannians, ...
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[PDF] Algebraic Groups I. Homework 1 1. This exercise studies the ...
(i) Elements of the graded symmetric algebra Sym(V ∗) are called polynomial functions on V . ... (i) Prove that A is represented by an affine space over k.
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[PDF] Chapter 1 Affine algebraic geometry
We have a space and a ring of k-valued functions on it. For each ideal in the ring we have its zero locus, and these form the closed sets for a topology, the ...
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[PDF] 1. Affine Varieties ∑ ∑ ∑
The affine space An. K will be the ambient space for our zero loci of ... So the quotient ring K[x1,...,xn]/I(X) can be thought of as the ring of polynomial ...
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[PDF] 14.1 Affine morphisms
Oct 24, 2013 · By taking the morphisms to be k-algebra homomorphisms, we can consider the category of affine algebras. In order to prove that the category of ...
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[PDF] lecture notes for algebraic geometry
Definition 3.3 (Coordinate ring). Let X ⊂ An be an affine variety and I := I(X). The ring. Γ(X) := k[x1,...,xn]/I is called the coordinate ring of X. In a ...
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[PDF] FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 21
Jan 11, 2008 · choice of embedding in affine space). Notice that in some sense, the cotangent space is more algebraically natural than the tangent space.
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[PDF] Affine Varieties and the Nullstellensatz - Purdue Math
Show that Hilbert's Nullstellensatz implies the weak Nullstellensatz. 0.3 ... This is called the coordinate ring of X. Lemma 0.3.1. If X ⇢ An k , O(X) ...
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[PDF] introduction to algebraic geometry, class 4
Morphisms from one affine algebraic set X ⊂ Am to another Y ⊂ An canonically correspond to morphisms of rings A(Y ) → A(X):. Homalg sets(X, Y ) ∼= Homrings over ...
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[PDF] 18.745 F20 Lecture 12: The Universal Enveloping Algebra of a Lie ...
The universal enveloping algebra of g, denoted. U(g), is the quotient of Tg by the ideal I generated by the elements xy − yx − [x, y], x, y ∈ g. Recall that any ...
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[PDF] An introduction to Lie algebroids and their enveloping algebras - arXiv
Sep 23, 2023 · The universal enveloping algebra of the Abelian Lie algebra V is isomorphic to the symmetric algebra of V : UpV q – dpV q. Remark: When ...<|control11|><|separator|>
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[PDF] A Primer of Commutative Algebra - James Milne
The symmetric algebra Sym.M / of an A-module M is the quotient of TM by the ... vector space. Each element of B can be expressed a linear combination ...
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[PDF] Calculus 214 - UC Berkeley math
Dec 14, 2008 · 1.1.4 Example: Derivations from the tensor algebra . . . . 13. 1.1.5 Variant: Derivations from the symmetric algebra . . . 14. 1.1.6 Variant ...
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[PDF] On the Structure of Hopf Algebras
This paper gives a comprehensive treatment of Hopf algebras and some ... the symmetric algebra generated by X. If X is free and generators {Xi}ji for. X are ...
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https://ocw.mit.edu/courses/18-745-lie-groups-and-lie-algebras-i-fall-2020/mit18_745_f20_lec12.pdf
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[PDF] Invariants of Hopf Algebras - Department of Mathematics
Jul 10, 1999 · Actions of Hopf algebras are a natural generalization of group actions. ... Hopf algebra Λ, then the symmetric algebra. S(V ) is a Λ-module ...