Symmetric AlgebraEdit

The symmetric algebra is a foundational construction in algebra that packages a vector space into a canonical commutative algebra. Given a vector space V over a field k (or more generally a module over a commutative ring R), the symmetric algebra S(V) (often called Sym(V)) is the free commutative algebra generated by V. Concretely, it is built to encode polynomial expressions in the generators that come from V, while enforcing commutativity of multiplication.

One way to construct S(V) is to start with the tensor algebra T(V) = ⊕_{n≥0} V^{⊗ n}, which carries no relations beyond those from tensor products. Then impose the basic commutativity relation by quotienting out the two-sided ideal I generated by all v ⊗ w − w ⊗ v for v, w ∈ V. The resulting algebra T(V)/I is the symmetric algebra S(V). This quotient presentation highlights its universal property: S(V) is the free commutative k-algebra generated by V. In particular, any linear map f: V → A into a commutative k-algebra A extends uniquely to an algebra homomorphism S(V) → A.

From a structural standpoint, S(V) carries a natural grading. It decomposes as S(V) = ⊕_{n≥0} S^n(V), where S^n(V) is the n-th symmetric power of V. Each S^n(V) consists of homogeneous elements of degree n and can be realized as the coinvariants of the natural action of the symmetric group S_n on V^{⊗ n}. When V is finite-dimensional, each S^n(V) is finite-dimensional, and the dimensions satisfy a familiar counting formula dim S^n(V) = binomial(n + dim V − 1, n). The algebra is thus a graded, commutative k-algebra with a distinguished degree-zero piece k.

The universal property of S(V) makes it the natural recipient for linear data from V. Given a linear map f: V → A into a commutative algebra A, there is a unique algebra homomorphism φ: S(V) → A extending f. This functorial viewpoint shows that S(−) is left adjoint to the forgetful functor from commutative algebras to vector spaces, hence it is the “free” way to generate a commutative algebra from a given generating set.

Relating to geometry, S(V) is tightly connected to polynomial functions. If V is finite-dimensional, the dual space V* acts by evaluation on V, and homogeneous components S^n(V*) correspond to homogeneous polynomial functions of degree n on V. Consequently, the entire algebra of polynomial functions on V is naturally isomorphic to Sym(V*) ≅ ⊕_{n≥0} S^n(V*). Conversely, Sym(V) can be viewed as the algebra generated by the coordinate directions coming from V, while Sym(V*) plays the role of the coordinate ring of polynomial functions on V. This duality underpins many constructions in algebraic geometry, where the spectrum of a polynomial algebra encodes affine spaces and more general varieties.

In the language of more advanced algebra, the symmetric algebra sits alongside other canonical constructions. For instance, over a Lie algebra g, the associated graded of the universal enveloping algebra U(g) is isomorphic to S(g) (the PBW theorem), linking the purely algebraic notion of symmetry with representations and enveloping structures. As a simple contrast, the exterior algebra Λ(V) imposes anti-commutativity and serves as the natural companion to S(V) in the study of alternating tensors and differential forms.

Beyond fields, S_R(M) can be defined for a commutative ring R and an R-module M. It remains the free commutative R-algebra generated by M, and it plays a central role in algebraic geometry over general bases, in deformation theory, and in the formulation of polynomial-like objects over rings that are not fields. The graded structure and the universal property persist in this broader setting, making Sym_R(M) a versatile tool in both algebra and geometry.

Examples help ground the abstraction. If V ≅ k^n with standard basis e_1, ..., e_n, then the symmetric algebra S(V) is freely generated by these basis elements and can be viewed as the polynomial algebra in n generators when one chooses to identify functions with symmetric tensors. On the other hand, the algebra of polynomial functions on V is naturally identified with Sym(V*), the symmetric algebra on the dual space, which in the finite-dimensional case is isomorphic to the familiar polynomial ring k[x_1, ..., x_n], with x_i corresponding to the dual basis element e_i*.

The symmetric algebra also appears in practical and theoretical contexts. In algebraic geometry, it provides the coordinate algebra of affine space and serves as a model for polynomial functions on vector spaces. In invariant theory, it forms the stage on which group actions on V induce actions on S(V) and where invariants can be studied, often through graded components. In differential geometry and mathematical physics, symmetric tensors—arising as elements of S^n(V)—encode symmetric multilinear forms, moment maps, and various field-theoretic objects.

See also - tensor algebra - polynomial ring - exterior algebra - universal property - polynomial function - affine space - Spec