Symmetric algebra

From Wikipedia, the free encyclopedia

In mathematics, the symmetric algebra S(V) on a vector space V over a field K is a certain commutative unital associative K-algebra containing V. In fact, it is the "most general" such algebra, a fact and characterization which can be expressed by a universal property.

It turns out that S(V) is in effect the same as the polynomial ring, over K, in indeterminates that are basis vectors for V. Therefore this construction only brings something extra when the "naturality" of looking at polynomials this way has some advantage.

It is possible to use the tensor algebra T(V) to describe the symmetric algebra S(V). In fact we pass from the tensor algebra to the symmetric algebra by forcing it to be commutative; if elements of V commute, then tensors in them must, so that we construct the symmetric algebra from the tensor algebra by taking the quotient algebra of T(V) by the ideal generated by all differences of products

v⊗w − w⊗v

for v and w in V.

Just as with a polynomial ring, there is a direct sum decomposition of S(V) as a graded algebra, into summands

S^k(V)

which consist of the linear span of the monomials in vectors of V of degree k, for k = 0, 1, 2, ... (with S⁰(V) = K and S¹(V)=V). The K-vector space S^k(V) is the k-th symmetric power of V. The case k = 2, for example, is the symmetric square. It has a universal property with respect to symmetric multilinear operators defined on V^k. The S^k are functors comparable to the exterior powers; here, though, the dimension grows with k; it is given by

$\operatorname{dim}(S^k(V)) = \binom{n+k-1}{k}$

where n is the dimension of V.

The construction of the symmetric algebra generalizes to the symmetric algebra S(M) of a module M over a commutative ring. If M is a free module over the ring R, then its symmetric algebra is isomorphic to the polynomial algebra over R whose indeterminates are a basis of M, just like the symmetric algebra of a vector space. However, that is not true if M is not free; then S(M) is more complicated.