Symmetric algebra
In mathematics, the symmetric algebra S(V) on a vector space V over a field K is the construction of a commutative K from K, in a way to have a universal property. It turns out that this is in effect the same as making the polynomial ring, over K, in indeterminates that are basis vectors for V. Therefore this construction only brings something extra, in case the naturality of looking at polynomials this way has some advantage. The construction is also a special case, that of bracket always being 0, of the universal enveloping algebra construction.
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 should take the quotient ring of T(V) by the ideal generated by all differences of products
- vw − wv
for v and w in V. Given the polynomial ring as model, one expects and can prove a direct sum decomposition of S(V) as a graded ring, into summands
- Sk
which consist the linear span of the monomials in vectors of V of degree k, for k = 0, 1, 2, ... . This gives rise to the symmetric powers of V, as functors comparable to the exterior powers; here though, of course, the dimension grows with k.