Symmetric algebra Guide, Meaning , Facts, Information and Description

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, 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, in case the naturality of looking at polynomials this way has some advantage. The construction of S(V) is also a special case, that of the Lie 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 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. 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, of course, the dimension grows with k.

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