symmetric algebra

Let $M$ be a module over a commutative ring $R$ . Form the tensor algebra $T(M)$ over $R$ . Let $I$ be the ideal of $T(M)$ generated by elements of the form

u\otimes v-v\otimes u

where $u,v\in M$ . Then the quotient algebra defined by

S(M):=T(M)/I

is called the symmetric algebra over the ring $R$ .

Remark. Let $R$ be a field, and $M$ a finite dimensional vector space over $R$ . Suppose $\{e_{1},e_{2},\ldots,e_{n}\}$ is a basis of $M$ over $R$ . Then $T(M)$ is nothing more than a free algebra on the basis elements $e_{i}$ . Alternatively, the basis elements $e_{i}$ can be viewed as non-commuting indeterminates in the non-commutative polynomial ring $R\langle e_{1},e_{2},\ldots,e_{n}\rangle$ . This then implies that $S(M)$ is isomorphic to the “commutative” polynomial ring $R[e_{1},e_{2},\ldots,e_{n}]$ , where $e_{i}e_{j}=e_{j}e_{i}$ .

Title	symmetric algebra
Canonical name	SymmetricAlgebra
Date of creation	2013-03-22 15:46:23
Last modified on	2013-03-22 15:46:23
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	4
Author	CWoo (3771)
Entry type	Definition
Classification	msc 15A78