symmetric algebra
Let be a module over a commutative ring . Form the tensor algebra over . Let be the ideal of generated by elements of the form
where .
Then the quotient algebra defined by
is called the symmetric algebra over the ring .
Remark. Let be a field, and a finite dimensional vector space over . Suppose is a basis of over . Then is nothing more than a free algebra
on the basis elements . Alternatively, the basis elements can be viewed as non-commuting indeterminates in the non-commutative polynomial ring
. This then implies that is isomorphic
to the “commutative
” polynomial ring , where .
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 |