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 .
|Date of creation||2013-03-22 15:46:23|
|Last modified on||2013-03-22 15:46:23|
|Last modified by||CWoo (3771)|