Clifford algebra is graded separable - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T05:43:24Z http://mathoverflow.net/feeds/question/116608 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/116608/clifford-algebra-is-graded-separable Clifford algebra is graded separable Sasha Pavlov 2012-12-17T13:50:06Z 2012-12-17T13:50:06Z <p>Let $D$ be an algebra of odd differential operators on a free module $V$, this algebra is isomorphic to the Clifford algebra $Cl(V^* \oplus V)$. Let $m$ denote multiplication map $$m : D\otimes D \to D.$$ I need an explicit formula for a bimodule splitting of this map or equivalently an element $z \in D\otimes D$ s.t. $az=za$ for any $a \in D$ and $m(z)=1$.</p> <p>It is possible to use isomorphism of algebras $Cl(V^* \oplus V) \cong End(\wedge V)$ and for $End(\wedge V)$ such splitting is given (up to sign) by the same formula as for matrix algebra. So, I know that such splitting exists and I want a nice formula in term of differential operators (or standard generators of Clifford algebra).</p>