What is the q-analogue of the Lefschetz decomposition? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T16:32:01Z http://mathoverflow.net/feeds/question/34260 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/34260/what-is-the-q-analogue-of-the-lefschetz-decomposition What is the q-analogue of the Lefschetz decomposition? Bruce Westbury 2010-08-02T13:22:07Z 2010-08-02T13:22:07Z <p>The representation theory behind the Lefschetz decomposition in Kahler geometry was summarised very neatly by Victor Protsak in his answer to <a href="http://mathoverflow.net/questions/29907" rel="nofollow">29907</a></p> <p>Let $W$ be a $2n$-dimensional symplectic vector space, $\bigwedge^\ast W$ its exterior algebra, and $\omega\in\bigwedge^2 W$ the invariant two-form. Exterior multiplication by $\omega$ and the contraction with $\omega$ define a pair of $Sp(W)$-equivariant graded linear transformations $L, \Lambda$ of $\bigwedge^\ast W$ into itself of degrees $2$ and $-2,$ and let $H=\deg-n$ be the graded degree $0$ map acting on $\bigwedge^k$ as multiplication by $k-n.$ Then $L,H,\Lambda$ form the standard basis of the Lie algebra $\mathfrak{sl_2}$ acting on $\bigwedge^\ast W$ and the actions of $Sp(W)$ and $\mathfrak{sl_2}$ are the commutants of each other.</p> <p>The representation theory is discussed in <a href="http://www.jstor.org/stable/2001418?origin=crossref" rel="nofollow">Remarks on classical invariant theory</a> by Roger Howe. This is also discussed in Griffiths and Harris "Principles of algebraic geometry" in Chapter 0, Section 7.</p> <p>My question is whether anyone knows the $q$-analogue of the representation theory (and just to be clear I mean explicitly and not in principle). More specifically, we have an inclusion of $U_q(A_{n-1})$ in $U_q(C_n)$ by ignoring the end node. The vector representation restricts to $U\oplus U^*$ where $U$ is also the vector representation. Then I want to define a $q$-analogue of the exterior algebra; ideally so it is a $U_q(C_n)$-module algebra. Then I want to construct the commutant, which ideally would be (a quotient of) $U_q(A_1)$.</p> <p>I hope that by now there is a consensus of the definition of $U_q$.</p>