Let $\mathfrak g$ be a semisimple Lie algebra. A $\mathfrak g$-algebra is an associative algebra $R$ on which $\mathfrak g$ acts by derivations. There are several significant examples. Let $V$ a finite dimensional $\mathfrak g$ module and take $R=\mathrm{End} V$ or $R=D(V)$ being the ring of derivations on $V$ . Again take $R=U(\mathfrak g)$. In all these cases $ S=U(\mathfrak g)\otimes R$ is again a $\mathfrak g$-algebra. Finally let $T$ denote the subalgebra of invariants of $S$.
For the first choice of $R$ above the representation theory of $T$ can be rather explicitly described in terms of Kazhdan-Lusztig polynomials. In the second case the simple $T$ modules can be described in terms of the simple $D(V)$ modules. In the third case it is shown that all simple $T$ modules are finite dimensional, despite the fact that $T$ is not a PI ring, except for the case $\mathfrak g =\mathfrak {sl}(2)$.
Seminar series
Date
Tue, 22 May 2018
Time
14:15 -
15:30
Location
L4
Speaker
Anthony Joseph
Organisation
Weizmann Institute