We calculate the first extension groups for finite-dimensional simple modules over an arbitrary generalized current Lie algebra, which includes the case of loop Lie algebras and their multivariable analogs.
The deformed current Lie algebra was introduced by the author to study the representation theory of cyclotomic q-Schur algebras at q=1. In this paper, we classify finite dimensional simple modules of deformed current Lie algebras.
The $(q, mathbf{Q})$-current algebra associated with the general linear Lie algebra was introduced by the second author in the study of representation theory of cyclotomic $q$-Schur algebras. In this paper, we study the $(q, mathbf{Q})$-current algebra $U_q(mathfrak{sl}_n^{langle mathbf{Q} rangle}[x])$ associated with the special linear Lie algebra $mathfrak{sl}_n$. In particular, we classify finite dimensional simple $U_q(mathfrak{sl}_n^{langle mathbf{Q} rangle}[x])$-modules.
In the present paper, using the technique of localization, we determine the center of the quantum Schr{o}dinger algebra $S_q$ and classify simple modules with finite-dimensional weight spaces over $S_q$, when $q$ is not a root of unity. It turns out that there are four classes of such modules: dense $U_q(mathfrak{sl}_2)$-modules, highest weight modules, lowest weight modules, and twisted modules of highest weight modules.
Let ${mathcal W}_n$ be the Lie algebra of polynomial vector fields. We classify simple weight ${mathcal W}_n$-modules $M$ with finite weight multiplicities. We prove that every such nontrivial module $M$ is either a tensor module or the unique simple submodule in a tensor module associated with the de Rham complex on $mathbb C^n$.
We construct a family of homomorphisms between Weyl modules for affine Lie algebras in characteristic p, which supports our conjecture on the strong linkage principle in this context. We also exhibit a large class of reducible Weyl modules beyond level one, for p not necessarily small.