Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new user$mathfrak{sl}_2$-Harish-Chandra modules for $mathfrak{sl}_2 ltimes L(4)$
254
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We use analogues of Enrights and Arkhipovs functors to determine the quiver and relations for a category of $mathfrak{sl}_2 ltimes L(4)$-modules which are locally finite (and with finite multiplicities) over $mathfrak{sl}_2$. We also outline serious obstacles to extend our result to $mathfrak{sl}_2 ltimes L(k)$, for $k>4$.
rate research
Read More
Let $V$ be the two-dimensional simple module and $M$ be a projective Verma module for the quantum group of $mathfrak{sl}_2$ at generic $q$. We show that for any $rge 1$, the endomorphism algebra of $Motimes V^{otimes r}$ is isomorphic to the type $B$ Temperley-Lieb algebra $rm{TLB}_r(q, Q)$ for an appropriate parameter $Q$ depending on $M$. The parameter $Q$ is determined explicitly. We also use the cellular structure to determine precisely for which values of $r$ the endomorphism algebra is semisimple. A key element of our method is to identify the algebras $rm{TLB}_r(q,Q)$ as the endomorphism algebras of the objects in a quotient category of the category of coloured ribbon graphs of Freyd-Yetter or the tangle diagrams of Turaev and Reshitikhin.
We classify the simple bounded weight modules of ${mathfrak{sl}(infty})$, ${mathfrak{o}(infty)}$ and ${mathfrak{sp}(infty)}$, and compute their annihilators in $U({mathfrak{sl}(infty}))$, $U({mathfrak{o}(infty))}$, $U({mathfrak{sp}(infty))}$, respectively.
Let $k$ be an algebraically closed field of characteristic $p > 2$, let $n in mathbb Z_{>0} $, and take $G$ to be one of the classical algebraic groups $mathrm{GL}_n(k)$, $mathrm{SL}_n(k)$, $mathrm{Sp}_n(k)$, $mathrm O_n(k)$ or $mathrm{SO}_n(k)$, with $mathfrak g = operatorname{Lie} G$. We determine the maximal $G$-stable closed subvariety $mathcal V$ of the nilpotent cone $mathcal N$ of $mathfrak g$ such that the $G$-orbits in $mathcal V$ are in bijection with the $G$-orbits of $mathfrak{sl}_2$-triples $(e,h,f)$ with $e,f in mathcal V$. This result determines to what extent the theorems of Jacobson--Morozov and Kostant on $mathfrak{sl}_2$-triples hold for classical algebraic groups over an algebraically closed field of ``small odd characteristic.
We prove a functorial correspondence between a category of logarithmic $mathfrak{sl}_2$-connections on a curve $X$ with fixed generic residues and a category of abelian logarithmic connections on an appropriate spectral double cover $pi : Sigma to X$. The proof is by constructing a pair of inverse functors $pi^{text{ab}}, pi_{text{ab}}$, and the key is the construction of a certain canonical cocycle valued in the automorphisms of the direct image functor $pi_ast$.
Let $n>1$ be an integer, $alphain{mathbb C}^n$, $bin{mathbb C}$, and $V$ a $mathfrak{gl}_n$-module. We define a class of weight modules $F^alpha_{b}(V)$ over $sl_{n+1}$ using the restriction of modules of tensor fields over the Lie algebra of vector fields on $n$-dimensional torus. In this paper we consider the case $n=2$ and prove the irreducibility of such 5-parameter $mathfrak{sl}_{3}$-modules $F^alpha_{b}(V)$ generically. All such modules have infinite dimensional weight spaces and lie outside of the category of Gelfand-Tsetlin modules. Hence, this construction yields new families of irreducible $mathfrak{sl}_{3}$-modules.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
