dorsal/arxiv
View SchemaDecomposition of the adjoint representation of the small quantum $sl_2$
| Authors | V. Ostrik |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9512026 |
| URL | https://arxiv.org/abs/q-alg/9512026 |
| DOI | 10.1007/s002200050109 |
| Journal | Commun.Math.Phys. 186 (1997) 253-264 |
Abstract
Given a finite type root datum and a primitive root of unity $q=\sqrt[l]{1}$, G.~Lusztig has defined in [Lu] a remarkable finite dimensional Hopf algebra $\fu$ over the cyclotomic field ${\Bbb Q}(\sqrt[l]{1})$. In this note we study the adjoint representation $\ad$ of $\fu$ in the simplest case of the root datum $sl_2$. The semisimple part of this representation is of big importance in the study of local systems of conformal blocks in WZW model for $\hat{sl}_2$ at level $l-2$ in arbitrary genus. The problem of distinguishing the semisimple part is closely related to the problem of integral representation of conformal blocks (see [BFS]). We find all the indecomposable direct summands of $\ad$ with multiplicities. It appears that $\ad$ is isomorphic to a direct sum of simple and projective modules. It can be lifted to a module over the (infinite dimensional) quantum universal enveloping algebra with divided powers $U_q(sl_2)$ which is also a direct sum of simples and projectives.
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"abstract": "Given a finite type root datum and a primitive root of unity $q=\\sqrt[l]{1}$,\nG.~Lusztig has defined in [Lu] a remarkable finite dimensional Hopf algebra\n$\\fu$ over the cyclotomic field ${\\Bbb Q}(\\sqrt[l]{1})$. In this note we study\nthe adjoint representation $\\ad$ of $\\fu$ in the simplest case of the root\ndatum $sl_2$. The semisimple part of this representation is of big importance\nin the study of local systems of conformal blocks in WZW model for $\\hat{sl}_2$\nat level $l-2$ in arbitrary genus. The problem of distinguishing the semisimple\npart is closely related to the problem of integral representation of conformal\nblocks (see [BFS]). We find all the indecomposable direct summands of $\\ad$\nwith multiplicities. It appears that $\\ad$ is isomorphic to a direct sum of\nsimple and projective modules. It can be lifted to a module over the (infinite\ndimensional) quantum universal enveloping algebra with divided powers\n$U_q(sl_2)$ which is also a direct sum of simples and projectives.",
"arxiv_id": "q-alg/9512026",
"authors": [
"V. Ostrik"
],
"categories": [
"q-alg",
"math.QA"
],
"doi": "10.1007/s002200050109",
"journal_ref": "Commun.Math.Phys. 186 (1997) 253-264",
"title": "Decomposition of the adjoint representation of the small quantum $sl_2$",
"url": "https://arxiv.org/abs/q-alg/9512026"
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