A Quantum Version of the Algebra of Distributions of SL2_2

  • Iván Ezequiel Angiono

    Universidad Nacional de Córdoba, Argentina
A Quantum Version of the Algebra of Distributions of SL$_2$ cover
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Let λ\lambda be a primitive root of unity of order \ell. We introduce a family of finite-dimensional algebras {Dλ,Nsl2}NN0\{\mathcal{D}_{\lambda, N}{\mathfrak{sl}_2}\}_{N\in\mathbb N_0} over the complex numbers, such that Dλ,Nsl2\mathcal{D}_{\lambda, N}{\mathfrak{sl}_2} is a subalgebra of Dλ,Msl2\mathcal{D}_{\lambda,M}{\mathfrak{sl}_2} if N<MN < M, and Dλ,N1sl2Dλ,Nsl2\mathcal{D}_{\lambda, N-1}{\mathfrak{sl}_2}\subset \mathcal{D}_{\lambda, N}{\mathfrak{sl}_2} is a uλ(sl2)\mathfrak u_{\lambda}(\mathfrak{sl}_2)-cleft extension.

The simple Dλ,Nsl2\mathcal{D}_{\lambda,N}{\mathfrak{sl}_2}-modules (LNp)0p<N+1(\mathcal{L}_{N}{p})_{0\le p<\ell^{N+1}} are highest weight modules, which admit a tensor product decomposition: the first factor is a simple uλ(sl2)\mathfrak u_{\lambda}(\mathfrak{sl}_2)-module and the second factor is a simple Dλ,N1sl2\mathcal{D}_{\lambda,N-1}{\mathfrak{sl}_2}-module. This factorization resembles the corresponding Steinberg decomposition, and the family of algebras resembles the presentation of the algebra of distributions of SL2_2 as a filtration by finite-dimensional subalgebras.

Cite this article

Iván Ezequiel Angiono, A Quantum Version of the Algebra of Distributions of SL2_2. Publ. Res. Inst. Math. Sci. 54 (2018), no. 1, pp. 141–161

DOI 10.4171/PRIMS/54-1-5