Singular Degenerations of Lie Supergroups of Type $D(2,1;a)$

Symmetry Integrability and Geometry-Methods and Applications

article

Kenji Iohara¹  Fabio Gavarini² 

[1] Université Claude Bernard Lyon 1, Institut Camille Jordan;Dipartimento di Matematica, Università di Roma ''Tor Vergata''

关键词: hyperbolic geometry;    hyperspherical geometry;    fundamental solution;    Helmholtz equation;    Gegenbauer series;    separation of variables;    addition theorems;    associated Legendre functions;    Ferrers functions 2010 Mathematics Subject C;   

DOI  :  10.3842/SIGMA.2018.137

来源: National Academy of Science of Ukraine

【 摘 要 】

The complex Lie superalgebras $\mathfrak{g}$ of type $D(2,1;a)$ - also denoted by $\mathfrak{osp}(4,2;a) $ - are usually considered for ''non-singular'' values of the parameter $a$, for which they are simple. In this paper we introduce five suitable integral forms of $\mathfrak{g}$, that are well-defined at singular values too, giving rise to ''singular specializations'' that are no longer simple: this extends the family of simple objects of type $D(2,1;a)$ in five different ways. The resulting five families coincide for general values of $a$, but are different at ''singular'' ones: here they provide non-simple Lie superalgebras, whose structure we describe explicitly. We also perform the parallel construction for complex Lie supergroups and describe their singular specializations (or ''degenerations'') at singular values of $a$. Although one may work with a single complex parameter $a$, in order to stress the overall $\mathfrak{S}_3$-symmetry of the whole situation, we shall work (following Kaplansky) with a two-dimensional parameter $\boldsymbol{\sigma} = (\sigma_1,\sigma_2,\sigma_3)$ ranging in the complex affine plane $\sigma_1 + \sigma_2 + \sigma_3 = 0$.

【 授权许可】

Unknown