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Multiparameter Quantum Supergroups, Deformations and Specializations

Gastón Andrés García, Fabio Gavarini, Margherita Paolini

Abstract

In this paper we introduce a multiparameter version of the quantum universal enveloping superalgebras introduced by Yamane in [H. Yamane, "Quantized enveloping algebras associated to simple Lie superalgebras and their universal $R$-matrices", Publ. Res. Inst. Math. Sci. 30 (1994), no. 1, 15-87]. For these objects we consider: - (1) their deformations by twist and by 2-cocycle (both of "toral type"); in particular, we prove that this family is stable under both types of deformations; - (2) their semiclassical limits, which are multiparameter Lie superbialgebras; - (3) the deformations by twist and by 2-cocycle (of "toral type") of these multiparameter Lie superbialgebras: in particular, we prove that this family is stable under these deformations, and that "quantization commutes with deformation".

Multiparameter Quantum Supergroups, Deformations and Specializations

Abstract

In this paper we introduce a multiparameter version of the quantum universal enveloping superalgebras introduced by Yamane in [H. Yamane, "Quantized enveloping algebras associated to simple Lie superalgebras and their universal -matrices", Publ. Res. Inst. Math. Sci. 30 (1994), no. 1, 15-87]. For these objects we consider: - (1) their deformations by twist and by 2-cocycle (both of "toral type"); in particular, we prove that this family is stable under both types of deformations; - (2) their semiclassical limits, which are multiparameter Lie superbialgebras; - (3) the deformations by twist and by 2-cocycle (of "toral type") of these multiparameter Lie superbialgebras: in particular, we prove that this family is stable under these deformations, and that "quantization commutes with deformation".

Paper Structure

This paper contains 24 sections, 47 theorems, 169 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Multiparameters and their realizations
  3. Cartan data, multiparameter matrices, and realizations
  4. Twist deformations of multiparameters and realizations
  5. 2-cocycle deformations of multiparameters and realizations
  6. Multiparameter Lie superbialgebras and their deformations
  7. Lie superbialgebras and their deformations
  8. Multiparameter Lie superbialgebras (=MpLSbA's)
  9. Deformations of MpLSbA's by toral twists
  10. Deformations of MpLSbA's by toral $2$--cocycles
  11. Formal Multiparameter QUE Superalgebras (=FoMpQUESA's)
  12. The Hopf (super)algebra setup and QUESA's
  13. Formal multiparameter QUESA's (=FoMpQUESA's)
  14. Deformations of FoMpQUESA's
  15. Deformations of FoMpQUESAs by toral twists
  16. ...and 9 more sections

Key Result

Lemma 2.1.4

(cf. GG3, Lemma 2.1.4)) Let $\, P \in M_n({\Bbbk[[\hbar]]}) \,$ be a matrix as above. If $\; {\mathcal{R}} := (\, \mathfrak{h} \, , \Pi \, , \Pi^\vee \,) \,$ is a straight realization of $\, P$, then the triple $\, (\, \mathfrak{h} \, , \Pi \, , \Pi^\vee_S ) \,$ --- with $\, \Pi^\vee_S \! := {\{S_i\

Figures (12)

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  • ...and 7 more figures

Theorems & Definitions (97)

  • Definition 2.1.2
  • Definition 2.1.3
  • Lemma 2.1.4
  • Proposition 2.1.5
  • Remark 2.1.6
  • Proposition 2.1.7
  • Lemma 2.1.8
  • proof
  • Lemma 2.1.9
  • Remark 2.1.10
  • ...and 87 more