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Graded Satake diagrams and super-symmetric pairs

D. Algethami, A. Mudrov, V. Stukopin

Abstract

We list classical spherical subalgebras in basic matrix Lie superalgebras which are quantizable to coideal subalgebras in the standard quantum supergroups, for any choice of Borel subalgebra. We classify the corresponding Satake-type diagrams and prove that each of them defines a family of proper spherical subalgebras.

Graded Satake diagrams and super-symmetric pairs

Abstract

We list classical spherical subalgebras in basic matrix Lie superalgebras which are quantizable to coideal subalgebras in the standard quantum supergroups, for any choice of Borel subalgebra. We classify the corresponding Satake-type diagrams and prove that each of them defines a family of proper spherical subalgebras.

Paper Structure

This paper contains 21 sections, 25 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Classical super-symmetric pairs
  3. Basic classical matrix Lie superalgebras
  4. Weyl operator and spherical data
  5. Decorated Dynkin diagrams and selection rules
  6. Super Satake diagrams
  7. General linear $\mathfrak{g}$
  8. Nonidentical $\tau$
  9. Identical $\tau$ and shaft spherical subalgebras
  10. Orthosymplectic $\mathfrak{g}$
  11. Black tail diagrams
  12. White tail diagrams
  13. Mixed coloured tail of $\mathfrak{D}$-shape
  14. No Satake diagrams with irregular $\mathfrak{l}$
  15. Non-trivial super-symmetric pairs
  16. ...and 6 more sections

Key Result

Lemma 2.3

The Weyl operator $w_\mathfrak{g}$ preserves the weight lattice $\Lambda$, and the root system $\mathrm{R}$. Furthermore, $w_\mathfrak{g}(\Pi)=-\Pi$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (55)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.6
  • proof
  • Lemma 2.7
  • proof
  • ...and 45 more