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The roots of exceptional modular Lie superalgebras with Cartan matrix

Sofiane Bouarroudj, Dimitry Leites, Alexander Lozhechnyk, Jin Shang

Abstract

For each of the exceptional Lie superalgebras with indecomposable Cartan matrix, we give the explicit list of its roots of and the corresponding Chevalley basis for one of the inequivalent Cartan matrices, the one corresponding to the greatest number of mutually orthogonal isotropic odd simple roots. Our main tools: Grozman's Mathematica-based code SuperLie, and Python.

The roots of exceptional modular Lie superalgebras with Cartan matrix

Abstract

For each of the exceptional Lie superalgebras with indecomposable Cartan matrix, we give the explicit list of its roots of and the corresponding Chevalley basis for one of the inequivalent Cartan matrices, the one corresponding to the greatest number of mutually orthogonal isotropic odd simple roots. Our main tools: Grozman's Mathematica-based code SuperLie, and Python.

Paper Structure

This paper contains 32 sections, 33 equations.

Table of Contents

  1. Roots and root vectors
  2. ${\mathfrak{osp}}(4|2; a)$, ${\mathfrak{ag}}(2)$, and ${\mathfrak{ab}}(3)$ for $p\geq 5$.
  3. $1{\mathfrak{brj}}(2;5)$ of $\mathop{\mathrm{sdim}}\nolimits 10|12$.
  4. $1{\mathfrak{brj}}(2;3)$ of $\mathop{\mathrm{sdim}}\nolimits 10|8$.
  5. $1{\mathfrak{el}}(5;5)$ of $\mathop{\mathrm{sdim}}\nolimits = 55|32$.
  6. $7{\mathfrak{el}}(5;3)$ of $\mathop{\mathrm{sdim}}\nolimits = 39|32$.
  7. $1{\mathfrak{g}}(1,6)$ of $\mathop{\mathrm{sdim}}\nolimits 21|14$.
  8. $2{\mathfrak{g}}(2,3)$ of $\mathop{\mathrm{sdim}}\nolimits 12/10|14$.
  9. $2{\mathfrak{g}}(2,6)$ of $\mathop{\mathrm{sdim}}\nolimits 36/34|20$.
  10. 7${\mathfrak{g}}(3,3)$ of $\mathop{\mathrm{sdim}}\nolimits 23/21|16$.
  11. $2{\mathfrak{g}}(3,6)$ of $\mathop{\mathrm{sdim}}\nolimits 36|40$.
  12. $6{\mathfrak{g}}(4,3)$ of $\mathop{\mathrm{sdim}}\nolimits 24|26$.
  13. $13{\mathfrak{g}}(8,3)$ of $\mathop{\mathrm{sdim}}\nolimits 55|50$.
  14. $2{\mathfrak{g}}(4,6)$ of $\mathop{\mathrm{sdim}}\nolimits 66|32$.
  15. $4{\mathfrak{g}}(6,6)$ of $\mathop{\mathrm{sdim}}\nolimits 78|64$.
  16. ...and 17 more sections