The number of independent traces and supertraces on symplectic reflection algebras

DOI

10.1080/14029251.2014.936755How to use a DOI?

Keywords

Symplectic reflection algebra; Cherednik algebra; trace; supertrace; invariant bilinear form

Abstract

It is shown that A:= H₁, _η (G), the sympectic reflection algebra over ℂ, has T_G independent traces, where T_G is the number of conjugacy classes of elements without eigenvalue 1 belonging to the finite group G ⊂ Sp(2N) ⊂ End(ℂ^2N) generated by the system of symplectic reflections.

Simultaneously, we show that the algebra A, considered as a superalgebra with a natural parity, has S_G independent supertraces, where S_G is the number of conjugacy classes of elements without eigenvalue -1 belonging to G.

We consider also A as a Lie algebra A^L and as a Lie superalgebra A^S.

It is shown that if A is a simple associative algebra, then the supercommutant [A^S, A^S] is a simple Lie superalgebra having at least S_G independent supersymmetric invariant non-degenerate bilinear forms, and the quotient [A^L, A^L]/([A^L, A^L] ∩ ℂ) is a simple Lie algebra having at least T_G independent symmetric invariant non-degenerate bilinear forms.

Copyright

© 2014 The Authors. Published by Atlantis Press and Taylor & Francis

Open Access

This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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TY  - JOUR
AU  - S.E. Konstein
AU  - I.V. Tyutin
PY  - 2021
DA  - 2021/01/06
TI  - The number of independent traces and supertraces on symplectic reflection algebras
JO  - Journal of Nonlinear Mathematical Physics
SP  - 308
EP  - 335
VL  - 21
IS  - 3
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2014.936755
DO  - 10.1080/14029251.2014.936755
ID  - Konstein2021
ER  -