The number of independent Traces and Supertraces on Symplectic Reflection Algebras

Abstract

It is shown that $A:=H_{1,\eta }(G)$, the Sympectic Reflection Algebra, has $T_G$ independent traces, where $T_G$ is the number of conjugacy classes of elements without eigenvalue 1 belonging to the finite group $G$ 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 $[A^L,A^L]/([A^L,A^L]\cap \mathbb{C})$ is a simple Lie algebra having at least $T_G$ independent symmetric invariant non-degenerate bilinear forms.

29 pages, LaTeX. arXiv admin note: substantial text overlap with arXiv:1211.6600

References in corpus (2)

• Klein operator and the Numbers of independent Traces and Supertraces on the Superalgebra of Observables of Rational Calogero Model based on the Root System
• Traces on the Algebra of Observables of Rational Calogero Model based on the Root System

Cited by in corpus (3)

• Connection between the ideals generated by traces and by supertraces in the superalgebras of observables of Calogero models
• The number of independent Traces and Supertraces on the Symplectic Reflection Algebra $H_{1,\nu }(\Gamma \wr S_N)$
• Supertraces on queerified algebras