The number of independent Traces and Supertraces on Symplectic Reflection Algebras

TL;DR

This paper investigates the structure of symplectic reflection algebras, revealing the number of independent traces and supertraces based on conjugacy classes, and explores their properties as Lie and Lie superalgebras.

Contribution

It establishes the counts of independent traces and supertraces on symplectic reflection algebras and analyzes their simplicity and invariant forms as Lie and Lie superalgebras.

Findings

Number of independent traces equals the number of conjugacy classes without eigenvalue 1.

Number of independent supertraces equals the number of conjugacy classes without eigenvalue -1.

Supercommutant forms a simple Lie superalgebra with multiple invariant bilinear forms.

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.