Connection between the ideals generated by traces and by supertraces in the superalgebras of observables of Calogero models

Abstract

If G is a finite Coxeter group, then symplectic reflection algebra H := H 1,η (G) has Lie algebra sl 2 of inner derivations and can be decomposed under spin:We show that if the ideals I i (i = 1, 2) of all the vectors from the kernel of degenerate bilinear forms B i (x, y) := sp i (x · y), where sp i are (super)traces on H, do exist, then I 1 = I 2 if and only if I 1 H 0 = I 2 H 0 .

Supertraces on queerified algebras

“…For a description of ideals in these algebras and superalgebras, see [KT1,KT2,KT3]. It is interesting to determine when these ideals are themselves simple algebras and superalgebras, and describe (super)traces on them and on their queerifications.…”

“…Inspired by [BLLS], where Lie algebras are queerified over an algebraically closed ground field of characteristic p = 2 in order to produce the complete list of simple finite-dimensional Lie superalgebras in characteristic p = 2, this new method -queerification -producing many new simple Lie superalgebras from associative algebras and superalgebras is applied in [L] over C to several infinite-dimensional algebras of interest in theoretical physics. A number of papers were devoted to description of traces on these algebras, and supertraces on these algebras considered as superalgebras, see [KS,KT1,KT2,KT3]. In this note, we describe the supertraces on the Lie queerifications of these algebras and superalgebras, having added one more type of examples.…”

Ideals generated by traces in the symplectic reflection algebra $H_{1,ν_1, ν_2}(I_2(2m))$. II

“…It is clear that the kernel of B tr is an ideal in H, we will denote it I tr . It is shown in [2] that for any degenerate trace tr, the ideal I tr is completely determined by I tr H 0 .…”

“…This paper is a continuation of [3]; we advise the reader to recall [3]. 2 The associative algebra H 1,ν 1 ,ν 2 (I 2 (n)) with n even 2.1 The group I 2 (2m) Definition 2.1. We denote the finite subgroup of O(2, R), generated by the root system I 2 (2m), also by I 2 (2m).…”