On the (super)cocenter of Cyclotomic Sergeev algebras

TL;DR

This paper investigates the structural properties of cyclotomic Sergeev algebras, revealing their symmetry and supersymmetry depending on the level, and provides bases and bounds related to their centers and traces.

Contribution

It establishes symmetry and supersymmetry properties of cyclotomic Sergeev algebras and provides explicit bases and bounds for their trace and center components.

Findings

Cyclotomic Sergeev algebra is symmetric at odd level.

It is supersymmetric at even level.

Provides an integral basis for the trace of the algebra.

Abstract

We show that cyclotomic Sergeev algebra $\mathfrak{h}_n^g$ is symmetric when the level is odd and supersymmetric when the level is even. We give an integral basis for ${\rm Tr}(\mathfrak{h}_n^g)_{\overline{0}}$, and recover Ruff's result on the rank of ${\rm Z}(\mathfrak{h}_n^g)_{\bar{0}}$ when the level is odd. We obtain a generating set of ${\rm SupTr}(\mathfrak{h}_n^g)_{\overline{0}}$, which gives an upper bound of the dimension of ${\rm Z}(\mathfrak{h}_n^g)_{\bar{0}}$ when the level is even.