On the (super)cocenter of Cyclotomic Sergeev algebras

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 $\mathrm{T}\mathrm{r}({\mathfrak{h}}_n^g{)}_{\bar{0}}$, and recover Ruff’s result on the rank of $\mathrm{Z}({\mathfrak{h}}_n^g{)}_{\bar{0}}$ when the level is odd. We obtain a generating set of $\mathrm{S}\mathrm{u}\mathrm{p}\mathrm{T}\mathrm{r}({\mathfrak{h}}_n^g{)}_{\bar{0}}$, which gives an upper bound of the dimension of $\mathrm{Z}({\mathfrak{h}}_n^g{)}_{\bar{0}}$ when the level is even.