The Z2-graded dimensions of the free Jordan superalgebra J(D1|D2)

Abstract

Let k be a field of characteristic 0. For a superspace $V=V_{\overline{0}}\oplus V_{\overline{1}}$ over k, we call the vector $({\dim }_k{V}_{\overline{0}},{\dim }_k{V}_{\overline{1}})$ the ($Z_2$-)graded dimension of V. Let $J\left(D_1\right|D_2)$ be the free Jordan superalgebra generated by $D_1$ even generators and $D_2$ odd generators. In this paper, we study the graded dimensions of the n-components of $J\left(D_1\right|D_2)$ and find the connection between them and the homology of Tits-Allison-Gao Lie superalgebra of $J\left(D_1\right|D_2)$ following the method given by I. Kashuba and O. Mathieu in [15], where they deal with the free Jordan algebra. And, four related conjectures on the free Jordan superalgebras and related Lie superalgebras are proposed in this article.