The ${\mathbb Z}_2$-graded dimensions of the free Jordan superalgebra $J(D_1|D_2)$

TL;DR

This paper investigates the graded dimensions of free Jordan superalgebras with specific generators and explores their connection to Lie superalgebra homology, proposing four conjectures based on these findings.

Contribution

It introduces a detailed study of graded dimensions of free Jordan superalgebras and links them to the homology of associated Lie superalgebras, extending previous work on free Jordan algebras.

Findings

Derived formulas for graded dimensions of $J(D_1|D_2)$

Established connections to Tits-Allison-Gao Lie superalgebra homology

Proposed four new conjectures in the theory of Jordan superalgebras

Abstract

Let $k$ be a field of characteristic $0$. For a superspace $V=V_\bar{0}\oplus V_\bar{1}$ over $k$, we call the vector $(\dim_k V_\bar{0} ,\dim_k V_\bar{1})$ the (${\mathbb Z}_2$-)graded dimension of $V$. Let $J(D_1|D_2)$ 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 $J(D_1|D_2)$ and find the connection between them and the homology of Tits-Allison-Gao Lie superalgebra of $J(D_1|D_2)$ following the method given by I.Kashuba and O.Mathieu in [KM], where they deal with the free Jordan algebra. And, four interesting conjectures of above contents are proposed in our paper.