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

2022/11/17 by Shikui Shang, Shang, Shikui
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Nonlinear Waves and Solitons #Quantum Algebra (math.QA) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2211.09393

openalex publication_date 2022/11/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let k be a field of characteristic 0. For a superspace V=V₀⊕ V₁ over k, we call the vector (dim_k V₀ ,dim_k V₁) the (\mathbb Z₂-)graded dimension of V. Let J(D₁|D₂) be the free Jordan superalgebra generated by D₁ even generators and D₂ odd generators. In this paper, we study the graded dimensions of the n-components of J(D₁|D₂) and find the connection between them and the homology of Tits-Allison-Gao Lie superalgebra of J(D₁|D₂) 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.

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