2023/12/27 by Shikui Shang, Shang, Shikui · 1 citation
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2312.16369
openalex publication_date 2023/12/27 · openalex created_date 2023/12/30 · openalex updated_date 2026/07/28
Let k be a field of characteristic 0. We introduce a pair of adjoint functors, Allison-Benkart-Gao functor \AG and Berman-Moody functor \BM, between the category of non-unital alternative algebras over k and the category \LieR of Lie algebras with compatible sl3(k)-actions. Surprisingly, when A is an alternative algebra without a unit, the Allison-Benkart-Gao Lie algebra \AG(A) is not isomorphic to the more well-known Steinberg Lie algebra st3(A) in general. Let A(D) be the free (non-unital) alternative algebra over D generators with the inner derivation algebra \innAD. A conjecture on the homology Hr(\AGAD) is proposed. Furthermore, consider the degree n component of A(D)n(resp. \innADn). The previous conjecture implies another conjecture on the dimensions on A(D)n and Inner A(D)n. Some evidences are given to support these conjectures. Finally, we prove the cyclicity of the alternative structure, namely that the symmetric group S1+D acts on the multilinear part of A(D), which plays an important role to connect the Lie algebra homology of \AGAD and the character of A(D).