The Green ring of a restricted enveloping algebra in characteristic 2

2025/11/17 by Nicolás Andruskiewitsch, Andruskiewitsch, Nicolás, Dirceu Bagio +5
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2511.13345

Abstract

Let \Bbbk be an algebraically closed field of characteristic 2 and let \mathfrakfsl(2) be the unique, up to isomorphism, 3-dimensional simple Lie algebra over \Bbbk. Denote by \mathfrakm the minimal 2-envelope of \mathfrakfsl(2) and by \mathfraku(\mathfrakm) its corresponding restricted enveloping algebra. The non-isomorphic finite-dimensional indecomposable \mathfraku(\mathfrakm)-modules were classified in \citeABDF. In this paper, the Green ring (or representation ring) for \mathfraku(\mathfrakm) is calculated. Also, the semisimplification of the representation category of \mathfraku(\mathfrakm) is determined.

Citations

• On the Drinfeld double of the restricted Jordan plane in characteristic 2
• A Prolog assisted search for new simple Lie algebras
• The Green rings of the generalized Taft Hopf algebras

Related

• Families of Green rings for abelian restricted Lie algebras
• Enveloping algebras that are principal ideal rings
• Realizing Enveloping Algebras via Varieties of Modules
• Restricted Irreducible Representations for the Non-graded Hamiltonian H(2; (1,1); Φ(1))
• The projective indecomposable modules for the restricted Zassenhaus algebras in characteristic 2