The Green Ring of a Restricted Enveloping Algebra in Characteristic 2

Abstract Let $$\Bbbk $$ k be an algebraically closed field of characteristic 2 and let $$\mathfrak {fsl}(2)$$ fsl ( 2 ) be the unique, up to isomorphism, 3-dimensional simple Lie algebra over $$\Bbbk $$ k . Denote by $$\mathfrak {m}$$ m the minimal 2-envelope of $$\mathfrak {fsl}(2)$$ fsl ( 2 ) and by $$\mathfrak {u}(\mathfrak {m})$$ u ( m ) its corresponding restricted enveloping algebra. The non-isomorphic finite-dimensional indecomposable $$\mathfrak {u}(\mathfrak {m})$$ u ( m ) -modules were classified in [1]. In this paper, the Green ring (or representation ring) for $$\mathfrak {u}(\mathfrak {m})$$ u ( m ) is calculated. Also, the semisimplification of the representation category of $$\mathfrak {u}(\mathfrak {m})$$ u ( m ) is determined.