In this paper we study the semisimplicity of ℬ_r(C,δ ) , the Brauer algebra of type C over ℂ . In contrast to the classical Brauer algebras, which are semisimple for all but finitely many integer values of the parameter δ , the Brauer algebra of type C may fail to be semisimple even for certain nonintegral values of δ . Assuming δ 0 , we exploit the cellular structure of the algebra to analyze its cell modules and establish a criterion for semisimplicity. In particular, we prove that ℬ_r(C,δ ) is semisimple provided that δ ^2+ξδ∉ℤ for all ξ∈{1,-1} .
In this paper, we introduce the cyclic property for basic algebras and provide a necessary condition for an algebra Λ to share the same support τ -tilting poset with a given algebra Γ having . Furthermore, we show that this necessary condition is also sufficient if Γ is either a preprojective algebra of type , a Nakayama algebra, a generalized Brauer tree algebra, or a τ -tilting finite algebra with radical square zero.
Let G be a finite abelian group and let A be a G-graded algebra with color involution *. We first establish a Wedderburn–Malcev decomposition in case A is finite-dimensional. Then, we prove several results for algebras with polynomial identities in this setting, including the Hook theorem, the Specht property, and, most notably, the Representability theorems. The latter assert that A satisfies the same * -identities as the Grassmann envelope of a finite-dimensional (G×ℤ_2) -graded algebra B with a suitable color involution ♯ related to * .
Let p be an odd prime. Let G = SL_2(𝔽_p) and let B denote the subgroup of upper triangular matrices of G. Finally, let 𝔽 be an algebraically closed field of characteristic p. The Green correspondence gives a bijection between the non-projective indecomposable 𝔽[G] modules and non-projective indecomposable 𝔽[B] modules, realised by restriction and induction. In this paper, after recalling a suitable description of the non-projective indecomposable modules for these group algebras, we explicitly describe the Green correspondence bijection. We do this by pinpointing the modules’ position on the Stable Auslanden-Reiten quivers. Finally, we obtain two corollaries in terms of this description: formula for lifting the 𝔽[B] module decomposition of an 𝔽[G] module, and a complete description of Ind_B^G and Res^G_B .
