Abstract We show that the Kazhdan–Lusztig category $KL_k$ of level-$k$ finite-length modules with highest-weight composition factors for affine Lie superalgebra $\widehat{\mathfrak{gl}(1|1)}$ has vertex algebraic braided tensor supercategory structure and its full subcategory $\mathcal{O}_k^{fin}$ objects semisimple Cartan subalgebra actions is a subcategory. every simple $\widehat{\mathfrak{gl...

The object of this paper is to study the relationship between certain projective modules and their endomorphism rings. Specifically, the basic problem is to describe the projective modules whose endomorphism rings are (von Neumann) regular, local semiperfect, or left perfect. Call a projective module regular if every cyclic submodule is a direct summand. Thus a ring is a regular module if it is...

Given a semisimple complex linear algebraic group G and a lower ideal I in positive roots of G, three objects arise: the ideal arrangement AI , the regular nilpotent Hessenberg variety Hess(N, I), and the regular semisimple Hessenberg variety Hess(S, I). We show that a certain graded ring derived from the logarithmic derivation module of AI is isomorphic to H∗(Hess(N, I)) and H∗(Hess(S, I)) , t...

1. Basics 2 1.1. Commutants 2 1.2. Opposite Rings 3 1.3. Units 3 1.4. Ideals 4 1.5. Modules 4 1.6. Division rings 5 1.7. Endomorphism rings 7 1.8. Monoid Rings 8 1.9. Noetherian and Artinian Rings 11 1.10. Simple rings 15 1.11. Prime ideals and prime rings 17 1.12. Notes 18 2. Wedderburn-Artin Theory 18 2.1. Semisimple modules and rings 18 2.2. Wedderburn-Artin I: Semisimplicity of Mn(D) 20 2.3...

Over many decades of development, there has been some evolution in the language and notation used for Weyl modules in the theory of semisimple algebraic groups in prime characteristic. Here we survey this evolution briefly, in the hope of clarifying what goes on in the literature. We also discuss briefly the related notion of tilting module. Since our emphasis is on rational representations, it...

Given a connected reductive complex algebraic group $G$ and spherical subgroup $H \subset G$, the extended weight monoid $\widehat \Lambda^+_G(G/H)$ encodes $G$-module structures on spaces of global sections all $G$-linearized line bundles $G/H$. Assuming that is semisimple simply $H$ specified by regular embedding in parabolic $P this paper we obtain description via set simple roots $G/H$ toge...

Let G be a finite group and R a strongly G-graded ring. The question of when R is semisimple (meaning in this paper semisimple artinian) has been studied by several authors. The most classical result is Maschke’s Theorem for group rings. For crossed products over fields there is a satisfactory answer given by Aljadeff and Robinson [3]. Another partial answer for skew group rings was given by Al...

We prove that a finite von Neumann algebra A is semisimple if the algebra of affiliated operators U of A is semisimple. When A is not semisimple, we give the upper and lower bounds for the global dimensions of A and U . This last result requires the use of the Continuum Hypothesis.

We make a first step towards a classification of simple generalized Harish-Chandra modules which are not Harish-Chandra modules or weight modules of finite type. For an arbitrary algebraic reductive pair of complex Lie algebras (g, k), we construct, via cohomological induction, the fundamental series F ·(p, E) of generalized Harish-Chandra modules. We then use F ·(p, E) to characterize any simp...