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We study a natural enlargement of the BGG Category O for a semisimple Lie algebra: the category of weight modules with trivial central character and finite-dimensional weight spaces supported on the root lattice. We give a geometric…

We classify anti-involutions of Lie superalgebra $\hsd$ preserving the principal gradation, where $\hsd$ is the central extension of the Lie superalgebra of differential operators on the super circle $S^{1|1}$. We clarify the relations…

We prove that the C*-algebra of a second-countable, \'etale, amenable groupoid is simple if and only if the groupoid is topologically principal and minimal. We also show that if G has totally disconnected unit space, then the associated…

A well-known and old result of Hazewinkel and Koszul states that the cohomology of a finite-dimensional Lie algebra is isomorphic, up to a suitable shift, to its twisted homology, a Lie-theoretical version of Poincare duality. This paper…

We extend Stone duality between generalized Boolean algebras and Boolean spaces, which are the zero-dimensional locally-compact Hausdorff spaces, to a non-commutative setting. We first show that the category of right-handed skew Boolean…

Gelfand-Naimark-Stone duality establishes a dual equivalence between the category ${\sf KHaus}$ of compact Hausdorff spaces and the category ${\boldsymbol{\mathit{uba}\ell}}$ of uniformly complete bounded archimedean $\ell$-algebras. We…

Let $A = \bigoplus_{i \geqslant 0} A_i$ be a graded locally finite $k$-algebra such that $A_0$ is an arbitrary finite-dimensional algebra satisfying some splitting condition. In this paper we develop a generalized Koszul theory generalizing…

Let A,B be finite dimensional G-graded algebras over an algebraically closed field K with char(K)=0, where G is an abelian group, and let Id_G(A) be the set of graded identities of A (res. Id_G(B)). We show that if A,B are G-simple then…

We show for an affine variety $X$, the derived category of quasi-coherent $D$-modules is equivalent to the category of DG modules over an explicit DG algebra, whose zeroth cohomology is the ring of Grothendieck differential operators…

Second-order superintegrable systems in dimensions two and three are essentially classified. With increasing dimension, however, the non-linear partial differential equations employed in current methods become unmanageable. Here we propose…

Let $G$ be an abelian group and $\mathbb{K}$ an algebraically closed field of characteristic zero. A. Valenti and M. Zaicev described the $G$-gradings on upper block-triangular matrix algebras provided that $G$ is finite. We prove that…

We describe the derived Picard groups and two-term silting complexes for quasi-hereditary algebras with two simple modules. We also describe by quivers with relations all algebras derived equivalent to a quasi-hereditary algebra with two…

We give a complete derived equivalence classification of all symmetric algebras of domestic representation type over an algebraically closed field. This completes previous work by R. Bocian and the authors, where in this paper we solve the…

In the present paper, we introduce two-dimensional categorified Hall algebras of smooth curves and smooth surfaces. A categorified Hall algebra is an associative monoidal structure on the stable $\infty$-category…

The geometric and algebraic theory of monomial ideals and multigraded modules is initiated over real-exponent polynomial rings and, more generally, monoid algebras for real polyhedral cones. The main results include the generalization of…

We use the category of linear complexes of tilting modules for the BGG category O, associated with a semi-simple complex finite-dimensional Lie algebra g, to reprove in purely algebraic way several known results about O obtained earlier by…

The authors developed in a recent paper natural dualities for finitely generated quasivarieties of Sugihara algebras. They thereby identified the admissibility algebras for these quasivarieties which, via the Test Spaces Method devised by…

We define graded, quasi-coherent $\mathcal{O}_S$-algebras over a given base derived scheme $S$, and show that these are equivalent to derived $\mathbb{G}_{m,S}$-schemes which are affine over $S$. We then use this $\mathbb{G}_{m,S}$-action…

Let $A$ and $B$ be finite-dimensional simple algebras with arbitrary signature over an algebraically closed field. Suppose $A$ and $B$ are graded by a semigroup $S$ so that the graded identitical relations of $A$ are the same as those of…

A classification theorem is obtained for a class of unital simple separable amenable Z-stable C*-algebras which exhausts all possible values of the Elliott invariant for unital stably finite simple separable amenable Z-stable C*-algebras.…