Journal•ISSN: 1609-3321
Moscow Mathematical Journal
Independent University of Moscow
About: Moscow Mathematical Journal is an academic journal published by Independent University of Moscow. The journal publishes majorly in the area(s): Lie algebra & Symplectic geometry. It has an ISSN identifier of 1609-3321. Over the lifetime, 685 publications have been published receiving 14961 citations.
Topics: Lie algebra, Symplectic geometry, Hypersurface, Invariant (mathematics), Holomorphic function
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TL;DR: In this paper, it was shown that coherent sheaves on smooth proper commutative and noncommutative varieties have strong generators, and hence are saturated, whereas a smooth compact analytic surface with no curves is not saturated.
Abstract: We give a sufficient condition for an Ext-finite triangulated category to be saturated. Saturatedness means that every contravariant cohomological functor of finite type to vector spaces is representable. The condition consists in the existence of a strong generator. We prove that the bounded derived categories of coherent sheaves on smooth proper commutative and noncommutative varieties have strong generators, and are hence saturated. In contrast, the similar category for a smooth compact analytic surface with no curves is not saturated. 2000 Math. Subj. Class. Primary 18E30.
709 citations
TL;DR: In this article, a formalism based on quantization of quadratichamil tonians and loop groups is proposed, which provides a convenient home for most known general results and conjectures about Gromov-Witten invariants of compact symplectic manifolds and Frobenius structures at higher genus.
Abstract: We describea formalism based on quantizationof quadratichamil- tonians and symplectic actions of loop groups which provides a convenient home for most of known general results and conjectures about Gromov-Witten invariants of compact symplectic manifolds and, more generally, Frobenius structures at higher genus. We state several results illustrating the formalism and its use. In particular, we establish Virasoro constraints for semisimple Frobenius structures and outline a proof of the Virasoro conjecture for Gro- mov - Witten invariants of complex projective spaces and other Fano toric manifolds. Details will be published elsewhere.
523 citations
TL;DR: In this article, a general structure theory of not necessarily semisimple finite tensor categories is proposed, which generalizes the results in the semi-simple case (i.e. for fusion categories), obtained recently in our joint work with D.Nikshych.
Abstract: We start the general structure theory of not necessarily semisimple finite tensor categories, generalizing the results in the semisimple case (i.e. for fusion categories), obtained recently in our joint work with D.Nikshych. In particular, we generalize to the categorical setting the Hopf and quasi-Hopf algebra freeness theorems due to Nichols-Zoeller and Schauenburg, respectively. We also give categorical versions of the theory of distinguished group-like elements in a finite dimensional Hopf algebra, of Lorenz's result on degeneracy of the Cartan matrix, and of the absence of primitive elements in a finite dimensional Hopf algebra in zero characteristic. We also develop the theory of module categories and dual categories for not necessarily semisimple finite tensor categories; the crucial new notion here is that of an exact module category. Finally, we classify indecomposable exact module categories over the simplest finite tensor categories, such as representations of a finite group in positive characteristic, representations of a finite supergroup, and representations of the Taft Hopf algebra.
428 citations
TL;DR: In this article, it was shown that H(n,m) is a cellular symmetric algebra and K(m,n) is an infinite dimensional quasi-hereditary algebra.
Abstract: This is the first of four articles studying some slight generalisations H(n,m) of Khovanov's diagram algebra, as well as quasi-hereditary covers K(n,m) of these algebras in the sense of Rouquier, and certain infinite dimensional limiting versions. In this article we prove that H(n,m) is a cellular symmetric algebra and that K(n,m) is a cellular quasi-hereditary algebra. In subsequent articles, we relate these algebras to level two blocks of degenerate cyclotomic Hecke algebras, parabolic category O and the general linear supergroup, respectively.
174 citations
TL;DR: In this article, the canonical basis for a special family of cluster algebras of rank 2 was constructed explicitly for semisimple algebraic groups, and the canonical bases of these groups were introduced and explicitly constructed.
Abstract: The main motivation for the study of cluster algebras initiated in math.RT/0104151, math.RA/0208229 and math.RT/0305434 was to design an algebraic framework for understanding total positivity and canonical bases in semisimple algebraic groups. In this paper, we introduce and explicitly construct the canonical basis for a special family of cluster algebras of rank 2.
162 citations