Moufang loops and toric surfaces
The toric surfaces for octonions and related objects are discussed.
math.AG↗
arXiv subjects
Vadim Schechtman
Publications and source records attributed to Vadim Schechtman.
PROBs and perverse sheaves I. Symmetric products
Algebraic structures involving both multiplications and comultiplications (such as, e.g., bialgebras or Hopf algebras) can be encoded using PROPs (categories with PROducts and Permutations) of Adams and MacLane. To encode such structures on objects of a braided monoidal category, we need PROBs (braided analogs of PROPs). Colored PROBs correspond to multi-sorted structures. In particular, we have a colored PROB B governing non-negatively graded bialgebras in braided categories. As a category, B splits into blocks B_n according to the grading. We relate B_n with the category P_n of perverse sheaves on the n-th symmetric product of the complex line, smooth with respect to the natural stratification by multiplicities. More precisely, we show that P_n is equivalent to the category of functors from B_n to vector spaces. This gives a natural quiver description of P_n.
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With Wronskian through the Looking Glass
In the work of Mukhin and Varchenko from 2002 there was introduced a Wronskian map from the variety of full flags in a finite dimensional vector space into a product of projective spaces. We establish a precise relationship between this map and the Plücker map. This allows us to recover the result of Varchenko and Wright saying that the polynomials appearing in the image of the Wronsky map are the initial values of the tau-functions for the Kadomtsev-Petviashvili hierarchy.
math.RT↗
Derived KZ equations
In this paper we strengthen the results of [SV] by presenting their derived version. Namely, we define a "derived Knizhnik - Zamolodchikov connection"\ and identify it with a "derived Gauss - Manin connection".
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Witt ring localized
The rings of $p$-typical Witt vectors are interpreted as spaces of vanishing cycles for some perverse sheaves over a disc. This allows to "localize"\ an isomorphism emerging in Drinfeld's theory of prismatization [Dr], Prop. 3.5.1, namely to express it as "an integral"\ of a standard exact triangle on the disc.
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Contingency Tables with Variable Margins (with an Appendix by Pavel Etingof)
Motivated by applications to perverse sheaves, we study combinatorics of two cell decompositions of the symmetric product of the complex line, refining the complex stratification by multiplicities. Contingency matrices, appearing in classical statistics, parametrize the cells of one such decomposition, which has the property of being quasi-regular. The other, more economical, decomposition, goes back to the work of Fox-Neuwirth and Fuchs on the cohomology of braid groups. We give a criterion for a sheaf constructible with respect to the ''contingency decomposition'' to be constructible with respect to the complex stratification. We also study a polyhedral ball which we call the stochastihedron and whose boundary is dual to the two-sided Coxeter complex (for the root system $A_n$) introduced by T.K. Petersen. The Appendix by P. Etingof studies enumerative aspects of contingency matrices. In particular, it is proved that the ''meta-matrix'' formed by the numbers of contingency matrices of various sizes, is totally positive.
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Shuffle algebras and perverse sheaves
We relate shuffle algebras, as defined by Nichols, Feigin-Odesskii and Rosso, to perverse sheaves on symmetric products of the complex line (i.e., on the spaces of monic polynomials stratified by multiplicities of roots). More precisely, we construct an equivalence between: (i) Braided Hopf algebras of a certain type. (ii) Factorizable collections of perverse sheaves on all the symmetric products. Under this eqiuvalence, the Nichols algebra associated to an object V corresponds to the collection of the intersection cohomology extensions of the local systems on the open configuration spaces associated to the tensor powers of V. Our approach is based on using real skeleta of complex configuration spaces.
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Positive Populations
A positive structure on the varieties of critical points of master functions for KZ equations is introduced. It comes as a combination of the ideas from classical works by G.Lusztig and a previous work by E.Mukhin and the second named author.
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Perverse sheaves over real hyperplane arrangements II
Let H be an arrangement of hyperplanes in R^n and Perv(C^n,H) be the category of perverse sheaves on C^n smooth with respect to the stratification given by complexified flats of H. We give a description of Perv(C^n,H) in terms of "matrix diagrams", i.e., diagrams formed by vector spaces E_{AB} labelled by pairs A,B of real faces of H (of all dimensions) or, equivalently, by the cells iA+B of a natural cell decomposition of C^n. A matrix diagram is formally similar to a datum describing a constructible (non-perverse) sheaf but with the direction of one half of the arrows reversed.
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Perverse schobers and birational geometry
Perverse schobers are conjectural categorical analogs of perverse sheaves. We show that such structures appear naturally in Homological Minimal Model Program which studies the effect of birational transformations such as flops, on the coherent derived categories. More precisely, the flop data are analogous to hyperbolic stalks of a perverse sheaf. In the first part of the paper we study schober-type diagrams of categories corresponding to flops of relative dimension 1, in particular we determine the categorical analogs of the (compactly supported) cohomology with coefficients in such schobers. In the second part we consider the example of a "web of flops" provided by the Grothendieck resolution associated to a reductive Lie algebra g and study the corresponding schober-type diagram. For g=sl(3) we relate this diagram to the classical space of complete triangles studied by Schubert, Semple and others.
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Solutions of KZ differential equations modulo $p$
We construct polynomial solutions of the KZ differential equations over a finite field $F_p$ as analogs of hypergeometric solutions.
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Rational differential forms on line and singular vectors in Verma modules over $\widehat {sl}_2$
We construct a monomorphism of the De Rham complex of scalar multivalued meromorphic forms on the projective line, holomorphic on the complement to a finite set of points, to the chain complex of the Lie algebra of $sl_2$-valued algebraic functions on the same complement with coefficients in a tensor product of contragradient Verma modules over the affine Lie algebra $\hat{sl}_2$. We show that the existence of singular vectors in the Verma modules (the Malikov-Feigin-Fuchs singular vectors) is reflected in the new relations between the cohomology classes of logarithmic differential forms.
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Coxeter element and particle masses
Let $\mathfrak{g}$ be a simple Lie algebra of rank $r$ over $\mathbb{C}$, $\mathfrak{h} \subset \mathfrak{g}$ a Cartan subalgebra. We construct a family of $r$ commuting Hermitian operators acting on $\mathfrak{h}$ whose eigenvalues are equal to the coordinates of the eigenvectors of the Cartan matrix of $\mathfrak{g}$.
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Vanishing cycles and Cartan eigenvectors
Using the vanishing cycles of simple singularities, we study the eigenvectors of Cartan matrices of finite root systems, and of q-deformations of these matrices.
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Perverse sheaves and graphs on surfaces
We give an explicit combinatorial description of the category Perv(S,N) of perverse sheaves on an oriented surface S (with boundary) with singularities at a given finite set N. The description is given in terms of any spanning graph K in S with the set of vertices N, so the category is defined entirely in terms of a ribbon graph. This description can be seen as an application, in the theory of perverse sheaves, of the idea of localization on a Lagrangian skeleton.
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Perverse Schobers
We suggest a possibility for a categorical generalization of the concept of a perverse sheaf, in which vector spaces are replaced by triangulated categories. We call such hypothetical objects perverse Schobers and consider several examples, giving a natural but ad hoc definition in each case. In the simplest case (perverse sheaves on a disk with one possible singular point) we propose, as a categorical analog, the data of a spherical functor.
math.AG↗
De Rham complex of a Gerstenhaber algebra
We introduce a notion of the De Rham complex of a Gerstenhaber algebra which produces a notion of a "quasi-BV structure", and allows to classify these structures, generalizing the classical results for polyvector fields.
math.AG↗
Perverse sheaves over real hyperplane arrangements
Let H be an arrangement of real hyperplanes in R^n. The complexification of H defines a natural stratification of C^n. We denote by Perv(C^n, H) the category of perverse sheaves on C^n smooth with respect to this stratification. We give a description of Perv(C^n, H) as the category of representations of an explicit quiver with relations, whose vertices correspond to real faces of H (of all dimensions). The relations are of monomial nature: they identify some pairs of paths in the quiver. They can be formulated in terms of the oriented matroid associated to H.
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