Quiver varieties and the quantum Knizhnik–Zamolodchikov equation

We show how equivariant volumes of tensor product quiver varieties of type A are given by matrix elements of vertex operators of centrally extended doubles of Yangians and how these elements satisfy the rational level-one quantum Knizhnik–Zamolodchikov equation in some cases.     

Kac’s conjecture from Nakajima quiver varieties

We prove a generating function formula for the Betti numbers of Nakajima quiver varieties. We prove that it is a q-deformation of the Weyl-Kac character formula. In particular this implies that the constant term of the polynomial counting the number of absolutely indecomposable representations of a quiver equals the multiplicity of a certain weight in the corresponding Kac-Moody algebra, which was conjectured by Kac in 1982.     

The Dual Quiver of the Auslander–Reiten Quiver of Path Algebras

Let Q be a finite quiver of type A _n , n ≥ 1, D _n , n ≥ 4, E ₆, E ₇ and E ₈, σ ∈ Aut(Q), k be an algebraic closed field whose characteristic does not divide the order of σ. In this article, we prove that the dual quiver [(G_Q)\tilde]\widetilde{\Gamma_{Q}} of the Auslander–Reiten quiver Γ _Q of kQ, the Auslander–Reiten quiver of kQ#kás?kQ\#k\langle\sigma\rangle, and the Auslander–Reiten quiver G_[(Q)\tilde]\Gamma_{\widetilde{Q}} of k[(Q)\tilde]k\widetilde{Q}, where [(Q)\tilde]\widetilde{Q} is the dual quiver of Q, are isomorphic.     

Geometric construction of highest weight crystals for quantum generalized Kac–Moody algebras

We present a geometric construction of highest weight crystals B(λ) for quantum generalized Kac–Moody algebras. It is given in terms of the irreducible components of certain Lagrangian subvarieties of Nakajima’s quiver varieties associated to quivers with edge loops.     

The Brill–Noether curve and Prym-Tyurin varieties

We prove that the Jacobian of a general curve C of genus $g=2a+1$ , with $a\ge 2$ , can be realized as a Prym-Tyurin variety for the Brill–Noether curve $W^{1}_{a+2}(C)$ . As consequence of this result we are able to compute the class of the sum of secant divisors of the curve C, embedded with a complete linear series $g^{a-1}_{3a-2}$ .     

On the Toda and Kac–van Moerbeke hierarchies

We provide a comprehensive treatment of the single and double commutation method as a tool for constructing soliton solutions of the Toda and Kac–van Moerbeke hierarchy on arbitrary background. In addition, we present a novel construction based on the single commutation method. As an illustration we compute the N-soliton solution of the Toda and Kac–van Moerbeke hierarchy. Received November 20, 1997; in final form July 7, 1998     

Character formula of Kac–Wakimoto type for generalized Kac–Moody algebras

We prove a character formula of Kac–Wakimoto type for generalized Kac–Moody algebras. A character formula of this type is a generalization of the Weyl–Kac character formula, and is proved by Kac–Wakimoto in the case of Kac–Moody algebras. We remark that the formula is a generalization of that of Kac–Wakimoto even in the case of Kac–Moody algebras of indefinite type.     

On Fano and weak Fano Bott–Samelson–Demazure–Hansen varieties

Let G be a simple algebraic group over the field of complex numbers. Fix a maximal torus T and a Borel subgroup B of G containing T. Let w be an element of the Weyl group W of G, and let $Z(\stackrel{?}{w})$ be the Bott–Samelson–Demazure–Hansen (BSDH) variety corresponding to a reduced expression $\stackrel{?}{w}$ of w with respect to the data $(G,B,T)$.In this article we give complete characterization of the expressions $\stackrel{?}{w}$ such that the corresponding BSDH variety $Z(\stackrel{?}{w})$ is Fano or weak Fano. As a consequence we prove vanishing theorems of the cohomology of tangent bundle of certain BSDH varieties and hence we get some local rigidity results.     

Some remarks on Chow varieties and Euler–Chow series

In this article we give a geometric explanation of the fact that the Betti numbers of the d-fold symmetric product of the proyective space of dimension n are the same as those of the Grassmanian of d-planes in the complex vector space of dimension n+d. In fact, we give a correspondence which is the graph of a rational morphism which induces an isomorphism, and whose matrix is the identity. We also prove some properties of Euler–Chow series and state some open problems related to this series.     

First order Feynman–Kac formula

We study the parabolic integral kernel for the weighted Laplacian with a potential. For manifolds with a pole we deduce formulas and estimates for the derivatives of the Feynman–Kac kernels and their logarithms, these are in terms of a ‘Gaussian’ term and the semi-classical bridge.     

Rank-One Isometries of Buildings and Quasi-Morphisms of Kac–Moody Groups

Given an irreducible non-spherical non-affine (possibly non-proper) building X, we give sufficient conditions for a group G < Aut(X) to admit an infinite-dimensional space of non-trivial quasi-morphisms. The result applies in particular to all irreducible (non-spherical and non-affine) Kac–Moody groups over integral domains. In particular, we obtain finitely presented simple groups of infinite commutator width, thereby answering a question of Valerii G. Bardakov [MK, Prob. 14.13]. Independently of these considerations, we also include a discussion of rank-one isometries of proper CAT(0) spaces from a rigidity viewpoint. In an appendix, we show that any homogeneous quasi-morphism of a locally compact group with integer values is continuous.     

Quantum symmetric Kac–Moody pairs

The present paper develops a general theory of quantum group analogs of symmetric pairs for involutive automorphism of the second kind of symmetrizable Kac–Moody algebras. The resulting quantum symmetric pairs are right coideal subalgebras of quantized enveloping algebras. They give rise to triangular decompositions, including a quantum analog of the Iwasawa decomposition, and they can be written explicitly in terms of generators and relations. Moreover, their centers and their specializations are determined. The constructions follow G. Letzter's theory of quantum symmetric pairs for semisimple Lie algebras. The main additional ingredient is the classification of involutive automorphisms of the second kind of symmetrizable Kac–Moody algebras due to Kac and Wang. The resulting theory comprises various classes of examples which have previously appeared in the literature, such as q-Onsager algebras and the twisted q-Yangians introduced by Molev, Ragoucy, and Sorba.     

Derivation of Feynman–Kac and Bloch–Torrey Equations in a Trapping Medium

Methodology and Computing in Applied Probability - We derive rigorously the fractional counterpart of the Feynman–Kac equation for a transport problem with trapping events characterized by...     

Quantizations of Kac–Moody algebras

We analyze the extent to which a quantum universal enveloping algebra of a Kac–Moody algebra g$\mathfrak{g}$ is defined by multidegrees of its defining relations. To this end, we consider a class of character Hopf algebras defined by the same number of defining relations of the same degrees as the Kac–Moody algebra g$\mathfrak{g}$. We demonstrate that if the generalized Cartan matrix A$A$ of g$\mathfrak{g}$ is connected then the algebraic structure, up to a finite number of exceptional cases, is defined by just one “continuous” parameter q$q$ related to a symmetrization of A$A$, and one “discrete” parameter m$\mathfrak{m}$ related to the modular symmetrizations of A$A$. The Hopf algebra structure is defined by n(n−1)/2$n(n-1)/2$ additional “continuous” parameters. We also consider the exceptional cases for Cartan matrices of finite or affine types in more detail, establishing the number of exceptional parameter values in terms of the Fibonacci sequence.     

Magnetic Energies and Feynman–Kac–Itô Formulas for Symmetric Markov Processes

Given a (conservative) symmetric Markov process on a metric space we consider related bilinear forms that generalize the energy form for a particle in an electromagnetic field. We obtain one bilinear form by semigroup approximation and another, closed one, by using a Feynman–Kac–Itô formula. If the given process is Feller, its energy measures have densities and its jump measure has a kernel, then the two forms agree on a core and the second is a closed extension of the first. In this case we provide the explicit form of the associated Hamiltonian.