Related papers: On the geometry of V-systems
We consider a class of trigonometric solutions of WDVV equations determined by collections of vectors with multiplicities. We show that such solutions can be restricted to special subspaces to produce new solutions of the same type. We find…
A special class of solutions to the generalised WDVV equations related to a finite set of covectors is considered. We describe the geometric conditions ($\vee$-conditions) on such a set which are necessary and sufficient for the…
The V-systems are special finite sets of covectors which appeared in the theory of the generalized Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations. Several families of V-systems are known but their classification is an open problem. We…
The idea of a $\bigvee$-system was introduced by Veselov in the study of rational solutions of the WDVV equations of associativity. These are algebraic/geometric conditions on the set of covectors that appear in rational solutions to the…
A special class of solutions to the generalised WDVV equations related to a finite set of covectors is investigated. Some geometric conditions on such a set which guarantee that the corresponding function satisfies WDVV equations are found…
We construct second order reductions of the generalized Witten-Dijkgraaf-Verlinde-Verlinde system based on simple Lie algebras. We discuss to what extent some of the symmetries of the WDVV system are preserved by the reduction.
The generalized Drinfeld-Sokolov construction of KdV systems is reviewed in the case of an arbitrary affine Lie algebra paying particular attention to Hamiltonian aspects and $\W$-algebras. Some extensions of known results as well as a new…
Rational solutions of the Witten-Dijkgraaf-Verlinde-Verlinde (or WDVV) equations of associativity are given in terms a configurations of vectors which satisfy certain algebraic conditions known as $\bigvee$-conditions. The simplest examples…
We consider a class of solutions of the WDVV equation related to the special systems of covectors (called $\vee$-systems) and show that the corresponding logarithmic Frobenius structures can be naturally restricted to any intersection of…
For a simple complex Lie algebra of finite rank and classical type, we fix a triangular decomposition and consider the simple Levi subalgebras associated to closed subsets of roots. We study the restriction of global and local Weyl modules…
It is shown that the description of certain class of representations of the holonomy Lie algebra associated to hyperplane arrangement $\Delta$ is essentially equivalent to the classification of $\vee$-systems associated to $\Delta.$ The…
We consider trigonometric solutions of WDVV equations and derive geometric conditions when a collection of vectors with multiplicities determines such a solution. We incorporate these conditions into the notion of trigonometric Veselov…
We present a new family of the locus configurations which is not related to $\vee$-systems thus giving the answer to one of the questions raised in \cite{V1} about the relation between the generalised quantum Calogero-Moser systems and…
We describe an algorithm for classifying the closed subsets of a root system, up to conjugation by the associated Weyl group. Such a classification of an irreducible root system is closely related to the classification of the regular…
The interaction of a Lie algebra $\LL,$ having a weight space decomposition with respect to a nonzero toral subalgebra, with its corresponding root system forms a powerful tool in the study of the structure of $\LL.$ This, in particular,…
A class of associative (super) algebras is presented, which naturally generalize both the symmetric algebra $Sym(V)$ and the wedge algebra $\wedge (V)$, where $V$ is a vector-space. These algebras are in a bijection with those subsets of…
This paper is dedicated to the differential Galois theory in the complex analytic context for Lie-Vessiot systems. Those are the natural generaliza- tion of linear systems, and the more general class of differential equations adimitting…
We describe the structure and different features of Lie algebras in the Verlinde category, obtained as semisimplification of contragredient Lie algebras in characteristic $p$ with respect to the adjoint action of a Chevalley generator. In…
We present a complete proof that solutions of the WDVV equations in Seiberg-Witten theory may be constructed from root systems. A generalization to weight systems is proposed.
Lie systems form a class of systems of first-order ordinary differential equations whose general solutions can be described in terms of certain finite families of particular solutions and a set of constants, by means of a particular type of…