Related papers: Duality for Bethe algebras acting on polynomials i…

Given a Lie group $G$, a compact subgroup $K$ and a representation $\tau\in\hat K$, we assume that the algebra of $\text{End}(V_\tau)$-valued, bi-$\tau$-equivariant, integrable functions on $G$ is commutative. We present the basic facts of…

A conformal partition function ${\cal P}_n^m(s)$, which arose in the theory of Diophantine equations supplemented with additional restrictions, is concerned with {\it self-dual symmetric polynomials} -- reciprocal ${\sf R}^{\{m\}}_ {S_n}$…

New identities on traces of representations of the Hecke algebra on the spaces of paths on graphs are presented. These identities are relevant in the computation of partition functions with fixed boundary conditions and of two-point…

A Lie-admissible algebra gives by anticommutativity a Lie algebra. In this work we study remarkable classes of Lie-admissible algebras such as Vinberg, PreLie algebras. We compute the corresponding binary quadratic operads and study their…

A general theorem due to Howe of dual action of a classical group and a certain non-associative algebra on a space of symmetric or alternating tensors is reformulated in a setting of second quantization, and familiar examples in atomic and…

We introduce n-particle quantum graphs with singular two-particle interactions in such a way that eigenfunctions can be given in the form of a Bethe ansatz. We show that this leads to a secular equation characterising eigenvalues of the…

Let $f$ be a Laurent polynomial in $d$ commuting variables with integer coefficients. Associated to $f$ is the principal algebraic $\mathbb{Z}^d$-action $\alpha_f$ on a compact subgroup $X_f$ of $\mathbb{T}^{\mathbb{Z}^d}$ determined by…

We consider the gl_N Gaudin model of a tensor power of the standard vector representation. The geometric Langlands correspondence in the Gaudin model relates the Bethe algebra of the commuting Gaudin Hamiltonians and the algebra of…

We interpret the equivariant cohomology algebra H^*_{GL_n\times\C^*}(T^*F_\lambda;\C) of the cotangent bundle of a partial flag variety F_\lambda parametrizing chains of subspaces 0=F_0\subset F_1\subset\dots\subset F_N =\C^n, \dim…

We study Lie bialgebra structures on \emph{flat metric Lie algebras}, that is, Lie algebras $(\mathfrak{g},\langle\cdot,\cdot\rangle)$ whose associated left-invariant Riemannian metric on the simply connected Lie group $G$ has zero…

We study quantum integrable models with GL(3) trigonometric $R$-matrix and solvable by the nested algebraic Bethe ansatz. Using the presentation of the universal Bethe vectors in terms of projections of products of the currents of the…

For a smooth (locally trivial) principal bundle in Ehresmann's sense, the relation between the commuting vertical and horizontal actions of the structural Lie group and the structural Lie groupoid (isomorphisms between vertical fibers) is…

Let G be a complex semisimple group and U its maximal unipotent subgroup. We study the algebra D(G/U) of algebraic differential operators on G/U and also its quasi-classical counterpart: the algebra of regular functions on the cotangent…

First we recall homology groups of prer Lie superalgebras. Then introducing double weighted chain spaces, we deal with pre Lie superalgebra of multi-vector fields with polynomial coefficients on n-dimensional number space. The bracket is…

We obtain an interesting realization of the de Rham cohomological operators of differential geometry in terms of the noncommutative q-superoscillators for the supersymmetric quantum group GL_{qp} (1|1). In particular, we show that a unique…

In this paper we use invariant theory to develop the notion of cohomological detection for Type I classical Lie superalgebras. In particular we show that the cohomology with coefficients in an arbitrary module can be detected on smaller…

This thesis is devoted to the study of joint spectral multipliers for a system of pairwise commuting, self-adjoint left-invariant differential operators L_1,...,L_n on a connected Lie group G. Under the assumption that the algebra generated…

Action Lie groupoids are used to model spaces of orbits of actions of Lie groups on manifolds. For each such action groupoid $M\rtimes H$ we construct a locally convex bialgebroid $\mathord{\mathrm{Dirac}}(M\rtimes H)$ with an antipode over…

We construct an action of the positive Witt algebra on the categorified quantum group associated to a simply-laced Lie algebra. In the type A case, we show that this action induces an action of the positive Witt algebra on…

In this article, we describe a computational study of the action of the two natural $U$-operators acting on $\Gamma$-invariant spaces of harmonic cocycles for $\mathrm{GL}_3$ for certain congruence subgroups $\Gamma$, in a positive…