Related papers: Generalized Casimir Operators
In this paper we introduce a notion of {\it generalized operad} containing as special cases various kinds of operad--like objects: ordinary, cyclic, modular, properads etc. We then construct inner cohomomorphism objects in their categories…
A family of vertex operators that generalizes those given by Jing for the Hall-Littlewood symmetric functions is presented. These operators produce symmetric functions related to the Poincare polynomials referred to as generalized Kostka…
For every simple Lie algebra $\mathfrak{g}$ we consider the associated Takiff algebra $\mathfrak{g}^{}_{\ell}$ defined as the truncated polynomial current Lie algebra with coefficients in $\mathfrak{g}$. We use a matrix presentation of…
In this paper an asymmetrical operator of generalised translation is introduced, the generalised modulus of smoothness is defined by its means and the direct and inverse theorems in approximation theory are proved for that modulus. ----- V…
For a quiver with weighted arrows we define gauge-theory K-theoretic W-algebra generalizing the definition of Shiraishi et al., and Frenkel and Reshetikhin. In particular, we show that the qq-character construction of gauge theory presented…
The universal Baxter operator is an element of the Archimedean spherical Hecke algebra H(G,K), K be a maximal compact subgroup of a Lie group G. It has a defining property to act in spherical principle series representations of G via…
This is an expanded version of the lectures given by the author on the 3rd school "Lie algebras, algebraic groups and invariant theory" in Togliatti, Russia. In these notes we explain the concept of a categorical Kac-Moody action by…
Let g be a simple Lie algebra and q transcendental. We consider the category C_P of finite-dimensional representations of the quantum loop algebra Uq(Lg) in which the poles of all l-weights belong to specified finite sets P. Given the data…
We introduce and systematically develop two classes of discrete integrable operators: those with $2\times 2$ matrix kernels and those possessing general differential kernels, thereby generalizing the discrete analogue previously studied. A…
It is shown that for inhomogeneous Lie algebras $\frak{g}=\frak{s}\overrightarrow{\oplus}_{\Lambda}(\dim \Lambda)L_{1}$ satisfying the condition $\mathcal{N}(\frak{g})=1$, the only Casimir operator can be explicitly constructed from the…
We describe a categorical g action, called a (g,theta) action, which is easier to check in practice. Most categorical g actions can be shown to be of this form. The main result is that a (g,theta) action carries actions of quiver Hecke…
The classical Gelfand--Naimark theorems provide important insight into the structure of general and of commutative C*-algebras. It is shown that these can be generalized to certain ordered *-algebras. More precisely, for $\sigma$-bounded…
We introduce an affinization of the quantum Kac-Moody algebra associated to a symmetric generalized Cartan matrix. Based on the affinization, we construct a representation of the quantum Kac-Moody algebra by vertex operators from bosonic…
We characterize generalized derivatives of the solution operator of the obstacle problem. This precise characterization requires the usage of the theory of so-called capacitary measures and the associated solution operators of relaxed…
Recent advances in the theory of complex symmetric operators are presented and related to current studies in non-hermitian quantum mechanics. The main themes of the survey are: the structure of complex symmetric operators, $C$-selfadjoint…
We present gauge invariant, self adjoint Einstein operators for mixed symmetry higher spin theories. The result applies to multi-forms, multi-symmetric forms and mixed antisymmetric and symmetric multi-forms. It also yields explicit action…
We construct a vertex operator realization for the simple current primary fields of WZW theories which are based on simply laced affine Lie algebras g. This is achieved by employing an embedding of the integrable highest weight modules of g…
We apply the techniques of quasideterminants to construct new families of Casimir elements for the general Lie superalgebra whose images under the Harish-Chandra isomorphism are respectively the elementary, complete and power sums…
We consider a natural generalisation of the class of hyperbolic Kac-Moody algebras. We describe in detail the conditions under which these algebras are Lorentzian. We also construct their fundamental weights, and analyse whether they…
We generalize the notion of a Kac-Moody Lie algebra to the setting of Deligne Categories. Then we derive the Kac-Weyl formula for the category $\mathcal{O}$ integrable representations for such an algebra. This paper generalizes results of…