Related papers: Cuspidal modules for the derivation Lie algebra ov…

In this paper we classify the irreducible quasifinite highest weight modules over the orthogonal and symplectic types Lie subalgebras of the Lie algebra of the matrix quantum pseudo differential operators. We also realize them in terms of…

The aim of this note is to introduce the notion of a $\operatorname{D}$-Lie algebra and to prove some elementary properties of $\operatorname{D}$-Lie algebras, the category of $\operatorname{D}$-Lie algebras, the category of modules on a…

The $n$-dimensional quantum torus $\Lambda$ is defined to be the $F$-algebra generated by variables $y_1, \cdots, y_n$ with the relations $y_iy_j = q_{ij}y_jy_i$ where $q_{ij}$ are suitable scalars from the base field. This algebra is also…

In this article, we determine the structure of the $p$-primary subgroup of the cuspidal rational torsion subgroup of the Jacobian $J_1(p^n)$ of the modular curve $X_1(p^n)$ for a regular prime $p$.

In this paper, we regard the smooth quadric threefold $Q_{3}$ as Lagrangian Grassmannian and search for fixed rational curves of low degree in $Q_{3}$ with respect to a torus action, which is the maximal subgroup of the special linear group…

We study quandle modules over quandle spaces $Q$, i.e. quandles endowed with geometric structures. In the case $Q$ is a regular $s$-manifold, we exhibit how modules over $Q$ are related with representations of Lie-Yamaguti algebras.

In this paper we classify and give Kazhdan-Lusztig type character formulas for equivariantly irreducible representations of Lie algebras of reductive algebraic groups over a field of large positive characteristic. The equivariance is with…

We establish a direct connection between the representation theories of Lie algebras and Lie superalgebras (of type A) via Fock space reformulations of their Kazhdan-Lusztig theories. As a consequence, the characters of finite-dimensional…

For a given symmetric quiver $Q$, we define a supercommutative quadratic algebra $\mathcal{A}_Q$ whose Poincar\'e series is related to the motivic generating function of $Q$ by a simple change of variables. The Koszul duality between…

We equip a family of algebras whose noncommutativity is of Lie type with a derivation based differential calculus obtained, upon suitably using both inner and outer derivations, as a reduction of a redundant calculus over the Moyal four…

This article presents a differential groupoid with ``coaction'' of the groupoid underlying the Quantum Euclidean Group (i.e. its $C^*$-algebra is the $C^*$-algebra of this quantum group). The dual of the Lie algebroid is a Poisson manifold…

The notion of crossed modules for Lie 2-algebras is introduced. We show that, associated to such a crossed module, there is a strict Lie 3-algebra structure on its mapping cone complex and a strict Lie 2-algebra structure on its…

The structure constants of quantum Lie algebras depend on a quantum deformation parameter q and they reduce to the classical structure constants of a Lie algebra at $q=1$. We explain the relationship between the structure constants of…

We study mirror symmetry (A-side vs B-side) in the framework of quantum differential systems. We focuse on the logarithmic and non-resonant case, which describes the geometric situation. We show that quantum differential systems provide a…

In this paper, we investigate the structure of highest weight modules over the quantum queer superalgebra $U_q(q(n))$. The key ingredients are the triangular decomposition of $U_q(q(n))$ and the classification of finite dimensional…

In this paper, we study the structures of Schur algebra and Lusztig algebra associated to partial flag varieties of affine type D. We show that there is a subalgebra of Lusztig algebra and the quantum groups arising from this subalgebras…

We survey some recent constructions of cluster algebra structures on coordinate rings of unipotent subgroups and unipotent cells of Kac-Moody groups. We also review a quantized version of these results.

A constructive approach to differential calculus on quantum principal bundles is presented. The calculus on the bundle is built in an intrinsic manner, starting from given graded (differential) *-algebras representing horizontal forms on…

An explicit vertex operator algebra construction is given of a class of irreducible modules for toroidal Lie algebras.

Let $\ell$ be a rational prime number and $K$ a number field. We prove that the logarithmic module $X_{d}$ attached to a $\mathbb{Z}_{\ell}^{d}$-extension $K_{d}$ of $K$ is a noetherian $\Lambda_{d}$-module. Moreover, under the…