Related papers: Modular vector fields attached to Dwork family: $\…

Let $\Gamma$ be a generic subgroup of the multiplicative group $\mathbb{C}^*$ of nonzero complex numbers. We define a class of Lie algebras associated to $\Gamma$, called twisted $\Gamma$-Lie algebras, which is a natural generalization of…

We study embeddings $J \rightarrow G$ of simple linear algebraic groups with the following property: the simple components of the $J$ module Lie($G$)/Lie($J$) are all minuscule representations of $J$. One family of examples occurs when the…

A Dwork family is a one-parameter monomial deformation of a Fermat hypersurface. In this paper we compute algebraically the invariant part of its Gauss-Manin cohomology under the action of certain subgroup of automorphisms. To achieve that…

A weakly complete vector space over $\mathbb{K}=\mathbb{R}$ or $\mathbb{K}=\mathbb{C}$ is isomorphic to $\mathbb{K}^X$ for some set $X$ algebraically and topologically. The significance of this type of topological vector spaces is…

Gaudin subalgebras are abelian Lie subalgebras of maximal dimension spanned by generators of the Kohno-Drinfeld Lie algebra t_n. We show that Gaudin subalgebras form a variety isomorphic to the moduli space of stable curves of genus zero…

In the 1990s, in work of Le Bruyn and Smith and in work of Le Bruyn and Van den Bergh, it was proved that point modules and line modules over the homogenization of the universal enveloping algebra of a finite-dimensional Lie algebra…

The universal enveloping algebra of any semisimple Lie algebra $\mathfrak{g}$ contains a family of maximal commutative subalgebras, called shift of argument subalgebras, parametrized by regular Cartan elements of $\mathfrak{g}$. For…

During the last decades algebraization of space turned out to be a promising tool at the interface between Mathematics and Theoretical Physics. Starting with works by Gel'fand-Kolmogoroff and Gel'fand-Naimark, this branch developed as from…

Let $\mathfrak{g}$ be a finite-dimensional complex Lie algebra and $\textrm{HLie}_{m}(\mathfrak{g})$ be the affine variety of all multiplicative Hom-Lie algebras on $\mathfrak{g}$. We use a method of computational ideal theory to describe…

In the present paper we describe a new class of Gelfand--Tsetlin modules for an arbitrary complex simple finite-dimensional Lie algebra g and give their geometric realization as the space of delta-functions" on the flag manifold G/B…

We present a family of selfinjective algebras of type D, which arise from the 3-preprojective algebras of type A by taking a $\mathbb{Z}_3$-quotient. We show that a subset of these are themselves 3-preprojective algebras, and that the…

The paper presents the complete classification of Automorphic Lie Algebras based on $\mathfrak{sl}_n (\mathbb{C})$, where the symmetry group $G$ is finite and the orbit is any of the exceptional $G$-orbits in $\overline{\mathbb{C}}$. A key…

We study {\em disemisimple} Lie algebras, i.e., Lie algebras which can be written as a vector space sum of two semisimple subalgebras. We show that a Lie algebra $\mathfrak{g}$ is disemisimple if and only if its solvable radical coincides…

For a simply connected (non-nilpotent) solvable Lie group $G$ with a lattice $\Gamma$ the de Rham and Dolbeault cohomologies of the solvmanifold $G/\Gamma$ are not in general isomorphic to the cohomologies of the Lie algebra $\mathfrak g$…

In the computation of the normal form of a colored network vector field, following the semigroup(oid) approach in [19], one would like to be able to say something about the structure of the Lie algebra of linear colored network vector…

Let $G$ be a simple complex Lie group with Lie algebra $\mf g$ and let $\af$ be the affine Lie algebra. We use intertwining operators and Knizhnik-Zamolodchikov equations to construct a family of $\N$-graded vertex operator algebras…

We give an affirmative answer to the question whether there exist Lie algebras for suitable closed subgroups of the unitary group $U(\mathcal{H})$ in a Hilbert space $\mathcal{H}$ with $U(\mathcal{H})$ equipped with the strong operator…

A Lie version of Turaev's $\overline{G}$-Frobenius algebras from 2-dimensional homotopy quantum field theory is proposed. The foundation for this Lie version is a structure we call a \textit{$\frak{g}$-quasi-Frobenius Lie algebra} for…

We investigate the existence of left-invariant closed G$_2$-structures on seven-dimensional non-solvable Lie groups, providing the first examples of this type. When the Lie algebra has trivial Levi decomposition, we show that such a…

We prove that Ad-semisimple conjugacy classes in a connected Lie group $G$ are closed embedded submanifolds of $G$. We also prove that if $\alpha:H\to G$ is a homomorphism of connected Lie groups such that the kernel of $\alpha$ is discrete…