Related papers: On the Ado Theorem for finite Lie conformal algebr…

We exhibit explicit orthogonal decompositions of every multidimensional restricted root space of a real semi-simple Lie algebra. We then show a link between this result and a radiality property of smooth functions on G-homogeneous spaces…

We show that if a countably generated Lie algebra $H$ does not contain isomorphic copies of certain finite-dimensional nilpotent Lie algebras $A$ and $B$ (satisfying some mild conditions), then $H$ embeds into a quotient of $A \ast B$ that…

In this paper we classify irreducible integrable representations of loop toroidal Lie algebras with finite dimensional weight spaces. In both the cases we classify modules, when a part of center acts non-trivially and trivially on modules.

We use Reznick's Theorem for positive homogeneous polynomials to prove an elliptic regularity result for representations of enveloping algebras of Lie algebras. This allows us to relax a technical condition for a sum of squares…

We develop structure theory of finite Lie conformal superalgebras.

We characterize co-Hopfian finitely generated torsion free nilpotent groups in terms of their Lie algebra automorphisms, and construct many examples of such groups.

We say that a Lie algebra $\gfr$ is quasi-state rigid if every Ad-invariant continuous Lie quasi-state on it is the directional derivative of a homogeneous quasimorphism. Extending work of Entov and Polterovich, we show that every reductive…

We show that a tensor product of irreducible, finite dimensional representations of a simple Lie algebra over a field of characteristic zero, determines the individual constituents uniquely. This is analogous to the uniqueness of prime…

The Lie algebra of planar vector fields with coefficients from the field of rational functions over an algebraically closed field of characteristic zero is considered. We find all finite-dimensional Lie algebras that can be realized as…

We present a new look at description of real finite-dimensional Lie algebras. The basic element turns out to be a pair $(F,v)$ consisting of a linear mapping $F\in End(V)$ and its eigenvector $v$. This pair allows to build a Lie bracket on…

We define the periodic Full Kostant-Toda on every simple Lie algebra, and show its Liouville integrability. More precisely we show that this lattice is given by a Hamiltonian vector field, associated to a Poisson bracket which results from…

A simple axiomatic characterization of the noncommutative Ito algebra is given and a pseudo-Euclidean fundamental representation for such algebra is described. It is proved that every quotient Ito algebra has a faithful representation in a…

In this paper we analyze the structure of some subalgebras of quantized enveloping algebras corresponding to unipotent and solvable subgroups of a simple Lie group G. These algebras have the non--commutative structure of iterated algebras…

For a smooth manifold $X$ equipped with a volume form, let $\dL$ be the Lie algebra of volume preserving smooth vector fields on $X$. A. Lichnerowicz proved that the abelianization of $\dL$ is a finite-dimensional vector space, and that its…

For every non-exceptional affine Lie algebra, we explicitly construct a positive geometric crystal associated with a fundamental representation. We also show that its ultra-discretization is isomorphic to the limit of certain perfect…

For every finitary set functor F we demonstrate that free algebras carry a canonical partial order. In case F is bicontinuous, we prove that the cpo obtained as the conservative completion of the free algebra is the free completely…

Let X be an affine variety and L be a solvable Lie subalgebra of Lie(Aut(X)) generated by a finite collection of locally finite Lie subalgebras. The authors of [arXiv:2507.09679] wondered whether L is itself locally finite. Here we present…

We investigate a certain class of solvable metric Lie algebras. For this purpose a theory of twofold extensions associated to an orthogonal representation of an abelian Lie algebra is developed. Among other things, we obtain a…

I give a short proof of the following algebraic statement: if a vertex algebra is simple, then its underlying Lie conformal algebra is either abelian, or it is an irreducible central extension of a simple Lie conformal algebra.

We observe several facts and make conjectures about commutative algebras satisfying the Jacobi identity. The central question is which of those algebras admit a faithful representation (i.e., in Lie parlance, satisfy the Ado theorem, or, in…