The talk is a survey of our recent results on the homotopy theory of operated algebras such as Rota-Baxter associative (or Lie) algebras and differential associative (or Lie) algebras etc. We make explicit the Kozul dual homotopy cooperads and the minimal models of the operads governing these operated algebras. As a consequence the L∞ structures on the deformation complexes are described as well.
A transposed Poisson algebra is a triple (L,⋅,[⋅,⋅]) consisting of a vector space L with two bilinear operations ⋅ and [⋅,⋅], such that (L,⋅) is a commutative associative algebra; (L,[⋅,⋅]) is a Lie algebra; and the 'transposed' Leibniz law holds: 2z⋅[x,y]=[z⋅x,y]+[x,z⋅y] for all x,y,z∈L. A transposed Poisson algebra structure on a Lie algebra (L,[⋅,⋅]) is a (commutative associative) multiplication ⋅ on L such that (L,⋅,[⋅,⋅]) is a transposed Poisson algebra. I will give an overview of my recent results in collaboration with Ivan Kaygorodov (Universidade da Beira Interior) on the classification of transposed Poisson structures on several classes of Lie algebras.
To any double Poisson algebra we produce a double Poisson vertex algebra using the jet algebra construction. We show that this construction is compatible with the representation functor which associates to any double Poisson (vertex) algebra and any positive integer a Poisson (vertex) algebra. We also consider related constructions, such as Poisson reductions and Hamiltonian reductions. This allows us to provide various interesting examples of double Poisson vertex algebras, in particular from double quivers.
I will present some interesting computations concerning polynomial and rational invariants of nilpotent Lie algebras. I will say more about standard filiform Lie algebras which appear to have the highest level of complication among the small-dimensional algebras. I will outline an implementable algorithm for the computation of generators of the field of rational invariants.
Let F be a finitely generated free algebra in a variety of algebras over a field of characteristic zero. A polynomial in F is called symmetric, if it is preserved under any permutation of the generators. The set S(F) of symmetric polynomials is a subalgebra of F. In this talk, we examine the algebras S(F), where F is the free metabelian associative, Lie, Leibniz, Poisson algebra or the free algebra generated by generic traceless matrices or the free algebra in the variety generated by Grassmann algebras.
In my talk I would like to discuss my joint articles with S. Sierra about the primitive ideals of universal enveloping U(W) and the symmetric algebra S(W) of Witt Lie algebra W and similar Lie algebras (including Virasoro Lie algebra). The key theorem in this setting is that every nontrivial quotient by a two-sided ideal of U(W) or S(W) has finite Gelfand-Kirillov dimension. Together with Sierra we enhanced this statement to the description of primitive Poisson ideals of S(W) in terms of certain points on the complex plane plus a few parameters attached to these points. In the end I will try to explain how all these concepts works for the ideals whose quotient has Gelfand-Kirillov dimension 2.
In this talk we will present recent results on the category of finite-dimensional modules for map superalgebras. Firstly, we will show a new description of certain irreducible modules. Secondly, we will use this new description to extract homological properties of the category of finite-dimensional modules for map superalgebras, most importantly, its block decomposition.
We say that an element x in a ring R is nilpotent last-regular if it is nilpotent of certain index n+1 and its last nonzero power xn is regular von Neumann, i.e., there exists another element y∈R such that xnyxn=xn. This type of elements naturally arise when studying certain inner derivations in the Lie algebra Skew(R,∗) of a ring R with involution ∗ whose indices of nilpotence differ when considering them acting as derivations on Skew(R,∗) and on the whole R. When moving to the symmetric Martindale ring of quotients Qms(R) of R we still obtain inner derivations with the same indices of nilpotence on Qms(R) and on the skew-symmetric elements Skew(Qms(R),∗) of Qms(R), but with the extra condition of being generated by a nilpotent last-regular element. This condition strongly determines the structure of Qms(R) and of Skew(Qms(R),∗). We will review the Jordan canonical form of nilpotent last-regular elements and show how to get gradings in associative algebras (with and without involution) when they have such elements.
The title matches that of a series of papers by various authors beginning in 1997, whose goal was the study and classification of such algebras over fields of positive characteristic. The original motivation came from group theory: the Leedham-Green and Newman coclass conjectures on pro-p groups from 1980 had all become theorems relatively recently, and subsequent results of Shalev and Zelmanov had raised interest in what one could say about Lie algebras of finite coclass. In positive characteristic, the simplest case of coclass one (i.e., 'Lie algebras of maximal class', also called 'filiform' in some quarters) appeared challenging even under the strong assumptions of those Lie algebras being infinite-dimensional and graded over the positive integers. I will review motivations and results of those studies, including some classifications obtained by Caranti, Newman, Vaughan-Lee. Then I will describe some generalizations recently established with three of my former PhD students.
The problem of determining centralizers in the enveloping algebras of Lie algebras is considered from both the algebraic and analytical perspectives. Applications of the procedure, such as the decomposition problem of the enveloping algebra of a simple Lie algebra, the labelling problem, and the construction of orthonormal bases of states are considered.
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