A result due to Joyal, Klyachko, and Stanley relates free Lie algebras to partition lattices. We will discuss the precise relationship and interpret the result in terms of the braid hyperplane arrangement. We will then extend this result to arbitrary (finite, real, and central) hyperplane arrangements, and do the same with several additional aspects of classical Hopf-Lie theory. The Tits monoid of an arrangement, and the notion of lune, play central roles in the discussion.
The Monster Lie algebra 𝔪 is an infinite-dimensional Lie algebra constructed by Borcherds as part of his programme to solve the Conway-Norton Monstrous Moonshine Conjecture. We describe how one may approach the problem of associating a Lie group analogue for 𝔪 and we outline some constructions.
The main purpose of this talk is to explain how the theory of torsors can be used to study problems in infinite dimensional Lie theory. I will not assume that the audience is familiar with torsors. Definitions and examples will be given. The main application in this case is to provide a general framework (relative sheaves of Lie algebras) that explains/justifies a known result about the derivations of multiloop algebras.
In this lecture I will explain a relationship between incidence theorems in elementary geometry and the theory of integrable systems, both classical and quantum. We will study geometric consistency relations between angles of 3-dimensional (3D) circular quadrilateral lattices, lattices whose faces are planar quadrilaterals inscribable into a circle. We show that these relations generate canonical transformations of a remarkable 'ultra-local' Poisson bracket algebra defined on discrete 2D surfaces consisting of circular quadrilaterals. Quantization of this structure allowed us to obtain new solutions of the tetrahedron equation (the 3D analogue of the Yang-Baxter equation) as well as reproduce all those that were previously known. These solutions generate an infinite number of non-trivial solutions of the Yang-Baxter equation and also define integrable 3D models of statistical mechanics and quantum field theory. The latter can be thought of as describing quantum fluctuations of lattice geometry.
n this talk we will discuss:
1. Tensor decompositions of locally matrix algebras and their parametrization by Steinitz numbers.
2. Automorphisms and derivations of locally matrix algebras.
3. Automorphisms and derivations of Mackey algebras and Mackey groups. In particular, we describe automorphisms of all infinite simple finitary torsion groups (in the classification of J.Hall) and derivations of all infinite-dimensional simple finitary Lie algebras (in the classification of A.Baranov and H.Strade).
Following Beilinson and Drinfeld, we describe vertex algebras as Lie algebras for a certain operad of n-ary chiral operations. This allows us to introduce the cohomology of a vertex algebra V as a
Lie algebra cohomology. When V is equipped with a good filtration, its associated graded is a Poisson vertex algebra. We relate the cohomology of V to the variational Poisson cohomology studied previously by De Sole and Kac.
All simple weak twisted modules over the Heisenberg-Virasoro vertex operator algebras, and all simple restricted modules over the mirror Heisenberg-Virasoro algebra will be given in this talk.
Classical free field realizations of affine Kac-Moody algebras (introduced by M.Wakimoto, B.Feigin and E.Frenkel) play an important role in quantum field theory. B.Cox initiated the study of free field realizations for the non-standard Borel subalgebras which led to an important class of intermediate (or parabolic) Wakimoto modules. A uniform construction of such realizations will be discussed based on a joint work with L.Krizka and P.Somberg.
In a joint work with D. López N. and L.-F. Préville-Ratelle in 2015 we introduce a family of non-symmetric operads Dyckm, which satisfies that:
1. Dyck0 is the operad of associative algebras,
2. Dyck1 is the operad Dend of dendriform algebras, introduced by J.-L. Loday,
3. the vector space spanned by the set of m-Dyck paths has a natural structure of free Dyckm algebra over one element,
4. for any k ≥ 1, there exist degeneracy operators si : Dyckm → Dyckm-1 and face operators dj: Dyckm → Dyckm+1, which defines a simplicial complex in the category of non-symmetric operads.
The main examples of Dyckm algebra are the vector spaces spanned by the m-simplices of certain combinatorial Hopf algebras, like the Malvenuto-Reutenauer algebras and the algebra of packed words.
A well-known result on associative algebras states that, as an 𝒮-module, the operad of Ass of associative algebras is the composition Ass = Com ∘ Lie, where Com is the operad of commutative algebras and Lie is the operad of Lie algebras. The version of this result for dendriform algebras is that Dend = Ass ∘ Brace, where Brace is the operad of brace algebras.
Our goal is to introduce the notion of m-brace algebra, for m ≥ 2, and prove that there exists a Poincaré-Birkoff-Witt Theorem in this context, stating that Dyckm = Ass ∘ m-Brace.
I intend to overview classifications of simple Lie (super)algebras of finite dimension and of polynomial growth. Various properties of complex Lie superalgebras resemble same of modular Lie algebras. I will encourage to consider these classifications without fanaticism: certain non-simple Lie (super)algebras, "close" to simple ones, are often "better" for us than simple ones.
Interesting features of deformations: semi-trivial deformations and (in super setting) odd parameters.
I'll formulate classification of finite-dimensional simple complex Lie superalgebras, odd parameters including.
I'll formulate a definition of Lie superalgebra suitable for any characteristic and classification of simple (finite-dimensional) Lie superalgebras over algebraically closed fields of characteristic 2. With a catch: modulo (a) classification of simple (finite-dimensional) Lie superalgebras (over the same field) and (b) classification of their gradings modulo 2. I'll mention conjectures on classification of modular Lie algebras and superalgebras.
Is it feasible to classify simple filtered Lie (super)algebras of polynomial growth? Interesting examples: deforms of the Poisson Lie (super)algebras, Lie (super)algebras of "matrices of complex size", etc.
Examples. Double extensions of simple Lie (super)algebras are definitely "more interesting" than the simple objects they extend.
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