I will give an overview of a programme investigating projective embeddings of (exceptional) geometries which Hendrik Van Maldeghem and I started in 2010.
Given a (quadratic) Jordan algebra J over a ring k, one obtains three Lie algebras, the derivation algebra, the structure algebra, and the Tits algebra. We are particularly interested in the case where J is an Albert algebra.
The nets giving a diagrammatic description of the category of (tensor products of) fundamental representations of 𝔰𝔩n form a cellular category. We can then ask about the natural inner form on certain cell modules. In this talk, we will calculate the determinant of some of these forms in terms of certain traces of clasps or magic weave elements (for which there is a conjectured formula due to Elias). The method appears moderately general and gives a result which is hopefully illuminating and applicable to other monoidal, cellular categories.
The upper triangular matrix algebras are important in Linear Algebra, and represent a powerful tool in Ring Theory. They also appear in the theory of PI algebras.
In addition to the usual associative product, one can consider the Lie bracket and also the symmetric (Jordan) product on the upper triangular matrices.
We discuss the group gradings on the upper triangular matrices viewed as an associative, Lie and Jordan algebra, respectively. Valenti and Zaicev proved that the associative gradings are, in a sense, given by gradings on the matrix units. Di Vincenzo, Valenti and Koshlukov classified such gradings. Later on, Yukihide and Koshlukov, described the Lie and the Jordan gradings. In this talk we recall some of these results as well as a new development in a rather general setting, obtained by Yukihide and Koshlukov.
Previous work constructed an analogue of the Springer resolution for the universal cover of the principal nilpotent orbit. In joint work with Precup and Russell, we showed that in type A this generalized Springer resolution is closely connected with Lusztig's generalized Springer correspondence. In this talk we discuss the geometry of the fibres of the generalized Springer resolution, and in particular, show that the fibres have an analogue of an affine paving.
Buildings have been introduced by Tits in order to study semisimple algebraic groups from a geometrical point of view. One of the most important results in the theory of buildings is the classification of thick irreducible spherical buildings of rank at least 3. In particular, any such building comes from an RGD-system. The decisive tool in this classification is the Extension theorem for spherical buildings, i.e. a local isometry extends to the whole building.
Twin buildings were introduced by Ronan and Tits in the late 1980s. Their definition was motivated by the theory of Kac-Moody groups over fields. Each such group acts naturally on a pair of buildings and the action preserves an opposition relation between the chambers of the two buildings. This opposition relation shares many important properties with the opposition relation on the chambers of a spherical building. Thus, twin buildings appear to be natural generalizations of spherical buildings with infinite Weyl group. Since the notion of RGD-systems exists not only in the spherical case, one can ask whether any twin building (satisfying some further conditions) comes from an RGD-system. In 1992 Tits proves several results that are inspired by his strategy in the spherical case and he discusses several obstacles for obtaining a similar Extension theorem for twin buildings. In this talk I will speak about the history and developments of the Extension theorem for twin buildings.
We provide a unified combinatorial framework to study orbits in affine flag varieties via the associated Bruhat-Tits buildings. We first formulate, for arbitrary affine buildings, the notion of a chimney retraction. This simultaneously generalizes the two well-known notions of retractions in affine buildings: retractions from chambers at infinity and retractions from alcoves. We then present a recursive formula for computing the images of certain minimal galleries in the building under chimney retractions, using purely combinatorial tools associated to the underlying affine Weyl group. Finally, for Bruhat-Tits buildings, we relate these retractions and their effect on certain minimal galleries to double coset intersections in the corresponding affine flag variety.
Previous work constructed an analogue of the Springer resolution for the universal cover of the principal nilpotent orbit. In joint work with Precup and Russell, we showed that in type A this generalized Springer resolution is closely connected with Lusztig's generalized Springer correspondence. In this talk we discuss the geometry of the fibres of the generalized Springer resolution, and in particular, show that the fibres have an analogue of an affine paving.
We consider tensor powers of the natural 𝔰𝔩n-representation, and we look for descriptions of highest weight vectors therein: We discuss explicit formulas for n=2, a recursion for n=3, and for bigger n we demonstrate how Jucys-Murphy elements allow us to compute highest weight vectors (both in theory and in practice using Sage).
We discuss non-abelian Poisson structures on affine and projective spaces over ℂ. We also construct a class of examples of non-abelian Poisson structures on ℂPn-1 for n ≥ 3. These non-abelian Poisson structures depend on a modular parameter τ ∈ ℂ and an additional discrete parameter k ∈ ℤ, where 1 ≤ k < n and k,n are coprime. The abelianization of these Poisson structures can be lifted to the quadratic elliptic Poisson algebras qn,k(τ).
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.
The famous Lehmer problem asks whether there is a gap between 1 and the Mahler measure of algebraic integers which are not roots of unity. Asked in 1933, this deep question concerning number theory has since then been connected to several other subjects. After introducing the concepts involved, we will briefly describe a few of these connections with the theory of linear groups. Then, we will discuss the equivalence of a weak form of the Lehmer conjecture and the 'uniform discreteness' of cocompact lattices in semisimple Lie groups (conjectured by Margulis). Joint work with Lam Pham.
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.