Tag - Lie algebras

Sergey Malev and Alexei Kanel-Belov: Evaluations of nonassociative polynomials on finite-dimensional algebras

5th August, 2021

Let p be a polynomial in several non-commuting variables with coefficients in an algebraically closed field K of arbitrary characteristic. It has been conjectured that for any n, for p multilinear, the image of p evaluated on the set Mn(K) of n by n matrices is either zero, or the set of scalar matrices, or the set sln(K) of matrices of trace 0, or all of Mn(K). In this talk we will discuss the generalization of this result for non-associative algebras such as Cayley-Dickson algebra (i.e. algebra of octonions), pure (scalar free) octonion Malcev algebra and basic low rank Jordan algebras.

Mikhail Kotchetov: Fine gradings on classical simple Lie algebras

8th July, 2021

Gradings by abelian groups have played an important role in the theory of Lie algebras since its beginning: the best known example is the root space decomposition of a semisimple complex Lie algebra, which is a grading by a free abelian group (the root lattice). Involutive automorphisms or, equivalently, gradings by the cyclic group of order 2, appear in the classification of real forms of these Lie algebras. Gradings by all cyclic groups were classified by V. Kac in the late 1960s and applied to the study of symmetric spaces and affine Kac-Moody Lie algebras.

In the past two decades there has been considerable interest in classifying gradings by arbitrary groups on algebras of different varieties including associative, Lie and Jordan. Of particular importance are the so-called fine gradings (that is, those that do not admit a proper refinement), because any grading on a finite-dimensional algebra can be obtained from them via a group homomorphism, although not in a unique way. If the ground field is algebraically closed and of characteristic 0, then the classification of fine abelian group gradings on an algebra (up to equivalence) is the same as the classification of maximal quasitori in the algebraic group of automorphisms (up to conjugation). Such a classification is now known for all finite-dimensional simple complex Lie algebras.

In this talk I will review the above mentioned classification and present a recent joint work with A. Elduque and A. Rodrigo-Escudero in which we classify fine gradings on classical simple real Lie algebras.

Waldemar Hołubowski: Normal subgroups in the group of column-finite infinite matrices

7th June, 2021

The classical result, due to Jordan, Burnside, Dickson, says that every normal subgroup of GLn(K) (K a field, n ≥ 3) which is not contained in the centre, contains SLn(K). A. Rosenberg gave description of normal subgroups of GL(V), where V is a vector space of any infinite cardinality dimension over a division ring. However, when he considers subgroups of the direct product of the centre and the group of linear transformations g such that g − idV has finite-dimensional range the proof is not complete. We fill this gap for countable-dimensional V giving a description of the lattice of normal subgroups in the group of infinite column-finite matrices indexed by positive integers over any field. Similar results for Lie algebras of matrices will be surveyed.

José María Pérez Izquierdo: Some aspects of the free non-associative algebra

15th April, 2021

The free nonassociative algebra provides a simple combinatorial context to extend some constructions from the associative setting. In this talk, based on joint work with J. Mostovoy and I. P. Shestakov, I will briefly discuss three of them related to nonassociative Lie theory: the embedding of the free loop as nonassociative formal power series, a nonassociative extension of the Baker-Campbell-Hausdorff formula and a nonassociative version of Solomon's descent algebra.

Daniele Rosso: Fixed rings of twisted generalized Weyl algebras

12th April, 2021

Twisted generalized Weyl algebras (TGWAs) are a large family of algebras that includes several algebras of interest for ring theory and representation theory, like Weyl algebras and quotients of the enveloping algebra of 𝔰𝔩2. In this work, we study invariants of TGWAs under diagonal graded automorphisms. Under certain conditions, we are able to show that the fixed ring of a TGWA by such an automorphism is again a TGWA. We apply this theorem to study properties of the fixed ring, such as the Noetherian property and simplicity. We also look at the behavior of simple weight modules for TGWAs when restricted to the action of the fixed ring.

Evgeny Mukhin: Supersymmetric analogues of partitions and plane partitions

8th April, 2021

We will explain combinatorics of various partitions arising in the representation theory of quantum toroidal algebras associated to Lie superalgebra 𝔤𝔩(m|n). Apart from being interesting in its own right, this combinatorics is expected to be related to crystal bases, fixed points of the moduli spaces of BPS states, equivariant K-theory of moduli spaces of maps, and other things.

Brian Boe: Complexity and Support Varieties for Type P Lie Superalgebras

5th April, 2021

We compute the complexity, z-complexity, and support varieties of the (thick) Kac modules for the Lie superalgebras of type P. We also show the complexity and the z-complexity have geometric interpretations in terms of support and associated varieties; these results are in agreement with formulas previously discovered for other classes of Lie superalgebras. Our main technical tool is a recursive algorithm for constructing projective resolutions for the Kac modules. The indecomposable projective summands which appear in a given degree of the resolution are explicitly described using the combinatorics of weight diagrams. Surprisingly, the number of indecomposable summands in each degree can be computed exactly: we give an explicit formula for the corresponding generating function. I wrote an iOS app to implement the combinatorics quickly and graphically, and I’ll be demoing live some of the interesting features of these resolutions.

Maria Gorelik: Depths and cores in the light of DS-functors

25th March, 2021

The Duflo-Serganova functors DS are tensor functors relating representations of different Lie superalgebras. In this talk I will consider the behaviour of various invariants, such as the defect, the dual Coxeter number, the atypicality and the cores, under the DS-functor. I will introduce a notion of depth playing the role of defect for algebras and atypicality for modules. I will mainly concentrate on examples of symmetrizable Kac-Moody and Q-type superalgebras.

Daniel Nakano: A new Lie theory for simple classical Lie superalgebras

15th March, 2021

In 1977, Kac classified simple Lie superalgebras over ℂ and showed they play an analogous role to simple Lie algebras over the complex numbers. For simple algebraic groups and their Lie algebras, the notions of a maximal torus, Borel subgroups and the Weyl groups provide a uniform method to treat the structure and representation theory for these groups and Lie algebras. Historically, much of the work for simple Lie superalgebras has involved dealing with these objects using a case by case analysis.

Fifteen years ago, Boe, Kujawa and the speaker introduced the important concept of detecting subalgebras for classical Lie superalgebras. These algebras were constructed by using ideas from geometric invariant theory. More recently, D. Grantcharov, N. Grantcharov, Wu and the speaker introduced the BBW parabolic subalgebras. Given a Lie superalgebra 𝔤, one has a triangular decomposition 𝔤=𝔫- ⨁ 𝔣 ⨁ 𝔫+ with 𝔟=𝔣 ⨁ 𝔫- where 𝔣 is a detecting subalgebra and 𝔟 is a BBW parabolic subalgebra. This holds for all classical 'simple' Lie superalgebras, and one can view 𝔣 as an analogue of the maximal torus, and 𝔟 like a Borel subalgebra. This setting also provide a useful method to define semisimple elements and nilpotent elements, and to compute various sheaf cohomology groups R• indBG (-).

The goal of my talk is to provide a survey of the main ideas of this new theory and to give indications of the interconnections within the various parts of this topic. I will also indicate how our ideas can further unify the study of the representation theory of classical Lie superalgebras.