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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 HoΕ‚ubowski: 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 β‰₯ 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 βˆ’ 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.

JosΓ© MarΓ­a PΓ©rez 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 𝔀𝔩(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 β„‚ 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 𝔀, one has a triangular decomposition 𝔀=𝔫- ⨁ 𝔣 ⨁ 𝔫+ with π”Ÿ=𝔣 ⨁ 𝔫- where 𝔣 is a detecting subalgebra and π”Ÿ 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 π”Ÿ 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 Rβ€’ 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.

David Galban: First and Second Cohomology Groups for BBW Parabolics for Lie Superalgebras

1st March, 2021

For semisimple Lie algebras, a well-known theorem of Kostant computes the cohomology groups of parabolic subalgebras, but it is unknown whether an analogue of Kostant’s theorem exists for Lie superalgebras. Seeking to provide the first calculations in this direction, in this talk, I will describe the cohomology groups for the subalgebra 𝔫+ relative to the BBW parabolic subalgebras constructed by D. Grantcharov, N. Grantcharov, Nakano and Wu. These classical Lie superalgebras have a triangular decomposition 𝔀 = 𝔫- + 𝔣 + 𝔫+, where 𝔣 is a detecting subalgebra as introduced by Boe, Kujawa and Nakano. I will show that there exists a Hochschild-Serre spectral sequence that collapses for all infinite families of classical simple Lie superalgebras. Using this, I will provide examples of computation of the first and second cohomologies for various 𝔫+.

Daniel Nakano: A new Lie theory for classical Lie superalgebras

4th December, 2020

In 1977, Kac classified simple Lie superalgebras over β„‚ 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 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 concept of a BBW parabolic subalgebra.

Given a Lie superalgebra 𝔀, one has a triangular decomposition 𝔀=𝔫– ⨁ 𝔣 ⨁ 𝔫+ with π”Ÿ = 𝔣 ⨁ 𝔫– where 𝔣 is a detecting subalgebra and π”Ÿ 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 π”Ÿ 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 βˆ™ 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 this treatment can further unify the study of the representation theory of classical Lie superalgebras.

This video was produced by the Universidade de SΓ£o Paulo, as part of the LieJor Online Seminar: Algebras, Representations, and Applications.

Andy Jenkins: The nilpotent cone for Lie superalgebras

19th October, 2020

Many aspects of the representation theory of a Lie algebra and its associated algebraic group are governed by the geometry of their nilpotent cone. In this talk, we will introduce an analogue of the nilpotent cone N for Lie superalgebras and show that for a simple classical Lie superalgebra the number of nilpotent orbits is finite. We will also show that the commuting variety X described by Duflo and Serganova, which has applications in the study of the finite-dimensional representation theory of Lie superalgebras, is contained in N. Consequently, the finiteness result on N generalizes and extends the work on the commuting variety.

Dimitar Grantcharov: Quantized enveloping superalgebra of type P

15th October, 2020

We will introduce a new quantized enveloping superalgebra corresponding to the periplectic Lie superalgebra 𝔭n. This quantized enveloping superalgebra is a quantization of a Lie bisuperalgebra structure on 𝔭n. Furthermore, we will define the periplectic q-Brauer algebra and see that it admits natural centralizer properties.

Pavel Kolesnikov: Derived algebras and their identities

9th July, 2020

In this talk we will consider a "differential counterpart" of the dendriform splitting procedure for operads. This problem has a very natural interpretation in the language of non-associative algebras. It is well-known that a (non-associative, in general) algebra equipped with a Rota-Baxter operator (a formalization of integration) gives rise to a system in a class of splitting algebras. The latter include dendriform (pre-associative), pre-Lie (left-symmetric), pre-Poisson, Zinbiel (pre-commutative) algebras, etc. What happens if we replace a Rota-Baxter operator with a derivation? The answer is well known for associative commutative algebras: the resulting class of systems obtained in this way coincides with the variety Nov of Novikov algebras. We will show in general that for an arbitrary binary operad Var the variety of derived Var-algebras coincides with the Manin white product of operads Var and Nov. If we allow the initial multiplication(s) to leave in the language of a derived algebra then the same sort of description can be obtained just by replacement of Nov with GD!, the Koszul dual to the operad of Gelfand-Dorfman algebras. We will also discuss similar statements for the "integral" case of Rota-Baxter operators.

Tomoyuki Arakawa: Urod algebra and translation for W-algebras

9th July, 2020

In 2016 Bershtein, Feigin and Litvinov introduced the Urod algebra, which gives a representation-theoretic interpretation of the celebrated Nakajima-Yoshioka blowup equations in the case that the sheaves are of rank two. In this talk we will introduce higher-rank Urod algebras. This is done by constructing translation functors for affine W-algebras.

Yuly Billig: Towards Kac-van de Leur Conjecture

2nd July, 2020

Superconformal algebras are graded Lie superalgebras of growth 1, containing a Virasoro subalgebra. They play an important role in Conformal Field Theory. In 1988 Kac and van de Leur made a conjectural list of simple superconformal algebras, which since has been amended with an exceptional superalgebra CK(6). It has been proposed to use conformal superalgebras to attack this conjecture, and Fattori and Kac established a classification of finite simple conformal superalgebras. It still needs to be proved that one can associate a finite conformal superalgebra to each simple superconformal algebra. In this talk we will show how to use the results of Billig-Futorny to prove that every simple superconformal algebra is polynomial, which implies that one can attach to it an affine conformal superalgebra. We will discuss the difference between finite and affine conformal algebras. We also introduce quasi-Poisson algebras and show how to use them to construct known simple superconformal algebras. Quasi-Poisson algebras may be viewed as a refinement of the notion of Novikov algebras. Quasi-Poisson algebras may be used for computations of automorphisms and twisted forms of superconformal algebras.

Vera Serganova: The celebrated Jacobson-Morozov theorem for Lie superalgebras via semisimplification functor for tensor categories

11th June, 2020

The famous Jacobson-Morozov theorem claims that every nilpotent element of a semisimple Lie algebra 𝔀 can be embedded into an 𝔰𝔩2-triple inside 𝔀. Let 𝔀 be a Lie superalgebra with reductive even part and x be an odd element of 𝔀 with non-zero nilpotent [x,x]. We give necessary and sufficient condition when x can be embedded in 𝔬𝔰𝔭(1|2) inside 𝔀. The proof follows the approach of Etingof and Ostrik and involves semisimplification functor for tensor categories. Next, we will show that for every odd x in 𝔀 we can construct a symmetric monoidal functor between categories of representations of certain superalgebras. We discuss some properties of these functors and applications of them to representation theory of superalgebras with reductive even part. (Joint work with Inna Entova-Aizenbud).

JoΓ£o Schwarz: Poisson birational equivalence and Coloumb branches of 3d N=4 SUSY gauge theories

14th May, 2020

In this talk we discuss a notion of birational equivalence suitable for Poisson affine varieties: namely, that their function fields are isomorphic as Poisson fields. Some very interesting questions on non-commutative birational geometry, such as the Gelfand-Kirillov Conjecture, make perfect sense in the quasi-classical limit, and naturally leads one to consider the Poisson birational class of the algebras they quantize. In this setting, we study the behaviour of Poisson birational equivalence on the quasi-classical limit of rings of differential operators. With this idea we solve a Poisson analogue of Noether's Problem, introduced by Julie Baudry and François Dumas, in a constructive fashion, for essentially all finite symplectic reflection groups. As applications of our method, we show the Poisson rationality of the Generalized Calogero-Moser spaces, introduced by Etingof and Ginzburg in 2002, and surprisngly for this author, all Coloumb branches of 3d, N=4 SUSY gauge theories - an important object in mathematical physics recently given a rigorous formulation by Nakajima in 2015, and later Nakajima, Braverman, Finkelberg in 2016.