Tag - Non-associative rings

Shavkat Ayupov: Local and 2-local derivations and automorphisms of Octonian algebras

15th April, 2021

The talk is devoted to description of local and 2-local derivations (respectively, automorphisms) on octonian algebras over fields with zero characteristics. We shall give a general form of local derivations on the real octonion algebra O(ℝ). This description implies that the space of all local derivations on O(ℝ) when equipped with Lie bracket is isomorphic to the Lie algebra 𝔰𝔬7(ℝ) of all real skew-symmetric 7 × 7-matrices. We also consider 2-local derivations on the octonion algebra O(F) over an algebraically closed field F and prove that every 2-local derivation on O(F) is a derivation. Further, we apply these results to problems for the simple 7-dimensional Malcev algebra. As a corollary we obtain that the real octonion algebra O(ℝ) and Malcev algebra M7(ℝ) are simple non-associative algebras which admit pure local derivations, that is, local derivations which are not derivation. Further, we shall give a general form of local automorphisms on the octonion algebra O(F) over a field F. This description implies that the group of all local automorphisms on O(F) is isomorphic to the group O7(F) of all orthogonal 7 × 7-matrices over F. We also consider 2-local automorphisms on the octonion algebra O(F) over an algebraically closed field F and prove that every 2-local automorphism on O(F) is an automorphism. As a corollary we obtain descriptions of local and 2-local automorphisms of seven dimensional simple Malcev 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.

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 𝔤 = 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 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 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 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 this treatment can further unify the study of the representation theory of classical Lie superalgebras.

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