Cartan and Killing described finite-dimensional simple Lie algebras (over fields of real or complex numbers) in terms of the distributions they preserve. The technique of root system and Dynkin (Coxeter) graphs was discovered several decades later. Two o the four series of simple infinite-dimensional Lie algebras of vector fields are Cartan prolongations of non-positive parts of simple finite-dimensional Lie algebras. For any $\mathbb{Z}$-grading of any simple finite-dimensional Lie algebra $\mathfrak{g}$ (bar the two series of examples), the Cartan prolongation of the non-positive part of $\mathfrak{g}$ returns $\mathfrak{g}$. This is not so for the exceptional Lie algebra $\mathfrak{g}_2$ in characteristic 5, whose Cartan prolongation is called Melikyan algebra. Recall that the Lie superalgebras appeared not in high energy physics in 1970s, but in topology, and either over $\mathbb{Z}$ as super Lie rings, or over finite fields. Lately, modular Lie (super)algebras became of interest... more
Cohomology in algebra, geometry, physics and statistics
usually takes place every Wednesday Institute of Mathematics of ASCR, Žitná 25, Praha 1Chair: Anton Galaev, Roman Golovko, Igor Khavkine, Alexei Kotov, Hong Van Le and Petr Somberg
In this seminar we shall discuss topics concerning constructions and applications of cohomology theory in algebra, geometry, physics and statistics. In particular we shall discuss in first four seminars the relations between vertex algebras and foliations on manifolds, Gelfand-Fuks cohomology on singular spaces, cohomology of homotopy Lie algebras. The expositions should be accessible for all participants.
Structures of G(2) type in super setting and in positive characteristic, and related curvature tensors
Dimitri Leites
NYUAD and Stockholm U.
Wednesday, 20. April 2022 - 13:30 to 14:30
ZOOM meeting
Combinatorics of multilinear differential operators, or still another explanation of the ubiquity of Lie and strongly homotopy Lie algebras
Martin Markl
Institute of Mathematics of the Czech Academy of Sciences
Wednesday, 13. April 2022 - 13:30 to 14:30
in IM building, ground floor +ZOOM meeting
As a motivation, we start with an analysis of the interplay between the classical Jacobi identity and differential operators, and
compare it with the effect of the associator. Moving to the `quantized' level, we compare the nature of the big bracket and
IBL(=infinitesimal Lie bialgebras)-infinity algebras with Terilla's quantization of associative algebras.
In the second part, we introduce a filtration mimicking combinatorial properties of multidifferential operators, and
the associated notion of tight operads. We then come back to Lie algebras and give another reason why they deserve
to be, along with commutative and associative algebras, recognized as one of the Three Graces.
The talk will be based on the paper "Calculus of multilinear differential operators, operator $L_\infty$-algebras and $IBL_\infty$-algebras"
of Denis Bashkirov and mine. Its preprint is available at
https://arxiv.org/abs/2108.12158... more
compare it with the effect of the associator. Moving to the `quantized' level, we compare the nature of the big bracket and
IBL(=infinitesimal Lie bialgebras)-infinity algebras with Terilla's quantization of associative algebras.
In the second part, we introduce a filtration mimicking combinatorial properties of multidifferential operators, and
the associated notion of tight operads. We then come back to Lie algebras and give another reason why they deserve
to be, along with commutative and associative algebras, recognized as one of the Three Graces.
The talk will be based on the paper "Calculus of multilinear differential operators, operator $L_\infty$-algebras and $IBL_\infty$-algebras"
of Denis Bashkirov and mine. Its preprint is available at
https://arxiv.org/abs/2108.12158... more
Symmetry and Separation of variables
Stepan Hudecek
Charles University
Wednesday, 6. April 2022 - 13:30 to 14:30
in IM rear building, ground floor, blue lecture room +ZOOM meeting
We present a condition under which a differential operator on a two dimensional manifold admits a so-called separated solution and the separation is non-trivial in a sense, that we explain. Along the way we "develop" definitions in order to make these propositions precise, such as of a symmetry generating an operator and of a function that does not depend on a set of variables with respect to a coordinate chart.
We are motivated by problems in Physics, where the separation of variables is often used, e.g., in specific problems of electromagnetic waves, quantum mechanics (hydrogen atom), or in general relativity. In mathematical Physics the notion of separation was studied in many works, including the works of Kalnins, Winternitz, Miller and Koornwinder. In a part of the Physics literature, the notion of the separation is studied without giving a definition of a separated solution.
In... more
Nondegenerate invariant symmetric bilinear forms on simple Lie superalgebras in characteristic 2
Andrey Krutov
Institute of Mathematics of the Czech Academy of Sciences
Wednesday, 30. March 2022 - 13:30 to 14:30
in IM building, front building, ground floor, konirna seminar room +ZOOM meeting
As is well-known, the dimension of the space spanned by the non-degenerate invariant symmetric bilinear forms (NISes) on any simple finite-dimensional Lie algebra or Lie superalgebra is equal to at most 1 if the characteristic of the algebraically closed ground field is not 2.
We prove that in characteristic 2, the superdimension of the space spanned by NISes can be equal to 0, or 1, or 0|1, or 1|1; it is equal to 1|1 if and only if the Lie superalgebra is a queerification (defined in arXiv:1407.1695) of a simple classically restricted Lie algebra with a NIS (for examples, mainly in characteristic distinct from 2, see arXiv:1806.05505).
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We shall meet this time at the seminar room KONIRNA, ground floor, front building of IM. We... more
We prove that in characteristic 2, the superdimension of the space spanned by NISes can be equal to 0, or 1, or 0|1, or 1|1; it is equal to 1|1 if and only if the Lie superalgebra is a queerification (defined in arXiv:1407.1695) of a simple classically restricted Lie algebra with a NIS (for examples, mainly in characteristic distinct from 2, see arXiv:1806.05505).
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We shall meet this time at the seminar room KONIRNA, ground floor, front building of IM. We... more