We shall study first split-quaternions and then split-octonions. For technical reasons split-quaternions will be constructed by applying the Cayley-Dickson construction to complex numbers and split-octonions by applying this construction to quaternions. Our attention will be devoted to special elements and their properties in these two algebras. We shall classify all subalgebras in split-quaternions and split-octonions and further describe the types of isotropic subspaces in them.
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.
On split-octonions
Jiri Vanzura
IM CAS
Wednesday, 20. November 2019 - 11:30 to 12:30
in Konirna seminar room, front IM building, ground floor
Koszul algebras and one-dependent random 0-1 sequences: CANCELLED because of DOD
Leonid Positselski
MI, CAS
Wednesday, 13. November 2019 - 11:30 to 12:30
in IM building, ground floor
Koszul algebras are a natural class of graded algebras with
quadratic relations, defined by a series of homological conditions.
To a Koszul algebra over with finite-dimensional components, one can
assign a one-dependent stochastic 0-1 sequence, which carries
information about the dimensions of the algebra's grading components.
... more
quadratic relations, defined by a series of homological conditions.
To a Koszul algebra over with finite-dimensional components, one can
assign a one-dependent stochastic 0-1 sequence, which carries
information about the dimensions of the algebra's grading components.
... more
Commutative cohomology in characteristic 2
Pasha Zusmanovich
University Ostrava
Wednesday, 6. November 2019 - 11:30 to 12:30
in Seminar room Konirna, front IM building, ground floor
Commutative Lie algebras are commutative algebras in characteristic 2 satisfying
the Jacobi identity. This class of algebras is broader than the class of Lie
algebras (which satisfy the stronger alternating identity [x,x] = 0).
The natural cohomology in this class is a variation of the usual
Chevalley-Eilenberg cohomology, where alternating cochains are
replaced by symmetric ones. This cohomology appears naturally in the
ongoing efforts to advance in classification of simple Lie algebras in
characteristic 2, and its properties resemble cohomology of Lie
superalgebras (in any characteristic) rather than the usual
Chevalley-Eilenberg cohomology. We will discuss some general results,
computations, and open questions. (Based on arXiv:1907.03690).
the Jacobi identity. This class of algebras is broader than the class of Lie
algebras (which satisfy the stronger alternating identity [x,x] = 0).
The natural cohomology in this class is a variation of the usual
Chevalley-Eilenberg cohomology, where alternating cochains are
replaced by symmetric ones. This cohomology appears naturally in the
ongoing efforts to advance in classification of simple Lie algebras in
characteristic 2, and its properties resemble cohomology of Lie
superalgebras (in any characteristic) rather than the usual
Chevalley-Eilenberg cohomology. We will discuss some general results,
computations, and open questions. (Based on arXiv:1907.03690).
Rigidity of Lagrangian intersection in conformal symplectic geometry.
Baptiste Chantraine
Université de Nantes
Tuesday, 29. October 2019 - 11:15 to 12:15
Konirna seminar room, front IM building, ground floor
In this talk I will review some of basic notion about conformal symplectic geometry and highlight similarities and differences with symplectic geometry. I will then explain how we can prove that in a conformal symplectic cotangent bundle lagrangian intersection of the 0-section with Hamiltonian deformations of itself are bounded bellow by the Novikov Betti numbers of the Lee form. This is joint work with Emmy Murphy.