Priit Lätt “Induced 3-Lie superalgebras and their applications in superspace” | University of Tartu

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Priit Lätt “Induced 3-Lie superalgebras and their applications in superspace”

Date: 
27.08.2020 - 11:00

On 27 August at 11:00 Priit Lätt will defend his doctoral thesis “Induced 3-Lie superalgebras and their applications in superspace” for obtaining the degree of Doctor of Philosophy (Mathematics).

Supervisor:
professor Viktor Abramov, University of Tartu

Opponents:
professor Abdenacer Makhlouf, Université de Haute Alsace (France)
​professor Sergei Silvestrov, Mälardalen University (Sweden)

Summary

The aim of the present thesis is to study the properties and characteristics of n-Lie superalgebras that are constructed using an operation from (n-1)-Lie superalgebras, especially in the case n=3. A regular Lie algebra can be extended to super- (or Z_2-graded) structures by introducing the notion of Lie superalgebra. Similarly n-Lie algebra, where binary operation is replcaed with n-ary multiplication law, can be extended to superstructures by making use of a graded Filippov-Jacobi identity, giving a notion of n-Lie superalgebra. In the dissertation a classification of low dimensional 3-Lie superalgebras is presented. We show that an n-Lie superalgebra equipped with a supertrace can be used to construct a (n+1)-Lie superalgebra, which is referred to as the induced (n+1)-Lie superalgebra. It is proved that one can construct induced 3-Lie superalgebras from commutative superalgebras by using involution, even degree derivation, or combination of both of them together. In the thesis a generalization of Nambu-Hamilton equation to a superspace is proposed, and it is shown that it induces a family of ternary Nambu-Poisson brackets of even degree functions on a superspace. Finally a representations of induced 3-Lie algebras and Lie superalgebras are constructed by means of a representation of the initial binary Lie algebra and trace or Lie superalgebra and supertrace, respectively. It is shown that the constructed induced representation of 3-Lie algebra is a representation by traceless matrices, that is, lies in the Lie algebra sl(V), where V is a representation space. For 2-dimensional representations the irreduciblility condition of the induced representation of induced 3-Lie algebra is found.

Location: 
Narva mnt 18 – 1020
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