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).
professor Viktor Abramov, University of Tartu
professor Abdenacer Makhlouf, Université de Haute Alsace (France)
professor Sergei Silvestrov, Mälardalen University (Sweden)
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.