Induced 3-Hom-Lie superalgebras

We construct 3-Hom-Lie superalgebras on a commutative Hom-superalgebra by means of involution and even degree derivation. We construct a representation of induced 3-Hom-Lie superalgebras by means of supertrace

Indutseeritud 3-Lie superalgebrad ja nende rakendused superruumis

Väitekirja elektrooniline versioon ei sisalda publikatsiooneKäesoleva doktoritöö eesmärk on uurida selliste n-Lie superalgerbrate omadusi, mis on konstrueeritud kasutades (n-1)-Lie superalgebra aluseks olevat (n-1)-aarset tehet, seda eriti juhul n=3. Tavalise Lie algebra mõistet on võimalik super- (või Z_2-gradueeritud) struktuuridele üle kanda kui toome sisse Lie superalgebra mõiste. Sarnaselt on võimalik n-Lie algebra, kus binaarne tehe on asendatud n-aarse tehtega, üldistada superstruktuuridele, kui kasutame Filippov-Jacobi samasuse gradueeritud analoogi, saades n-Lie superalgebra. Väitekirjas on esitatud madaladimensionaalsete 3-Lie superalgebrate klassifikatsioon. Lisaks näitame, et n-Lie superalgebra abil, mille tehtele leidub superjälg, saab konstrueerida (n+1)-Lie superalgebra, mida me nimetame indutseeritud (n+1)-Lie superalgebraks. Enamgi veel, on tõestatud, et kommutatiivse superalgebra korral on võimalik indutseerida erinevad 3-Lie superalgebra struktuurid kasutades involutsiooni, derivatsiooni või neid mõlemad korraga. Dissertatsioonis on toodud Nambu-Hamiltoni võrrandi üldistus superruumis jaoks, ja on näidatud, et selle abil on võimalik indutseerida ternaarsete Nambu-Poissoni sulgude pere superruumi paarisfunktsioonide jaoks. Järgnevalt on konstrueeritud indutseeritud 3-Lie superalgebrate indutseeritud esitused, kasutades selleks vastavalt kas esialgset binaarset Lie algebrat koos jäljega või Lie superalgebrat koos superjäljega. Töös on näidatud, et 3-Lie algebra indutseeritud esitus on sisestatav jäljeta maatriksite Lie algebrasse sl(V), kus sümboliga V on tähistatud esituse ruum. Kahedimensionaalse indutseeritud esituse korral on esitatud tingimused, mida vastav esitus peab rahuldama, et ta oleks taandumatu.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.https://www.ester.ee/record=b536058

3-Lie Superalgebras Induced by Lie Superalgebras

We show that given a Lie superalgebra and an element of its dual space, one can construct the 3-Lie superalgebra. We apply this approach to Lie superalgebra of ( m , n ) -block matrices taking a supertrace of a matrix as the element of dual space. Then we also apply this approach to commutative superalgebra and the Lie superalgebra of its derivations to construct 3-Lie superalgebra. The graded Lie brackets are constructed by means of a derivation and involution of commutative superalgebra, and we use them to construct 3-Lie superalgebras