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

Manin-Olshansky triples for Lie superalgebras

Following V. Drinfeld and G. Olshansky, we construct Manin triples (\fg, \fa, \fa^*) such that \fg is different from Drinfeld's doubles of \fa for several series of Lie superalgebras \fa which have no even invariant bilinear form (periplectic, Poisson and contact) and for a remarkable exception. Straightforward superization of suitable Etingof--Kazhdan's results guarantee then the uniqueness of $q$-quantization of our Lie bialgebras. Our examples give solutions to the quantum Yang-Baxter equation in the cases when the classical YB equation has no solutions. To find explicit solutions is a separate (open) problem. It is also an open problem to list (\`a la Belavin-Drinfeld) all solutions of the {\it classical} YB equation for the Poisson superalgebras \fpo(0|2n) and the exceptional Lie superalgebra \fk(1|6) which has a Killing-like supersymmetric bilinear form but no Cartan matrix

Cohomology and Support Varieties for Lie Superalgebras II

In \cite{BKN} the authors initiated a study of the representation theory of classical Lie superalgebras via a cohomological approach. Detecting subalgebras were constructed and a theory of support varieties was developed. The dimension of a detecting subalgebra coincides with the defect of the Lie superalgebra and the dimension of the support variety for a simple supermodule was conjectured to equal the atypicality of the supermodule. In this paper the authors compute the support varieties for Kac supermodules for Type I Lie superalgebras and the simple supermodules for $\mathfrak{gl}(m|n)$. The latter result verifies our earlier conjecture for $\mathfrak{gl}(m|n)$. In our investigation we also delineate several of the major differences between Type I versus Type II classical Lie superalgebras. Finally, the connection between atypicality, defect and superdimension is made more precise by using the theory of support varieties and representations of Clifford superalgebras.Comment: 28 pages, the proof of Proposition 4.5.1 was corrected, several other small errors were fixe