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 Mathematics , 1998, Abstract: Let $\Gamma$ be a countable abelian semigroup and $A$ be a countable abelian group satisfying a certain finiteness condition. Suppose that a group $G$ acts on a $(\Gamma \times A)$-graded Lie superalgebra ${\frak L}=\bigoplus_{(\alpha,a) \in \Gamma\times A} {\frak L}_{(\alpha,a)}$ by Lie superalgebra automorphisms preserving the $(\Gamma\times A)$-gradation. In this paper, we show that the Euler-Poincar\'e principle yields the generalized denominator identity for ${\frak L}$ and derive a closed form formula for the supertraces $\text{str}(g|{\frak L}_{(\alpha,a)})$ for all $g\in G$,$(\alpha,a) \in \Gamma\times A$. We discuss the applications of our supertrace formula to various classes of infinite dimensional Lie superalgebras such as free Lie superalgebras and generalized Kac-Moody superalgebras. In particular, we determine the decomposition of free Lie superalgebras into a direct sum of irreducible $GL(n) \times GL(k)$-modules, and the supertraces of the Monstrous Lie superalgebras with group actions. Finally, we prove that the generalized characters of Verma modules and the irreducible highest weight modules over a generalized Kac-Moody superalgebra ${\frak g}$ corresponding to the Dynkin diagram automorphism $\sigma$ are the same as the usual characters of Verma modules and irreducible highest weight modules over the orbit Lie superalgebra $\breve{\frak g}={\frak g}(\sigma)$ determined by $\sigma$.
 Dimitry Leites Mathematics , 2002, Abstract: In 1960's I. Gelfand posed a problem: describe indecomposable representations of any simple infinite dimensional Lie algebra of polynomial vector fields. Here, by applying the elementary technique of Gelfand and Ponomarev, a toy model of the problem is solved: finite dimensional indecomposable representations of vect(0|2), the Lie superalgebra of vector fields on the (0|2)-dimensional superspace, are described. Since vect (0|2) is isomorphic to sl(1|2) and osp(2|2), their representations are also described. The result is generalized in two directions: for sl(1|n) and osp(2|2n). Independently and differently J. Germoni described indecomposable representation of the series sl(1|n) and several individual Lie superalgebras. Partial results for other simple Lie superalgebras without Cartan matrix are reviewed. In particular, it is only for vect(0|2) and sh(0|4) that the typical irreducible representations can not participate in indecomposable modules; for other simple Lie superalgebras without Cartan matrix (of series vect(0|n), svect(0|n)$, svect(0|n)', spe(n) for n>2 and sh(0|m) for m>4) one can construct indecomposable representations with arbitrary composition factors. Several tame open problems are listed, among them a description of odd parameters  Alexander Sergeev Mathematics , 1998, Abstract: The description of irreducible finite dimensional representations of finite dimensional solvable Lie superalgebras over complex numbers given by V.~Kac is refined. In reality these representations are not just induced from a polarization but twisted, as infinite dimensional representations of solvable Lie algebras. Various cases of irreducibility (general and of type Q) are classified.  Mathematics , 2013, Abstract: This paper aims to characterize the minimal dimensions and super-dimensions of faithful representations for the Heisenberg Lie superalgebras over an algebraically closed field of characteristic zero.  Maria Gorelik Mathematics , 2000, Abstract: The category of representations with a strongly typical central character of a basic classical Lie superalgebra is proven to be equivalent to the category of representations of its even part corresponding to an appropriate central character. For a Lie superalgebra$osp(1,2l)$the category of representations with a "generic" weakly atypical central character is described.  Cuiling Luo Mathematics , 2010, Abstract: In this paper, various polynomial representations of strange classical Lie superalgebras are investigated. It turns out that the representations for the algebras of type P are indecomposable, and we obtain the composition series of the underlying modules. As modules of the algebras of type Q, the polynomial algebras are decomposed into a direct sum of irreducible submodules.  Deniz Kus Mathematics , 2015, Abstract: We study the category of finite--dimensional representations for a basic classical Lie superalgebra$\Lg=\Lg_0\oplus \Lg_1$. For the ortho--symplectic Lie superalgebra$\Lg=\mathfrak{osp}(1,2n)$we show that certain objects in that category admit a fusion flag, i.e. a sequence of graded$\Lg_0[t]$--modules such that the successive quotients are isomorphic to fusion products. Among these objects we find fusion products of finite--dimensional irreducible$\Lg\$--modules, truncated Weyl modules and Demazure type modules. Moreover, we establish a presentation for these types of fusion products in terms of generators and relations of the enveloping algebra.