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In this paper we study extensions between finite-dimensional simple modules over classical Lie superalgebras gl (m | n) , osp (M | 2 n) and q m. We consider a simplified version of the extension graph which is produced from the Ext 1 -graph by identifying representations obtained by parity change and removal of the loops. We give a necessary condition for a pair of vertices to be connected and show that this condition is sufficient in most of the cases. This condition implies that the image of a finite-dimensional simple module under the Duflo-Serganova functor has indecomposable isotypical components. This yields semisimplicity of Duflo-Serganova functor for F in (gl (m | n)) and for F in (osp (M | 2 n)). [ABSTRACT FROM AUTHOR]

We construct universal monoidal categories of topological tensor supermodules over the Lie superalgebras gl (V ⊕ Π V) and osp (V ⊕ Π V) associated with a Tate space V. Here V ⊕ Π V is a Z / 2 Z -graded topological vector space whose even and odd parts are isomorphic to V. We discuss the purely even case first, by introducing monoidal categories T ˆ gl (V) , T ˆ o (V) and T ˆ sp (V) , and show that these categories are anti-equivalent to respective previously studied categories T gl (V) , T o (V) , T sp (V). These latter categories have certain universality properties as monoidal categories, which consequently carry over to T ˆ gl (V) , T ˆ o (V) and T ˆ sp (V). Moreover, the categories T o (V) and T sp (V) are known to be equivalent, and this implies the equivalence of the categories T ˆ o (V) and T ˆ sp (V). After introducing a supersymmetric setting, we establish the equivalence of the category T ˆ gl (V) with the category T ˆ gl (V ⊕ Π V) , and the equivalence of both categories T ˆ o (V) and T ˆ sp (V) with T ˆ osp (V ⊕ Π V). [ABSTRACT FROM AUTHOR]

This paper is to study vertex operator superalgebras which are strongly generated by their weight-2 and weight- 3 2 homogeneous subspaces. Among the main results, it is proved that if such a vertex operator superalgebra V is simple, then V (2) has a canonical commutative associative algebra structure equipped with a non-degenerate symmetric associative bilinear form and V (3 2) is naturally a V (2) -module equipped with a V (2) -valued symmetric bilinear form and a non-degenerate (C -valued) symmetric bilinear form, satisfying a set of conditions. On the other hand, assume that A is any commutative associative algebra equipped with a non-degenerate symmetric associative bilinear form and assume that U is an A -module equipped with a symmetric A -valued bilinear form and a non-degenerate (C -valued) symmetric bilinear form, satisfying the corresponding conditions. Then we construct a Lie superalgebra L (A , U) and a simple vertex operator superalgebra L L (A , U) (ℓ , 0) for every nonzero number ℓ such that L L (A , U) (ℓ , 0) (2) = A and L L (A , U) (ℓ , 0) (3 2) = U. [ABSTRACT FROM AUTHOR]

We study a class of modules, called Chari-Venkatesh modules, for the current superalgebra sl (1 | 2) [ t ]. This class contains other important modules, such as graded local Weyl, truncated local Weyl and Demazure-type modules. We prove that Chari-Venkatesh modules can be realized as fusion products of generalized Kac modules. In particular, this proves Feigin and Loktev's conjecture, that fusion products are independent of their fusion parameters, in the case where the fusion factors are generalized Kac modules. As an application of our results, we obtain bases, dimension and character formulas for Chari-Venkatesh modules. [ABSTRACT FROM AUTHOR]

The research is motivated by a construction of groups of oscillating growth by Kassabov and Pak [25] and a description of possible growth functions of finitely generated associative algebras by Bell and Zelmanov [9]. In this paper we address both, the question of possible growth functions in case of Lie algebras, and the Kurosh problem, because our examples of restricted Lie algebras have a nil p -mapping, which is an analogue of nillity for associative algebras or periodicity for groups. Namely, for any field of positive characteristic, we construct a family of 3-generated restricted Lie algebras of intermediate oscillating growth. We call them Phoenix algebras because, for infinitely many periods of time, the algebra is "almost dying" by having a quasi-linear growth, namely the lower Gelfand-Kirillov dimension is one, more precisely, the growth is of type n ( ln ⁡ ⋯ ln ︸ q times n) κ , where q ∈ N , κ > 0 are constants. On the other hand, for infinitely many n the growth function has a rather fast intermediate behavior of type exp ⁡ (n / (ln ⁡ n) λ) , λ being a constant determined by characteristic, for such periods the algebra is "resuscitating". Moreover, the growth function is bounded and oscillating between these two types of behavior. These restricted Lie algebras have a nil p -mapping, thus addressing the Kurosh problem as well. [ABSTRACT FROM AUTHOR]

The Grigorchuk and Gupta-Sidki groups play fundamental role in modern group theory. They are natural examples of self-similar finitely generated periodic groups. The first author constructed their analogue in case of restricted Lie algebras of characteristic 2 [50] , Shestakov and Zelmanov extended this construction to an arbitrary positive characteristic [68]. Thus, we have examples of finitely generated restricted Lie algebras with a nil p -mapping. In characteristic zero, similar examples of Lie and Jordan algebras do not exist by results of Martinez and Zelmanov [43] and [78]. The first author constructed analogues of the Grigorchuk and Gupta-Sidki groups in the world of Lie superalgebras of arbitrary characteristic, the virtue of that construction is that Lie superalgebras have clear monomial bases [51] , they have slow polynomial growth. As an analogue of periodicity, Z 2 -homogeneous elements are ad-nilpotent. A recent example of a Lie superalgebra is of linear growth, of finite width 4, just infinite but not hereditary just infinite [13]. By that examples, an extension of the result of Martinez and Zelmanov [43] for Lie superalgebras of characteristic zero is not valid. Now, we construct a just infinite fractal 3-generated Lie superalgebra Q over arbitrary field, which gives rise to an associative hull A , a Poisson superalgebra P , and two Jordan superalgebras J and K , the latter being a factor algebra of J. In case char K ≠ 2 , A has a natural filtration, which associated graded algebra has a structure of a Poisson superalgebra such that gr A ≅ P , also P admits an algebraic quantization using a deformed superalgebra A (t). The Lie superalgebra Q is finely Z 3 -graded by multidegree in the generators, A , P are also Z 3 -graded, while J and K are Z 4 -graded by multidegree in four generators. By virtue of our construction, these five superalgebras have clear monomial bases and slow polynomial growth. We describe multihomogeneous coordinates of bases of Q , A , P in space as bounded by "almost cubic paraboloids". We determine a similar hypersurface in R 4 that bounds monomials of J and K. Constructions of the paper can be applied to Lie (super)algebras studied before to obtain Poisson and Jordan superalgebras as well. The algebras Q , A , and the algebras without unit P o , J o , K o are direct sums of two locally nilpotent subalgebras and there are continuum such decompositions. Also, Q = Q 0 ¯ ⊕ Q 1 ¯ is a nil graded Lie superalgebra, so, Q again shows that an extension of the result of Martinez and Zelmanov for Lie superalgebras of characteristic zero is not valid. In case char K = 2 , Q has a structure of a restricted Lie algebra with a nil p -mapping. The Jordan superalgebra K is nil finely Z 4 -graded, in contrast with non-existence of such examples (roughly speaking, analogues of the Grigorchuk group) of Jordan algebras in characteristic distinct from 2 [78]. Also, K is of slow polynomial growth, just infinite, but not hereditary just infinite. We call the superalgebras Q , A , P , J , K fractal because they contain infinitely many copies of themselves. [ABSTRACT FROM AUTHOR]

Inspired by [12] , we give shuffle algebra realization of positive part of quantum affine superalgebra U v (D ˆ (2 , 1 ; θ)) associated to any simple root systems. We also determine the shuffle algebra associated to sl ˆ (2 | 1) with odd root system when v is a primitive root of unity of even order, generalizing results in [4]. [ABSTRACT FROM AUTHOR]

The generalized quantum group U (ϵ) of type A is an affine analogue of quantum group associated to a general linear Lie superalgebra gl M | N. We prove that there exists a unique R matrix on the tensor product of fundamental type representations of U (ϵ) for arbitrary parameter sequence ϵ corresponding to a non-conjugate Borel subalgebra of gl M | N. We give an explicit description of its spectral decomposition, and then as an application, construct a family of finite-dimensional irreducible U (ϵ) -modules which have subspaces isomorphic to the Kirillov-Reshetikhin modules of usual affine type A M − 1 (1) or A N − 1 (1). [ABSTRACT FROM AUTHOR]

Abstract Let Γ be a group acting on a scheme X and on a Lie superalgebra g , both defined over an algebraically closed field of characteristic zero k. The corresponding equivariant map superalgebra M (g , X) Γ is the Lie superalgebra of Γ-equivariant regular maps from X to g. In this paper we complete the classification of finite-dimensional irreducible M (g , X) Γ -modules when g is a finite-dimensional simple Lie superalgebra, X is of finite type and Γ is a finite abelian group acting freely on the rational points of X , by classifying these M (g , X) Γ -modules in the case where g is a periplectic Lie superalgebra. We also describe extensions between irreducible modules in terms of homomorphisms and extensions between modules for certain finite-dimensional Lie superalgebras. As an application, one obtains the block decomposition of the category of finite-dimensional M (g , X) Γ -modules in terms of blocks and spectral characters of finite-dimensional Lie superalgebras. [ABSTRACT FROM AUTHOR]

A Lie (super)algebra with a non-degenerate invariant symmetric bilinear form will be called a NIS-Lie (super)algebra. The double extension of a NIS-Lie (super)algebra is the result of simultaneously adding to it a central element and an outer derivation so that the larger algebra has also a NIS. Affine loop algebras, Lie (super)algebras with symmetrizable Cartan matrix over any field, Manin triples, symplectic reflection (super)algebras are among the Lie (super)algebras suitable to be doubly extended. We consider double extensions of Lie superalgebras in characteristic 2, and concentrate on peculiarities of these notions related with the possibility for the bilinear form, the center, and the derivation to be odd. Two Lie superalgebras we discovered by this method are indigenous to the characteristic 2. [ABSTRACT FROM AUTHOR]

We prove a determinant formula for a parabolic Verma module of a contragredient finite-dimensional Lie superalgebra, previously conjectured by the second author. Our determinant formula generalizes the previous results of Jantzen for a parabolic Verma module of a (non-super) Lie algebra, and of Kac concerning a (non-parabolic) Verma module for a Lie superalgebra. The resulting formula is expected to have a variety of applications in the study of higher-dimensional supersymmetric conformal field theories. We also discuss irreducibility criteria for the Verma module. [ABSTRACT FROM AUTHOR]

The Grigorchuk and Gupta–Sidki groups play fundamental role in modern group theory. They are natural examples of self-similar finitely generated periodic groups. The author constructed their analogue in case of restricted Lie algebras of characteristic 2 [31] , Shestakov and Zelmanov extended this construction to an arbitrary positive characteristic [41] . Thus, we have examples of (self-similar) finitely generated restricted Lie algebras with a nil p -mapping. In characteristic zero, similar examples of Lie algebras do not exist by a result of Martinez and Zelmanov [27] . The goal of the present paper is to construct analogues of the Grigorchuk and Gupta–Sidki groups in the world of Lie superalgebras of an arbitrary characteristic. We construct two examples R , Q of finitely generated self-similar Lie superalgebras over a field K of an arbitrary characteristic and study their properties and properties of their associative hulls. In case char K = 2 , these examples turn into restricted Lie algebras. The virtue of the present construction is that the Lie superalgebras have clear monomial bases. These Lie superalgebras have slow polynomial growth and are multigraded by multidegree in the generators. The Lie superalgebra R is Z 2 -graded, while Q has a multidegree Z 3 -gradation and a Z 2 -gradation. Both algebras R and Q have similar constructions, computations for R are simpler, but Q enjoys some more specific interesting properties. The Z 3 -components of Q lie inside an elliptic paraboloid in space, they are at most one-dimensional, thus, the Z 3 -grading of Q is fine. In the Z 2 -gradation of Q , all components Q n m , n , m ∈ Z , are infinite dimensional except for Q 00 = { 0 } . The Z 2 -gradation also yields a continuum of different decompositions into a direct sum of two locally nilpotent subalgebras Q = Q + ⊕ Q − . In both examples, ad a is nilpotent, a being even or odd with respect to the Z 2 -grading as Lie superalgebras. This property is an analogue of the periodicity of the Grigorchuk and Gupta–Sidki groups. In particular, Q is a nil finely-graded Lie superalgebra, which shows that an extension of a theorem due to Martinez and Zelmanov [27] for the Lie superalgebras of characteristic zero is not valid. Both Lie superalgebras are self-similar, contain infinitely many copies of themselves, let us also call them fractal due to this property. [ABSTRACT FROM AUTHOR]

The main result of this paper is the computation of the Lie superalgebras of holomorphic vector fields on complex flag supermanifolds, introduced by Yu.I. Manin. We prove that with several exceptions any holomorphic vector field is fundamental with respect to the natural action of the Lie superalgebra gl m | n ( C ) . [ABSTRACT FROM AUTHOR]

It is well known that if the ground field K has characteristic zero and G is a connected algebraic group, defined over K , then the Lie algebra Lie ( G ′ ) of the commutant G ′ of G coincides with the commutant Lie ( G ) ′ of Lie ( G ) . We show that this result is no longer true in the category of algebraic supergroups. We also construct a reductive supergroup H = X ⋊ G , where X and G are connected, reduced and abelian supergroups, such that X u ≠ 1 and ( H ev ) u is non-trivial connected (super)group. Quasi-reductive supergroups have been introduced in [10] . We prove that a supergroup H is quasi-reductive if and only if the largest even (super)subgroup of the solvable radical R ( H ) is a torus, H ˜ = H / R ( H ) contains a normal supersubgroup U , which is quasi-isomorphic to a direct product of normal supersubgroups U i , and H ˜ / U is a triangulizable supergroup with odd unipotent radical. Moreover, for every i , Lie ( U i ) = U i ⊗ Sym ( n i ) are such that either n i = 0 and U i is a classical simple Lie superalgebra, or n i = 1 and U i is a simple Lie algebra. [ABSTRACT FROM AUTHOR]

The complexity of the simple and the Kac modules over the general linear Lie superalgebra gl ( m | n ) of type A was computed by Boe, Kujawa, and Nakano in [2] . A natural continuation to their work is computing the complexity of the same family of modules over the ortho-symplectic Lie superalgebra osp ( 2 | 2 n ) of type C . The two Lie superalgebras are both of Type I which will result in similar computations. In fact, our geometric interpretation of the complexity agrees with theirs. We also compute a categorical invariant, z -complexity, introduced in [2] , and we interpret this invariant geometrically in terms of a specific detecting subsuperalgebra. In addition, we compute the complexity and the z -complexity of the simple modules over the Type II Lie superalgebras osp ( 3 | 2 ) , D ( 2 , 1 ; α ) , G ( 3 ) , and F ( 4 ) . [ABSTRACT FROM AUTHOR]

An algebra L over a field F , in which product is denoted by [ , ] , is called a Lie type algebra if for all elements a , b , c ∈ L there exist α , β ∈ F (depending on a , b , c ) such that α ≠ 0 and [ [ a , b ] , c ] = α [ a , [ b , c ] ] + β [ [ a , c ] , b ] . Examples of Lie type algebras include associative algebras, Lie algebras, Leibniz algebras, etc. It is proved that if a Lie type algebra L admits an automorphism of finite order n with finite-dimensional fixed-point subalgebra of dimension m , then L has a soluble ideal of finite codimension bounded in terms of n and m and of derived length bounded in terms of n . [ABSTRACT FROM AUTHOR]

Abstract: We construct an action of the Lie superalgebras involving one parameter μ on the exterior algebra with infinitely many variables and show this representation to be irreducible if and only if μ is nonzero. We also discuss the module structure and irreducibility over a Lie subsuperalgebra graded by the root system . [Copyright &y& Elsevier]

Abstract: It is well known that non-constant holomorphic functions do not exist on a connected compact complex manifold. This statement is false for a supermanifold with a connected compact reduction. In this paper we study the question under what conditions non-constant holomorphic functions do not exist on a connected compact homogeneous complex supermanifold. We describe also the vector bundles determined by split homogeneous complex supermanifolds. As an application, we compute the algebra of holomorphic functions on the classical flag supermanifolds introduced in Manin (1997) . [Copyright &y& Elsevier]

Abstract: We consider some questions about subnormal subgroups of a group in the setting of Lie superalgebras. In particular, the analogues of Nilpotence Join Theorem and Rosebladeʼs Theorem for Lie superalgebras are proved. [Copyright &y& Elsevier]

Abstract: We define regular Kac–Moody superalgebras and classify them using integrable modules. We give conditions for irreducible highest weight modules of regular Kac–Moody superalgebras to be integrable. This paper is a major part of the proof for the classification of finite-growth contragredient Lie superalgebras. [Copyright &y& Elsevier]

Abstract: We describe the simple Lie superalgebras arising from the unital structurable superalgebras of characteristic 0 and construct four series of the unital simple structurable superalgebras of Cartan type. We give a classification of simple structurable superalgebras of Cartan type over an algebraically closed field F of characteristic 0. Together with the Faulkner theorem on the classification of classical such superalgebras, it gives a classification of the simple structurable superalgebras over F. [Copyright &y& Elsevier]

Abstract: Fock space realisations of unitary highest weight representations of compact and noncompact real forms of the quantum general linear superalgebra are constructed. We decompose all tensor powers of the Fock spaces into direct sums of irreducible unitary submodules for compact or noncompact real forms of the quantum superalgebra, obtaining multiplicities of the irreducibles and also explicit formulae for their highest weight vectors. A generalised Howe duality of type is also established, which is used in the study of unitary representations and should also be of independent interest. [Copyright &y& Elsevier]

Abstract: In this paper we use the Etingof–Kazhdan quantization of Lie bi-superalgebras to investigate some interesting questions related to Drinfeld–Jimbo type superalgebra associated to a Lie superalgebra of classical type. It has been shown that the D-J type superalgebra associated to a Lie superalgebra of type A-G, with the distinguished Cartan matrix, is isomorphic to the E-K quantization of the Lie superalgebra. The first main result in the present paper is to extend this to arbitrary Cartan matrices. This paper also contains two other main results: (1) a theorem stating that all highest weight modules of a Lie superalgebra of type A-G can be deformed to modules over the corresponding D-J type superalgebra and (2) a super version of the Drinfeld–Kohno theorem. [Copyright &y& Elsevier]

Abstract: The second author gave a formula for the elements of the enveloping algebra of a Lie superalgebra defined by Gorelik under an appropriate unimodularity assumption. We show that this formula is a particular case of a formula for the Jacobian of the exponential map of a symmetric superspace. [Copyright &y& Elsevier]

Abstract: We classify all linearly compact simple Jordan superalgebras over an algebraically closed field of characteristic zero. As a corollary, we deduce the classification of all linearly compact unital simple generalized Poisson superalgebras. [Copyright &y& Elsevier]

Abstract: A noncommutative Jordan algebra of a specific type is attached to any -balanced Freudenthal Kantor triple system, in such a way that the triple product in this system is determined by the binary product in the algebra. Over fields of characteristic zero, the simple noncommutative Jordan algebras of this type are classified. [Copyright &y& Elsevier]

The Fock space of m+p bosonic and n+q fermionic quantum oscillators forms a unitarizable module of the general linear superalgebra glm+p&z.sfnc; n+q. Its tensor powers decompose into direct sums of infinite-dimensional irreducible highest-weight glm+p&z.sfnc; n+q-modules. We obtain an explicit decomposition of any tensor power of this Fock space into irreducibles, and develop a character formula for the irreducible glm+p&z.sfnc; n+q-modules arising in this way. [Copyright &y& Elsevier]