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This paper performed an investigation into the s-embedding of the Lie superalgebra (→ S 1 ∣ 1 ) , a representation of smooth vector fields on a (1,1)-dimensional super-circle. Our primary objective was to establish a precise definition of the s-embedding, effectively dissecting the Lie superalgebra into the superalgebra of super-pseudodifferential operators ( S ψ D ⊙ ) residing on the super-circle S 1 | 1 . We also introduce and rigorously define the central charge within the framework of (→ S 1 ∣ 1 ) , leveraging the canonical central extension of S ψ D ⊙ . Moreover, we expanded the scope of our inquiry to encompass the domain of fuzzy Lie algebras, seeking to elucidate potential connections and parallels between these ostensibly distinct mathematical constructs. Our exploration spanned various facets, including non-commutative structures, representation theory, central extensions, and central charges, as we aimed to bridge the gap between Lie superalgebras and fuzzy Lie algebras. To summarize, this paper is a pioneering work with two pivotal contributions. Initially, a meticulous definition of the s-embedding of the Lie superalgebra (→ S 1 | 1 ) is provided, emphasizing the representationof smooth vector fields on the (1,1)-dimensional super-circle, thereby enriching a fundamental comprehension of the topic. Moreover, an investigation of the realm of fuzzy Lie algebras was undertaken, probing associations with conventional Lie superalgebras. Capitalizing on these discoveries, we expound upon the nexus between central extensions and provide a novel deformed representation of the central charge. [ABSTRACT FROM AUTHOR]

Two one-parameter families of positively graded Lie superalgebras generated by two elements and two relations that are narrow in the sense of Zelmanov and Shalev are considered. The first family contains the positive part R+ of the Ramond algebra, while the second one contains the positive part NS+ of the Neveu–Schwarz algebra. The results of the article are super analogues of Benoist's theorem on defining the positive part of the Witt algebra by generators and relations. [ABSTRACT FROM AUTHOR]

For classical Lie superalgebras of type I, we provide necessary and sufficient conditions for a Verma supermodule $\Delta (\lambda)$ to be such that every nonzero homomorphism from another Verma supermodule to $\Delta (\lambda)$ is injective. This is applied to describe the socle of the cokernel of an inclusion of Verma supermodules over the periplectic Lie superalgebras $\mathfrak {pe} (n)$ and, furthermore, to reduce the problem of description of $\mathrm {Ext}^1_{\mathcal O}(L(\mu),\Delta (\lambda))$ for $\mathfrak {pe} (n)$ to the similar problem for the Lie algebra $\mathfrak {gl}(n)$. Additionally, we study the projective and injective dimensions of structural supermodules in parabolic category $\mathcal O^{\mathfrak {p}}$ for classical Lie superalgebras. In particular, we completely determine these dimensions for structural supermodules over the periplectic Lie superalgebra $\mathfrak {pe} (n)$ and the orthosymplectic Lie superalgebra $\mathfrak {osp}(2|2n)$. [ABSTRACT FROM AUTHOR]

As is well-known, the dimension of the space spanned by the non-degenerate invariant symmetric bilinear forms (NISes) on any simple finite-dimensional Lie algebra or Lie superalgebra is equal to at most 1 if the characteristic of the algebraically closed ground field is not 2. We prove that in characteristic 2, the superdimension of the space spanned by NISes can be equal to 0, or 1, or 0 | 1 , or 1 | 1 ; it is equal to 1 | 1 if and only if the Lie superalgebra is a queerification (defined in arXiv:1407.1695) of a simple classically restricted Lie algebra with a NIS (for examples, mainly in characteristic ≠2, see arXiv:1806.05505). We give examples of NISes on deformations (with both even and odd parameters) of several simple finite-dimensional Lie superalgebras in characteristic 2. We also recall examples of multiple NISes on simple Lie algebras over non-closed fields. [ABSTRACT FROM AUTHOR]

In this paper, we study the shape and the reduced shape algebra for the Lie superalgebra sl (m , n) . We define quasistandard Young tableaux, and they give a combinatorial basis for the reduced shape algebra, called the diamond cone. There is a natural bijective map from the set of semistandard tableaux with shape λ to the set of quasistandard tableaux with smaller shape: the diamond cone is compatible with the stratification of the reduced shape algebra. [ABSTRACT FROM AUTHOR]

The Grigorchuk and Gupta–Sidki groups play a 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 [V. M. Petrogradsky, Examples of self-iterating Lie algebras, J. Algebra302(2) (2006) 881–886], Shestakov and Zelmanov extended this construction to an arbitrary positive characteristic [I. P. Shestakov and E. Zelmanov, Some examples of nil Lie algebras, J. Eur. Math. Soc. (JEMS)10(2) (2008) 391–398]. Now, we construct a family of so called clover 3-generated restricted Lie algebras T (Ξ) , where a field of positive characteristic is arbitrary and Ξ an infinite tuple of positive integers. All these algebras have a nil p -mapping. We prove that 1 ≤ GKdim T (Ξ) ≤ 3. We compute Gelfand–Kirillov dimensions of clover restricted Lie algebras with periodic tuples and show that these dimensions for constant tuples are dense on [ 1 , 3 ]. We construct a subfamily of nil restricted Lie algebras T (Ξ q , κ) , with parameters q ∈ ℕ , κ ∈ ℝ + , having extremely slow quasi-linear growth of type: γ T (Ξ q , κ) (m) = m ln ⋯ ln ︸ q m κ + o (1) , as m → ∞. The present research is motivated by construction by Kassabov and Pak of groups of oscillating growth [M. Kassabov and I. Pak, Groups of oscillating intermediate growth. Ann. Math. (2)177(3) (2013) 1113–1145]. As an analogue, we construct nil restricted Lie algebras of intermediate oscillating growth in [V. Petrogradsky, Nil restricted Lie algebras of oscillating intermediate growth, preprint (2020), arXiv:2004.05157]. We call them Phoenix algebras because, for infinitely many periods of time, the algebra is "almost dying" by having a "quasi-linear" growth as above, for infinitely many n it has a rather fast intermediate growth of type exp (n / (ln n) λ) , for such periods the algebra is "resuscitating". The present construction of three-generated nil restricted Lie algebras of quasi-linear growth is an important part of that result, responsible for the lower quasi-linear bound in that construction. [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]

In this article, we find a family S L n , m , in any arbitrary dimensions, of cohomologically rigid solvable Lie superalgebras with nilradical the model filiform Lie superalgebra L n , m . Moreover, we exhibit a family of cohomologically rigid solvable Lie superalgebras with nilradical the model nilpotent Lie superalgebra of generic characteristic sequence. Both cases correspond to solvable Lie superalgebras of maximal dimension for a given nilradical. Contrariwise, we will show that the family of Lie superalgebras S L n , m can be deformed if defined over a field of odd characteristic. [ABSTRACT FROM AUTHOR]

The main topic of this paper is to compute the first relative cohomology group of the Lie algebra of smooth vector fields Vect (R) , with coefficients in the space of trilinear differential operators that acts on tensor densities, D λ ¯ , μ , vanishing on the Lie algebra s l (2) , where λ ¯ = (λ 1 , λ 2 , λ 3) ∈ R 3 and μ ∈ R. [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]

Under suitable conditions the skewfield of fractions of a superalgebra which is a noetherian domain is canonically provided with a structure of superalgebra. This gives rise to a notion of rational equivalence in the category of superalgebras. We study from the point of view of this rational equivalence some low dimensional examples of quantum enveloping algebras of Lie superalgebras. [ABSTRACT FROM AUTHOR]

The inverses of indecomposable Cartan matrices are computed for finite-dimensional Lie algebras and Lie superalgebras over fields of any characteristic, and for hyperbolic (almost affine) complex Lie (super)algebras. This enables one to express the fundamental weights in terms of simple roots corresponding to the Cartan matrix. We discovered three yet inexplicable new phenomena, of which (a) and (b) concern hyperbolic (almost affine) complex Lie (super)algebras: (a) several of the inverses of Cartan matrices have all their elements negative (not just non-positive, as they should be according to an a priori characterization due to Zhang Hechun); (b) the 0s only occur on the main diagonals of the inverses; (c) the determinants of inequivalent Cartan matrices of the simple Lie (super)algebra of any dimension may differ (in any characteristic). We interpret most of the results of Wei and Zou (2017) [31] as inverses of the Gram matrices of non-degenerate invariant symmetric bilinear forms on the (super)algebras considered, not of Cartan matrices, and give more adequate references. In particular, the inverses of Cartan matrices of simple Lie algebras were already published, starting with Dynkin's paper in 1952, see also Table 2 in Springer's book by Onishchik and Vinberg (1990). [ABSTRACT FROM AUTHOR]

Let k be a field of characteristic not two or three. We classify up to isomorphism all finite-dimensional Lie superalgebras over k, where is a three-dimensional simple Lie algebra. If denotes the center of the result is the following: either or or [ABSTRACT FROM AUTHOR]

In the 1990s, in work of Le Bruyn and Smith and in work of Le Bruyn and Van den Bergh, it was proved that point modules and line modules over the homogenization of the universal enveloping algebra of a finite-dimensional Lie algebra describe useful data associated to the Lie algebra. In particular, in the case of the Lie algebra sl2(C), there is a correspondence between Verma modules and certain line modules that associates a pair (h, φ), where h is a 2-dimensional Lie subalgebra of sl2(C) and φ ∊ h* satisfies φ([h, h]) = 0, to a particular type of line module. In this article, we prove analogous results for the Lie superalgebra sl(1|1) and for a color Lie algebra associated to the Lie algebra sl2. [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]

We review recent results obtained at the intersection of the theory of quantum deformed Calogero-Moser-Sutherland systems and the theory of Lie superalgebras. We begin with a definition of admissible deformations of root systems of basic classical Lie superalgebras. For classical series, we prove the existence of Lax pairs. Connections between infinite-dimensional Calogero-Moser-Sutherland systems, deformed quantum CMS systems, and representation theory of Lie superalgebras are discussed. [ABSTRACT FROM AUTHOR]

To a finite type knot invariant, a weight system can be associated, which is a function on chord diagrams satisfying so-called 4-term relations. In the opposite direction, each weight system determines a finite type knot invariant. In particular, a weight system can be associated to any metrized Lie algebra, and any metrized Lie superalgebra. However, computation of these weight systems is complicated. In the recent paper by the present author, an extension of the gl (N) -weight system to arbitrary permutations is defined, which allows one to develop a recurrence relation for an efficient computation of its values. In addition, the result proves to be universal, valid for all values of N and allowing thus to define a unifying gl -weight system taking values in the ring of polynomials in infinitely many variables C 0 = N , C 1 , C 2 , .... In the present paper, we extend this construction to the weight system associated to the Lie superalgebra gl (m | n). Then we prove that the gl (m | n) -weight system is equivalent to the gl -one, under the substitution C 0 = m − n. [ABSTRACT FROM AUTHOR]

We give explicit formulas proving that the following Lie (super)algebras are restricted: known exceptional simple vectorial Lie (super)algebras in characteristic 3, deformed Lie (super)algebras with indecomposable Cartan matrix, simple subquotients over an algebraically closed field of characteristic 3 of these (super)algebras (under certain conditions), and deformed divergence-free Lie superalgebras of a certain type with any finite number of indeterminates in any characteristic. [ABSTRACT FROM AUTHOR]

We develop a biorthogonal formalism for non-Hermitian multimode and multiphoton Jaynes-Cummings models. For these models, we define supersymmetric generators, which are especially convenient for diagonalizing the Hamiltonians. The Hamiltonian and its adjoint are expressed in terms of supersymmetric generators having the Lie superalgebra properties. The method consists in using a similarity dressing operator that maps onto spaces suitable for diagonalizing Hamiltonians even in an infinite-dimensional Hilbert space. We then successfully solve the eigenproblems related to the Hamiltonian and its adjoint. For each model, the eigenvalues are real, while the eigenstates do not form a set of orthogonal vectors. We then introduce the biorthogonality formalism to construct a consistent theory. [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 [27], Shestakov and Zelmanov extended this construction to an arbitrary positive characteristic [39]. There are a few more examples of self-similar finitely generated restricted Lie algebras with a nilp-mapping, but, as a rule, that algebras have no clear basis and require technical computations. Now we construct a familyL(Ξ) of 2-generated restricted Lie algebras of slow polynomial growth with a nilp-mapping, where a field of positive characteristic is arbitrary andΞan infinite tuple of positive integers. Namely,GKdimL(Ξ)≤2 for all such algebras. The algebras are constructed in terms of derivations of infinite divided power algebra Ω. We also study their associative hullsA⊂End(Ω). AlgebrasLandAareℤ2-graded by a multidegree in the generators. IfΞis periodic thenL(Ξ) is self-similar. As a particular case, we construct a continuum subfamily of non-isomorphic nil restricted Lie algebrasL(Ξα),α∈ℝ+, with extremely slow growth. Namely, they have Gelfand-Kirillov dimension one but the growth is not linear. For this subfamily, the associative hullsAhave Gelfand-Kirillov dimension two but the growth is not quadratic. The virtue of the present examples is that they have clear monomial bases. [ABSTRACT FROM PUBLISHER]

In the present paper, we define the diamond cone for the Lie superalgebra spo ( 2 m , 1 ) , considering the (covariant) tensor representation of spo ( 2 m , 1 ) . The diamond cone is no more indecomposable. Nevertheless, we give a basis for each indecomposable component, using quasistandard Young tableaux for spo ( 2 m , 1 ) . We realize a bijection between the set of semistandard tableaux with shape λ and the set of quasistandard tableaux with shape μ ≤ λ . This gives the compatibility of the diamond cone with the natural stratification of the reduced shape algebra. [ABSTRACT FROM AUTHOR]

This paper determines the minimal dimensions of faithful representations for abelian Lie superalgebras of finite dimensions over an algebraically closed field of characteristic zero. In particular, we also obtain the maximal dimensions of abelian subalgebras for the general linear Lie superalgebras. [ABSTRACT FROM PUBLISHER]

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]

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]

In this paper, we develop a general theory on Leibniz central extensions of Lie superalgebras and apply it to determine the second Leibniz cohomology groups for several classes of Lie superalgebras, including classical Lie superalgebras, Neveu-Schwarz superalgebras, differentiably simple Lie superalgebras, and affine (toroidal) Kac-Moody Lie superalgebras. [ABSTRACT FROM AUTHOR]

Let G be a Lie group, G its Lie algebra and T*G its cotangent bundle. On T*G, we consider the Lie group structure obtained by performing a left trivialization and endowing the resulting trivial bundle G ×G* with the semi-direct product, using the co-adjoint action of G on the dual space G* of G. We investigate the group of automorphisms of the Lie algebra D:= T*G of T*G. More precisely, we fully characterize the Lie algebra of all derivations of D, exhibiting a finer decomposition into components made of well known spaces. Further, we specialize to the cases where G has a bi-invariant Riemannian or pseudo-Riemannian metric, with the semi-simple and compact cases investigated as particular cases. [ABSTRACT FROM AUTHOR]

Given a quasi-reductive algebraic supergroup G , we use the theory of semisimplifications of symmetric monoidal categories to define a symmetric monoidal functor Φ x : Rep (G) → Rep (O S p (1 | 2)) associated to any given element x ∈ Lie (G) 1 ¯ . For nilpotent elements x , we show that the functor Φ x can be defined using the Deligne filtration associated to x. We use this approach to prove an analogue of the Jacobson-Morozov Lemma for algebraic supergroups. Namely, we give a necessary and sufficient condition on odd nilpotent elements x ∈ Lie (G) 1 ¯ which define an embedding of supergroups O S p (1 | 2) → G so that x lies in the image of the corresponding Lie algebra homomorphism. [ABSTRACT FROM AUTHOR]

The principal filtration of the infinite-dimensional odd contact Lie superalgebra over a field of characteristic p > 2 is proved to be invariant under the automorphism group by investigating ad-nilpotent elements and determining certain invariants such as subalgebras generated by some ad-nilpotent elements. Then, it is proved that two automorphisms coincide if and only if they coincide on the −1 component with respect to the principal grading. Finally, all the odd contact superalgebras are classified up to isomorphisms. [ABSTRACT FROM AUTHOR]

Abstract: We give a new combinatorial model for the crystals of integrable highest weight modules over the classical Lie algebras of type B and C in terms of classical Young tableaux. We then obtain a new description of its Littlewood–Richardson rule and a maximal Levi branching rule in terms of classical Littlewood–Richardson tableaux, which extends in a bijective way the well-known stable formulas at large ranks. We also show that this tableau model admits a natural superization and it produces the characters of irreducible highest weight modules over orthosymplectic Lie superalgebras, which correspond to the integrable highest weight modules over the classical Lie algebras of type B and C under the Cheng–Lam–Wangʼs super duality. [Copyright &y& Elsevier]

At the previous congress (CRM 6), we reviewed the constructions of Yang-Baxter operators from associative algebras, and presented some (colored) bialgebras and Yang-Baxter systems related to them. The current talk deals with Yang-Baxter operators from (G, θ)-Lie algebras (structures which unify the Lie algebras and Lie superalgebras). Thus, we produce solutions for the constant and the spectral-parameter Yang-Baxter equations, Yang-Baxter systems, etc. Attempting to present the general framework we review the work of other authors and we propose problems, applications and directions of study. [ABSTRACT FROM AUTHOR]

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]

The classification of simple structurable superalgebras of Cartan type over an algebraically closed field of characteristic 0 has been done recently by Pozhidaev and Shestakov [8]. Here we present an easy construction and quick classification of the algebras using a superalgebra version of results due to Elduque and Okubo [5] relating Lie algebras with S4-actions and structurable algebras. [ABSTRACT FROM AUTHOR]

Abstract: The diamond cone for a semi simple Lie algebra is a quotient of the shape algebra of . If is the nilpotent factor of the Iwasawa decomposition of , we get an indecomposable -module. If or , particular basis in , were defined, using the notion of quasi standard Young tableaux. In the present paper, we define the diamond cone for the Lie superalgebra , starting with the covariant tensor representations of . The diamond cone is no more indecomposable, but we give basis for each its indecomposable component, using quasi standard Young tableaux for . [ABSTRACT FROM AUTHOR]

In this paper, we study the operator forms of the classical Yang–Baxter equation (CYBE) in Lie superalgebras. We introduce the notion of super $\mathcal{O}$-operators and give their relationship with the (standard) tensor form. There are close relationships between the CYBE in Lie superalgebras and left-symmetric superalgebras which can be interpreted through the super $\mathcal{O}$-operators. In particular, there are natural and explicit solutions of the CYBE in Lie superalgebras constructed from left-symmetric superalgebras. [ABSTRACT FROM AUTHOR]

A Lie superalgebra is constructed from which establishes an isospectral problems. By solving the zero curvature equation, a resulting super hierarchies of the Guo hierarchy are obtained. By making use of the super identity, the Hamiltonian structures of the above super integrable hierarchies are generated, this method can be used to other superhierarchy. [ABSTRACT FROM AUTHOR]

We consider the δ-derivations of classical Lie superalgebras and prove that these superalgebras admit nonzero δ-derivations only when δ = 0, ½, 1. The structure of ½-derivations for classical Lie superalgebras is completely determined. [ABSTRACT FROM AUTHOR]

For the Kac-Moody superalgebra associated with the loop superalgebra with values in a finite-dimensional Lie superalgebra g, we show what its quadratic Casimir element is equal to if the Casimir element for g is known (if g has an even invariant supersymmetric bilinear form). The main tool is the Wick normal form of the even quadratic Casimir operator for the Kac-Moody superalgebra associated with g; this Wick normal form is independently interesting. If g has an odd invariant supersymmetric bilinear form, then we compute the cubic Casimir element. In addition to the simple Lie superalgebras g = g(A) with a Cartan matrix A for which the Shapovalov determinant was known, we consider the Poisson Lie superalgebra poi(0|n) and the related Kac-Moody superalgebra. [ABSTRACT FROM AUTHOR]

It is proved that the exponential growth rate of identities in any superalgebra with a nilpotent commutator subalgebra over a field of zero characteristic is an integer. [ABSTRACT FROM AUTHOR]

Abstract: A new combinatorial interpretation of the Howe dual pair acting on an infinite-dimensional Fock space is presented. The character of a quasi-finite irreducible highest weight representation of occurring in the Fock space is realized in terms of certain bitableaux of skew shapes. We study a general combinatorics of these bitableaux, including the Robinson–Schensted–Knuth correspondence and the Littlewood–Richardson rule, and then its dual relation with rational semistandard tableaux for . This result also explains other Howe dual pairs , where is a Lie superalgebra. [Copyright &y& Elsevier]

The purpose of this contribution, is to initiate a classification of Lie superalgebras (LS) of dimension five, over the base field ℝ of real numbers. We use the “ graded skew-symmetry” and the “ graded Jacobi identity” in order to get restrictions for the commutators and anticommutators of an arbitrary five-dimensional Lie superalgebra L = L 0⊕ L 1 [ABSTRACT FROM AUTHOR]

In this Letter, we will prove a super-analogue of a well-known result by Kontsevich which states that the homology of a certain complex generated by isomorphism classes of oriented graphs can be calculated as the Lie algebra homology of an infinite-dimensional Lie algebra of symplectic vector fields. [ABSTRACT FROM AUTHOR]

A class of representations of a Lie superalgebra (over a commutative superring) in its symmetric algebra is studied. As an application we get a direct and natural proof of a strong form of the Poincare´–Birkhoff–Witt theorem, extending this theorem to a class of nilpotent Lie superalgebras. Other applications are presented. Our results are new already for Lie algebras. [Copyright &y& Elsevier]

We give a representation-theoretical interpretation of the Gelfand–Tsetlin pattern for strict partitions. Using the Howe duality involving a pair of the queer Lie superalgebras and an analogue of the Littlewood–Richardson rule for Schur Q-functions, we show that such patterns give the branching rule for the irreducible tensor representations of the queer Lie superalgebra. [ABSTRACT FROM AUTHOR]

A Lie superalgebra g is called a complete if C(g)=0 and Der g=ad g. This paper provides sufficient conditions for a Lie superalgebra to be complete and uses this result to construct families of such algebras. The basic results are a generalization of D.J. Meng [Commun. Algebra 22 (1994) 5509]. [Copyright &y& Elsevier]

We prove that there exists no maximal BCH-invertible subalgebra of P(n) that can be imagined as and called skew with respect to Z-gradation of P(n). [Copyright &y& Elsevier]