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Throughout this paper, we show on one hand, that there are nilpotent and solvable Lie superalgebras with infinitely many local superderivations which are not standard superderivations. On the other hand, we show that every local superderivation is a superderivation on the maximal-dimensional solvable Lie superalgebras with model filiform or model nilpotent nilradical. Moreover, we extend the latter result for Leibniz superalgebras by showing that every local superderivation is a superderivation on the maximal-dimensional solvable Leibniz superalgebras with model filiform or model nilpotent non-Lie nilradical. Agencia Estatal de Investigación PID2020- 115155GB-I00 Junta de Andalucía FEDER-UCA18-107643 Junta de Extremadura GR18001 Junta de Extremadura IB18032

The Rueda's algebras R ( f , ς ) (simply R) are a class of algebras similar to the universal enveloping algebra of sl 2 . We study the category H of R-modules whose objects are free of rank 1 when restricted to R 0 = C [ H ] . We classify the isomorphism classes of objects in H and determine the simplicity of these modules. As a result, we also give an explicit description of submodule structures and obtain new simple non-weight modules over R. In particular, we recover some results about U ( h ) -free modules over the Lie algebra sl 2 obtained by J. Nilsson and over the Lie superalgebra osp ( 1 | 2 ) obtained by Y. Cai and K. Zhao.

All results concern characteristic 2. Two procedures that to every simple Lie algebra assign simple Lie superalgebras, most of the latter new, are offered. We prove that every simple finite-dimensional Lie superalgebra is obtained as the result of one of these procedures, so we classified all simple finite-dimensional Lie superalgebras modulo non-existing at the moment classification of simple finite-dimensional Lie algebras. This result concerns Lie superalgebras considered naively, as vector spaces. To obtain classification of simple Lie superalgebras in the category of supervarieties, one should list the non-isomorphic deforms (results of deformations) with odd parameter. This problem is open bar several examples described in arXiv~0807.3054. For Lie algebras, in addition to the known ---"classical" --- restrictedness, we introduce a (2,4)-structure on the two non-alternating series: orthogonal and of Hamiltonian vector fields. For Lie superalgebras, the classical restrictedness of Lie algebras has two analogs: $(2|4)$- and $(2|2)$-structures, one more analog --- a $(2,4)|4$-structure on Lie superalgebras is the analog of (2,4)-structure on Lie algebras.

In this paper, a finite-dimensional Lie superalgebra K n , m over a field of prime characteristic is constructed. Then, we study some properties of K n , m . Moreover, we prove that K n , m is an extension of a simple Lie superalgebra, and if m = n − 1 , then it is isomorphic to a subalgebra of a restricted Lie superalgebra.

It is well known that superderivation of a Lie superalgebra is certain generalization of derivation of a Lie algebra. This paper is devoted to investigate the structure and dimension of superderiva...

Let $${{\cal D}_{\lambda ,\mu }}$$ be the space of linear differential operators on weighted densities from $${{\cal F}_\lambda }$$ to $${{\cal F}_\mu }$$ as module over the orthosymplectic Lie superalgebra $$\mathfrak{osp}(3\left| 2 \right.)$$ , where $${{\cal F}_\lambda }$$ , $$\lambda \in \mathbb{C}$$ is the space of tensor densities of degree λ on the supercircle S1∣3. We prove the existence and uniqueness of projectively equivariant quantization map from the space of symbols to the space of differential operators. An explicite expression of this map is also given.

In this paper, we study 2-local superderivations on the Lie superalgebra of Block type S(q), which is an infinite dimensional Lie superalgebra with an outer derivation. We prove that all 2-local su...

The present paper is devoted to studying 2-local superderivations on the super Virasoro algebra and the super W(2,2) algebra. It is proved that all 2-local superderivations on the super Virasoro al...

Let (V, ω) be an orthosymplectic ℤ2-graded vector space and let 𝔤:= 𝔤𝔬𝔰𝔭 (V, ω) denote the Lie superalgebra of similitudes of (V, ω). It is known that as a 𝔤-module, the space (V ) of superpolynomials on V is completely reducible, unless dim $$ {V}_{\overline{\mathrm{o}}} $$ and dim $$ {V}_{\overline{1}} $$ are positive even integers and dim $$ {V}_{\overline{\mathrm{O}}}\le \dim\ {V}_{\overline{1}} $$ . When (V ) is not a completely reducible 𝔤-module, we construct a natural basis $$ {\left\{{D}_{\leftthreetimes}\right\}}_{\leftthreetimes \in \mathcal{T}} $$ of “Capelli operators” for the algebra (V ) 𝔤 of 𝔤 -invariant superpolynomial superdifferential operators on V , where the index set 𝒯 is the set of integer partitions of length at most two. We compute the action of the operators $$ {\left\{{D}_{\leftthreetimes}\right\}}_{\leftthreetimes \in \mathcal{T}} $$ on maximal indecomposable components of (V ) explicitly, in terms of Knop–Sahi interpolation polynomials. Our results show that, unlike the cases where (V ) is completely reducible, the eigenvalues of a subfamily of the {D⋋} are not given by specializing the Knop–Sahi polynomials. Rather, the formulas for these eigenvalues involve suitably regularized forms of these polynomials. This is in contrast with what occurs for previously studied Capelli operators. In addition, we demonstrate a close relationship between our eigenvalue formulas for this subfamily of Capelli operators and the Dougall–Ramanujan hypergeometric identity. We also transcend our results on the eigenvalues of Capelli operators to the Deligne category Rep (Ot). More precisely, we define categorical Capelli operators $$ {\left\{{D}_{t,\leftthreetimes}\right\}}_{\leftthreetimes \in \mathcal{T}} $$ that induce morphisms of indecomposable components of symmetric powers of Vt, where Vt is the generating object of Rep (Ot). We obtain formulas for the eigenvalue polynomials associated to the $$ {\left\{{D}_{t,\leftthreetimes}\right\}}_{\leftthreetimes \in \mathcal{T}} $$ that are analogous to our results for the operators $$ {\left\{{D}_{\leftthreetimes}\right\}}_{\leftthreetimes \in \mathcal{T}} $$ .

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.

Let A m , n be the tensor product of the Laurent polynomial algebra in m even variables and the exterior algebra in n odd variables over the complex field C , and the Witt superalgebra W m , n be the Lie superalgebra of superderivations of A m , n . In this paper, we classify the simple weight W m , n modules with finite-dimensional weight spaces with respect to the standard Cartan subalgebra of W m , 0 . Every such module is either a simple quotient of a tensor module or a module of highest weight type.

In this paper we give an in-deph analysis of the nilpotency index of nilpotent homogeneous inner superderivations in associative prime superalgebras with and without superinvolution. We also presen...

Let $ 0\rightarrow \mathfrak{a} \rightarrow \mathfrak{e} \rightarrow \mathfrak{g} \rightarrow 0$ be an abelian extension of the Lie superalgebra $\mathfrak{g}$. In this article we consider the problems of extending endomorphisms of $\mathfrak{a}$ and lifting endomorphisms of $\mathfrak{g}$ to certain endomorphisms of $\mathfrak{e}$. We connect these problems to the cohomology of $\mathfrak{g}$ with coefficients in $\mathfrak{a}$ through construction of two exact sequences, which is our main result, involving various endomorphism groups and the second cohomology. The first exact sequence is obtained using the Hochschild-Serre spectral sequence corresponding to the above extension while to prove the second we rather take a direct approach. As an application of our results we obtain descriptions of certain automorphism groups of semidirect product Lie superalgebras., Comment: 17 pages

Let $\mathfrak {g}$ be a complex contragredient Lie superalgebra, and let D be its distinguished extended Dynkin diagram. Let $\text {Aut}(\mathfrak {g})$ be the automorphism group of $\mathfrak {g}$ , and let $\text {Int}(\mathfrak {g})$ be its identity component. We prove that $\text {Aut}(\mathfrak {g})/\text {Int}(\mathfrak {g}) \cong \text {Aut}(\mathrm {D})$ .

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...

We generalize the P(N)-graded Lie superalgebras of Martinez-Zelmanov. This generalization is not so restrictive but sufficient enough so that we are able to have a classification for this generalized P(N)-graded Lie superalgebras. Our result is that the generalized P(N)-graded Lie superalgebra L is centrally isogenous to a matrix Lie superalgebra coordinated by an associative superalgebra with a super-involution. Moreover, L is P(N)-graded if and only if the coordinate algebra R is commutative and the super-involution is trivial. This recovers Martinez-Zelmanov’s theorem for type P(N). We also obtain a generalization of Kac’s coordinatization via Tits-Kantor-Koecher construction. Actually, the motivation of this generalization comes from the Fermionic-Bosonic module construction.

We classify irreducible representations of finite W-algebra for the queer Lie superalgebra Q ( n ) associated with the principal nilpotent coadjoint orbits. We use this classification and our previous results to obtain a classification of irreducible finite-dimensional representations of the super Yangian Y Q ( 1 ) .

We classify Jet modules for the Lie (super)algebras L = W ⋉ ( g ⊗ C [ t , t − 1 ] ) , where W is the Witt algebra and g is a Lie superalgebra with an even diagonlizable derivation. Then we give a conceptional method to classify all simple Harish-Chandra modules for L and the map superalgebras, which are of the form L ⊗ R , where R is a Noetherian unital supercommutative associative superalgebra.

In 1986, Van de Leur introduced and classified affine Lie superalgebras. An affine Lie superalgebra is defined as the quotient of certain Lie superalgebra G defined by generators and relations, corresponding to a symmetrizable generalized Cartan matrix, over the so-called radical of G . Because of the interesting applications of affine Lie (super)algebras in combinatorics, number theory and physics, it is very important to recognize how far a Lie (super)algebra is to be an affine Lie (super)algebra. In this regard, we determine affine Lie superalgebras axiomatically.

We determine the Verma multiplicities of standard filtrations of projective modules in integral atypical blocks in the BGG category O for the orthosymplectic Lie superalgebras osp ( 3 | 4 ) by way of translation functors. We then explicitly determine the composition factor multiplicities of Verma modules using BGG reciprocity.

We study the eigenspace decomposition of a basic classical Lie superalgebra under the adjoint action of a toral subalgebra, thus extending results of Kostant. In recognition of Kostant's contribution we refer to the eigenfunctions appearing in the decomposition as Kostant roots. We then prove that Kostant root systems inherit the main properties of classical root systems. Our approach is combinatorial in nature and utilizes certain graphs naturally associated with Kostant root systems. In particular, we reprove Kostant's results without making use of the Killing form.

We define the degenerate two boundary affine Hecke-Clifford algebra H d , and show it admits a well-defined q ( n ) -linear action on the tensor space M ⊗ N ⊗ V ⊗ d , where V is the natural module for q ( n ) , and M , N are arbitrary modules for q ( n ) , the Lie superalgebra of Type Q. When M and N are irreducible highest weight modules parameterized by a staircase partition and a single row, respectively, this action factors through a quotient of H d . We then construct explicit modules for this quotient, H p , d , using combinatorial tools such as shifted tableaux and the Bratteli graph. These modules belong to a family of modules which we call calibrated. Using the relations in H p , d , we also classify a specific class of calibrated modules. The irreducible summands of M ⊗ N ⊗ V ⊗ d coincide with the combinatorial construction, and provide a weak version of the Schur-Weyl type duality.

We prove denominator identities for the periplectic Lie superalgebra p ( n ) , thereby completing the problem of finding denominator identities for all simple classical finite-dimensional Lie superalgebras.

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 ) .

We study the notion of isoclinism on central extensions of Lie superalgebras and discuss some conditions under which central extensions are isoclinic as well as some results on isoclinic homomorphi...

We prove the unrolled superalgebra $\mathcal{U}_{\xi}^{H}\mathfrak{sl}(2|1)$ has a completion which is a ribbon superalgebra in a topological sense where $\xi$ is a root of unity of odd order. Using this ribbon superalgebra we construct its universal invariant of links. We use it to construct an invariant of $3$-manifolds of Hennings type.

Let L be a contragredient Lie superalgebra. A symmetric pair of L is a pair $$({\mathfrak {g}},{\mathfrak {g}}^\sigma )$$ , where $${\mathfrak {g}}$$ is a real form of L, and $$\sigma $$ is a $${\mathfrak {g}}$$ -involution with invariant subalgebra $${\mathfrak {g}}^\sigma $$ . We show that a symmetric pair carries invariant symplectic forms if and only if $${\mathfrak {g}}^\sigma $$ has a 1-dimensional center. Furthermore, the symplectic form is pseudo-Kahler if and only if the center of $${\mathfrak {g}}^\sigma $$ is compact. As an application, we classify the symplectic symmetric pairs, as well as the pseudo-Kahler symmetric pairs.

In this paper, both canonical and noncanonical polynomial representations of strange classical Lie superalgebra P(n) are investigated. It turns out all these representations are undecomposable, and their composition series are obtained.

We classify all simple bounded highest weight modules of a basic classical Lie superalgebra $\mathfrak g$. In particular, our classification leads to the classification of the simple weight modules with finite weight multiplicities over all classical Lie superalgebras. We also obtain some character formulas of strongly typical bounded highest weight modules of $\mathfrak g$., Comment: 22 pages

Using the approach developed by Gautam and Toledano Laredo, we introduce analogues of the category $$ \mathfrak{O} $$ for representations of the Yangian $$Y_\hbar(A(m,n))$$ of a special linear Lie superalgebra and the quantum loop superalgebra $$U_q(LA(m,n))$$ . We investigate the relation between them and conjecture that these categories are equivalent.

Let g be a restricted Lie superalgebra of Cartan type W(n), S(n)or H(n) over an algebraically closed field k of prime characteristic p > 3, in the sense of modular version of Kac’s definition in 1977. In this note, we show that the restricted representation category over g has only one block (reckoning parities in). This phenomenon is very different from the case of characteristic zero.

Based on the description of irreducible finite-dimensional representations of the Yangian of special linear Lie superalgebra and quantum loop special linear superalgebra we introduce the counterparts of the categories $$\mathfrak{O}$$ for representations of these quantum superalgebras. The theorem of equivalence of these representation categories is announced.

In this article, Lie super-bialgebra structures on the Lie superalgebra of Witt type W are investigated. They are shown to be triangular coboundary. As a byproduct, we obtain that the first cohomol...

We introduce a class of novel ℤ2 × ℤ2-graded color superalgebras of infinite dimension. It is done by realizing each member of the class in the universal enveloping algebra of a Lie superalgebra which is a module extension of the Virasoro algebra. Then the complete classification of central extensions of the ℤ2 × ℤ2-graded color superalgebras is presented. It turns out that infinitely many members of the class have nontrivial extensions. We also demonstrate that the color superalgebras (with and without central extensions) have adjoint and superadjoint operations.

Let L be a Lie superalgebra over a field of characteristic different from 2, 3 and write ID∗(L) for the Lie superalgebra consisting of superderivations mapping L to L2 and the central elements to z...

A Lie superalgebra H satisfying H2=Z(H) is called generalized Heisenberg Lie superalgebra. If d(H):=(m∣n) is the minimum number of generators required to describe H, then in this article we intend ...

Supermanifolds are known to admit both differential forms and integral forms, thus any appropriate super analogue of the de Rham theory should take both types of forms into account. However, the cohomology of Lie superalgebras studied so far in the literature involves only differential forms when interpreted as a de Rham theory for Lie supergroups. Thus a new cohomology theory of Lie superalgebras is needed to fully incorporate differential-integral forms, and we investigate such a theory here. This new cohomology is defined by a BRST complex of Lie superalgebra modules, and includes the standard Lie superalgebra cohomology as a special case. General properties expected of a cohomology theory are established for the new cohomology, and examples of the new cohomology groups are computed.

In this paper, we determine the derivations of affine Lie superalgebras. As the nature of affine Lie superalgebra of type A(l,l)(1) is slightly different from the nature of the other types, we stud...

Super-Hopf algebra structure on the function algebra on the extended quantum symplectic superspace $\text{SP}_q^{2|1}$, denoted by ${\mathbb F}({\text{SP}}_q^{2|1})$, is defined. A quantum Lie superalgebra derived from ${\mathbb F}({\text{SP}}_q^{2|1})$ is explicitly obtained.

Drinfeld Yangian of a queer Lie superalgebra is defined as the quantization of a Lie bisuperelgebra of twisted polynomial currents. An analogue of the new system of generators of Drinfeld is being constructed. It is proved for the partial case of Lie superalgebra $$sq_{1}$$ that this so defined Yangian and the Yangian, introduced earlier by M. Nazarov using the Faddeev–Reshetikhin–Takhtadzhjan approach, are isomorphic.

For each of the exceptional (not entering infinite series) finite-dimensional modular Lie superalgebras with indecomposable Cartan matrix, we give the explicit list of its roots, and the corresponding Chevalley basis, for one of its inequivalent Cartan matrices, namely the one corresponding to the greatest number of mutually orthogonal isotropic odd simple roots (this number, called the defect of the Lie superalgebra, is important in the representation theory). Our main tools: Grozman’s Mathematica-based code SuperLie, Python, and A. Lebedev’s help.

We study non-standard Verma type modules over the Kac-Moody queer Lie superalgebra 𝔮(n)(2). We give a sufficient condition under which such modules are irreducible. We also give a classification of all irreducible diagonal ℤ-graded modules over certain Heisenberg Lie superalgebras contained in 𝔮(n)(2).

We investigate projective modules of the reduced enveloping superalgebra $ U_{\chi }(\frak {g}) $ of the Lie superalgebra $ \frak {g}$ of Witt type with p-character χ of height at most one. We give a formula for the Cartan invariants of $ U_{\chi }(\frak {g}) $ which implies that $ U_{\chi }(\frak {g}) $ has only one block.

We study the truncation functors and show the existence of projective cover with a finite Verma flag of each irreducible module in parabolic BGG category O over infinite rank Lie algebra of types a , b , c , d . Moreover, O is a Koszul category. As a consequence, the corresponding parabolic BGG category O ‾ over infinite rank Lie superalgebra of types a , b , c , d through the super duality is also a Koszul category.

The Bannai–Ito algebra BI(n) is viewed as the centralizer of the action of $$\mathfrak {osp}(1|2)$$ in the n-fold tensor product of the universal algebra of this Lie superalgebra. The generators of this centralizer are constructed with the help of the universal R-matrix of $$\mathfrak {osp}(1|2)$$. The specific structure of the $$\mathfrak {osp}(1|2)$$ embeddings to which the centralizing elements are attached as Casimir elements is explained. With the generators defined, the structure relations of BI(n) are derived from those of BI(3) by repeated action of the coproduct and using properties of the R-matrix and of the generators of the symmetric group $${\mathfrak {S}}_n$$.

We compute the first differential cohomology of the affine Lie superalgebra $$\mathfrak {aff}(n|1)$$ with coefficients in the superspace of linear differential operators acting on the space of weighted densities on the (1, n)-dimensional real superspace. We also compute the same, but $$\mathfrak {aff}(n-1|1)$$-relative cohomology. We explicitly give 1-cocycles spanning these cohomology groups.

Let F be an algebraically closed field of prime characteristic, and W(m, n, 1) be the simple restricted Lie superalgebra of Witt type over F, which is the Lie superalgebra of superderivations of the superalgebra $$\mathfrak{A}(m;1)\otimes\wedge(n)$$, where $$\mathfrak{A}(m;1)$$ is the truncated polynomial algebra with m indeterminants and ∧(n) is the Grassmann algebra with n indeterminants. In this paper, the author determines the character formulas for a class of simple restricted modules of W(m, n, 1) with atypical weights of type I.

Given a Lie superalgebra and an even ad-nilpotent element of index ≤3, one can obtain a Jordan superalgebra attached to that element; inspired by that construction we build a Jordan superpair attac...