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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.

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...

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 $$ {\left\{{D}_{\leftthreetimes}\right\}}_{\leftthreetimes \in \mathcal{T}} $$ .

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 $\text {Aut}(\mathfrak {g})/\text {Int}(\mathfrak {g}) \cong \text {Aut}(\mathrm {D})$ .

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 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 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.

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.

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 ...

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...

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...

Let $$(g,~[-,-],~\omega )$$ be a finite-dimensional complex $$\omega $$ -Lie superalgebra. In this paper, we introduce the notions of derivation superalgebra $${\mathrm{Der}}(g)$$ and the automorphism group $${\mathrm{Aut}}(g)$$ of $$(g,~[-,-],~\omega )$$ . We study $${\mathrm{Der}}^{\omega }(g)$$ and $${\mathrm{Aut}}^{\omega }(g)$$ , which are superalgebra of $${\mathrm{Der}}(g)$$ and subgroup of $${\mathrm{Aut}}(g)$$ , respectively. For any 3-dimensional or 4-dimensional complex $$\omega $$ -Lie superalgebra g, we explicitly calculate $${\mathrm{Der}}(g)$$ and $${\mathrm{Aut}}(g)$$ , and obtain Jordan standard forms of elements in the two sets. We also study representation theory of $$\omega $$ -Lie superalgebras and give a conclusion that all nontrivial non- $$\omega $$ -Lie 3-dimensional and 4-dimensional $$\omega $$ -Lie superalgebras are multiplicative, as well as we show that any irreducible respresentation of the 4-dimensional $$\omega $$ -Lie superalgebra $$P_{2,k}(k\ne 0,-1)$$ is 1-dimensional.

In this paper, we introduce the notions of non-abelian tensor and exterior products of two ideal graded crossed submodules of a given crossed module of Lie superalgebras. We also study some of their basic properties and their connection with the second homology of crossed modules of Lie superalgebras.

We show that the blocks of category $$\mathcal {O}$$ for the Lie superalgebra $${\mathfrak {q}}_n({\mathbb {C}})$$ associated to half-integral weights carry the structure of a tensor product categorification for the infinite rank Kac-Moody algebra of type $$\hbox {C}_\infty $$ . This allows us to prove two conjectures formulated by Cheng, Kwon and Wang. We then focus on the full subcategory consisting of finite-dimensional representations, which we show is a highest weight category with blocks that are Morita equivalent to certain generalized Khovanov arc algebras.

Inspired by the work of Le Bruyn and Smith in (Proc. Amer. Math. Soc. 118(3): 725–730, 1993) and the work of Shelton and Vancliff in (Comm. Alg. 30(5): 2535-2552, 2002), we analyze certain graded algebras related to the Lie algebra $\mathfrak {sl}(2,\Bbbk)$ using geometric techniques in the spirit of Artin, Tate and Van den Bergh. In particular, we discuss the point schemes and line schemes of certain quadratic quantum $\mathbb {P}^{3}$ s associated to the Lie superalgebra $\mathfrak {sl}(1|1)$ , to a quantized enveloping algebra, $\mathcal {U}_q(\mathfrak {sl}(2,\Bbbk))$ , of $\mathfrak {sl}(2,\Bbbk)$ , and to a color Lie algebra $\mathfrak {sl}_k(2,\Bbbk)$ , respectively. The geometry we consider identifies certain normal elements in the universal enveloping algebra of $\mathfrak {sl}(1|1)$ and in $\mathcal {U}_q(\mathfrak {sl}(2,\Bbbk))$ .

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 s l 2 ( C ) \mathfrak {sl}_2(\mathbb {C}) , there is a correspondence between Verma modules and certain line modules that associates a pair ( h , ϕ ) (\mathfrak {h},\,\phi ) , where h \mathfrak {h} is a 2 2 -dimensional Lie subalgebra of s l 2 ( C ) \mathfrak {sl}_2(\mathbb {C}) and ϕ ∈ h ∗ \phi \in \mathfrak {h}^* satisfies ϕ ( [ h , h ] ) = 0 \phi ([\mathfrak {h}, \, \mathfrak {h}]) = 0 , to a particular type of line module. In this article, we prove analogous results for the Lie superalgebra s l ( 1 | 1 ) \mathfrak {sl}(1|1) and for a color Lie algebra associated to the Lie algebra s l 2 \mathfrak {sl}_2 .

We construct HNN-extensions of Lie superalgebras and prove that every Lie superalgebra embeds into any of its HNN-extensions. Then as an application we show that any Lie superalgebra with at most countable dimension embeds into a two-generator Lie superalgebra.

We give a diagrammatic calculus for the intertwiners between permutation supermodules of the Sergeev superalgebra over the complex numbers. We also give a diagrammatic basis for the space o...

We compute the Lie superalgebras of holomorphic vector fields on isotropic flag supermanifolds of maximal type corresponding to the Lie superalgebra osp2m−1|2n(C).

Let W m | n be the (finite) W-algebra attached to the principal nilpotent orbit in the general linear Lie superalgebra gl m | n ( C ) . In this paper we study the Whittaker coinvariants functor, which is an exact functor from category O for gl m | n ( C ) to a certain category of finite-dimensional modules over W m | n . We show that this functor has properties similar to Soergel's functor V in the setting of category O for a semisimple Lie algebra. We also use it to compute the center of W m | n explicitly, and deduce consequences for the classification of blocks of O up to Morita/derived equivalence.

Let W denote the generalized Witt Lie superalgebra W(m,n;t_) over a field of characteristic p>2. Utilizing the abelian subalgebra T and the weight space decomposition of W with respect to T, we sho...

Crystal basis theory for the queer Lie superalgebra was developed by Grantcharov et al. and it was shown that semistandard decomposition tableaux admit the structure of crystals for the queer Lie superalgebra or simply $\mathfrak{q}$-crystal structure. In this paper, we explore the $\mathfrak{q}$-crystal structure of primed tableaux (semistandard marked shifted tableaux) and that of signed unimodal factorizations of reduced words of type $B$. We give the explicit odd Kashiwara operators on primed tableaux and the forms of the highest and lowest weight vectors. We also give the explicit algorithms for odd Kashiwara operators on signed unimodal factorizations of reduced words of type $B$., 25 pages; Typos corrected

To an arbitrary Lie superalgebra L we associate its Jordan double \({\mathcal Jor}(L)\), which is a Jordan superalgebra. This notion was introduced by the second author before (Shestakov in Sib Adv Math 9(2):83–99, 1999). Now we study further applications of this construction. First, we show that the Gelfand–Kirillov dimension of a Jordan superalgebra can be an arbitrary number \(\{0\}\cup [1,+\,\infty ]\). Thus, unlike the associative and Jordan algebras (Krause and Lenagan in Growth of algebras and Gelfand–Kirillov dimension, AMS, Providence, 2000; Martinez and Zelmanov in J Algebra 180(1):211–238, 1996), one hasn’t an analogue of Bergman’s gap (1, 2) for the Gelfand–Kirillov dimension of Jordan superalgebras. Second, using the Lie superalgebra \({\mathbf {R}}\) of de Morais Costa and Petrogradsky (J Algebra 504:291–335, 2018), we construct a Jordan superalgebra \({\mathbf {J}}={\mathcal Jor}({{\mathbf {R}}})\) that is nil finely \({\mathbb {Z}}^3\)-graded (moreover, the components are at most one-dimensional), the field being of characteristic not 2. This example is in contrast with non-existence of such examples (roughly speaking, analogues of the Grigorchuk and Gupta–Sidki groups) of Lie algebras in characteristic zero (Martinez and Zelmanov in Adv Math 147(2):328–344, 1999) and Jordan algebras in characteristic not 2 (Zelmanov, E., A private communication). Also, \({\mathbf {J}}\) is just infinite but not hereditary just infinite. A similar Jordan superalgebra of slow polynomial growth was constructed before Petrogradsky and Shestakov (Fractal nil graded Lie, associative, poisson, and Jordan superalgebras. arXiv:1804.08441, 2018). The virtue of the present example is that it is of linear growth, of finite width 4, namely, its \(\mathbb N\)-gradation by degree in the generators has components of dimensions \(\{0,2,3,4\}\), and the sequence of these dimensions is non-periodic. Third, we review constructions of Poisson and Jordan superalgebras of Petrogradsky and Shestakov (2018) starting with another example of a Lie superalgebra introduced in Petrogradsky (J Algebra 466:229–283, 2016). We discuss the notion of self-similarity for Lie, associative, Poisson, and Jordan superalgebras. We also suggest the notion of a wreath product in case of Jordan superalgebras.

By using the Lie superalgebra B(0, n ), we give a Cartan matrix of affine Lie superalgebra B(0, n)(1), and the relations of their generators. We also give an example of super AKNS hierarchy associated with B(0, n). As an application of affine Lie superalgebra B(0, n)(1), we obtain two kinds of Lie-superalgebraic interpretations of super AKNS hierarchy. We also get conservation laws associated with super AKNS hierarchy.

Using the approach of Gautam and Toledano Laredo, we construct an explicit isomorphism of the Yangian Yħ(A(m, n)) of the special linear Lie superalgebra and the quantum loop superalgebra Uħ(LA(m, n)).

Suppose the ground field $\mathbb{F}$ is an algebraically closed field of characteristic different from 2, 3. We determine the Betti numbers and make a decomposition of the associative superalgebra of the cohomology for the model filiform Lie superalgebra. We also describe the associative superalgebra structures of the (divided power) cohomology for some low-dimensional filiform Lie superalgebras., Comment: 30 pages