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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.

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.

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 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.

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.

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.

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.

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.

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

We prove that the Duflo-Serganova functor D S x attached to an odd nilpotent element x of osp ( m | 2 n ) is semisimple, i.e. sends a semisimple representation M of osp ( m | 2 n ) to a semisimple representation of osp ( m − 2 k | 2 n − 2 k ) where k is the rank of x. We prove a closed formula for D S x ( L ( λ ) ) in terms of the arc diagram attached to λ.

Let OSp ( V ) be the orthosymplectic supergroup on an orthosymplectic vector superspace V of superdimension ( m | 2 n ) . Lehrer and Zhang showed that there is a surjective algebra homomorphism F r r : B r ( m − 2 n ) → End OSp ( V ) ( V ⊗ r ) , where B r ( m − 2 n ) is the Brauer algebra of degree r with parameter m − 2 n . The second fundamental theorem of invariant theory in this setting seeks to describe the kernel Ker F r r of F r r as a 2-sided ideal of B r ( m − 2 n ) . In this paper, we show that Ker F r r ≠ 0 if and only if r ≥ r c : = ( m + 1 ) ( n + 1 ) , and give a basis and a dimension formula for Ker F r r . We show that Ker F r r as a 2-sided ideal of B r ( m − 2 n ) is generated by Ker F r c r c for any r ≥ r c , and we provide an explicit set of generators for Ker F r c r c . These generators coincide in the classical case with those obtained in recent papers of Lehrer and Zhang on the second fundamental theorem of invariant theory for the orthogonal and symplectic groups. As an application we obtain the necessary and sufficient conditions for the endomorphism algebra End osp ( V ) ( V ⊗ r ) over the orthosymplectic Lie superalgebra osp ( V ) to be isomorphic to B r ( m − 2 n ) .

We give an elementary and explicit proof of the first fundamental theorem of invariant theory for the orthosymplectic supergroup by generalising the geometric method of Atiyah, Bott and Patodi to the supergroup context. We use methods from super-algebraic geometry to convert invariants of the orthosymplectic supergroup into invariants of the corresponding general linear supergroup on a different space. In this way, super Schur–Weyl–Brauer duality is established between the orthosymplectic supergroup of superdimension (m|2n) and the Brauer algebra with parameter m − 2n. The result may be interpreted either in terms of the group scheme OSp(V) over \({{\mathbb C}}\), where V is a finite dimensional super space, or as a statement about the orthosymplectic Lie supergroup over the infinite dimensional Grassmann algebra \({\Lambda}\). We take the latter point of view here, and also state a corresponding theorem for the orthosymplectic Lie superalgebra, which involves an extra invariant generator, the super-Pfaffian.

We consider the 1 | 2 -dimensional real superspace R 1 | 2 endowed with its standard contact structure defined by the 1-form α . The conformal Lie superalgebra K ( 2 ) acts on R 1 | 2 as the Lie superalgebra of contact vector fields; it contains the Mobius superalgebra o s p ( 2 | 2 ) . We classify o s p ( 2 | 2 ) -invariant superskew-symmetric binary differential operators from K ( 2 ) ∧ K ( 2 ) to D λ , μ vanishing on o s p ( 2 | 2 ) , where D λ , μ is the superspace of linear differential operators between the superspaces of weighted densities. This result allows us to compute the second differential o s p ( 2 | 2 ) -relative cohomology of K ( 2 ) with coefficients in D λ , μ . We study generic formal o s p ( 2 | 2 ) -trivial deformations of the K ( 2 ) -module structure on the direct sum of the superspaces of weighted densities. This work is the simplest superization of a result by Bouarroudj (2007).

We prove a determinant formula for a parabolic Verma module of a 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., 24 pages

In this paper, we used the free fields of Wakimoto to construct a class of irreducible representations for the general linear Lie superalgebra gl m | n ( C ) . The structures of the representations over the general linear Lie superalgebra and the special linear Lie superalgebra are studied in this paper. Then we extend the construction to the affine Kac–Moody Lie superalgebra gl m | n ˆ ( C ) on the tensor product of a polynomial algebra and an exterior algebra with infinitely many variables involving one parameter μ , and we also obtain the necessary and sufficient condition for the representations to be irreducible. In fact, the representation is irreducible if and only if the parameter μ is nonzero.

In this paper, the notion of unitary vertex operator superalgebra is introduced. It is proved that the vertex operator superalgebras associated to the unitary highest weight representations for the Neveu-Schwarz Lie superalgebra, Heisenberg superalgebras and to positive definite integral lattices are unitary vertex operator superalgebras. The unitary structures are then used to study the structures of vertex operator superalgebras, it is proved that any unitary vertex operator superalgebra is a direct sum of strong CFT type unitary simple vertex operator superalgebras. The classification of unitary vertex operator superalgebras generated by the subspaces with conformal weights less than or equal to $1$ is also considered., Comment: 28 pages

In this paper the authors introduce a class of parabolic subalgebras for classical simple Lie superalgebras associated to the detecting subalgebras introduced by Boe, Kujawa and Nakano. These parabolic subalgebras are shown to have good cohomological properties governed by the Bott-Borel-Weil theorem involving the zero component of the Lie superalgebra in conjunction with the odd roots. These results are later used to verify an open conjecture given by Boe, Kujawa and Nakano pertaining to the equality of various support varieties.

We derive the solutions of the boundary Yang–Baxter equation associated with a supersymmetric nineteen vertex model constructed from the three-dimensional representation of the twisted quantum affine Lie superalgebra U q [ osp ( 2 | 2 ) ( 2 ) ] . We found three classes of solutions. The type I solution is characterized by three boundary free-parameters and all elements of the corresponding reflection K-matrix are different from zero. In the type II solution, the reflection K-matrix is even (every element of the K-matrix with an odd parity is null) and it has only one boundary free-parameter. Finally, the type III solution corresponds to a diagonal reflection K-matrix with two boundary free-parameters.

We show that every block of category OO for the general linear Lie superalgebra glm|n(k)glm|n(k) is equivalent to some corresponding block of category OO for the queer Lie superalgebra qm+n(k)qm+n(k). This implies the truth of the Kazhdan–Lusztig conjecture for the so-called type A blocks of category OO for the queer Lie superalgebra as formulated by Cheng, Kwon and Wang.

Starting from a 3 × 3 matrix-valued spectral problem associated with a Lie superalgebra sl(2|1), a generalized super Ablowitz–Kaup–Newell–Segur (AKNS) hierarchy is derived. The resulting super AKNS hierarchy has a super bi-Hamiltonian structure by the supertrace identity.

We establish a maximal parabolic version of the Kazhdan-Lusztig conjecture \cite[Conjecture 5.10]{CKW} for the BGG category $\mathcal{O}_{k,\zeta}$ of $\mathfrak{q}(n)$-modules of "$\pm \zeta$-weights", where $k\leq n$ and $\zeta\in\mathbb{C} \setminus \frac{1}{2} \mathbb{Z}$. As a consequence, the irreducible characters of these $\mathfrak{q}(n)$-modules in this maximal parabolic category are given by the Kazhdan-Lusztig polynomials of type $A$ Lie algebras. As an application, closed character formulas for a class of $\mathfrak{q}(n)$-modules resembling polynomial and Kostant modules of the general linear Lie superalgebras are obtained., Comment: 26 pages

Let F be an algebraically closed field of prime characteristic p > 2 , and n ∈ N + . Let Z ( n ) be the Zassenhaus superalgebra defined over F , which, as the simplest non-restricted simple Lie superalgebra, is the superversion of the Zassenhaus algebra. More precisely, Z ( n ) is the Lie superalgebra of the special super-derivations of the superalgebra Π ( n ) . Here Π ( n ) is the tensor product of the divided power algebra of one variable and the Grassmann superalgebra of one variable. In this paper we study generalized restricted simple modules over the Zassenhaus superalgebra Z ( n ) . Classification of isomorphism classes of generalized restricted simple modules and their dimensions are precisely determined. A sufficient and necessary condition for irreducibility of generalized restricted Kac modules is provided.

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.

The “Capelli problem” for the symmetric pairs ( gl × gl , gl ) , ( gl , o ) , and ( gl , sp ) is closely related to the theory of Jack polynomials and shifted Jack polynomials for special values of the parameter (see [12] , [15] , [14] , [18] ). In this paper, we extend this connection to the Lie superalgebra setting, namely to the supersymmetric pairs ( g , k ) : = ( gl ( m | 2 n ) , osp ( m | 2 n ) ) and ( gl ( m | n ) × gl ( m | n ) , gl ( m | n ) ) , acting on W : = S 2 ( C m | 2 n ) and C m | n ⊗ ( C m | n ) ⁎ . To achieve this goal, we first prove that the center of the universal enveloping algebra of the Lie superalgebra g maps surjectively onto the algebra PD ( W ) g of g -invariant differential operators on the superspace W, thereby providing an affirmative answer to the “abstract” Capelli problem for W. Our proof works more generally for gl ( m | n ) acting on S 2 ( C m | n ) and is new even for the “ordinary” cases ( m = 0 or n = 0 ) considered by Howe and Umeda in [9] . We next describe a natural basis { D λ } of PD ( W ) g , that we call the Capelli basis. Using the above result on the abstract Capelli problem, we generalize the work of Kostant and Sahi [12] , [15] , [20] by showing that the spectrum of D λ is given by a polynomial c λ , which is characterized uniquely by certain vanishing and symmetry properties. We further show that the top homogeneous parts of the eigenvalue polynomials c λ coincide with the spherical polynomials d λ , which arise as radial parts of k -spherical vectors of finite dimensional g -modules, and which are super-analogues of Jack polynomials. This generalizes results of Knop and Sahi [14] . Finally, we make a precise connection between the polynomials c λ and the shifted super Jack polynomials of Sergeev and Veselov [25] for special values of the parameter. We show that the two families are related by a change of coordinates that we call the “Frobenius transform”.

The present work is regarding filiform Lie superalgebras which is an important type of nilpotent Lie superalgebras. In general, classifying nilpotent Lie superalgebras is at present an open and unsolved problem. Throughout the present work we contribute to the resolution of this wide problem by classifying filiform Lie superalgebras of low dimensions, in particular less or equal to 7 . Furthermore we would establish a method that could be applied to obtain similar results for higher dimensions. Thus, this method would mainly consist in using infinitesimal deformations of the model filiform Lie superalgebra.

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.

We revisit the problem of consistent free propagation of higher-spin fields in nontrivial backgrounds, focusing on symmetric tensor(-spinor)s. The Fierz-Pauli equations for massive fields in flat space form an involutive system, whose algebraic consistency owes to certain gauge identities. The zero mass limit of the former leads directly to massless higher-spin equations in the transverse-traceless gauge, where both the field and the gauge parameter have their respective involutive systems and gauge identities. In nontrivial backgrounds, it is the preservation of these gauge identities and symmetries that ensures the correct number of propagating degrees of freedom. With this approach we find consistent sets of equations for massive and massless higher-spin bosons and fermions in certain gravitational/electromagnetic backgrounds. We also present the involutive system of partially massless fields, and give an explicit form of their gauge transformations. We consider the Lie superalgebra of the operators on symmetric tensor(-spinor)s in flat space, and show that in AdS space the algebra closes nonlinearly and requires a central extension., 62 pages, references added; to appear in Nuclear Physics B

In this paper, we use the quantization technique by a Drinfel’d twist to quantize the Lie superalgebra of Witt type, whose Lie super-bialgebra structures were recently determined by Yue and Zhu, and we obtain a new class of non-commutative and non-cocommutative Hopf superalgebras.

Let $e$ be an arbitrary even nilpotent element in the general linear Lie superalgebra $\mathfrak{gl}_{M|N}$ and let $\mathcal{W}_e$ be the associated finite $W$-superalgebra. Let $Y_{m|n}$ be the super Yangian associated to the Lie superalgebra $\mathfrak{gl}_{m|n}$. A subalgebra of $Y_{m|n}$, called the shifted super Yangian and denoted by $Y_{m|n}(\sigma)$, is defined and studied. Moreover, an explicit isomorphism between $\mathcal{W}_e$ and a quotient of $Y_{m|n}(\sigma)$ is established., Comment: 56 pages, version 2, comments welcome. Minor revision, added one section and fixed some typos

We realize the simple Lie superalgebra G ( 3 ) as supersymmetry of various geometric structures, most importantly super-versions of the Hilbert–Cartan equation (SHC) and Cartan's involutive PDE system that exhibit G ( 2 ) symmetry. We provide the symmetries explicitly and compute, via the first Spencer cohomology groups, the Tanaka–Weisfeiler prolongation of the negatively graded Lie superalgebras associated with two particular choices of parabolics. We discuss non-holonomic superdistributions with growth vector ( 2 | 4 , 1 | 2 , 2 | 0 ) deforming the flat model SHC, and prove that the second Spencer cohomology group gives a binary quadratic form, thereby indicating a “square-root” of Cartan's classical binary quartic invariant for generic rank 2 distributions in a 5-dimensional space. Finally, we obtain super-extensions of Cartan's classical submaximally symmetric models, compute their symmetries and observe a supersymmetry dimension gap phenomenon.

Over a field of characteristic p > 2 , the first cohomology of the special linear Lie superalgebra sl ( 2 , 1 ) with coefficients in all χ-reduced Kac modules and simple modules is determined by use of the weight space decompositions of these modules relative to the standard Cartan subalgebra of sl ( 2 , 1 ) .

In this paper, by defining a new vocabulary for Jacobi theta functions, namely “spectral theta functions”, we show that how remarkably the N = 2 superconformal characters are extracted by imposing a residue calculation on s l ( 2 | 1 ) affine Lie superalgebra characters in their non-unitary (non-trivial) cases. Therefore, to have more versatility in conformal quantum field theory context, we finalize our study by a complete prescription by which, all contents of well-known s l ( 2 | 1 ) characters could be uniquely translated to N = 2 superconformal characters. In this regard, we firstly establish the key spectral theta function’s formula as well as deriving quasi-periodic and differentiation formulae of spectral theta functions. Then, both the mentioned algebraic characters in their non-unitary cases are studied to explore a relation between these algebras. Finally we come up with our residue calculation method to obtain a total reduction from s l ( 2 | 1 ) characters to N = 2 characters, as a novel and applicable idea in mathematical physics.

Two descriptions of the dual −1 Hahn algebra are presented and shown to be related under Howe duality. The dual pair involved is formed by the Lie algebra o ( 4 ) and the Lie superalgebra osp ( 1 | 2 ) .

We show that the normalized supercharacters of principal admissible modules, associated to each integrable atypical module over the affine Lie superalgebra s l ˆ 2 | 1 can be modified, using Zwegers' real analytic corrections, to form an SL 2 ( Z ) -invariant family of functions. Using a variation of Zwegers' correction, we obtain a similar result for osp ˆ 3 | 2 . Applying the quantum Hamiltonian reduction, this leads to new families of positive energy modules over the N = 2 (resp. N = 3 ) superconformal algebras with central charge c = 3 ( 1 − 2 m + 2 M ) , where m ∈ Z ≥ 0 , M ∈ Z ≥ 2 , gcd ⁡ ( 2 m + 2 , M ) = 1 if m > 0 (resp. c = − 3 2 m + 1 M , where m ∈ Z ≥ 0 , M ∈ Z ≥ 2 gcd ⁡ ( 4 m + 2 , M ) = 1 ), whose modified supercharacters form an SL 2 ( Z ) -invariant family of functions.

In this paper we study finite W-algebras for basic classical superalgebras and Q(n) associated to the regular even nilpotent coadjoint orbits. We prove that this algebra satisfies the Amitsur-Levitzki identity and therefore all its irreducible representations are finite-dimensional. In the case of Q(n) we give an explicit description of the W-algebra in terms of generators and relation and realize it as a quotient of the super-Yangian of Q(1)., 37 pages

We investigate the second cohomology space associated with the embedding of the orthosymplectic Lie superalgebra o s p ( n | 2 ) on the ( 1 , n ) -dimensional superspace R 1 | n in the Lie superalgebra S Ψ D O ( n ) of superpseudodifferential operators with smooth coefficients, where n = 0 , 1 , 2 . We show that this space is purely even and we give explicit expressions of the basis cocycles.

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 ) .

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 Xu≠1 and (Hev)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 Ui, and H˜/U is a triangulizable supergroup with odd unipotent radical. Moreover, for every i, Lie(Ui)=Ui⊗Sym(ni) are such that either ni=0 and Ui is a classical simple Lie superalgebra, or ni=1 and Ui is a simple Lie algebra.

The interaction of a Lie algebra L , having a weight space decomposition with respect to a nonzero toral subalgebra, with its corresponding root system forms a powerful tool in the study of the structure of L . This, in particular, suggests a systematic study of the root system apart from its connection with the Lie algebra. Although there have been a lot of researches in this regard on Lie algebra level, such an approach has not been considered on Lie superalgebra level. In this work, we introduce and study extended affine root supersystems which are a generalization of both affine reflection systems and locally finite root supersystems. Extended affine root supersystems appear as the root systems of the super version of extended affine Lie algebras and invariant affine reflection algebras including affine Lie superalgebras. This work provides a framework to study the structure of this kind of Lie superalgebras refereed to as extended affine Lie superalgebras.

We introduce a new Lie algebra of infinite N × N matrices which have nonzero entries only in finite number of rows. We show that its subalgebra of matrices with trace 0 is uncountably dimensional simple Lie algebra.

We construct an action of the Lie superalgebras glm|nˆ(Cq) on the tensor product of the polynomial algebra and the exterior algebra with infinitely many variables. This construction is involving one parameter μ and we show that it is 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 A(m−1,n−1).

We identify the universal central extension of g = A ⊗ k , where k is a finite dimensional perfect Lie superalgebra equipped with a nondegenerate homogeneous invariant supersymmetric bilinear form κ which is invariant under all derivations and A is a unital supercommutative associative (super)algebra.