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Let U be an associative unital algebra containing a non-trivial idempotent e. We consider U as a superalgebra whose Z 2 -grading is induced by e. This paper aims to describe Lie superderivations of U . In particular, we characterize the general form of Lie superderivations of U and apply it to present the necessary and sufficient conditions for a Lie superderivation on U to be proper. Similar results have been presented for triangular algebras as superalgebras, wherein their Z 2 -grading is also obtained concerning standard idempotent. The main result is subsequently applied to full matrix algebras and upper triangular matrix algebras. [ABSTRACT FROM AUTHOR]

We exhibit a class C of finite dimensional algebras with superinvolution over an algebraically closed field of characteristic zero, with the remarkable property that each member of C generates a minimal variety of algebras with superinvolution. This sums up to the fact that any affine minimal variety of algebras with superinvolution is generated by a suitable member of C , thus providing a complete characterization of the affine minimal varieties of algebras with superinvolution. [ABSTRACT FROM AUTHOR]

Let A be an associative superalgebra endowed with a pseudoautomorphism p. In this paper we generalize the Wedderburn-Malcev Theorem in this setting and we prove that the sequence of p-codimensions of A is polynomially bounded if and only if the variety generated by A does not contain the group algebra of Z 2 and the algebra of 2 × 2 upper-triangular matrices with suitable pseudoautomorphisms. [ABSTRACT FROM AUTHOR]

Let M 1 , 2 (F) be the algebra of 3 × 3 matrices with orthosymplectic superinvolution ⁎ over a field F of characteristic zero. We study the ⁎-identities of this algebra through the representation theory of the group H n = (Z 2 × Z 2) ∼ S n. We decompose the space of multilinear ⁎-identities of degree n into the sum of irreducibles under the H n -action in order to study the irreducible characters appearing in this decomposition with non-zero multiplicity. Moreover, by using the representation theory of the general linear group, we determine all the ⁎-polynomial identities of M 1 , 2 (F) up to degree 3. [ABSTRACT FROM AUTHOR]

In this paper, we define the affine super Yangian ... ith a coproduct structure. We also obtain an evaluation homomorphism, that is, an algebra homomorphism from ... to the completion of the universal enveloping algebra of ... [ABSTRACT FROM AUTHOR]

Yu. A. Medvedev [Algebra and Logic, 19, No. 3, 191—201 (1980)] constructed an example of alternative algebra that he used to prove that a certain variety of alternative algebras possess the non-Specht property over a field of characteristic 2. Though his result concerned the characteristic 2 case, the example was claimed to be alternative over an arbitrary field, and it was later used by V. T. Filippov in a series of papers. Unfortunately, Medvedev's example is in fact not alternative in any characteristic. Therefore, whether the variety considered by Medvedev has the non-Specht property is still not clear. Moreover, the results of Filippov's papers, in which Medvedev's example was used, also become questionable. We construct new examples and employ them to prove that the results of Filippov remain true. [ABSTRACT FROM AUTHOR]

To any superalgebra A is attached a numerical sequence c n s u p (A) , n ≥ 1 , called the sequence of supercodimensions of A. In characteristic zero its asymptotics are an invariant of the superidentities satisfied by A. It is well-known that for a PI-superalgebra such sequence is exponentially bounded and e x p s u p (A) = lim n → ∞ ⁡ c n s u p (A) n is an integer that can be explicitly computed. Here we introduce a notion of fundamental superalgebra over a field of characteristic zero. We prove that if A is such an algebra, then (1) C 1 n t e x p s u p (A) n ≤ c n s u p (A) ≤ C 2 n t e x p s u p (A) n , where C 1 > 0 , C 2 , t are constants and t is a half integer that can be explicitly written as a linear function of the dimension of the even part of A and the nilpotency index of the Jacobson radical. We also give a characterization of fundamental superalgebras through the representation theory of the symmetric group. As a consequence we prove that if A is a finitely generated PI-superalgebra, then the inequalities in (1) still hold and t is a half integer. It follows that lim n → ∞ ⁡ log n ⁡ c n s u p (A) e x p s u p (A) n exists and is a half integer. [ABSTRACT FROM AUTHOR]

We construct the super Koszul complex of a free supercommutative A-module V of rank p|q and prove that its homology is concentrated in a single degree and it yields an exact resolution of A. We then study the dual of the super Koszul complex and show that its homology is concentrated in a single degree as well and isomorphic to Π p + q A , with Π the parity changing functor. Finally, we show that, given an automorphism of V, the induced transformation on the only non-trivial homology class of the dual of the super Koszul complex is given by the multiplication by the Berezinian of the automorphism, thus relating this homology group with the Berezinian module of V. [ABSTRACT FROM AUTHOR]

We consider supersymmetry in five dimensions, where the fermionic parameters are a 2‐form under SL(5). Supermultiplets are investigated using the pure spinor superfield formalism, and are found to be closely related to infinite‐dimensional extensions of the supersymmetry algebra: the Borcherds superalgebra B(E4), the tensor hierarchy algebra S(E4) and the exceptional superalgebra E(5, 10). A theorem relating B(E4) and E(5, 10) to all levels is given. In this article supersymmetry in five dimensions is considered, where the fermionic parameters are a 2‐form under SL(5). Supermultiplets are investigated using the pure spinor superfield formalism, and are found to be closely related to infinite‐dimensional extensions of the supersymmetry algebra: the Borcherds superalgebra B(E4), the tensor hierarchy algebra S(E4) and the exceptional superalgebra E(5, 10). A theorem relating B(E4) and E(5, 10) to all levels is given. [ABSTRACT FROM AUTHOR]

We give a new proof of the fact that the Euler supercharacters of the Lie superalgebra osp(2m + 1, 2n) can be obtained as a certain limit of the super Jacobi polynomials. The known proof was not direct one and it was mostly based on calculations. In this paper we propose more simple and more conceptional proof. The main idea is to use the Pieri formulae from the beginning. It turns out that the super Jacobi polynomials and their specialisations can be uniquely characterised by two properties. The first one is that they are eigenfunctions of CMS operator and the second one is that they satisfy the Pieri formulae. As by product we get some interesting identities involving a Young diagram and rational functions. We hope that our approach can be useful in many similar cases.

We develop a three‐dimensional N=4 theory of rigid supersymmetry describing the dynamics of a set of hypermultiplets (ΛIαα′α̇′,ϕIαA) on a curved AdS3 worldvolume background, whose supersymmetry is captured by the supergroup D2(2,1;α). To unveil some remarkable features of this model, we perform two twists, involving the SL(2,R) factors of the theory. After the first twist, our spacetime Lagrangian exhibits a Chern‐Simons term associated with an odd one‐form field, together with a fermionic "gauge‐fixing", in the spirit of the Rozansky‐Witten model. The second twist allows to interpret the D2(2,1;α) setup as a framework capable of describing massive Dirac particles. In particular, this yields a generalisation of the Alvarez‐Valenzuela‐Zanelli model of "unconventional supersymmetry". We comment on specific values of the combination α+1, which in our model is related to a sort of gauging in the absence of dynamical gauge fields. The authors develop a three‐dimensional N = 4 theory of rigid supersymmetry describing the dynamics of a set of hypermultiplets on a curved AdS3 worldvolume background, whose supersymmetry is captured by the supergroup D2(2, 1; α). To unveil some remarkable features of this model, two twists are performed, involving the SL(2,R) factors of the theory. After the first twist, the spacetime Lagrangian used exhibits a Chern‐Simons term associated with an odd one‐form field, together with a fermionic "gauge‐fixing", in the spirit of the Rozansky‐Witten model. The second twist allows to interpret the D2(2, 1; α) setup as a framework capable of describing massive Dirac particles. In particular, this yields a generalisation of the Alvarez‐Valenzuela‐Zanelli model of "unconventional supersymmetry". The authors comment on specific values of the combination α + 1, which in the model presented here is related to a sort of gauging in the absence of dynamical gauge fields. [ABSTRACT FROM AUTHOR]

Let V be a variety of superalgebras with graded involution and let c n gri (V) be its sequence of ⁎-graded codimensions. We say that V has polynomial growth n k if asymptotically c n gri (V) ≈ a n k , for some a ≠ 0. Furthermore, V is minimal of polynomial growth n k if c n gri (V) grows as n k and any proper subvariety of V has polynomial growth n t , with t < k. In this paper, we classify superalgebras with graded involution generating minimal varieties of quadratic growth. [ABSTRACT FROM AUTHOR]

Let F be a field of characteristic zero. By a φ -algebra we mean a superalgebra or an algebra with involution over F. In the last years, the sequence of φ -codimensions { c n φ (A) } n ≥ 1 of a φ -algebra A has been extensively studied. In this paper, we classify varieties generated by unitary φ -algebras having quadratic growth of φ -codimensions. As a consequence we obtain that a unitary φ -algebra with quadratic growth is T φ -equivalent to a finite direct sum of minimal unitary φ -algebras with at most quadratic growth of the φ -codimensions. In addition, we explicit all quadratic functions describing the φ -codimension sequence of a unitary φ -algebra. [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]

In the paper, we study Lie n superderivations and generalized Lie n superderivations of superalgebras, using the theory of functional identities in superalgebras. We prove that if A = A0 ⊕ A1 is a prime superalgebra with deg(A1) ≥ 2n + 5, n ≥ 2, then any Lie n superderivation of A is the sum of a superderivation and a linear mapping, and any generalized Lie n superderivation of A is the sum of a generalized superderivation and a linear mapping.

An analog in characteristic 2 for the Grassmann algebra G was essential in a counterexample to the long standing Specht conjecture. We define a generalization G of the Grassmann algebra, which is well-behaved over arbitrary commutative rings C, even when 2 is not invertible. This lays the foundation for a supertheory over arbitrary base ring, allowing one to consider general deformations of superalgebras. The construction is based on a generalized sign function. It enables us to provide a basis of the non-graded multilinear identities of the free superalgebra with supertrace, valid over any ring. We also show that all identities of G follow from the Grassmann identity, and explicitly give its co-modules, which turn out to be generalizations of the sign representation. In particular, we show that the nth co-module is a free C-module of rank 2n−1. [ABSTRACT FROM AUTHOR]

The coefficients of the super Jacobi polynomials of B C 1 , n -type are rational functions in three parameters k , p , q. At the point (− 1 , 0 , 0) these coefficients may have poles. Let us set q = 0 and consider a pair (k , p) as the point of A 2. If we apply blow up procedure at the point (− 1 , 0) then we get a new family of polynomials. This family depends on one parameter t ∈ P. If t = ∞ then we get Euler supercharacters for Lie supergroup O S P (2 , 2 n). The supercharacters of finite dimensional irreducible modules can be also obtained by a specialization of the parameter t. But in such a case the specialization depends on the highest weight of the corresponding irreducible module. [ABSTRACT FROM AUTHOR]

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 s l (2 , 𝕜) 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 ℙ 3 s associated to the Lie superalgebra s l (1 | 1) , to a quantized enveloping algebra, U q (s l (2 , 𝕜)) , of s l (2 , 𝕜) , and to a color Lie algebra s l k (2 , 𝕜) , respectively. The geometry we consider identifies certain normal elements in the universal enveloping algebra of s l (1 | 1) and in U q (s l (2 , 𝕜)) . [ABSTRACT FROM AUTHOR]

By a φ -variety V , we mean a supervariety or a ⁎-variety generated by an associative algebra over a field F of characteristic zero. In this case, we consider its sequence of φ -codimensions c n φ (V) and say that V is minimal of polynomial growth n k if c n φ (V) grows like n k , but any proper φ -subvariety grows like n t with t < k. In this paper, we deal with minimal φ -varieties generated by unitary algebras and prove that for k ≤ 2 there is only a finite number of them. We also explicit a list of finite dimensional algebras generating such minimal φ -varieties. For k ≥ 3 , we show that the number of minimal φ -varieties can be infinity and we classify all minimal φ -varieties of polynomial growth n k by giving a recipe for the construction of their T φ -ideals. [ABSTRACT FROM AUTHOR]

Let ℱ μ be the space of μ -densities over S 1 , we compute the space H diff 1 (𝔞 𝔣 𝔣 (1) ; 𝒟 λ ̲ , μ) where λ ̲ = (λ 1 , ... , λ n) ∈ ℂ n and 𝒟 λ ̲ , μ is the space of n -ary linear differential operators from ℱ λ 1 ⊗ ⋯ ⊗ ℱ λ n to ℱ μ . We also compute the simplest super analog spaces H diff 1 (𝔞 𝔣 𝔣 (1 | 1) ; 𝔇 λ ̲ , μ). [ABSTRACT FROM AUTHOR]

The class of generalized Lie-type color algebras contains the ones of generalized Lie-type algebras, of n-Lie algebras and superalgebras, commutative Leibniz n-ary algebras and superalgebras, among others. We focus on the class of generalized Lie-type color algebras L admitting a quasi-multiplicative basis, with restrictions neither on the dimensions nor on the base field F and study its structure. We state that if L admits a quasi-multiplicative basis then it decomposes as L = U ⊕ (∑ J k) with any J k a well described color gLt-ideal of L admitting also a quasi-multiplicative basis, and U a linear subspace of V . Also the minimality of L is characterized in terms of the connections and it is shown that the above direct sum is by means of the family of its minimal color gLt-ideals, admitting each one a μ-quasi-multiplicative basis inherited by the one of L . [ABSTRACT FROM AUTHOR]

In this paper, the third root of the Lie algebra su(2) based on the permutation group S3 is formulated in the Hopf algebra formalism. [ABSTRACT FROM AUTHOR]

We introduce the class of conservative superalgebras, in particular, the superalgebra of bilinear operations on a superspace V. Moreover, we show that each conservative superalgebra modulo its maximal Jacobian ideal is embedded into for a certain superspace V. [ABSTRACT FROM AUTHOR]

In this paper we investigate the functional identity in a prime associative superalgebras. We prove the following result. Suppose that there exists a nonzero additive mapping f = f0 + f1, on a prime associative superalgebra with char(R) ≠ 2, satisfying the relation [f (x), y2] = 0 for all x, y ∈ ℋ(). If is prime algebra then [f (), ] = 0 or [ , ] = 0. 0 is prime algebra then [f (), ] = 0 and [ , 0] = 0 or A is trivial. More- over, if C1 = 0 then f0(1) = 0 and f1(0) = 0. [ABSTRACT FROM AUTHOR]

Over the (1 , 2) -dimensional supercircle S 1 | 2 , we investigate the first 𝔬 𝔰 𝔭 (2 | 2) -relative cohomology space associated with the embedding of the Lie superalgebra 𝒦 (2) of contact vector fields in the Lie superalgebra 𝒮 Ψ 𝒟 𝒪 (S 1 | 2) of superpseudodifferential operators with smooth coefficients, where 𝔬 𝔰 𝔭 (2 | 2) is the orthosymplectic Lie superalgebra. Likewise, we study the same problem for the affine Lie superalgebra 𝔞 𝔣 𝔣 (2 | 1) instead of 𝔬 𝔰 𝔭 (2 | 2). We classify generic formal 𝔬 𝔰 𝔭 (2 | 2) -trivial deformations of the 𝒦 (2) -module structure on the superspace of the supercommutative algebra 𝒮 𝒫 (2) of pseudodifferential symbols on S 1 | 2 . [ABSTRACT FROM AUTHOR]

Over the (1 , 1) -dimensional supercircle S 1 | 1 , we investigate the first 𝔬 𝔰 𝔭 (1 | 2) -relative cohomology space associated with the embedding of the Lie superalgebra 𝒦 (1) of contact vector fields in the Lie superalgebra 𝒮 Ψ 𝒟 𝒪 (S 1 | 1) of superpseudodifferential operators with smooth coefficients, where 𝔬 𝔰 𝔭 (1 | 2) is the orthosymplectic Lie superalgebra. Likewise, we study the same problem for the affine Lie superalgebra 𝔞 𝔣 𝔣 (1 | 1) instead of 𝔬 𝔰 𝔭 (1 | 2). We classify 𝔞 𝔣 𝔣 (1 | 1) -trivial deformations of the standard embedding of the Lie superalgebra 𝒦 (1) into the Lie superalgebra 𝒮 Ψ 𝒟 𝒪 (S 1 | 1). This approach leads to the deformations of the central charge induced on 𝒦 (1) by the canonical central extension of 𝒮 Ψ 𝒟 𝒪 (S 1 | 1). [ABSTRACT FROM AUTHOR]

Some standard theorems on Noetherian schemes are generalized to the case of Noetherian superschemes. [ABSTRACT FROM AUTHOR]

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 Möbius 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). [ABSTRACT FROM AUTHOR]

In this study, fractional supersymmetric iso(1,1) based on the permutation groups S3 , is obtained in the Hopf algebra formulation. This algebra is denoted by U2,3(iso(1,1)). [ABSTRACT FROM AUTHOR]

Over the -dimensional real superspace, we compute the cohomology space of the affine Lie superalgebra with coefficient in a large class of -modules . We apply our results to the module of weight densities and the module of linear differential operators acting on a superspace of weighted densities. We study nontrivial deformations of the natural action of the Lie superalgebra on the direct sum of the superspaces of weighted densities. [ABSTRACT FROM AUTHOR]

Let F be a field of characteristic zero. We study two minimal superalgebras A and B having the same superexponent but such that T ( A) ⫋ T ( B), thus providing the first example of a minimal superalgebra generating a non minimal supervariety. We compare the structures and codimension sequences of A and B. [ABSTRACT FROM AUTHOR]