Simple Superalgebras Research Articles

Classifying simple superalgebras with automorphism and pseudoautomorphism

Let F be an algebraically closed field of characteristic zero. We give a complete classification of finite dimensional simple superalgebras over F endowed with a graded automorphism or a pseudoautomorphism.

ℤ2-gradings and minimal varieties of ℤ2-graded PI algebras

Let [Formula: see text] be an algebraically closed field of characteristic zero and let [Formula: see text] be a minimal superalgebra with semisimple part [Formula: see text], where each [Formula: see text] is a simple superalgebra. We say that [Formula: see text] is [Formula: see text]-special if [Formula: see text] and there exists [Formula: see text] such that [Formula: see text], with [Formula: see text], whereas, for the remaining indices [Formula: see text], [Formula: see text]. In [1], Avelar et al. classified the [Formula: see text]-special minimal superalgebras and described some necessary and sufficient conditions for the varieties generated by such superalgebras to be minimal of [Formula: see text]-exponent [Formula: see text]. In this paper, we give continuity to a such study and we made explicit that the [Formula: see text]-gradings play an important role in the minimality of some varieties. We provide sufficient conditions, depending on the [Formula: see text]-grading of [Formula: see text], for the variety generated by [Formula: see text] to be minimal. Moreover, we characterize two families of [Formula: see text]-special minimal superalgebras and we give necessary and sufficient conditions (depending on their [Formula: see text]-gradings) for the varieties generated by them to be minimal.

On a family of vertex operator superalgebras

This paper is to study vertex operator superalgebras which are strongly generated by their weight-2 and weight-32 homogeneous subspaces. Among the main results, it is proved that if such a vertex operator superalgebra V is simple, then V(2) has a canonical commutative associative algebra structure equipped with a non-degenerate symmetric associative bilinear form and V(32) is naturally a V(2)-module equipped with a V(2)-valued symmetric bilinear form and a non-degenerate (C-valued) symmetric bilinear form, satisfying a set of conditions. On the other hand, assume that A is any commutative associative algebra equipped with a non-degenerate symmetric associative bilinear form and assume that U is an A-module equipped with a symmetric A-valued bilinear form and a non-degenerate (C-valued) symmetric bilinear form, satisfying the corresponding conditions. Then we construct a Lie superalgebra L(A,U) and a simple vertex operator superalgebra LL(A,U)(ℓ,0) for every nonzero number ℓ such that LL(A,U)(ℓ,0)(2)=A and LL(A,U)(ℓ,0)(32)=U.

Quantum coordinate ring in WZW model and affine vertex algebra extensions

In this paper, we construct various simple vertex superalgebras which are extensions of affine vertex algebras, by using abelian cocycle twists of representation categories of quantum groups. This solves the Creutzig and Gaiotto conjectures (Creutzig and Gaiotto in Comm Math Phys 379:785–845, 2020, Conjecture 1.1 and 1.4) in the case of type ABC. If the twist is trivial, the resulting algebras correspond to chiral differential operators in the chiral case, and to WZW models in the non-chiral case.

Super orbifold theory

Let V be a simple vertex operator superalgebra and G a finite automorphism group of V containing the canonical automorphism σ such that VG is regular. It is proved that every irreducible VG-module occurs in an irreducible g-twisted V-module for some g∈G. Moreover, the quantum dimensions of irreducible VG-modules are determined, a global dimension formula for V in terms of twisted modules is obtained and a super quantum Galois theory is established. In addition the S-matrix of VG is computed.

Interacting SPT phases are not morita invariant

The tenfold way provides a strong organizing principle for invertible topological phases of matter. Mathematically, it is intimately connected with K-theory via the fact that there exist exactly ten Morita classes of simple real superalgebras. This connection is physically unsurprising, since weakly interacting topological phases are classified by K-theory. We argue that when strong interactions are present, care has to be taken when formulating the exact ten symmetry groups present in the tenfold way table. We study this phenomenon in the example of class D by providing two possible mathematical interpretations of a class D symmetry. These two interpretations of class D result in Morita equivalent but different symmetry groups. As K-theory cannot distinguish Morita-equivalent protecting symmetry groups, the two approaches lead to the same classification of topological phases on the weakly interacting side. However, we show that these two different symmetry groups yield different interacting classifications in spacetime dimension 2+1. We use the approach to interacting topological phases using bordism groups, reducing the relevant classification problem to a spectral sequence computation.

G-twisted associative algebras of vertex operator superalgebras

Let V be a vertex operator superalgebra and G a finite automorphism group of V. Let σ be the order 2 automorphism of V associated with the superstructure of V. A sequence of associative algebras AG,n(V) are constructed to study the twisted representations of V for nonnegative n∈1T′Z where T′ is the order of group generated by G and σ. This result which generalizes many previous results on Zhu's algebras is then used to investigate the super orbifold theory. If V is a simple vertex operator superalgebra and S is a finite set of inequivalent irreducible twisted V-modules which is closed under the action of G, a duality theorem of Schur-Weyl type is obtained for the actions of certain finite dimensional semisimple associated algebra Aα(G,S) and VG on the direct sum of twisted V-modules in S. In particular, for any g∈G every irreducible g-twisted V-module is completely reducible VG-module.

BERSHADSKY–POLYAKOV VERTEX ALGEBRAS AT POSITIVE INTEGER LEVELS AND DUALITY

We study the simple Bershadsky–Polyakov algebra 𝒲k = 𝒲k(sl3, fθ) at positive integer levels and classify their irreducible modules. In this way, we confirm the conjecture from [9]. Next, we study the case k = 1. We discover that this vertex algebra has a Kazama–Suzuki-type dual isomorphic to the simple affine vertex superalgebra Lk′ (osp(1|2)) for k′ = –5=4. Using the free-field realization of Lk′ (osp(1|2)) from [3], we get a free-field realization of 𝒲k and their highest weight modules. In a sequel, we plan to study fusion rules for 𝒲k.

Analogs of Bol operators on superstrings

The Bol operators are unary differential operators between spaces of weighted densities on the 1-dimensional manifold invariant under projective transformations of the manifold. On the [Formula: see text]-dimensional supermanifold (superstring) [Formula: see text], we classify analogs of Bol operators invariant under the simple maximal subalgebra [Formula: see text] of the same rank as its simple ambient superalgebra [Formula: see text] of vector fields on [Formula: see text] and containing all elements of negative degree of [Formula: see text] in a [Formula: see text]-grading. We also consider the Lie superalgebras of vector fields [Formula: see text] preserving a contact structure on the superstring [Formula: see text]. We have discovered many new operators.

Low-dimensional commutative power-associative superalgebras

The aim of this work is to provide a concrete list of non-isomorphic commutative power-associative superalgebras up to dimension 4 over an algebraically closed field of characteristic prime to 30. As a byproduct, we exhibit an example of a simple non-Jordan power-associative superalgebra whose even part is not semisimple.

$$\frac{1}{2}$$-derivations of Lie algebras and transposed Poisson algebras

A relation between $\frac{1}{2}$-derivations of Lie algebras and transposed Poisson algebras was established. Some non-trivial transposed Poisson algebras with a certain Lie algebra (Witt algebra, algebra $\mathcal{W}(a,-1)$, thin Lie algebra and solvable Lie algebra with abelian nilpotent radical) were constructed. In particular, we constructed an example of the transposed Poisson algebra with associative and Lie parts isomorphic to the Laurent polynomials and the Witt algebra. On the other side, it was proven that there are no non-trivial transposed Poisson algebras with Lie algebra part isomorphic to a semisimple finite-dimensional algebra, a simple finite-dimensional superalgebra, the Virasoro algebra, $N=1$ and $N=2$ superconformal algebras, or a semisimple finite-dimensional $n$-Lie algebra.

SU(2/1) superchiral self-duality: a new quantum, algebraic and geometric paradigm to describe the electroweak interactions

We propose an extension of the Yang-Mills paradigm from Lie algebras to internal chiral superalgebras. We replace the Lie algebra-valued connection one-form A, by a superalgebra-valued polyform tilde{A} mixing exterior-forms of all degrees and satisfying the chiral self-duality condition tilde{A} =^{ast }{tilde{A}}_{chi } , where χ denotes the superalgebra grading operator. This superconnection contains Yang-Mills vectors valued in the even Lie subalgebra, together with scalars and self-dual tensors valued in the odd module, all coupling only to the charge parity CP-positive Fermions. The Fermion quantum loops then induce the usual Yang-Mills-scalar Lagrangian, the self-dual Avdeev-Chizhov propagator of the tensors, plus a new vector-scalar-tensor vertex and several quartic terms which match the geometric definition of the supercurvature. Applied to the SU(2/1) Lie-Kac simple superalgebra, which naturally classifies all the elementary particles, the resulting quantum field theory is anomaly-free and the interactions are governed by the super-Killing metric and by the structure constants of the superalgebra.

Central orders in simple finite dimensional superalgebras

The well-known Formanek's module finiteness theorem states that every unital prime PI-algebra (i.e. a central order in a matrix algebra by Posner's theorem) embeds into a finitely generated module over its center. An analogue of this theorem for alternative and Jordan algebras was earlier proved by V.N.Zhelyabin and the author. In this paper we discuss this problem for associative, classical Jordan and some alternative superalgebras.

Simple Right-Alternative Superalgebras with Semisimple Even Part

We classify the unital finite-dimensional simple right-alternative superalgebras with semisimple even part and prove that each of these superalgebras is either a simple associative matrix Wall algebra, or a simple alternative Shestakov superalgebra, or an asymmetric double, or an abelian superalgebra of type Bn∣n, n ≥ 2, or B2∣2(v). Furthermore, we obtain a description of right-alternative superalgebras with simple even part; every such superalgebra either is simple or has the odd part with zero product.

Conformal embeddings in affine vertex superalgebras

This paper is a natural continuation of our previous work on conformal embeddings of vertex algebras [6], [7], [8]. Here we consider conformal embeddings in simple affine vertex superalgebra Vk(g) where g=g0¯⊕g1¯ is a basic classical simple Lie superalgebra. Let Vk(g0¯) be the subalgebra of Vk(g) generated by g0¯. We first classify all levels k for which the embedding Vk(g0¯) in Vk(g) is conformal. Next we prove that, for a large family of such conformal levels, Vk(g) is a completely reducible Vk(g0¯)–module and obtain decomposition rules. Proofs are based on fusion rules arguments and on the representation theory of certain affine vertex algebras. The most interesting case is the decomposition of V−2(osp(2n+8|2n)) as a finite, non simple current extension of V−2(Dn+4)⊗V1(Cn). This decomposition uses our previous work [10] on the representation theory of V−2(Dn+4).We also study conformal embeddings gl(n|m)↪sl(n+1|m) and in most cases we obtain decomposition rules.

Linearly generated singular superalgebras

A simple right alternative superalgebra is called singular if its even part has zero multiplication. The first example of a singular superalgebra was given in [1]. In [2], a classification is given up to an isomorphism of 5-dimensional singular superalgebras. In this paper, the concepts of algebraically generated and linearly generated singular superalgebras are introduced. It is proved that an algebraically generated singular superalgebra with a 2-dimensional even part is linearly generated. It is also proved that every linearly generated singular superalgebra is isomorphic to a superalgebra associated with a suitable non-degenerate symmetric matrix. If the base field is algebraically closed, then for each admissible dimension of type 4n+1≥5 there is only one such superalgebra, called the standard one. Lie superalgebras of differentiations and automorphism groups of standard superalgebras are described. Examples of singular superalgebras in which the even and odd parts are infinite-dimensional are also indicated.

On the classification of a family of minimal superalgebras

Let F be an algebraically closed field of characteristic zero and let be an n-tuple of simple superalgebras such that and there exists such that , with , whereas for all remaining indices. In this paper, we classify the minimal F-superalgebras such that its graded semisimple part is given by . Moreover, we obtain necessary and sufficient conditions to the supervariety generated by such minimal superalgebra be or not minimal of superexponent

Kronecker factorization theorems for alternative superalgebras

We consider alternative superalgebras that contain some central simple alternative superalgebras of finite dimension.We prove that the classical Kronecker Factorization Theorem of Kaplansky-Jacobson is valid in arbitrary characteristic, that is, we describe the alternative algebras that contain the Cayley-Dickson algebra O. We generalize this result and prove a Kronecker Factorization Theorem for alternative superalgebras whose even part contains O. In addition, we prove a Kronecker Factorization Theorem for alternative superalgebras that contain the associative superalgebra M(1|1)(F). As a corollary of this result, we respond to an analogue of Jacobson's problem for alternative superalgebras, that is, we describe the alternative superalgebras that contain the associative superalgebra M(1|1)(F). Finally, we study the representations of the simple alternative superalgebras O(4|4) and O[u]. We classify the bimodules over these superalgebras and prove some analogues of the Kronecker Factorization Theorem for alternative superalgebras that contain O(4|4) or O[u].

Simple Right-Alternative Unital Superalgebras Over an Algebra of Matrices of Order 2

We classify simple right-alternative unital superalgebras over a field of characteristic not 2, whose even part coincides with an algebra of matrices of order 2. It is proved that such a superalgebra either is a Wall double W2|2(ω), or is a Shestakov superalgebra S4|2(σ) (characteristic 3), or is isomorphic to an asymmetric double, an 8-dimensional superalgebra depending on four parameters. In the case of an algebraically closed base field, every such superalgebra is isomorphic to an associative Wall double M2[√1], an alternative 6-dimensional Shestakov superalgebra B4|2 (characteristic 3), or an 8-dimensional Silva–Murakami–Shestakov superalgebra.

The factorability of T2-ideals of minimal supervarieties

Giambruno and Zaicev stated that, over a field of characteristic zero, T-ideals of minimal varieties of a fixed exponent have the factoring property. In the present article, we describe necessary and sufficient conditions for the factorability of T2-ideals of minimal supervarieties of a fixed superexponent. In light of the characterization of minimal supervarieties of a fixed superexponent given by Di Vincenzo, da Silva, and Spinelli, the crucial point is the study of the factorability of T2-ideals of the upper-block triangular matrix algebras equipped with an elementary -grading, where are simple superalgebras. We obtain necessary and sufficient conditions for the isomorphism between two superalgebras . We also show that the concept of -regularity establishes a nice connection between the factorability of the T2-ideal of and the number of isomorphism classes of .