Finite-dimensional Superalgebras Research Articles

$$\frac{1}{2}$$-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.

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.

Superalgebras with superinvolution or graded involution with colengths sequence bounded by 3

Let [Formula: see text] be a superalgebra with superinvolution or graded involution over a field of characteristic zero and let [Formula: see text], [Formula: see text], be the [Formula: see text]-cocharacter of [Formula: see text]. The ∗-colengths sequence, [Formula: see text], [Formula: see text], is the sum of the multiplicities in the decomposition of the [Formula: see text]-cocharacter [Formula: see text], for all [Formula: see text]. The main purpose of this paper is to classify the superalgebras with superinvolution with ∗-colengths sequence bounded by three. Moreover, we shall extend to the general case, the analogous result proved by do Nascimento and Vieira in [Superalgebras with graded involution and star-graded colength bounded by 3, Linear Multilinear Algebra 67(10) (2019) 1999–2020] for finite-dimensional superalgebras with graded involution.

Star-polynomial identities: Computing the exponential growth of the codimensions

Can one compute the exponential rate of growth of the ⁎-codimensions of a PI-algebra with involution ⁎ over a field of characteristic zero? It was shown in [2] that any such algebra A has the same ⁎-identities as the Grassmann envelope of a finite dimensional superalgebra with superinvolution B. Here, by exploiting this result we are able to provide an exact estimate of the exponential rate of growth exp⁎(A) of any PI-algebra A with involution. It turns out that exp⁎(A) is an integer and, in case the base field is algebraically closed, it coincides with the dimension of an admissible subalgebra of maximal dimension of B.

Asymptotics for Graded Capelli Polynomials

The finite dimensional simple superalgebras play an important role in the theory of PI-algebras in characteristic zero. The main goal of this paper is to characterize the T 2-ideal of graded identities of any such algebra by considering the growth of the corresponding supervariety. We consider the T 2-ideal Γ M+1,L+1 generated by the graded Capelli polynomials C a p M+1[Y,X] and C a p L+1[Z,X] alternanting on M+1 even variables and L+1 odd variables, respectively. We prove that the graded codimensions of a simple finite dimensional superalgebra are asymptotically equal to the graded codimensions of the T 2-ideal Γ M+1,L+1, for some fixed natural numbers M and L. In particular $$c^{sup}_{n}(\Gamma_{k^{2}+l^{2}+1,2kl+1})\simeq c^{sup}_{n}(M_{k,l}(F))$$ and $$c^{sup}_{n}(\Gamma_{s^{2}+1,s^{2}+1})\simeq c^{sup}_{n}(M_{s}(F\oplus tF)).$$

Graded cocharacters of minimal subvarieties of supervarieties of almost polynomial growth

The sequence of graded codimensions of a minimal supervariety of polynomial growth grows like nt, but any proper subvariety grows like ns with s<t. Such supervarieties are the building blocks of general supervarieties of polynomial growth. In this paper, we exhibit the decompositions of the graded cocharacters of all minimal subvarieties of supervarieties of almost polynomial growth and compute their graded colengths. Furthermore, for a finite dimensional superalgebra A such that its sequence of graded codimensions grows like nt we exhibit an upper bound for the nth graded colength of A which depends only on t and on two parameters related to A such as its dimension and the index of nilpotence of its Jacobson radical J(A).

Exploring three-dimensional higher-spin supergravity based on sl(N|N − 1) Chern-Simons theories

We investigate various aspects of higher-spin anti-de Sitter supergravity in three dimensions as described by Chern-Simons theory based on the finite-dimensional superalgebra sl(N |N - 1), with the particular case of N = 3 as our prime example. This class of theories serves as a natural supersymmetrization of the higher-spin gravity theory based on sl(N) Chern-Simons theories. We demonstrate explicitly that the asymptotic symmetry algebra contains the N = 2 superconformal algebra in each sector. The appropriate Killing spinor equations are derived and used to classify existing and new classical solutions. We also discuss holonomy conditions, higher-spin black holes and conical defect spacetimes in this class of theories.

Elementary Superalgebras

Let Λ be a finite-dimensional superalgebra over a field K. A characterization of an elementary superalgebra Λ is given by a quiver and a weight function. It is shown that Λ is elementary if and only if its Hochschild extension is elementary. Furthermore, if Λ is elementary of finite global dimension and {e1, …, en} is a complete set of gr-primitive orthogonal idempotents of Λ, then the following equalities hold: [Formula: see text] where ΦΛ is the Coxeter matrix of Λ, tr is the trace function of a matrix, HHi(Λ) and HHi(Λ) are the i-th Hochschild homology and cohomology, respectively.

Supervarieties of small graded colength

Let A be a superalgebra over a field F of characteristic zero, and let χngr(A), n=1,2,…, be the sequence of graded cocharacters of A. For every n≥1, let lngr(A) denote the nth graded colength of A, counting the number of Z2≀Sn-irreducibles appearing in χngr(A). In this article, we classify the finite dimensional superalgebras A such that the sequence of graded colengths, lngr(A),n=1,2,…, is bounded by two. Moreover, we prove that there is a finite number of superalgebras A1,…,Aq such that lngr(A)≤2 if and only if A1,…,Aq∉vargr(A).

Graded Morita Equivalence of Clifford Superalgebras

This note uses a variation of graded Morita theory for finite dimensional superalgebras to determine explicitly the graded basic superalgebras for all real and complex Clifford superalgebras. As an application, the Grothendieck groups of the category of left $${\mathbb{Z}_2}$$ -graded modules over all real and complex Clifford superalgebras are described explicitly.

Super-W∞ asymptotic symmetry of higher-spin AdS3 supergravity

We consider (2+1)-dimensional (N, M)-extended higher-spin anti-de Sitter supergravity and study its asymptotic symmetries. The theory is described by the Chern-Simons action based on a real, infinite-dimensional higher-spin superalgebra. We specify consistent boundary conditions on the higher-spin super-gauge connection corresponding to asymptotically anti-de Sitter spacetimes. We then determine the residual gauge transformations that preserve these asymptotic conditions and compute their Poisson bracket algebra. We find that the asymptotic symmetry is enhanced from the higher-spin superalgebra to some (N,M)-extended super-W(infinity) nonlinear superalgebra. The latter has the same classical central charge as pure Einstein gravity. Special attention is paid to the (1,1)-case. Truncation to the bosonic sector yields the previously found W(infinity) algebra, while truncation to the underlying finite-dimensional superalgebra reproduces the N-extended superconformal algebra (in its nonlinear version for N>2). We discuss string theory realization of these higher-spin anti-de Sitter supergravity theories as well as relations to previous treatments of super-W(infinity) in the literature.

Holographic space-time from the Big Bang to the de Sitter era

I review the holographic theory of space-time and its applications to cosmology. Much of this has appeared before, but this discussion is more unified and concise. I also include some material on work in progress, whose aim is to understand compactification in terms of finite-dimensional super-algebras. This is an expanded version of a lecture I gave at the conference on Liouville Quantum Gravity and Statistical Systems, in memory of Alexei Zamolodchikov, at the Poncelet Institute in Moscow, 21–24 June 2008.

Superspecies and their representations

Superspecies are introduced to provide a construction of all finite-dimensional superalgebras. All acyclic superspecies are classified according to their graded representation types.

Codimension growth and minimal superalgebras

A celebrated theorem of Kemer (1978) states that any algebra satisfying a polynomial identity over a field of characteristic zero is PI-equivalent to the Grassmann envelope G(A) of a finite dimensional superalgebra A. In this paper, by exploiting the basic properties of the exponent of a PI-algebra proved by Giambruno and Zaicev (1999), we define and classify the minimal superalgebras of a given exponent over a field of characteristic zero. In particular we prove that these algebras can be realized as block-triangular matrix algebras over the base field. The importance of such algebras is readily proved: A is a minimal superalgebra if and only if the ideal of identities of G(A) is a product of verbally prime T-ideals. Also, such superalgebras allow us to classify all minimal varieties of a given exponent i.e., varieties V such that exp(V) = d > 2 and exp(U) < d for all proper subvarieties U of V. This proves in the positive a conjecture of Drensky (1988). As a corollary we obtain that there is only a finite number of minimal varieties for any given exponent. A classification of minimal varieties of finite basic rank was proved by the authors (2003). As an application we give an effective way for computing the exponent of a T-ideal given by generators and we discuss the problem of what functions can appear as growth functions of varieties of algebras.

Quasi-exactly solvable Lie superalgebras of differential operators

In this paper, we study Lie superalgebras of $2\times 2$ matrix-valued first-order differential operators on the complex line. We first completely classify all such superalgebras of finite dimension. Among the finite-dimensional superalgebras whose odd subspace is nontrivial, we find those admitting a finite-dimensional invariant module of smooth vector-valued functions, and classify all the resulting finite-dimensional modules. The latter Lie superalgebras and their modules are the building blocks in the construction of QES quantum mechanical models for spin 1/2 particles in one dimension.

Multiparametric quantization of the special linear superalgebra

It is shown how using the graded Faddeev–Reshetikhin–Takhtajan (FRT) formalism, one can multiparametrize the quantum superalgebra slq(n‖m). Since the naive Serre–Chevalley relations are not adequate for the presentation of this algebra, the quantization is carried out in the Cartan–Weyl basis. The final results can be generalized to any finite dimensional superalgebra with a symmetrizable Cartan matrix.