The Variety of Jordan Superalgebras of dimension four and even part of dimension two

Non-necessarily associative algebras have been extensively studied. Among the main classes are the Lie and Jordan algebras. It is essential to know examples of a given algebraic system, particularly in low dimensions.

In relation to finite-dimensional algebras over an arbitrary field, it is natural to study their algebraic classification, that is, to determine all that are not isomorphic. It is also of interest to determine the deformations between low-dimensional algebras in a variety defined by polynomial identities, in particular finding the rigid ones. The irreducible components within the regarded variety related to Zariski topology are constituted by the closure of their orbits under the action of the linear group.

We now briefly describe the progress of the geometric classification of Jordan algebras and superalgebras. In $2004$, I. Kashuba and I. Shestakov [9] investigated the variety ${\rm Jor}_{3}$ of Jordan algebras of dimension $3$, for $\mathbb{F}$ algebraically closed of characteristic $\neq 2,3$, listing all $\operatorname{GL}_{3}(\mathbb{F})$-orbits of ${\rm Jor}_{3}$ and establishing its $5$ irreducible components. In $2006$, I. Kashuba [10] studied the variety ${\rm Jor}_{n}$ of Jordan unitary algebras of dimension $n$, for an algebraically closed field $\mathbb{F}$ with $\textrm{char\,}\mathbb{F}\neq 2$, as well as infinitesimal deformations of Jordan algebras, establishing the list of $\operatorname{GL}_{n}(\mathbb{F})$-orbits on ${\rm Jor}_{n}$, $n=4,5$, under the “change of basis” action, finding the number of irreducible components as being $3$ and $6$ respectively, and a list of rigid algebras was included. In $2014$, I. Kashuba and M. E. Martin [11] studied the variety ${\rm Jor}_{4}$ of four-dimensional Jordan algebras, for an algebraically closed field $\mathbb{F}$ of characteristic $\neq 2$, describing its irreducible components and proving that ${\rm Jor}_{4}$ is the union of Zariski closures of the orbits of 10 rigid algebras. In $2016$, I. Kashuba and M. E. Martin [12] studied the variety ${\rm Jor}_{3}$ of three-dimensional Jordan algebras over the field of real numbers, establishing the list of $26$ non-isomorphic Jordan algebras and describing the irreducible components of ${\rm Jor}_{3}$, proving that it is the union of Zariski closures of the orbits of eight rigid algebras. In $2018$, I. Kashuba and M. E. Martin [13] also investigated the variety ${\rm JorN}_{5}$ of five-dimensional nilpotent Jordan algebras structures over an algebraically closed field, showing that ${\rm JorN}_{5}$ is the union of five irreducible components, four of them correspond to the Zariski closure of the $\operatorname{GL}_{5}(\mathbb{F})$-orbits of four rigid algebras and the other one is the Zariski closure of an union of orbits of an infinite family of algebras, none of them being rigid.

In the $\mathbb{Z}_{2}$-graded context, in $2019$, M. A. Alvarez et al. [2] described degenerations of three-dimensional Jordan superalgebras over the field of complex numbers $\mathbb{C}$. In particular, they describe all irreducible components in the corresponding varieties. Recently, we [7] described the variety of Jordan superalgebras of dimension $4$ whose even part is a Jordan algebra of dimension $1$ or $3$, proving that the variety is the union of $11$ and $21$ rigid superalgebras, respectively, and in both cases the irreducible components of the varieties were also described.

In this article, we deal with the geometric classification of Jordan superalgebras of dimension four with even and odd parts of dimension 2 over an algebraically closed field $\mathbb{F}$ of characteristic $0$. In Section 2, we set up notation and terminology, we indicated some useful invariants to guarantee the non-existence of a deformation between two superalgebras, and we introduce the notion of Jordan superalgebras. Section 3 provides a necessary condition for a superalgebra to be rigid and some examples are given. In particular, a rigid Jordan superalgebra whose second cohomology group does not vanish is constructed. In Section 4 we describe deformations between Jordan superalgebras of type (2,2) and characterize the irreducible components of such variety. As a direct consequence, we achieve the geometric classification of associative and supercommutative superalgebras of dimension four and type $(2,2)$, resulting in 6 irreducible components. Additionally, we determine that the subvariety of nilpotent Jordan superalgebras of dimension four and type $(2,2)$ has 3 irreducible components.

Let $\mathbb{F}$ be any field of characteristic different from two. A superalgebra $A$ is just a $\mathbb{Z}_{2}$-graded algebra over $\mathbb{F}$, that is, $A$ is decomposed into a direct sum of subspaces $A=A_{0}\oplus A_{1}$ such that $A_{i}A_{j}\subseteq A_{i+j}$, for $i,j\in\mathbb{Z}_{2}$. Notice that $A_{0}$ is a subalgebra and $A_{1}$ is a $A_{0}$-bimodule. The elements in $A_{0}\setminus\{0\}$ (respectively, in $A_{1}\setminus\{0\}$) are called even (respectively, odd) and the elements in $A_{i}\setminus\{0\}$, $i\in\mathbb{Z}_{2}$, are said to be homogeneous and of degree $i$. The degree of a homogeneous element $x$ is denoted by $|x|=i$. The pair $(\dim_{\mathbb{F}}(A_{0}),\dim_{\mathbb{F}}(A_{1}))$ is said to be the type of $A$.

The morphisms in this category are defined as follows. Let $A$ and $A^{\prime}$ be superalgebras. A linear map $\Phi\colon A\to A^{\prime}$ is a morphism of superalgebras if $\Phi$ is even (i.e., $\Phi(A_{i})\subset A^{\prime}_{i}$, for $i\in\mathbb{Z}_{2}$) and $\Phi(xy)=\Phi(x)\Phi(y)$, for all $x,y\in A$.

We now introduce the concept of the Grassmann envelope. Let $\mathcal{G}={\rm alg}\{1,e_{i}|1\leq i,\;e_{i}e_{j}=-e_{j}e_{i}\}$ be the Grassmann algebra. Then, $\mathcal{G}=\mathcal{G}_{0}\oplus\mathcal{G}_{1}$ is an associative $\mathbb{Z}_{2}$-graded algebra, where $\mathcal{G}_{0}$ (respectively, $\mathcal{G}_{1}$) are subspaces spanned by the products of even (respectively, odd) length. In addition, $\mathcal{G}$ is supercommutative, that is, $uv=(-1)^{|u||v|}vu$, for all homogeneous elements $u,v\in\mathcal{G}$. The Grassmann envelope of a superalgebra $A=A_{0}\oplus A_{1}$ is the algebra

where the multiplication is defined by $(x\otimes u)(y\otimes v):=xy\otimes uv$, for all $x\otimes u\in\mathcal{G}_{i}\otimes A_{i}$, $y\otimes v\in\mathcal{G}_{j}\otimes A_{j}$, and $i,j\in\mathbb{Z}_{2}$.

Let $\mathcal{V}$ be a variety of algebras over $\mathbb{F}$ defined by a set of multilinear identities $\{P_{\lambda}\}_{\lambda\in\Lambda}$. A superalgebra $A=A_{0}\oplus A_{1}$ is called a $\mathcal{V}$-superalgebra if its Grassmann envelope $\mathcal{G}(A)$ lies in $\mathcal{V}$. In particular, if $A=A_{0}\oplus A_{1}$ is a $\mathcal{V}$-superalgebra, then the algebra $A_{0}$ lies in $\mathcal{V}$ and $A_{1}$ is a $A_{0}$-bimodule in the class $\mathcal{V}$.

Let $\{P_{\lambda}\}_{\lambda\in\Lambda}$ be the set of multilinear identities that defines the variety $\mathcal{V}$. In general, the corresponding set of superidentities $\{P^{s}_{\lambda}\}_{\lambda\in\Lambda}$ which defines the variety of $\mathcal{V}$-superalgebras can be obtained following the Kaplansky Rule: If two homogeneous adjacent elements $x,y$ are exchanged, then the corresponding term is multiplied by $(-1)^{|x||y|}$.

The set of all $\mathcal{V}$-superalgebras of type $(m,n)$ defines an affine variety in $\mathbb{F}^{m^{3}+3mn^{2}}$ and it will be denoted by $\mathcal{VS}^{(m,n)}$. In fact, let $V=V_{0}\oplus V_{1}$ be a $\mathbb{Z}_{2}$-graded vector space with a fixed homogeneous basis $\left\{e_{1},\dots,e_{m},f_{1},\dots,f_{n}\right\}$. Our goal is to provide a superalgebra structure for $V$. To this end, the possible multiplication tables are given by

where $\alpha_{ij}^{k},\beta_{ij}^{k},{\beta^{\prime}}_{ij}^{k},\gamma_{ij}^{k}\in\mathbb{F}$, for all $i,j$ and $k$. Notice that every set of structure constants must satisfy the polynomial identities $P^{s}_{\lambda}$, for $\lambda\in\Lambda$. It follows that each point $(\alpha_{ij}^{k},\beta_{ij}^{k},{\beta^{\prime}}_{ij}^{k},\gamma_{ij}^{k})\in\mathcal{VS}^{(m,n)}$ represents, in the fixed basis, a $\mathcal{V}$-superalgebra $A$ over $\mathbb{F}$ of type $(m,n)$.

Since superalgebras morphisms are even maps, there is a natural “change of basis” action of the group $G=\operatorname{GL}_{m}(\mathbb{F})\times\operatorname{GL}_{n}(\mathbb{F})$ on $\mathcal{VS}^{(m,n)}$ which gives a one-to-one correspondence between $G$-orbits $A^{G}$ on $\mathcal{VS}^{(m,n)}$ and the isomorphism classes of $\mathcal{V}$-superalgebras of type $(m,n)$. Let $\overline{A^{G}}$ be the Zariski closure of $A^{G}$. The superalgebra $A\in\mathcal{VS}^{(m,n)}$ is said to be a deformation of $B\in\mathcal{VS}^{(m,n)}$ if $B^{G}\subseteq\overline{A^{G}}$. We indicate it by $A\to B$. The superalgebra $A\in\mathcal{VS}^{(m,n)}$ is called (geometrically) rigid if the orbit $A^{G}$ is Zariski-open set in $\mathcal{VS}^{(m,n)}$. Hence, if $A$ is rigid, then $\overline{A^{G}}$ is an irreducible component of the variety $\mathcal{VS}^{(m,n)}$ and any deformation $C$ of $A$ satisfies $C\simeq A$.

The one-parameter family of deformations technique is used in order to describe the deformations on $\mathcal{V}$-superalgebras of type $(m,n)$, and it consists of the following. Let $A\in\mathcal{VS}^{(m,n)}$, denote the Laurent polynomials in the variable $t$ by $\mathbb{F}(t)$, and take $g(t)\in\operatorname{Mat}_{m}(\mathbb{F}(t))\times\operatorname{Mat}_{n}(\mathbb{F}(t))$. For any $0\neq t\in\mathbb{F}$, assume that $g(t)\in G$. If $A_{t}=(\alpha_{ij}^{k}(t),\beta_{ij}^{k}(t),{\beta^{\prime}}_{ij}^{k}(t),\gamma_{ij}^{k}(t))$ is the $\mathcal{V}$-superalgebra resulting from the application of the change of basis $g(t)$ to $A$, then $A$ is a deformation of the $\mathcal{V}$-superalgebra $B=(\alpha_{ij}^{k}(0),\beta_{ij}^{k}(0),{\beta^{\prime}}_{ij}^{k}(0),\gamma_{ij}^{k}(0))$. In particular, a $\mathcal{V}$-superalgebra $A$ is rigid if any $g(t)$ satisfying the above conditions defines the $\mathcal{V}$-superalgebra $A_{t}$ isomorphic to $A$ for every $t\in\mathbb{F}$. In other words, the following result holds.

The following results will be useful to guarantee the non-existence of a deformation between two superalgebras of $\mathcal{VS}^{(m,n)}$. They were demonstrated in [1] and [2] for the case of Lie and Jordan superalgebras, respectively, but the proofs can be easily adapted to any other variety.

In this article, we consider the variety $\mathcal{V}$ of Jordan algebras over an algebraically closed field $\mathbb{F}$ of $\textrm{char\,}\mathbb{F}=0$, i.e., the variety of algebras defined by the multilinear identities $xy=yx$ and

The last one is known as Jordan identity. We will work with the corresponding variety of $\mathcal{V}$-superalgebras, called Jordan superalgebras, whose set of superidentities, obtained through Kaplansky’s Rule, are: the supercommutativity $xy=(-1)^{|x||y|}yx$, and the Jordan superidentity:

where $x,y,z,w$ are homogeneous elements. The variety of all Jordan superalgebras of type $(m,n)$ will be denoted by $\mathcal{JS}^{(m,n)}$.

In the next section we describe a powerful machinery to find rigid superalgebras. This technique is valid for other varieties, such as associative and Lie superalgebras. In order to be precise, we introduce it only for Jordan superalgebras.

The most known sufficient condition for a superalgebra to be rigid is given in terms of its cohomology group. We say that the second cohomology group $H^{2}(\mathcal{J},\mathcal{J})$ of a Jordan superalgebra $J$ with coefficients in itself vanishes if for every bilinear map $h\colon\mathcal{J}\times\mathcal{J}\to\mathcal{J}$ satisfying

and

for all homogeneous elements $a,b,c,d$, where

there exists a linear map $\mu\colon\mathcal{J}\to\mathcal{J}$ such that

For the precise definition and properties of this group for Jordan superalgebras, we refer the reader to [5] and [6].

The last implication applies equally well to any category of algebras or superalgebras defined by superidentities with appropriate modifications. In fact, this result was originally obtained in [3] for associative and Lie algebras; for the case of Jordan algebras, see [10].The proof for Jordan superalgebras is analogous.

In positive characteristic, it has been shown that the reverse implication of Proposition 3.1 is false for finite-dimensional associative algebras. M. Gerstenhaber and S. Schack [4] constructed rigid high-dimensional associative algebras $A$ over a field of positive characteristic such that $H^{2}(A,A)\neq 0$. Furthermore, Richardson [16] showed that there exist complex Lie algebras in every even dimension greater than $16$ that are rigid but the second cohomology group does not vanish. The question of the existence of a rigid Jordan algebra $\mathcal{J}$ satisfying $H^{2}(\mathcal{J},\mathcal{J})\neq 0$ is still an open problem. In what follows, we will show that the converse does not hold for Jordan superalgebras. We will show an example of a Jordan superalgebra that is rigid (as shown in Theorem 4.5) but whose second cohomology group does not vanish.

Analogously, we say that the even part of the second cohomology group $H^{2}(\mathcal{J},\mathcal{J})$ of a Jordan superalgebra $\mathcal{J}$ with coefficients in itself vanishes (i.e., $(H^{2}(\mathcal{J},\mathcal{J}))_{0}=0$) if for every even bilinear map $h_{0}:\mathcal{J}\times\mathcal{J}\to\mathcal{J}$ (this means that $h_{0}(e_{i},e_{j}),h_{0}(f_{i},f_{j})\in J_{0}$ and $h_{0}(e_{i},f_{j}),h_{0}(f_{i},e_{j})\in\mathcal{J}_{1}$ for $i,j=1,2$) satisfying (1) and (3) there exists an even linear map $\mu_{0}:\mathcal{J}\to\mathcal{J}$ (i.e., $\mu_{0}(\mathcal{J}_{i})\subseteq\mathcal{J}_{i}$ for $i=1,2$) such that

In [1], the authors proved that it suffices for the even part of the second cohomology group of a Lie superalgebra to vanish in order to conclude that it is rigid. The proof for Jordan superalgebras is analogous.

We leave open the natural question that arises from Proposition 3.5, namely whether there exists a rigid Jordan superalgebra $\mathcal{J}$ such that $(H^{2}(\mathcal{J},\mathcal{J}))_{0}\neq 0$.

In this section, we investigate the variety $\mathcal{JS}^{(2,2)}$ over an algebraically closed field of characteristic zero. I. Hernández et al. [8] have provided 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 consequence of this classification, we see that there exist, up to isomorphism, $72$ Jordan superalgebras of type $(2,2)$ and an one parameter family over an algebraically closed field of characteristic zero (see [8], Tables 6, 8, 9, 10, 12, and 14). Table LABEL:table:JSA_(2,2) gives representatives for isomorphism classes and some additional useful information, namely the dimension of the automorphism group of each superalgebra, and we indicate by “A” if the superalgebra is associative and “NA” otherwise. The superscript “N” in $(2,2)_{i}^{N}$ indicates that the superalgebra is nilpotent.