Journal of Algebra 399 (2014) 277–289

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Journal of Algebra

www.elsevier.com/locate/jalgebra

Deformations of Jordan algebras of dimension four

Iryna Kashuba ∗,1, María Eugenia Martin 2

Instituto de Matemática e Estatística, Universidade de São Paulo, R. do Matão 1010, 05508-090, São Paulo, Brazil

a r t i c l e i n f o a b s t r a c t

Article history:Received 25 May 2013Available online 1 November 2013Communicated by Alberto Elduque

MSC:14J1017C5517C10

Keywords:Jordan algebraDeformation of algebrasRigidity

We study the variety Jor4 of four-dimensional Jordan algebrason k4, for k an algebraically closed field of characteristic �= 2. Wedescribe its irreducible components and prove that Jor4 is theunion of Zariski closures of the orbits of 10 rigid algebras.

© 2013 Elsevier Inc. All rights reserved.

1. Introduction

The goal of this paper is to study the variety of Jordan algebras of dimension four over an alge-braically closed field k of characteristic �= 2 and to describe its irreducible components. Namely, let Vbe a k-vector space of finite dimension n, then the bilinear maps Homk(V × V,V) = V∗ ⊗ V∗ ⊗ V forma vector space of dimension n3 and it has the structure of an affine variety kn3

. The algebras satisfyingthe Jordan identity form a Zariski-closed affine subset of kn3

which we will call the variety of Jordanalgebras of dimension n, Jorn .

The problem of finding procedure for determining generic points of Jorn or equivalently irre-ducible components of the algebraic variety, could be formulated geometrically as follows: the lineargroup G = GL(V) operates on Jorn by conjugation, decomposing Jorn into G-orbits which correspondto the classes of isomorphic Jordan algebras. If an algebra J lies in the Zariski closure of the orbit

* Corresponding author.E-mail addresses: kashuba@ime.usp.br (I. Kashuba), eugenia@ime.usp.br (M.E. Martin).

1 The author was supported by CNPq (308090/2010-1).2 The author was supported by the CAPES scholarship for PhD program of Mathematics, IME-USP.

0021-8693/$ – see front matter © 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.jalgebra.2013.09.040

278 I. Kashuba, M.E. Martin / Journal of Algebra 399 (2014) 277–289

of a (non-isomorphic) algebra J ′ in the variety, then we will say that J ′ is a deformation of J .An algebra J whose orbit J G is Zariski-open in Jorn is called rigid. They are of particular interestsince the Zariski closure of the orbits of rigid algebra gives an irreducible component of the variety.

Note that analogously one defines the varieties of Lie and associative algebras. Denote them Lien

and Assocn respectively. The geometry of both is rather complicated. In 1890 E. Study in [1] con-sidered complex associative algebras of dimension four and showed that it is impossible to find thegeneric algebra, or equivalently, that Assoc4 has more than one irreducible component.

In 1964, M. Gerstenhaber in his work [2] turned the geometric definition of deformation intoanalytical one, namely he introduced the notion of formal (infinitesimal) deformation between asso-ciative algebras. Let A be an n-dimensional associative algebra and consider an n3-tuple g = {gijk(t)}of power series in the variable t , such that g(t) defines the associative multiplication on At = V ⊗ k(t)and gijk(0) coincide with the structure constants of A then we say that A has been deformed into At .One can check that the analytical definition implies the geometric one. In particular an algebra Ais rigid if any gijk(t) satisfying the above conditions defines the algebra At isomorphic to A for ev-ery t ∈ k. Also, it was shown that if the second cohomology group of A ∈ Assocn with coefficientsin A, H2(A, A), is trivial then A is a rigid algebra. Four years later, the analytical deformation theoryintroduced by Gerstenhaber for associative algebras was extended to Lie algebras by Nijenhuis andRichardson in [3].

There were two further papers of F. Flanigan, first [4, 1968], where he compared the structure ofthe deformed algebra At with the structure of original algebra A, see Fact 8 for the analogous theoremfor Jordan algebras. Also in [5, 1969] he showed that the irreducible component of Assocn is eitherthe closure of the orbit of a rigid algebra or the closure of the union of the orbits of non-isomorphicalgebras, called semi-rigid there.

In [6, 1974] P. Gabriel described all generic algebras for the closed subvariety of unital associativealgebras of Assocn for n � 4. Also this is a good reference to check the basic characteristics which arealmost constant on the irreducible components such as the dimension of the radical of algebras, thedimension of the automorphism group, etc. Further in [7, 1979], Gabriel’s student G. Mazzola pre-sented the inclusion diagram of the orbits of five-dimensional unital associative algebras and provedthat there are ten irreducible components (generic algebras) in this subvariety of Assoc5. The otherquestions considered by Mazzola dealt with the variety of unital commutative algebras inside ofAssocn , see [8, 1980] and the references therein. Seven years later, in [9] were described the com-ponents of Lien for n � 6. There are further recent developments in this theory when the variety ofrepresentations of algebras is considered instead of the variety of algebras.

As to the variety Jorn the references are rather recent. In [10, 2005] all deformations in Jor3were described, in [11, 2006] the closed subvarieties of unital Jordan algebras of Jorn , n � 5, wereconsidered, the irreducible components being described. Also, some properties and facts known forAssocn and Lien were extended to the case of Jordan algebras. Finally, in [12, 2011] infinitesimaldeformations were used to study nilpotent Jordan algebras of dimension four. The goal of this paperis to generalize the results of [11] and [12] and to obtain a complete description of the irreduciblecomponents of Jor4.

For the standard terminology on Jordan algebras, the reader is referred to the books ofR. Schafer [13] and N. Jacobson [14], for concepts from deformation theory see [2]. In the follow-ing, we will work over an algebraically closed field k of characteristic �= 2 and, furthermore, all Jordanalgebras are assumed to be of finite dimension over k.

The paper is organized as follows. Section 2 contains some preliminaries on Jordan algebras, alsowe provide the list of Jordan algebras of dimension less or equal to four over an algebraically closedfield k. In Section 3, we define the variety of Jordan algebras Jorn and write down its basic propertiesand useful characteristics of Jorn , while in Section 4, we apply them to establish the list of G-orbitsconstructing deformations between the algebras in Jorn for n � 4.

2. Jordan algebras of small dimensions

In this section we present the basic concepts of Jordan algebras as well as we provide the list ofJordan algebras of dimension less than or equal to four.

I. Kashuba, M.E. Martin / Journal of Algebra 399 (2014) 277–289 279

Table 1Indecomposable two-dimensional Jordan algebras.

B Multiplication table Observation

B1 e21 = e1, e1n1 = n1, n2

1 = 0 associative

B2 e21 = e1, e1n1 = 1

2 n1, n21 = 0 non-associative

B3 n21 = n2, n1n2 = 0, n2

2 = 0 nilpotent, associative

A Jordan k-algebra is a commutative algebra J with a multiplication “·” satisfying the Jordanidentity

((a · a) · b

) · a = (a · a) · (b · a), for any a,b ∈ J . (1)

We will also use the linearized version of (1)

(a,b, c · d) + (d,b, c · a) + (c,b,a · d) = 0, (2)

where a,b, c,d ∈J and (a,b, c) = (a · b) · c − a · (b · c).Since J is a finite-dimensional Jordan algebra it can be decomposed as J =Jss ⊕ Rad(J ), where

Rad(J ) is the radical of J , the unique maximal nilpotent ideal and Jss is a semi-simple subalgebraof J . Any finite-dimensional semi-simple algebra contains an identity element e. Suppose that e =∑r

i=1 ei , where ei are orthogonal idempotents, then we have the Peirce decomposition of J relative toe1, e2, . . . , er

J =⊕

0�i� j�r

Pi j, (3)

where Pii = {x ∈ J | x · ei = δi>0x}, P0 j = {x ∈ J | x · e j = 12 x} for j �= 0 and Pi j = {x ∈ J | x · ei =

x · e j = 12 x} for 0 �= i �= j.

We recall the list of four-dimensional non-isomorphic Jordan algebras, obtained in [15]. To simplifythe notations we start with the indecomposable algebras of dimension less than four. Henceforth, forconvenience we drop · and denote a multiplication in J simply as ab. The elements ei will denote thebase of Jss (usually pairwise orthogonal idempotents, except for e3 in T5 and in J9 and e3, e4 ∈ J2)and ni are generators of the radical Rad(J ).

There are two non-isomorphic one-dimensional Jordan algebras: the simple algebra ke, with e2 = eand the nilpotent algebra kn, with n2 = 0.

From the list of non-isomorphic two-dimensional Jordan algebras in [15], we choose the only threenon-isomorphic indecomposable algebras (see Table 1).

In [10] were considered three-dimensional Jordan algebras together with their deformations. Usingtheir list we obtain 10 indecomposable algebras (see Table 2).

In [15] all four-dimensional Jordan algebras are described. In Table 3 we list the 73 non-isomorphicJordan algebras. Also for each algebra we calculate the dimension of its automorphism group Aut(J ),its annihilator Ann(J ) = {a ∈J | a ·J = 0}, its second power J 2 and its radical Rad(J ).

3. Variety of Jordan algebras and its properties

In this section we define the algebraic variety of Jordan algebras Jorn and explain the methodsused in Section 4 to describe the components of Jor4 as well as deformation of Jordan algebras.

Let V be an n-dimensional k-vector space and fix e1, e2, . . . , en some basis of V. To introduce on Vthe Jordan algebra structure (J , ·) it is enough to specify n3 structure constants ck

i j ∈ k, namely,

280 I. Kashuba, M.E. Martin / Journal of Algebra 399 (2014) 277–289

Table 2Indecomposable three-dimensional Jordan algebras.

T Multiplication table Observation

T1 e21 = e1, n2

1 = n2, n22 = 0,

e1n1 = n1, e1n2 = n2, n1n2 = 0unitaryassociative

T2 e21 = e1, n2

1 = 0, n22 = 0,

e1n1 = n1, e1n2 = n2, n1n2 = 0unitaryassociative

T3 n21 = n2, n2

2 = 0, n23 = 0,

n1n2 = n3, n1n3 = 0, n2n3 = 0associativenilpotent

T4 n21 = n2, n2

2 = 0, n23 = n2,

n1n2 = 0, n1n3 = 0, n2n3 = 0associativenilpotent

T5 e21 = e1, e2

2 = e2, e23 = e1 + e2,

e1e2 = 0, e1e3 = 12 e3, e2e3 = 1

2 e3

unitarysemi-simple

T6 e21 = e1, n2

1 = 0, n22 = 0,

e1n1 = 12 n1, e1n2 = n2, n1n2 = 0

T7 e21 = e1, n2

1 = 0, n22 = 0,

e1n1 = 12 n1, e1n2 = 1

2 n2, n1n2 = 0

T8 e21 = e1, n2

1 = n2, n22 = 0,

e1n1 = 12 n1, e1n2 = 0, n1n2 = 0

T9 e21 = e1, n2

1 = n2, n22 = 0,

e1n1 = 12 n1, e1n2 = n2, n1n2 = 0

T10 e21 = e1, e2

2 = e2, n21 = 0,

e1e2 = 0, e1n1 = 12 n1, e2n1 = 1

2 n1

unitary

Table 3Four-dimensional Jordan algebras.

J Multiplication table dim Aut(J ) dim Ann(J ) dimJ 2 dim Rad(J ) Observation1

J1 T5 ⊕ ke4 1 0 4 0 USJ2 e2

j = 0, e3e4 = 12 (e1 + e2), ei e j = 1

2 e j ,i = 1,2 and j = 3,4

3 0 4 0 US

J3 ke1 ⊕ ke2 ⊕ ke3 ⊕ ke4 0 0 4 0 USAJ4 B1 ⊕ ke2 ⊕ ke3 1 0 4 1 UAJ5 ke1 ⊕ ke2 ⊕ ke3 ⊕ kn1 1 1 3 1 AJ6 B2 ⊕ ke2 ⊕ ke3 2 0 4 1J7 T10 ⊕ ke3 2 0 4 1 UJ8 T5 ⊕ kn1 2 1 3 1J9 e2

3 = e1 + e2, ei e3 = 12 e3, ein1 = 1

2 n1,i = 1,2

4 0 4 1 U

J10 B2 ⊕ ke2 ⊕ kn2 3 1 3 2J11 T10 ⊕ kn2 3 1 3 2J12 T7 ⊕ ke2 6 0 4 2J13 B2 ⊕B2 4 0 4 2J14 T6 ⊕ ke2 3 0 4 2J15 B1 ⊕B2 3 0 4 2J16 e1n2 = 1

2 n2, ein1 = 12 n1, i = 1,2 4 0 4 2

J17 e1n2 = n2, ein1 = 12 n1, i = 1,2 3 0 4 2 U

J18 ein j = 12 n j , i, j = 1,2 6 0 4 2 U

J19 ke1 ⊕ ke2 ⊕ kn1 ⊕ kn2 4 2 2 2 AJ20 B1 ⊕ ke2 ⊕ kn2 2 1 3 2 AJ21 T2 ⊕ ke2 4 0 4 2 UAJ22 B1 ⊕B1 2 0 4 2 UAJ23 T8 ⊕ ke2 2 1 4 2J24 T9 ⊕ ke2 2 0 4 2J25 n2

1 = e1n2 = n2, ein1 = 12 n1, i = 1,2 2 0 4 2 U

J26 B3 ⊕ ke1 ⊕ ke2 2 1 3 2 AJ27 T1 ⊕ ke2 2 0 4 2 UA

I. Kashuba, M.E. Martin / Journal of Algebra 399 (2014) 277–289 281

Table 3 (continued)

J Multiplication table dim Aut(J ) dim Ann(J ) dimJ 2 dim Rad(J ) Observation1

J28 B2 ⊕ kn2 ⊕ kn3 6 2 2 3J29 T6 ⊕ kn3 4 1 3 3J30 T7 ⊕ kn3 7 1 3 3J31 e1n3 = 1

2 n3, e1ni = ni , i = 1,2 6 0 4 3J32 e1n3 = n3, e1ni = 1

2 ni , i = 1,2 7 0 4 3J33 e1ni = 1

2 ni , i = 1,2,3 12 0 4 3J34 ke1 ⊕ kn1 ⊕ kn2 ⊕ kn3 9 3 1 3 AJ35 B1 ⊕ kn2 ⊕ kn3 5 2 2 3 AJ36 e1ni = ni , i = 1,2,3 9 0 4 3 UAJ37 T2 ⊕ kn3 5 1 3 3 AJ38 T3 ⊕ ke1 3 1 3 3 AJ39 n2

1 = n2, n1n2 = n3, e1ni = ni , i = 1,2,3 3 0 4 3 UAJ40 B3 ⊕ ke1 ⊕ kn3 5 2 2 3 AJ41 T4 ⊕ ke1 4 1 2 3 AJ42 n2

1 = n2, e1ni = ni , i = 1,2,3 5 0 4 3 UAJ43 n2

1 = n1n3 = n2, e1ni = ni , i = 1,2,3 4 0 4 3 UAJ44 T8 ⊕ kn3 4 2 3 3J45 n2

1 = n23 = n2, e1n1 = 1

2 n1 3 1 3 3J46 B2 ⊕B3 4 1 3 3J47 B1 ⊕B3 3 1 3 3 AJ48 n2

3 = n1, e1ni = 12 ni , i = 2,3 5 1 4 3

J49 n23 = n2n3 = n1, e1ni = 1

2 ni , i = 2,3 4 1 4 3

J50 n1n2 = n3, e1ni = 12 ni , i = 2,3 5 0 3 3

J51 T9 ⊕ kn3 3 1 3 3J52 n2

1 = n3, e1n1 = 12 n1, e1n2 = n2 3 1 4 3

J53 n21 = n2 + n3, e1n1 = 1

2 n1, e1n2 = n2 2 1 4 3J54 T1 ⊕ kn3 3 1 3 3 AJ55 n2

3 = n2, e1n3 = 12 n3, e1ni = ni , i = 1,2 4 0 4 3

J56 n21 = n2, e1n3 = 1

2 n3, e1ni = ni , i = 1,2 4 0 4 3

J57 n21 = n2

3 = n2, e1n3 = 12 n3, e1ni = ni ,

i = 1,23 0 4 3

J58 n23 = e1n1 = n1, e1ni = 1

2 ni , i = 2,3 5 0 4 3

J59 n23 = n2n3 = e1n1 = n1, e1ni = 1

2 ni , i = 2,3 4 0 4 3

J60 e1n1 = n1, n1n2 = n3, e1ni = 12 ni , i = 2,3 5 0 4 3

J61 n21 = n2, n2

2 = n1n3 = n4, n1n2 = n3 4 1 3 4 ANJ62 n2

1 = n24 = n2, n1n2 = n3 3 1 2 4 N

J63 n21 = n2, n2

4 = −n2 − n3, n1n2 = n2n4 = n3 4 1 2 4 NJ64 n2

1 = n2, n24 = −n2, n1n2 = n2n4 = n3 5 1 2 4 N

J65 n21 = n2, n1n2 = n2n4 = n3 4 1 2 4 N

J66 n21 = n2, n2

4 = n1n2 = n3 5 1 2 4 ANJ67 T3 ⊕ kn4 6 2 2 4 ANJ68 B3 ⊕B3 6 2 2 4 ANJ69 n2

1 = n2, n1n3 = n4 7 2 2 4 ANJ70 n2

1 = n3n4 = n2 7 1 1 4 ANJ71 T4 ⊕ kn4 8 2 1 4 ANJ72 B3 ⊕ kn3 ⊕ kn4 10 3 1 4 ANJ73 kn1 ⊕ kn2 ⊕ kn3 ⊕ kn4 16 4 0 4 AN

1 In this column (A) stays for J associative, (U) for J unitary, (S) for J semi-simple and (N) for J nilpotent algebra.

ei · e j =n∑

k=1

cki jek, i, j = 1,2, . . . ,n.

The choice of cki j has to reflect the fact that algebra J is commutative and satisfies the Jordan identity.

Thus we obtain:

282 I. Kashuba, M.E. Martin / Journal of Algebra 399 (2014) 277–289

cki j = ck

ji,

n∑a=1

cai j

n∑b=1

cbklc

pab −

n∑a=1

cakl

n∑b=1

cbjacp

ib +n∑

a=1

calj

n∑b=1

cbkic

pab

−n∑

a=1

caki

n∑b=1

cbjacp

lb +n∑

a=1

cakj

n∑b=1

cbilc

pab −

n∑a=1

cail

n∑b=1

cbjacp

kb = 0, (4)

for all i, j,k, l, p ∈ {1,2, . . . ,n}. Observe that the second equation is obtained using the linearizedversion of Jordan identity (2).

The polynomial equations (4) cut out an algebraic variety Jorn in kn3and a point (ck

i j) ∈ Jorn

represents an n-dimensional k-algebra J , along with a particular choice of basis (which gives thestructure constants ck

i j).The general linear group G = GL(V) operates on Jorn changing basis of J

g(J , ·) → (J , ·g), x ·g y = g(

g−1x · g−1 y)

(5)

where J ∈ Jorn , g ∈ G and x, y ∈ V. Denote by J G the set of all images of J under the action of G,i.e.

J G = {g(J , ·) ∈ Jorn

∣∣ g ∈ G}

the G-orbit of J . Then G-orbits J G are in one-to-one correspondence with the set of isomorphismclasses of n-dimensional Jordan algebras. Indeed, two sets of structure constants generate two iso-morphic algebras if and only if there exists an element of G which sends one algebra into another asin (5).

We can consider the inclusion diagram of the Zariski closure of the orbits of elements in Jorn . Wesay that an algebra J1 is a deformation of J2 or that J1 dominates J2 or still that J2 is a specializationof J1 and denote this by J1 →J2, if the orbit J G

2 is contained in the Zariski closure of the orbit J G1 .

A natural problem is to classify the laws whose orbits are Zariski-open in Jorn , we will call suchalgebras rigid. In terms of deformation if J1 is a deformation of a rigid algebra J then J G

1 ∩J G �= ∅and therefore J1 � J . The rigid algebras are of particular interest. Indeed, Jorn as any affine vari-ety could be decomposed into its irreducible components. Then if J ∈ Jorn is rigid, there exists anirreducible component T such that J G ∩ T is a non-empty open in T , thus J G contains T .

The most known sufficient condition for an algebra to be rigid is given in terms of its cohomologygroup. We say that the second cohomology group H2(J ,J ) of a Jordan algebra J with coefficientsin itself vanishes if for every bilinear mapping h :J ×J →J satisfying

h(a,b) = h(b,a),(h(a,a)b

)a + h

(a2,b

)a + h

(a2b,a

) = a2h(b,a) + h(a,a)(ba) + h(a2,ba

)(6)

for all a,b ∈J there exists a linear mapping μ :J →J such that

h(a,b) = μ(ab) − aμ(b) − μ(a)b. (7)

For the precise definition of this group for Jordan algebras we refer to [14].

Proposition 3.1. If J ∈ Jorn, the orbit J G is open as soon as the second Hochschild cohomology groupH2(J ,J ) = 0. In particular it follows that any semi-simple Jordan algebra is rigid.

I. Kashuba, M.E. Martin / Journal of Algebra 399 (2014) 277–289 283

It was originally obtained in [2] for associative and Lie algebras, for the case of Jordan algebras theproof is analogous, see [11].

Example 3.2. Consider B2 ∈Jor2 and let h : B2 ×B2 → B2 be a bilinear map satisfying (6), then

h(e1, e1) = αe1, h(e1,n1) = βe1 + α

2n1, h(n1,n1) = 2βn1

for any α,β ∈ k. Put μ(e1) = −αe1 + n1 and μ(n1) = −2βe1 + n1 and extend by linearity toμ :B2 → B2. We have that (7) holds and then H2(B2,B2) = 0 which implies B2 is rigid.

Following [2], we will describe the deformations of Jordan algebras using ‘one parameter family ofdeformations’. Let

g(t) ∈ Matn(k[t]) (8)

and suppose that g(t) is non-degenerate whenever t �= 0, i.e. g(t) ∈ G. We denote by Jt the algebra(isomorphic to J for any t �= 0) obtained from J by the basis transformation g(t) and let ck

i j(t)denote the corresponding structure constants. Consider the Jordan algebra J1 defined by structureconstants ck

i j(0) with respect to the same basis e1, . . . , en then J is a deformation of J1.

Example 3.3. The deformation J1 →J25 is given by the matrix

g(t) =⎡⎢⎣

1 0 0 10 1 0 00 0 t 0t2 0 0 0

⎤⎥⎦ ,

the basis of (J1)t is At = e1 + e4, Bt = e2, Ct = te3 and Dt = t2e1, where {e1, e2, e3, e4} is the ba-sis given in Section 2. Observe that for any t �= 0, det g(t) = −t3 �= 0 and then (J1)t � J1, while(J1)0 �J25.

Simple but useful observation gives sufficient condition of existence of deformation between de-composable algebras.

Fact 1. Let Ji and J ′i be in J orni , for i = 1,2. If Ji →J ′

i , then J1 ⊕J2 →J ′1 ⊕J ′

2 .

Indeed, if the deformations Ji → J ′i are described by matrices gi(t) for i = 1,2, then in the

appropriate basis g(t) =[

g1(t) 00 g2(t)

]gives the deformation J1 ⊕J2 →J ′

1 ⊕J ′2.

Example 3.4. In Jor2, the deformation ke1 ⊕ ke2 → B1 is given by the matrix

g(t) =[

1 10 t

],

the basis of (ke1 ⊕ ke2)t is At = e1 + e2 and Bt = te2. Observe that for any t �= 0, det g(t) = t �= 0 andthen (ke1 ⊕ ke2)t � ke1 ⊕ ke2, while a simple calculation shows that (ke1 ⊕ ke2)0 � B1. It followsfrom Fact 1 that J4 →J22 since J4 = B1 ⊕ ke2 ⊕ ke3 and J22 = B1 ⊕B1.

The non-existence of deformation for a given pair of algebras will follow from the violation of oneof the conditions below.

284 I. Kashuba, M.E. Martin / Journal of Algebra 399 (2014) 277–289

Fact 2. If J →J1 then dim Aut(J ) � dim Aut(J1), where

Aut(J ) = {g ∈ G

∣∣ (J , ·g) has the same structure constants cki j that J

}.

For the proof just combine that dim(J G \ J G) < dimJ G with dim Aut(J ) = n2 − dimJ G. Notethat this inequality is strict whenever the deformation is non-trivial.

Further we will describe several integer characteristics of Jordan algebras which are locally con-stant on Jorn .

Fact 3. If J →J1 , then dim Rad(J )� dim Rad(J1).

The proof was given in [6] for associative algebras. It uses the upper-semi-continuity argumentfrom [2, II.4] and can be easily extended to the case of Jordan algebras, see [11].

Fact 4. If J →J1 , then dim Ann(J ) � dim Ann(J1).

Proof. We construct the matrix Pn2,n , where each line consists of n coordinates of the product ei · e j ,{1 � i j � n2}. Then dim Ann(J ) � s is equivalent to the fact that rk(Pn2,n) � n − s + 1, therefore{J ∈Jorn | dim Ann(J )� s} is closed. �Fact 5. If J →J1 then dimJ r � dimJ r

1 , for any positive integer r.

Proof. Let Ωn,r be the set of all non-associative words in e1, . . . , en of length r, put l(r) = |Ωn,r |.Analogously to the previous proof consider the matrix Pl(r),n , where each line of Pl(r),n consistsof n coordinates of the corresponding word from Ωn,r with respect to e1, . . . , en . Then dimJ r � sis equivalent to the fact that all minors of degree s + 1 are zeros, therefore {J ∈Jorn | dimJ r � s} isclosed. �Fact 6. Any deformation of a non-associative algebra is again non-associative.

The set of n-dimensional associative Jordan algebras is Jorn ∩ Assocn and thus is a closed variety.Using the same argument in J G we obtain the following fact.

Fact 7. Let J →J1 , and suppose that A1 is an associative subalgebra of J1 of maximal dimension s then anyassociative subalgebra of J has dimension less or equals to s.

Finally, we will use the fact that deformation preserves the action of the semi-simple subalge-bra Jss on J . Namely,

Fact 8. For a non-nilpotent Jordan algebra J consider the pair (Jss,Γ (J )), where Γ (J ) is the adjoint rep-resentation of Jss on J . Then for any irreducible component T of Jorn, (Jss,Γ (J )) is constant on an opensubset of T . If J1 ∈ T then (J1)ss is a subalgebra of Jss and Γ (J1) is the restriction of Γ (J ) on (J1)ss .

For the proof see [11]. As a corollary we get

Fact 9. If J →J1 , then (J1)ss is a subalgebra of Jss and therefore the deformation preserves the correspond-ing Peirce decomposition.

I. Kashuba, M.E. Martin / Journal of Algebra 399 (2014) 277–289 285

4. The algebraic variety Jor4

In this section we will determine the irreducible components of Jor4. First we will describe thegeneric algebras of dimension up to three.

In Jor1, the only rigid algebra is the simple one ke and it is clear that ke → kn. There are sixnon-isomorphic two-dimensional Jordan algebras, two of them are rigid: the semi-simple ke1 ⊕ ke2and the indecomposable non-associative B2, see Example 3.2. All associative two-dimensional algebrasmay be deformed into ke1 ⊕ ke2: ke1 ⊕ ke2 → ke1 ⊕ kn1 by Fact 1, from Example 3.4 follows thatke1 ⊕ ke2 → B1. To see that B1 → B3, for t �= 0 choose the basis At = te1 + n1 and Bt = −t2e1of B1, then the structural constants of this basis specialize to those of B3 when t = 0. The basistransformation At = te1 + n1 and Bt = t2e1 of ke1 ⊕ kn1 shows that ke1 ⊕ kn1 → B3. Finally tocomplete the geometric description of Jor2 we observe that any B is a deformation of kn1 ⊕ kn2 via

transformation g(t) =[

t 00 t

].

In [10] I. Kashuba and I. Shestakov determined that Jor3 is a connected affine variety of dimensionnine with 5 irreducible components given by the Zariski closures of the orbits of the algebras ke1 ⊕ke2 ⊕ ke3, T5, T7, T9 and B2 ⊕ ke2.

We are now ready to enunciate the principal theorem of this section.

Theorem 4.1. The variety Jor4 has 10 irreducible components given by Zariski closures of the orbits of thefollowing algebras

Ω = {J1, J2, J3, J6, J12, J13, J16, J24, J33, J59}.

Proof. We will divide the proof into two parts: first, we show that the algebras in the Ω are rigidand, second, that every structure from J1 to J73 in the algebraic classification can be deformed intoone of the above algebras.

There are three semi-simple algebras J1, J2 and J3 thus they are rigid.Further there is no algebra in Jor4, which dominates J6: by Fact 3 the only possible candidates

to specialize into J6 are J1 to J9, by Fact 2 we may exclude J2, J7 and J9 from this list, while thealgebras J5 and J8 do not dominate J6 due to Fact 4. A non-associative algebra cannot be deformedinto an associative one, therefore J3 � J6 and J4 � J6. Finally the Peirce decomposition of J1, (3),relative to e1, e2, e4 produces the Peirce component P12 �= 0, while for J6 and its orthogonal idem-potents e1, e2, e3 if 0 �= i < j, Pi j = 0 thus J1 does not dominate J6 by Fact 9. This proves that J6 isrigid.

Consider J ∈ Jor4, J → J12, then by Fact 8 for a suitable basis (J12)ss is a subalgebra of Jsssuch that the action of the semi-simple part is preserved under deformation. Choose the basis of J{a,b, c,d} where products are a2 = a, b2 = b, ac = 1

2 c, ad = 12 d and ab = bc = bd = 0 and, using the

multiplication table for the Peirce decomposition (see [14, Chapter III.1]), we have c2 = αa, d2 = βa,cd = γ a for some α,β,γ ∈ k. Since the basis satisfies the linearized version of Jordan identity (2) weobtain that α = β = γ = 0. Thus J �J12 and J12 is rigid.

We check that J13 is rigid. Since dim Rad(J13) = 2 we have that if Ji → J13 then 1 � i � 27.Further Ji has to be non-associative thus i �= 3,4,22 and 27, while using the dimension argumentfor the automorphism group (resp. annihilator) we deduce that J13 does not deform into J9, J12,J16, J18, J19 and J21 (resp. J5, J8, J10, J11, J20, J23 and J26). The algebras J14, J15 and J17could not specialize in J13 by Fact 8, just compare the Peirce components relative to two idempotentsin the semi-simple parts. The algebra J2 does not dominate J13 since in the Peirce decompositionof J2 relative to e1, e2 we obtain non-trivial P12. Finally, note that J13 does not have any associa-tive subalgebra of dimension three thus no one of the remaining algebras could dominate J13 byFact 7.

For the proof of the rigidness of J16 we use the same arguments as for J13. First observe thatthe dimensions of Rad, Aut and Ann of J16 are the same as of J13 and J16 is also non-associativewhat shrinks the list of possible deformations to � = {J1, J2, J6, J7, J14, J15, J17, J24, J25}.Analogously to show that the algebras J14, J15 and J17 do not dominate J16 we use Fact 8. To see

286 I. Kashuba, M.E. Martin / Journal of Algebra 399 (2014) 277–289

that the algebra J2 does not dominate J16 we note that J16 has non-trivial Peirce component P01.Finally, the algebra J16 does not have any associative subalgebra of dimension three thus no one ofthe remaining algebras is a deformation of J16.

Observe that there is no algebra in Jor4 specializing into J24: J24 is non-associative and bothdimensions of annihilator and of radical coincide with those of J16, while dim AutJ24 = 2 thereforethe possible algebra J to dominate J24 belongs to � and dim AutJ < 2, what leaves only J1. Finally,the action of any two idempotents of J1 is not preserved in J24, thus J1 �J24 and J24 is rigid.

For J33 we will calculate the second cohomology group. Let h :J33 ×J33 →J33 be a bilinear mapsatisfying (6), then

h(e1, e1) = αe1, h(e1,ni) = βie1 + α

2ni, h(ni,n j) = β jni + βin j

for any α,βi ∈ k and 1 � i, j � 3. Define a linear mapping μ : J33 → J33 via μ(e1) = −αe1 andμ(ni) = −2βie1 + ni for i = 1,2,3, then (7) holds and H2(J33,J33) = 0, which implies J33 is rigid.

Finally, suppose that J ∈ Jor4 and J → J59 then by Fact 8 one may choose a basis such that(J59)ss ⊆ Jss and the action of (J59)ss should be preserved, so J could only be J12, J13, J32, J58and J60 but then dim Aut(J ) � 4, therefore J59 is rigid.

It remains to show that for any algebra J ∈ Jor4 there exists an algebra J ′ ∈ Ω such that J ′ isa deformation of J . In what follows all transformations are given using the basis from Section 2.

We show that J1 dominates J7, J8, J10, J11, J17, J23, J25, J28, J44, J45, J46, J52, J53, J62,J63, J64 and J65. Example 3.3 gives J1 →J25. Further J25 →J17, indeed for t �= 0 choose the basisAt = e1, Bt = e2, Ct = tn1 and Dt = n2 of J25, in particular, C2

t = t2 Dt and J25 →J17 as t → 0.The specialization J1 → J53 is given as one parameter family of J1, via basis At = e1 + e4,

Bt = te3, Ct = t2e1 and Dt = t2e2, and J53 →J62 via At = te1 +tn2 −tn3, Bt = t2n2 +t2n3, Ct = −t3n3and Dt = tn1. Moreover from [12] we get deformations of nilpotent algebras: J62 → J65, J62 → J63and J63 →J64.

Further, combining the results for three-dimensional algebras in [10] with Fact 1 we obtain thefollowing: J1 → J23, J1 → J7, J7 → J10, J1 → J8 and J8 → J11. To show that J11 → J46, takethe basis transformation At = e1, Bt = n1, Ct = te2 − n2 and Dt = tn2, t �= 0, of J11. Since Bt Ct = t Bt

2and C2

t = Dt + tCt we get the structure J46 when t tends to zero.For the deformation J8 → J45, consider the basis At = e1, Bt = te3, Ct = t2e2 and Dt = te2 + n1

of J8, then when t = 0 we get J45. The basis At = e1, Bt = n1, Ct = n2 and Dt = tn3 of J45 does thetrick for J45 →J44, while J44 →J28 taking At = e1, Bt = tn1, Ct = n2 and Dt = n3 in J44.

Finally, to see that J23 → J52, consider the basis transformation At = e1 + e2, Bt = n1, Ct = te2and Dt = n2 of J23.

Further we show that J2 dominates J9, J18, J48 and J49. To show that J2 → J9, for t �= 0 takethe basis At = e1, Bt = e2, Ct = e3 + e4 and Dt = te4 of J2. Since C2

t = At + Bt , Ct Dt = t( At+Bt2 )

and D2t = 0 we obtain the algebra J9 when t = 0. For the deformation J9 → J18, it is sufficient to

consider the basis At = e1, Bt = e2, Ct = te3 and Dt = n1 of J9. Then, since C2t = t2(At + Bt), when t

tends to zero we obtain J18. Analogously J2 → J49 with the basis At = e1, Bt = te2, Ct = 2te3 andDt = te3 + e4 and J49 →J48 with the basis At = e1, Bt = n1, Ct = tn2 and Dt = n3.

All associative structures, namely J4, J5, J19, J20, J21, J22, J26, J27, J34, J35, J36, J37, J38,J39, J40, J41, J42, J43, J47, J54, J61, J66, J67, J68, J69, J70, J71, J72 and J73 are dominatedby J3.

Using both Fact 1 and the geometric description of Jor2 we obtain J3 →J4, J3 →J5, J5 →J20,J5 → J26, J26 → J19, J19 → J35, J40 → J34, J72 → J73, J4 → J22, and J26 → J47. From Fact 1and the geometric description of Jor3 given in [10] we have J4 → J27, J27 → J21, J21 → J37,J20 → J54, J38 → J41, J41 → J40 and J71 → J72. Finally from [12] we have the following defor-mations between nilpotent algebras J61 →J67, J66 →J68, J66 →J70, J68 →J69 and J69 →J71.

To see that J20 → J38, for t �= 0 choose the basis transformation At = e2, Bt = te1 + n1 + n2,Ct = tn1 − tn2 and Dt = t2n1 of J20. Analogously the dominance J38 → J66 is given by the basisAt = tn1, Bt = t2n2, Ct = t3n3 and Dt = te1 − t2n3 of J38. For the deformation J47 → J61 take t �= 0

I. Kashuba, M.E. Martin / Journal of Algebra 399 (2014) 277–289 287

and consider the basis At = te1 + n1 + n2, Bt = tn1 − tn2 + n3, Ct = t2n1 − tn3 and Dt = t2n3 of J47,then when t = 0 we get J61.

To show that J22 → J39, for t �= 0 use the basis At = e1 + e2, Bt = te2 + n1 + n2, Ct = −tn1 + tn2and Dt = t2n2 of J22. Since B2

t = Ct + t Bt and Bt Dt = t Dt we get the structure J39 when t tendsto zero. In the case of J39 → J43 we consider the basis At = e1, Bt = −tn1, Ct = −t2n3 and Dt =tn2 + tn3 of J39. Analogously, to show that J43 → J42 it is sufficient to consider the basis At = e1,Bt = n1, Ct = n2 and Dt = tn1 − tn3 of J43.

Finally, to complete the dominance of J3 it remains to prove that J42 → J36. For t �= 0 takeAt = e1, Bt = tn1, Ct = n2 and Dt = n3 of J42.

The dominance of the rigid algebras J6 over J15 and J12 over J30 follows from Fact 1. We nextshow that J12 → J32, to get that consider the basis At = e1 + e2, Bt = n1, Ct = n2 and Dt = te2, fort �= 0, of J12. Since D2

t = t Dt we get the algebra J32 when t = 0.Now, we show that J13 dominates J60. For t �= 0, it is sufficient to consider the basis At = e1 + e2,

Bt = −2te1, Ct = n1 + n2 and Dt = tn2 of J13 and then when t tends to zero we obtain J60. Analo-gously J16 dominates J50, use the basis At = e1, Bt = te2, Ct = 2n1 + n2 and Dt = tn1 of J16.

The algebras J14, J29, J31, J51, J55, J56 and J57 are dominated by J24. Once again combiningFact 1 and the deformations obtained in [10] we obtain that J24 → J51, J51 → J29 and J24 → J14.In the case of J14 →J56 and J24 →J57, use for both algebras the basis At = e1 + e2, Bt = t2e1 +n2,Ct = t2n2 and Dt = tn1, then when t = 0 we get J56 and J57 respectively. It remains to prove thatJ56 → J31 and J57 → J55. Analogously, consider the basis At = e1, Bt = tn1, Ct = n2 and Dt = n3,t �= 0 of J56 and J57, since in both cases B2

t = t2Ct , for t = 0 we get J31 and J55, respectively.To complete the geometric description of the variety Jor4 it remains to show that J59 domi-

nates J58. For t �= 0, it is sufficient to consider the basis: At = e1, Bt = n1, Ct = tn2 and Dt = n3of J59. Since Ct Dt = t Bt it is clear that the structural constants of this basis specialize to those of J58when t = 0.

The orbits of Jor4 are represented in Fig. 1. �4.1. Final remarks

1. Observe that for any J ∈Jor4, the basis transformation

g(t) =⎡⎢⎣

t 0 0 00 t 0 00 0 t 00 0 0 t

⎤⎥⎦ ,

gives J →J73 then the orbit J G73 belongs to any irreducible component of Jor4 and hence Jor4

is a connected affine variety. Since the dimension of a variety is the maximum of the dimensionsof its irreducible components, we obtain that

dimJor4 = dimJ G3 = 42 − dim Aut(J3) = 16.

2. The basis transformations in the proof of the theorem, as well as the arrows on Fig. 1 do notprovide the complete list of deformations in Jor4, for any non-rigid algebra J we found at leastone rigid which dominates J . For further examples of deformation of Jor4 see [16].

3. Any algebra J in Ω in the main theorem satisfies the sufficient condition for rigidness of Propo-sition 3.1, namely H2(J ,J ) = 0. Moreover we checked that if I is a Jordan algebra in Jor5and I is a direct sum of rigid algebras of smaller dimensions then H2(I,I) = 0 and therefore Iis rigid.

4. There is only one irreducible component J G3 which contains all associative algebras of Jor4,

though the thickest one.

288 I. Kashuba, M.E. Martin / Journal of Algebra 399 (2014) 277–289

Fig.

1.Th

eor

bits

ofJ

or4

.

I. Kashuba, M.E. Martin / Journal of Algebra 399 (2014) 277–289 289

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