Variety of Jordan algebras in small dimensions
The variety J orn of Jordan unitary algebra structures on k n , k an algebraically closed field with char k 6= 2 , is studied, as well as infinitesimal deformations of Jordan algebras. Also we establish the list of GLn-orbits on J orn, n = 4, 5 under the action of structural transport. The numb...
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irk-123456789-1573852019-06-21T01:27:27Z Variety of Jordan algebras in small dimensions Kashuba, I. The variety J orn of Jordan unitary algebra structures on k n , k an algebraically closed field with char k 6= 2 , is studied, as well as infinitesimal deformations of Jordan algebras. Also we establish the list of GLn-orbits on J orn, n = 4, 5 under the action of structural transport. The numbers jor₄ and jor₅ of irreducible components are 3 and 6 respectively; a list of generic structures is included. 2006 Article Variety of Jordan algebras in small dimensions / I. Kashuba // Algebra and Discrete Mathematics. — 2006. — Vol. 5, № 2. — С. 62–76. — Бібліогр.: 15 назв. — англ. 1726-3255 2000 Mathematics Subject Classification: 14J10, 17C55, 17C10.. http://dspace.nbuv.gov.ua/handle/123456789/157385 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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The variety J orn of Jordan unitary algebra
structures on k
n , k an algebraically closed field with char k 6= 2 ,
is studied, as well as infinitesimal deformations of Jordan algebras.
Also we establish the list of GLn-orbits on J orn, n = 4, 5 under
the action of structural transport. The numbers jor₄ and jor₅ of
irreducible components are 3 and 6 respectively; a list of generic
structures is included. |
| format |
Article |
| author |
Kashuba, I. |
| spellingShingle |
Kashuba, I. Variety of Jordan algebras in small dimensions Algebra and Discrete Mathematics |
| author_facet |
Kashuba, I. |
| author_sort |
Kashuba, I. |
| title |
Variety of Jordan algebras in small dimensions |
| title_short |
Variety of Jordan algebras in small dimensions |
| title_full |
Variety of Jordan algebras in small dimensions |
| title_fullStr |
Variety of Jordan algebras in small dimensions |
| title_full_unstemmed |
Variety of Jordan algebras in small dimensions |
| title_sort |
variety of jordan algebras in small dimensions |
| publisher |
Інститут прикладної математики і механіки НАН України |
| publishDate |
2006 |
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http://dspace.nbuv.gov.ua/handle/123456789/157385 |
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Variety of Jordan algebras in small dimensions / I. Kashuba // Algebra and Discrete Mathematics. — 2006. — Vol. 5, № 2. — С. 62–76. — Бібліогр.: 15 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT kashubai varietyofjordanalgebrasinsmalldimensions |
| first_indexed |
2025-07-14T09:49:23Z |
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2025-07-14T09:49:23Z |
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1837615351329718272 |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Number 2. (2006). pp. 62 – 76
c© Journal “Algebra and Discrete Mathematics”
Variety of Jordan algebras in small dimensions
Iryna Kashuba
Communicated by I. P. Shestakov
Abstract. The variety J orn of Jordan unitary algebra
structures on k
n , k an algebraically closed field with chark 6= 2 ,
is studied, as well as infinitesimal deformations of Jordan algebras.
Also we establish the list of GLn-orbits on J orn, n = 4, 5 under
the action of structural transport. The numbers jor4 and jor5 of
irreducible components are 3 and 6 respectively; a list of generic
structures is included.
1. Introduction
In his survey [1] of the classification theory of finite dimensional associa-
tive algebras E. Study gave a list of isomorphism classes of associative
algebras up to dimension four. He considered all algebras of a given
dimension as an algebraic variety and he showed that for associative al-
gebras of dimension more than three it is impossible to find a generic
algebra, or equivalently that corresponding variety has more than one
irreducible components. In modern language the problem can be formu-
lated as follows. Let k be an algebraically closed field of characteristic
6= 2, V an n-dimensional k-vector space and e1, e2, . . . , en a basis of
V . In order to endow V with the k-algebra structure we specify n3
structure constants ck
ij ∈ k,
ei · ej =
n∑
k=1
ck
ijek.
Equivalently, an algebra structure on V is given by bilinear map, i.e. an
element of V ∗ ⊗ V ∗ ⊗ V , which we consider together with its natural
2000 Mathematics Subject Classification: 14J10, 17C55, 17C10..
I. Kashuba 63
structure of an algebraic variety over k. Moreover for any class of al-
gebras defined by identities, the corresponding multiplication maps form
an algebraically closed subset Algn of V ∗ ⊗ V ∗ ⊗ V . The linear group
GL(V ) operates on Algn by so-called ’transport of structure’ action,
which is induced by the natural action of GL(V ) on V ∗ ⊗ V ∗ ⊗ V
(g,J )(x, y) → gJ (g−1x, g−1y) (1)
for any J ∈ Algn , g ∈ GL(V ) and x, y ∈ V . For any J ∈ Algn we
denote by J GL(V ) the orbit of J under GL(V ) and we can consider the
inclusion diagram of the Zariski closure of orbits of elements in Algn .
More precisely we define for J1, J2 ∈ Algn that J1 deforms to J2 if
J2 lies in the Zariski closure of the orbit of J1 (J1 → J2 ). Algebraic de-
formation theory was introduced for associative algebras by Gerstenhaber
[2], [3], and was extended to Lie algebras by Nijenhius and Richardson
[4]. The corresponding varieties of associative algebras Assocn and of
Lie algebras Lien are of fundamental importance in the theory of algebra
and their deformations. Flanigan in [9] referred to the study of Assocn
as ’algebraic geography’.
The geometry of both Assocn and Lien is rather complicated. It is
known that the number of irreducible components increases exponentially
with n [5], [6] and their dimensions are at most 4
27n3 + O(r8/3) for
Assocn and 2
27n3+O(r8/3) for Lien [6]. However, a complete description
of the orbits and degeneration partial order is only known for n ≤ 5 in
associative case [5] and n ≤ 4 for Lie algebras [7]. Many but not all
components of Assocn are orbit closures [9]. An algebra J such that
the closure of J GL(V ) is a component is called rigid. Every semisimple
associative or Lie algebra is known to be rigid [2], [6].
In this paper we introduce the variety of Jordan unitary algebras
J orn . We extend some properties and facts mentioned above for Assocn
and Lien to the case of Jordan algebras. In Section 2, we consider
infinitesimal deformations of Jordan algebras. We show that semisimple
Jordan algebras are rigid and also prove analogue of the "straightening-
out theorem" of Flanigan [10]. In Section 3, we define an algebraic
variety J orn and write down explicitly some routine properties as well
as some invariants. Finally in Section 4, we will use them to establish
the list of GL n-orbits on J orn for n = 4, 5 .
64 Variety of Jordan algebras in small dimensions
2. Infinitesimal deformations of Jordan algebras
Recall, that a Jordan k-algebra is an algebra J with a multiplication ” ·”,
satisfying the following relations for any a, b ∈ J
a · b = b · a
((a · a) · b) · a = (a · a) · (b · a).
(2)
In particular any commutative associative algebra is Jordan.
Let V be a vector space associated to J and K = k((t)) be the
quotient field of the ring of formal series in one variable t. If we put
VK = V ⊗k K then any bilinear function f : V × V → V , in particular
a multiplication in J can be extended to a function VK × VK → VK .
Let ft be a bilinear function VK × VK → VK of the following for
ft(a, b) = a · b + tF1(a, b) + t2F2(a, b) + . . . ,
where each Fi is a bilinear function defined over k . We suppose that ft
is a Jordan product i.e. it satisfies (2). We may consider the algebra Jt =
(VK , ft) as the generic element of a ’one parameter family of deformations
of J ’ as in [2]. The condition (2) for ft is equivalent to having for all
a, b ∈ V and all ν = 0, 1, 2
Fν(a, b) = Fν(b, a)∑
cλ+µ+γ=ν
λ>0 µ>0 γ>0
Fγ(Fµ(a, a), Fλ(b, a)) − Fγ(Fµ(Fλ(a, a), b), a) = 0 (3)
In particular for ν = 0 we obtain (2) for original multiplication and
for ν = 1 the relations (3) could be written in the form:
F1(a
2, b · a) − F1(a
2 · b, a) + a2 · F1(b, a) − F1(a
2, b) · a+
+ F1(a, b) · (b · a) − (F1(a, a) · b) · a = 0 (4)
Due to Theorem II.8.12, [8] any function satisfying (4) defines a null
extension of a Jordan algebra J by Jordan bimodule J , i.e. any such
function is 2-cocycle of Jordan algebra J with coefficients in J .
From [2], two one parameter families of the deformations ft and
gt are called equivalent if gt(a, b) = Φ−1
t ft(Φta,Φtb), where Φt is an
automorphism of VK of the form:
Φt(x) = x + tφ1(x) + t2φ2(x) + . . . . (5)
Moreover if gt(a, b) = a · b+ tG1(a, b)+ . . . and ft and gt are equivalent
then G1(a, b) = F1(a, b)+a ·φ1(b)+b ·φ1(a)−φ1(a ·b). If gt is equivalent
to ft(x, y) = x · y it is called a trivial deformation. It is obvious that the
obtained algebra Jt = (VK , gt) is isomorphic to J ⊗k K.
I. Kashuba 65
Proposition 2.1. Any one-parameter family ft of deformations of J
is equivalent to gt(a, b) = a · b + tnFn(a, b) + tn+1Fn+1(a, b) + . . . , with
Fn(a, b) being the non-split extension of J by Jordan bimodule J .
Proof. Let ft(a, b) = a · b + tnFn(a, b) + . . . . Writing (3) for ν = n we
obtain that Fn defines null extension of J by J . Suppose that Fn
corresponds to split extension, i.e. as in [8], II.8 Fn(a, b) = φ(a · b) −
φ(a) · b− a · φ(b) for some φ : J → J . Choosing Φt(a) = a + tnφ(a) we
obtain that
Φ−1
t ft(Φta,Φtb) = ab + tn+1Fn+1(a, b) + . . . .
Since
ft(Φta,Φtb) = ft(a + tnφ(a), b + tnφ(b)) =
= ft(a, b) + tnft(φ(a), b) + tn(a, φ(b)) + t2nft(φ(a), φ(b)) =
= ab + tnFn(a, b) + tnφ(a)b + tnaφ(b) + tn+1(. . . )
and
Φt(ab + tn+1Fn+1(a, b) + . . . ) = ab + tnφ(ab) + tn+1(. . . ).
As we already mentioned in the introduction that one of the impor-
tant problems in geometric classification is to describe rigid algebras
(i.e. algebras which have only trivial deformations). Using last result we
obtain the following class of rigid algebras.
Corollary 2.2. If any null extension of J by J is split then J is rigid.
In particular, from [12] follows that any semisimple Jordan algebra is
rigid. Indeed, in this case the set of equivalence classes of null extension
of J by any Jordan bimodule M is trivial, in other words any extension
is split.
Finally let us discuss the structure of the deformed algebra Jt and
compare it with that of J (or to be precise, with that of JK ). Any
finite dimensional Jordan algebra admits Levi-Maltsev decomposition
J = S(J ) + RadJ , where S(J ) is a semisimple subalgebra of J and
RadJ its radical. If ft is the generic element of a one parameter family
of deformation of J then, as in [9], one can construct a deformation
equivalent to the given one such that the new deformation preserves the
original multiplication in S(JK) as well as action of S(JK) on RadJ .
Indeed, suppose that we are given the generic element of a one param-
eter family of deformations of J algebra Jt given by multiplication
ft(x, y) = xy + tF1(x, y) + t2F2(x, y) . . . .
66 Variety of Jordan algebras in small dimensions
Theorem 2.3. There exists a one parameter family of deformations of
J with the generic element gt(x, y) which is equivalent to ft such that
the radical of algebra Jt = (VK , gt) is RadJt = R ⊗k K, where R is a
nilpotent ideal in J . Moreover, for all x, y ∈ S(JK) gt(x, y) = x ·y and
for all x ∈ S(JK), z ∈ RadJK gt(x, z) = x · z .
Proof. First we consider a k-basis of Rad(Jt, ft) = {ξ1, ξ2, . . . , ξr}. We
can always choose ξi = zi + ai
1t + ai
2t
2 + . . . with zi, ai
k ∈ J such
that z1, . . . , zr are linearly independent. Moreover, by definition we can
choose ξi in a such way that zi are k-linearly independent. Indeed, if for
any ξ ∈ RadJt ξ = z + t(. . . ) we always can write a linear combination
z = α1z1 + α2z2 + . . . αszs for some s < r then
RadJt ∋ t−1(ξ − α1ξ1 − . . . αkξk) = b + t(. . . )
and b is again a linear combination of z1, . . . , zs . Repeating it we ob-
tain that any ξ is a linear combination of ξ1, . . . , ξs what contradicts
to the fact that dim Rad(ft) = r > k. Since ξi belongs to the radical
Rad(Jt, ft), zi also generates a nilpotent ideal and in particular zi be-
longs to the radical RadJ . Now extend z1, . . . , zr to a k-basis of J
and define Φt in JK in the form id + tφ1 + . . . such that Φt(ξi) = zi.
The multiplication Φt · ft · (Φ−1
t × Φ−1
t ) satisfies the first part of the
theorem.
To prove the second part we introduce a deformation of the homo-
morphism, as in [9]. If f : A → B is a homomorphism of associative
algebras then a deformation of f is given by K-algebra homomorphism
ft : AK → BK in the form ft = f + tF1 + t2F2 + . . . with Fi : A → B
k-linear. The deformations ft is called trivial if there exists an automor-
phism βt : BK → BK such that ft = βt · f . We say that f is rigid if all
the deformation of ft are trivial.
Further, for any Jordan algebra I we denote by U(I) its universal
enveloping algebra. The associative algebra U(I) is a factor algebra of
the associative free algebra F (I) by the ideal generated by right mul-
tiplication maps Ra, a ∈ I, see [8], II.8. We define an application:
g : U(S(J )) ⊗ U(S(J )) → End(U(J )) , g(x, x′)(y) = x ∗ y ∗ x′ for all
x, x′ ∈ U(S(J )) and y ∈ U(J ) , where U(S(J )) , U(J ) are universal
enveloping algebras for S(J ) and J respectively and ∗ is a multiplica-
tion in U(J ) . Now we consider the universal enveloping algebra U(Jt)
of Jt with multiplication
m(a, b) = t−rF−r(a, b) + · · · + F0(a, b) + tF1(a, b) + . . . ,
for a, b ∈ U(Jt) . The multiplication in associative free algebra F (Jt)
is k-linear therefore for elements of J Fi is 0 for i < 0. Using k-
linearity we obtain that m(a, b) = a∗b+tF1(a, b)+ . . . . Finally we define
I. Kashuba 67
gt : U(S(Jt))⊗U(S(Jt)) → End(U(Jt)) , gt(w, w′)(v) = m(m(w, v), w′)
for all w, w′ ∈ U(S(Jt)) and v ∈ U(Jt) . The homomorphism gt is a
deformation of g. On the other hand U(S(J )) is a semisimple algebra
and therefore from [10] it follows that gt must be trivial. Therefore we
obtain an automorphism βt : U(Jt) → U(Jt) such that gt = βt · g
which induce a K−linear automorphism Ωt of JK in the form Ωt(x) =
x + tω1(x) + . . . such that the composition Ω−1
t · ft · (Ωt × Ωt) satisfies
the second part of the theorem. We denote this multiplication by gt .
Roughly speaking, this theorem proves that the radical of JK shrinks
under deformation to that of Jt while a semisimple part of Jt contains
semisimple part of JK and absorb part of its radical. In particular,
remark that the dimension of the radical does not increase under defor-
mation.
3. Algebraic variety J orn
As we already mentioned in introduction, J orn is an algebraic subvariety
in affine space A
n3
≃ V ∗ ⊗ V ∗ ⊗ V . For any chosen set of structure
constants ck
ij ∈ k the product defined by it defines a Jordan algebra if it
satisfies the identities (2), i.e.
ck
ij = ck
ji,∑n
a=1 ca
ij
∑n
b=1 cb
klc
p
ab −
∑n
a=1 ca
kl
∑n
b=1 cb
jac
p
ib +
∑n
a=1 ca
lj
∑n
b=1 cb
kic
p
ab−∑n
a=1 ca
ki
∑n
b=1 cb
jac
p
lb +
∑n
a=1 ca
kj
∑n
b=1 cb
ilc
p
ab −
∑n
a=1 ca
il
∑n
b=1 cb
jac
p
kb = 0,
(6)
for all i, j, k, l, p ∈ {1, 2, . . . , n}. The first one guaranties that the product
is commutative and the second one provided by linearization of Jordan
identity. Moreover, since we consider only unitary algebras we can choose
e1 as the identity element of J . In addition to (6) last condition trans-
lates into
cj
1i = cj
i1 = δij i, j = 1, . . . , n. (7)
Equations (7) and (6) cut out an algebraic variety J orn in k
n3
=
V ∗ ⊗ V ∗ ⊗ V . A point (ck
ij) ∈ J orn represents n-dimensional unitary
k-algebra J , along with a particular choice of basis (which gives the
structural constants ck
ij ). A change of basis in J gives rise to a pos-
sible different point of J orn or equivalently GL(V ) operates on Algn .
The set of different GL(V )-orbits of this action is in one-to-one corre-
spondence with the isomorphism classes of n-dimensional Jordan alge-
bras. Recall from the introduction that J1 is called a deformation of J2
68 Variety of Jordan algebras in small dimensions
(J1 → J2 ) if J
GL(V )
2 ⊂ J orn is contained in the Zariski closure of the
orbit J
GL(V )
1 .
Lemma 3.1. If J ∈ J orn and I is a subvariety of J orn then J ∈ I
implies:
n2 − dim (Aut(J )) ≤ dim(I).
In particular for J1 ∈ J orn and J → J1 we have
dim Aut(J ) ≤ dim Aut(J1).
Proof. From [13], p. 98 follows that dim I ≤ dimJ GL(V ). To finish the
proof use J GL(V ) = GL(V )/Aut(J ).
This lemma gives a partial order on the set of GL-orbits of Jordan
algebras. The following lemma is the basic tool for construction of defor-
mation between two algebras.
Lemma 3.2. If there exists a curve Γ in J orn which generically lies in
I and which cuts J GL(V ) in special point than J ∈ Ī .
The proof follows directly from the definition. To illustrate the lemma
let us consider J or2 the variety of unitary 2-dimensional Jordan alge-
bras. There are only two non-isomorphic unitary Jordan algebras in this
dimension both are associative algebras J1 = k × k and J2 = k[x]/x2.
We choose a basis e1, e2 corresponding to primitive idempotents of J1.
And consider the transformation f1 = e1 + e2 and f2 = te2 , for some
parameter t. Then for any t ∈ k
∗ the new algebra J ′ is isomorphic to
J1. For t = 0 we get the following multiplication f2
1 = f1 , f2
2 = tf2 = 0
and f1 · f2 = f2 which is J2. Hence J1 → J2 , all algebras of J or2
belong to the closure of J
GL(V )
1 and therefore J or2 is irreducible sub-
variety of A
8
k
of dimension 4. Let Commn denote the algebraic variety
of commutative associative unitary algebras of dimension n. As we just
showed J or2 = Comm2. In general, any commutative associative al-
gebra is Jordan algebra and therefore Commn is a closed subvariety in
J orn. If J1 ∈ J orn is associative algebra and J1 → J2 then J2 is also
associative.
Proposition 3.3. The following sets are closed in Zariski topology in
J orn :
1. {J ∈ J orn|dim RadJ ≥ s}
2. {J ∈ J orn|dimJm ≥ s}
I. Kashuba 69
for all positive integers m, s.
Proof. The proof for the first set is given in [11]. To prove the second
statement choose a basis {e1, . . . , en} of algebra J ∈ J orn and consider
the set Ω of all commutative words of the length m in n variables
{e1, . . . , en}. We denote by Pn,m its cardinality. Any such word can be
written
wl(e1, . . . , en) = f l
1e1 + · · · + f l
nen,
where f l
i is a polynomial in structure constants ck
ij of J . Further let
A be the matrix of dimension Pm,n × n where to every word from Ω
corresponds the line (f l
1, . . . , f
l
n) . dimJm ≤ r The fact that dimJm ≤
r is equivalent to the fact that all minors of degree s + 1 are zeros,
therefore we obtain a finite number of equations for structure constants
and consequently the set defined by these identities is Zariski closed.
4. Varieties J or4 and J or5
In this section we study the variety J orn for n = 4, 5. Let Commn
be a variety define by n-dimensional commutative associative algebras
with the identity. Obviously, Commn is a closed subset in J orn and
therefore is affine subvariety of J orn . In [14] Mazzola proved that for
n ≤ 6 Commn is irreducible affine variety and its only component is a
Zariski closure of the orbit of semisimple associative commutative algebra
k × · · · × k. Thus to complete a geometric classification for J orn for
n ≤ 6 it is enough to deal with non-associative algebras.
Example 4.1. Consider 3-dimensional unitary non-associative Jordan
algebras. In fact, there are only two non-isomorphic non-associative Jor-
dan algebras in J or3: simple Jordan algebra J1 = {e1, e2, a} with e1, e2
orthogonal idempotents and e1 · a = e2 · a = 1
2a , a2 = e1 + e2 and a Jor-
dan algebra with one-dimensional radical J2 = {e1, e2, a} with the same
multiplication table except for a2 = 0. Consider the following transfor-
mation of J1 : f1 = e1 , f2 = e2 and f3 = ta . For t = 0 we obtain the
multiplication table of J2 and therefore J1 → J2 . Hence J or3 con-
sists of two irreducible components: one comes from Jordan associative
algebra k × k × k and the other one comes from J1 .
4.1. J or4
Let first J be a 4-dimensional Jordan algebra. From [15] we obtain the
following list of non-isomorphic Jordan, non-associative unitary algebras
of dimension 4. Here c := c1 + c2 .
70 Variety of Jordan algebras in small dimensions
1. dim RadJ = 0 :
J1 =
c1 c2 a c3
c1 c1 0 1
2a 0
c2 0 c2
1
2a 0
a 1
2a 1
2a c 0
c3 0 0 0 c3
J2 =
c1 c2 a b
c1 c1 0 1
2a 1
2b
c2 0 c2
1
2a 1
2b
a 1
2a 1
2a 0 1
2c
b 1
2b 1
2b 1
2c 0
dim AutJ1 = 1 dim AutJ2 = 1
2. dim RadJ = 1 :
J3 =
c1 c2 c3 b
c1 c1 0 0 1
2b
c2 0 c2 0 1
2b
c3 0 0 c3 0
b 1
2b 1
2b 0 0
J4 =
c1 c2 a b
c1 c1 0 1
2a 1
2b
c2 0 c2
1
2a 1
2b
a 1
2a 1
2a c 0
b 1
2b 1
2b 0 0
dim AutJ3 = 2 dim AutJ4 = 4
3. dim RadJ = 2 :
J5 =
c1 c2 a b
c1 c1 0 1
2a b
c2 0 c2
1
2a 0
a 1
2a 1
2a 0 0
b b 0 0 0
J6 =
c1 c2 a b
c1 c1 0 1
2a b
c2 0 c2
1
2a 0
a 1
2a 1
2a b 0
b b 0 0 0
dim AutJ5 = 3 dim AutJ6 = 2
J5 =
c1 c2 a b
c1 c1 0 1
2a 1
2b
c2 0 c2
1
2a 1
2b
a 1
2a 1
2a 0 0
b 1
2b 1
2b 0 0
dim AutJ7 = 6
4. When dim RadJ = 3 all 4-dimensional Jordan unitary algebras are
associative.
Theorem 4.2. The irreducible components of J or4 are the Zariski clo-
sures of the orbits of algebras Ω = {J1,J2}.
Proof. The proof consists of two parts. First, we show that any Jordan
algebra J from the above list is dominated by either algebra J1 or J2 .
Second, we will show that both J1 and J2 are rigid. To show that
J1 → J3 we construct transformation of J1 as C1t = c1 , C2t = c2 ,
I. Kashuba 71
C3t = c3 , At = ta . Then it is clear that the structure constants of this
basis specialize to those of J3. Analogously, J6 → J5 with C1t = c1 ,
C2t = c2 , At = ta , Bt = b ; J2 → J4 with C1t = c1 , C2t = c2 ,
At = a + b , Bt = tb ; J4 → J7 with C1t = c1 , C2t = c2 , At = ta ,
Bt = b ; and J1 → J6 with C1t = c1+c3 , C2t = c2 , At = ta , C3t = t2c .
The second part of the proof is trivial in this case since both J1 and
J2 are semisimple and therefore rigid by Corollary 2.3.
Therefore we obtain that J or4 \ Comm4 is affine variety with 2
irreducible components. Finally, J or4 consists of three irreducible com-
ponents, two provided by non-associative Jordan semisimple algebra and
one by the associative commutative semisimple algebra k × k × k × k .
4.2. J or5
Now let J be a 5-dimensional Jordan unitary algebra. Again, from [15]
we obtain the following list of non-associative Jordan unitary algebras of
dimension 5 . Here c := c1 + c2 .
1. dim RadJ = 0 :
J1 =
c1 c2 a c3 c4
c1 c1 0 1
2a 0 0
c2 0 c2
1
2a 0 0
a 1
2a 1
2a c 0 0
c3 0 0 0 c3 0
c4 0 0 0 0 c4
J2 =
c1 c2 a b c3
c1 c1 0 1
2a 1
2b 0
c2 0 c2
1
2a 1
2b 0
a 1
2a 1
2a 0 1
2c 0
b 1
2b 1
2b 1
2c 0 0
c3 0 0 0 0 c3
dim AutJ1 = 1 dim AutJ2 = 3
J3 =
c1 c2 a b d
c1 c1 0 1
2a 1
2b 0
c2 0 c2
1
2a 1
2b 0
a 1
2a 1
2a c 0 0
b 1
2b 1
2b 0 c 0
d 1
2d 1
2d 0 0 c
dim AutJ3 = 6
72 Variety of Jordan algebras in small dimensions
2. dim RadJ = 1 :
J4 =
c1 c2 a c3 c4
c1 c1 0 1
2a 0 0
c2 0 c2
1
2a 0 0
a 1
2a 1
2a 0 0 0
c3 0 0 0 c3 0
c4 0 0 0 0 c4
J5 =
c1 c2 a c3 b
c1 c1 0 1
2a 0 0
c2 0 c2
1
2a 0 0
a 1
2a 1
2a c 0 0
c3 0 0 0 c3 b
c4 0 0 0 b 0
dim AutJ4 = 2 dim AutJ5 = 2
J6 =
c1 c2 a b c3
c1 c1 0 1
2a 1
2b 0
c2 0 c2
1
2a 1
2b 0
a 1
2a 1
2a c 0 0
b 1
2b 1
2b 0 0 0
c3 0 0 0 0 c3
J7 =
c1 c2 a b d
c1 c1 0 1
2a 1
2b 1
2d
c2 0 c2
1
2a 1
2b 1
2d
a 1
2a 1
2a 0 1
2c 0
b 1
2b 1
2b 1
2c 0 0
d 1
2d 1
2d 0 0 0
dim AutJ6 = 4 dim AutJ7 = 7
3. dim RadJ = 2 :
J8 =
c1 c2 a c3 b
c1 c1 0 1
2a 0 0
c2 0 c2
1
2a 0 0
a 1
2a 1
2a 0 0 0
c3 0 0 0 c3 b
c4 0 0 0 b 0
J9 =
c1 c2 a b c3
c1 c1 0 1
2a b 0
c2 0 c2
1
2a 0 0
a 1
2a 1
2a b 0 0
b b 0 0 0 0
c3 0 0 0 0 c3
dim AutJ8 = 3 dim AutJ9 = 2
J10 =
c1 c2 a b c3
c1 c1 0 1
2a b 0
c2 0 c2
1
2a 0 0
a 1
2a 1
2a 0 0 0
b b 0 0 0 0
c3 0 0 0 0 c3
J11 =
c1 c2 a b c3
c1 c1 0 1
2a 1
2b 0
c2 0 c2
1
2a 1
2b 0
a 1
2a 1
2a 0 0 0
b 1
2b 1
2b 0 0 0
c3 0 0 0 0 c3
dim AutJ10 = 3 dim AutJ11 = 6
J12 =
c1 c2 c3 a b
c1 c1 0 0 1
2a 1
2b
c2 0 c2 0 1
2a 0
c3 0 0 c3 0 1
2b
a 1
2a 1
2a 0 0 0
b 1
2b 0 1
2b 0 0
J13 =
c1 c2 a b d
c1 c1 0 1
2a 1
2b 1
2d
c2 0 c2
1
2a 1
2b 1
2d
a 1
2a 1
2a c 0 0
b 1
2b 1
2b 0 0 0
d 1
2d 1
2d 0 0 0
dim AutJ12 = 3 dim AutJ13 = 6
I. Kashuba 73
4. dim RadJ = 3 :
J14 =
c1 c2 a b d
c1 c1 0 a b 1
2d
c2 0 c2 0 0 1
2d
a a 0 0 0 0
b b 0 0 0 0
d 1
2d 1
2d 0 0 a
J 0
14 =
c1 c2 a b d
c1 c1 0 a b 1
2d
c2 0 c2 0 0 1
2d
a a 0 0 0 0
b b 0 0 0 0
d 1
2d 1
2d 0 0 0
dim AutJ14 = 4 dim AutJ 0
14 = 5
J15 =
c1 c2 a b d
c1 c1 0 a b 1
2d
c2 0 c2 0 0 1
2d
a a 0 b 0 0
b b 0 0 0 0
d 1
2d 1
2d 0 0 b
J 0
15 =
c1 c2 a b d
c1 c1 0 a b 1
2d
c2 0 c2 0 0 1
2d
a a 0 b 0 0
b b 0 0 0 0
d 1
2d 1
2d 0 0 0
dim AutJ15 = 3 dim AutJ 0
15 = 4
J16 =
c1 c2 a b d
c1 c1 0 a 0 1
2d
c2 0 c2 0 b 1
2d
a a 0 0 0 0
b 0 b 0 0 0
d 1
2d 1
2d 0 0 a + b
J
(1,0)
16 =
c1 c2 a b d
c1 c1 0 a 0 1
2d
c2 0 c2 0 b 1
2d
a a 0 0 0 0
b 0 b 0 0 0
d 1
2d 1
2d 0 0 a
dim AutJ16 = 2 dim AutJ
(1,0)
16 = 3
J 0
16 =
c1 c2 a b d
c1 c1 0 a 0 1
2d
c2 0 c2 0 b 1
2d
a a 0 0 0 0
b 0 b 0 0 0
d 1
2d 1
2d 0 0 0
J17 =
c1 c2 a b d
c1 c1 0 a 1
2b 1
2d
c2 0 c2 0 1
2b 1
2d
a a 0 0 0 b
b 1
2b 1
2b 0 0 0
d 1
2d 1
2d b 0 0
dim AutJ 0
16 = 4 dim AutJ17 = 5
J18 =
c1 c2 a b d
c1 c1 0 a 1
2b 1
2d
c2 0 c2 0 1
2b 1
2d
a a 0 0 0 0
b 1
2b 1
2b 0 a a
d 1
2d 1
2d 0 a a
J19 =
c1 c2 a b d
c1 c1 0 a 1
2b 1
2d
c2 0 c2 0 1
2b 1
2d
a a 0 0 0 0
b 1
2b 1
2b 0 0 0
d 1
2d 1
2d 0 0 0
dim AutJ18 = 4 dim AutJ19 = 7
5. For dim RadJ = 4 all Jordan algebras are associative commutative.
The geometric classification in this case is the following:
74 Variety of Jordan algebras in small dimensions
Theorem 4.3. The irreducible components of J or5 are the Zariski clo-
sures of the orbits of algebras: Ω = {J1,J2,J3,J18,J12}.
Proof: Again, we show that for any Jordan algebra J from the above
list exist an algebra J̃ ∈ Ω such that J̃ is a deformation of J .
First, we establish deformations in the above list of algebras by con-
structing transformations which specialize the structural constant of one
algebra into another one.
1. J1 → J4 with C1t = c1 , C2t = c2 , C3t = c3 , C4t = c4 , At = ta ;
2. J1 → J5 with C1t = c1 , C2t = c2 , C3t = c3 + c4 , C4t = tc4 ,
At = a ;
3. J4 → J8 with C1t = c1 , C2t = c2 , C3t = c3 + c4 , C4t = tc4 ,
At = a ;
4. J2 → J6 with C1t = c1 , C2t = c2 , C3t = c3 , At = a + b ,
Bt = ta + b ;
5. J6 → J11 with C1t = c1 , C2t = c2 , C3t = c3 , At = ta , Bt = b ;
6. J3 → J7 with C1t = c1 , C2t = c2 , At = 1
2a + b , Bt = 1
2a − bi ,
Dt = td ;
7. J7 → J13 with C1t = c1 , C2t = c2 , At = a + b , Bt = ta − bi ,
Dt = d ;
8. J9 → J10 with C1t = c1 , C2t = c2 , C3t = c3 , At = ta , Bt = b ;
9. J14 → J 0
14 with C1t = c1 , C2t = c2 , At = a , Bt = b , Dt = td ;
10. J15 → J 0
15 with C1t = c1 , C2t = c2 , At = a , Bt = b , Dt = td ;
11. J16 → J 1,0
16 with C1t = c1 , C2t = c2 , At = t2a , Bt = b ,
Dt = td ;
12. J 1,0
16 → J 0
16 with C1t = c1 , C2t = c2 , At = a , Bt = b , Dt = td ;
13. J18 → J19 with C1t = c1 , C2t = c2 , At = a , Bt = b , Dt = td ;
14. J15 → J14 with C1t = c1 , C2t = c2 , At = ta , Bt = b , Dt = d ;
15. J1 → J16 with C1t = c1+c3 , C2t = c2+c4 , At = ta , C3t = t2c1 ,
C4t = t2c2 ;
16. J1 → J9 with C1t = c1 +c3 , C2t = c2 , At = ta , C3t = t2c1 + c2 ,
C4t = c4 ;
I. Kashuba 75
17. J9 → J17 with C1t = c1 + c3 , C2t = c2 , At = a + c1 , Bt = b ,
C3t = c3 + tb .
Now we prove that each algebra in Ω has only trivial transformations.
First, we observe that it suffices to consider only deformation between
non-associative algebras. Further, since J1,J2,J3 are semisimples, they
are rigid by Corollary 2.2. As the dimension of the radical does not
increase under deformation we get that J1,J2 are the only candidates
to deform into J12 e J18 . But J12 can not be deformed in J1 , for J1
contain a associative subalgebra of dimension 4 . By Theorem (2.3) the
algebra J12 could not be deformed in J2 since the action of semisimple
part of radical is preserved under deformation. Finally, the algebra J18
is quadratic when both J1 and J2 are not.
We obtain that J or5\Comm5 is an affine algebraic variety consisting
of 5 irreducible components. Then J or5 is an algebraic variety with six
irreducible components, each of them is the closure of one of the Jordan
algebras from {k × k × k × k × k, Ω}.
5. Acknowledgments
The author is very grateful to her supervisor Prof. I. Shestakov for in-
troducing this subject and many helpful discussions. The author was
supported by the Fundação de Amparo á Pesquisa do Estado de São
Paulo (99/12223-5).
References
[1] E. Study Ueber Systeme complexer Zahlen und ihre Anwendung in der Theorie der
Transformationsgruppen, Monatsh. f. Math. I. (1890), 283-355.
[2] M. Gerstenhaber, On the deformation of rings and algebras, Ann. of Math. 79,
(1964), 59-103.
[3] M. Gerstenhaber, On the deformation of rings and algebras. II, Ann. Math. (2)
84, (1966), 1-19.
[4] A. Nijenhuis, R. Richardson, Commutative algebra cohomology and deformations
of Lie and associative algebras, J. Algebra 9, (1968), 42-105.
[5] G. Mazzola, The algebraic and geometric classification of associative algebras of
dimension 5, Manuscripta Math. 27, (1979), 81-101.
[6] Yu. Neretin, An estimate for the number of parameters defining an n-dimensional
algebra, Izv.Acad.Nauk SSSR Ser. Mat. 51, (1987), 306-318.
[7] D. Burde, C. Steinhoff, Classification of orbit closures of 4-dimensional complex
Lie algebras, J. Algebra 214, (1999) 729-739.
[8] N. Jacobson, Structure and representations of Jordan algebras, AMS Colloq. Publ.
39, AMS, Providence 1968.
[9] F. J. Flanigan, Algebraic geography: varieties of structure constants, Pacific J.
Math. 27, (1968), 71-79.
76 Variety of Jordan algebras in small dimensions
[10] F. J. Flanigan, Straightening out and semi-regidity in associative algebra, Trans.
Amer.Math.Soc. 138, (1969), 415-425.
[11] P. Gabriel, Finite representation type is open, proceedings of the international
conference on representations of algebras, Carleton University, Ottawa, Ontario
1974, Lecture Notes in Mathematics 488, (1975), 132-155.
[12] N. Glassman, Cohomology of non-associative algebras, Doctoral Dissertation, Yale
univ., New Haven, (1963).
[13] A. Borel, Linear algebraic groups, New York - Amsterdam: W. A. Benjamin, Inc.,
(1969) 398p.
[14] G. Mazzola, Hochschild-cocycles and generic finite schemes, Comment. Math.
Helv. 55, (1980), 267-293.
[15] H. Wesseler, Der Klassification der Jordan-Algebren niedrider Dimension, Staat-
sexamensarbeit für das Lehramt am Gymnasium, Münster, (1978) 165p.
Contact information
I. Kashuba Instituto de Matemática e Estat́ıstica, Uni-
versidade de São Paulo, R. do Matao 1010,
São Paulo 05311-970, Brazil
E-Mail: kashuba@ime.usp.br
Received by the editors: 05.09.2006
and in final form 29.09.2006.
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