On the representation type of Jordan basic algebras
A finite dimensional Jordan algebra J over a field k is called \textit{basic} if the quotient algebra J/RadJ is isomorphic to a direct sum of copies of k. We describe all basic Jordan algebras J with (RadJ)²=0 of finite and tame representation type over an algebraically closed field of characteris...
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nasplib_isofts_kiev_ua-123456789-1559202025-02-09T13:52:38Z On the representation type of Jordan basic algebras Kashuba, I. Ovsienko, S. Shestakov, I. A finite dimensional Jordan algebra J over a field k is called \textit{basic} if the quotient algebra J/RadJ is isomorphic to a direct sum of copies of k. We describe all basic Jordan algebras J with (RadJ)²=0 of finite and tame representation type over an algebraically closed field of characteristic 0. Supported by the Brazilian FAPESP Proc. 2016/08740-1 and CNPq Proc.307998/2016-9. Supported by the Brazilian FAPESP Proc. 2014/09310-5 and CNPq Proc.303916/2014-1. 2017 Article On the representation type of Jordan basic algebras / I. Kashuba, S. Ovsienko, I. Shestakov // Algebra and Discrete Mathematics. — 2017. — Vol. 23, № 1. — С. 47-61. — Бібліогр.: 15 назв. — англ. 1726-3255 2010 MSC:16G60, 17C55, 17C99. https://nasplib.isofts.kiev.ua/handle/123456789/155920 en Algebra and Discrete Mathematics application/pdf Інститут прикладної математики і механіки НАН України |
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A finite dimensional Jordan algebra J over a field k is called \textit{basic} if the quotient algebra J/RadJ is isomorphic to a direct sum of copies of k.
We describe all basic Jordan algebras J with (RadJ)²=0 of finite and tame representation type over an algebraically closed field of characteristic 0. |
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Kashuba, I. Ovsienko, S. Shestakov, I. |
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Kashuba, I. Ovsienko, S. Shestakov, I. On the representation type of Jordan basic algebras Algebra and Discrete Mathematics |
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On the representation type of Jordan basic algebras |
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On the representation type of Jordan basic algebras |
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On the representation type of Jordan basic algebras |
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On the representation type of Jordan basic algebras |
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On the representation type of Jordan basic algebras |
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on the representation type of jordan basic algebras |
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Інститут прикладної математики і механіки НАН України |
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On the representation type of Jordan basic algebras / I. Kashuba, S. Ovsienko, I. Shestakov // Algebra and Discrete Mathematics. — 2017. — Vol. 23, № 1. — С. 47-61. — Бібліогр.: 15 назв. — англ. |
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Algebra and Discrete Mathematics |
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AT kashubai ontherepresentationtypeofjordanbasicalgebras AT ovsienkos ontherepresentationtypeofjordanbasicalgebras AT shestakovi ontherepresentationtypeofjordanbasicalgebras |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 23 (2017). Number 1, pp. 47–61
c© Journal “Algebra and Discrete Mathematics”
On the representation type
of Jordan basic algebras
Iryna Kashuba∗, Serge Ovsienko and Ivan Shestakov∗∗
Abstract. A finite dimensional Jordan algebra J over a
field k is called basic if the quotient algebra J/ Rad J is isomorphic
to a direct sum of copies of k. We describe all basic Jordan algebras
J with (Rad J)2 = 0 of finite and tame representation type over an
algebraically closed field of characteristic 0.
1. Introduction
Jordan algebras were first introduced by P. Jordan, J. von Neumann
and E. Wigner in the early 1930’s in the search of a new algebraic setting
for quantum mechanics [10]. A Jordan algebra J is a commutative algebra
such that for any a, b ∈ J
(a2 · b) · a = a2 · (b · a).
In their fundamental paper the authors classified all finite-dimensional
formally real algebras. In particular, they showed that any simple formally
real finite-dimensional Jordan algebra is either an algebra of Hermitian
matrices H(A) over a composition algebra A, or a so-called Jordan algebra
of non-degenerated bilinear form J(V, f); for more details see [9, Corollary
∗Supported by the Brazilian FAPESP Proc. 2016/08740-1 and CNPq Proc.
307998/2016-9.
∗∗Supported by the Brazilian FAPESP Proc. 2014/09310-5 and CNPq Proc.
303916/2014-1.
2010 MSC: 16G60, 17C55, 17C99.
Key words and phrases: Jordan algebra, Jordan bimodule, Representation type,
Quiver of an algebra.
48 On the representation type of Jordan basic algebras
V.6.2]. Later on, A.Albert developed a structure theory of finite dimen-
sional Jordan algebras over arbitrary field of characteristic 6= 2 [1] (see
also the book [9]).
The fundamentals of representation theory of Jordan algebras were
developed by N.Jacobson [8, 9]. He introduced Jordan bimodules, defined
their enveloping algebras and described representations of simple Jordan
algebras. Jacobson used Eilenberg’s definition for bimodules in the variety
of algebras, as in [2]. In this approach, the notion of the universal multi-
plicative enveloping algebra proved to be very useful. Jacobson introduced
the universal multiplicative enveloping algebra U(J) of a Jordan algebra
J as a quotient of the tensor algebra T (J) by the certain ideal defined by
Jordan representations of J . The fundamental property of the universal
algebra U(J) is that the category of Jordan bimodules J-bimod over J is
isomorphic to the category U(J)-mod of left modules over the (associative)
algebra U(J). Moreover, Jacobson showed that for a unital Jordan algebra
J we have the following decomposition of U(J)
U(J) = U0(J) ⊕ U 1
2
(J) ⊕ U1(J),
where each direct summand is an ideal of U(J), with U0(J) being a one-
dimensional algebra which corresponds to the trivial J-module, U 1
2
(J) is
the universal associative enveloping algebra, corresponding to so-called
special or one-sided bimodules, and finally U1(J) is the unital univer-
sal multiplicative enveloping algebra, corresponding to unital bimodules.
(For more details look Proposition 3.3 and Proposition 5.1, [12] ). As a
consequence, we have a decomposition
J-bimod = J-bimod0 ⊕ J-bimod1
2
⊕ J-bimod1
of the category J-bimod into a direct sum of the three corresponding
subcategories.
Furthermore, Jacobson showed that for any finite-dimensional Jordan
algebra J the algebra U(J) is of finite dimension as well. Finally, he
described the enveloping algebras U 1
2
(J) and U1(J) and the irreducible
J-bimodules for all finite dimensional simple Jordan algebra J . It turned
out that for any simple algebra J both U 1
2
(J) and U1(J) are semi-simple
algebras, thus both have only finite number of irreducible non-isomorphic
Jordan bimodules and are completely reducible. Since the categories
U(J)-mod and J-bimod are isomorphic, the same is valid for J-bimod.
To describe the universal enveloping algebras for algebras of Hermitian
I . Kashuba, S. Ovsienko, I. Shestakov 49
matrices, Jacobson used the Coordinatization Theorem and classification
of bimodules over composition algebras, [9, VII, 1-2], while for a Jordan
algebra of bilinear form J = J(V, f) the enveloping algebras U 1
2
(J) and
U1(J) turned out to be the Clifford algebra and the meson algebra defined
by a bilinear form f , respectively, see[9, VII,4-5].
Until recent time, there were no results which would describe bimodules
for any class of Jordan algebras other then semi-simple algebras. In 2002
Serge Ovsienko was visiting São Paulo University, and the authors decided
to look on representations of non-semisimple finite-dimensional Jordan
algebras. One of the most known tools in the representation theory of
finite-dimensional associative algebra A is to construct a quiver Q with
relations R such that A-mod is Morita equivalent to the path algebra
k[Q] modulo the ideal generated by relations R. Moreover, all finite-
dimensional associative algebras can be divided into three types: finite,
tame and wild. For the first two classes one can provide a complete
description of indecomposable finite-dimensional left modules. We define
a representation type of Jordan algebra as being a representation type of
the corresponding associative universal enveloping algebra. Our objective
was to describe all finite and tame Jordan algebras.
We started with considering basic Jordan algebra J of small dimension,
calculating U(J) as a quotient tensor algebra modulo relations (2)–(3).
Our first examples and basic relations between Jordan algebra and its
quiver were result of intensive repeated calculations. Ovsienko suggested
to consider a class of algebras which proved to be very useful in the case
of associative algebras, namely algebras J with (Rad J)2 = 0. Although
long but straightforward method allowed us to describe finite and tame
basic Jordan algebras in this class.
Unfortunately, for arbitrary non-basic Jordan algebra J to calculate
U(J) and then find basic algebra Morita equivalent to U(J) was much
harder task, and we were forced to find (and fortunately found) new meth-
ods, relaying on Jordan theory rather than associative one. Generalizing
Jacobson’s Coordinatization Theorem, we described Jordan algebras J
such that (Rad J)2 = 0, semi-simple part of J is a direct sum of algebras
of Hermitian matrices of order > 1, for which U 1
2
(J) is of finite or tame
representation type, see [11]. In [9], Kashuba and Serganova, using a
connection between Jordan and Lie algebras (the famous Tits-Kantor-
Koecher construction), classified Jordan algebras J , such that semi-simple
part of J is a direct sum of algebras of bilinear form and U1(J) is of finite
or tame representation type.
50 On the representation type of Jordan basic algebras
In this paper we describe all basic Jordan algebras J with (Rad J)2 = 0
of finite and tame representation type by calculating the basic algebra U(J)
and then constructing its quiver. It was a starting point of our project
in 2002–2003, which eventually was not included in [11]. By writing it
we want to recall all joy and pleasure we had while working with Sergey
Adamovich Ovsienko.
In Section 2 we present both definition and basic properties of Jordan
bimodules over J and define the universal enveloping algebra U(J). We
also define the representation type of Jordan algebra and recall how to
construct the quiver with relations corresponding to a basic associative
algebra. In Section 3, we start with the technical lemma which describes
relations between the elements of J in U(J) based on the Peirce decom-
position of J . Then we describe quivers for two basic Jordan algebras
J ii and J ij such that both algebras have the dimension of the radical
equals to one (one could think of these examples as "cells"). We finish Sec-
tion 3 with the theorem classifying finite and tame basic Jordan algebras
with (Rad J)2 = 0. Finally, Section 4 is the Appendix where we collect
the results from the representation theory which we use to determine
representation type of quivers.
2. Jordan bimodules and the universal
enveloping algebras
We work over an algebraically closed field k of characteristic 0. Let J
be a Jordan algebra over k, M be a k-vector space endowed with a pair
of linear mapping r : M ⊗k J → M , m ⊗ a 7→ m · a and l : J ⊗k M → M ,
a ⊗ m 7→ a · m. Then we consider k-algebra Ω = J ⊕ M with the following
multiplication ∗ : Ω × Ω → Ω
(a1 + m1) ∗ (a2 + m2) = a1 · a2 + a1 · m2 + m1 · a2, (1)
for a1, a2 ∈ J , m1, m2 ∈ M . We will say that M is a Jordan bimodule over
J if Ω is a Jordan algebra with respect to ∗. Following [8], the (two-sided)
action of the Jordan algebra J on the bimodule M can be rewritten
as a (one-sided) action of the associative algebra U(J), the universal
multiplicative envelope of J . The algebra U(J) can be constructed in the
following way. Consider the free associative k-algebra F 〈J〉 generated
by the vector space J , and let I be the ideal of F 〈J〉 generated by the
elements
abc + cba + (a · c) · b − a(b · c) − b(a · c) − c(a · b), (2)
I . Kashuba, S. Ovsienko, I. Shestakov 51
a(b · c) − (b · c)a + b(a · c) − (a · c)b + c(a · b) − (a · b)c, (3)
where a, b, c ∈ J and a · b, ab denote the products of a, b in J and in
F 〈J〉, respectively. Put U(J) := F 〈J〉/I. Then one may endow every
left U(J)-module M with the canonical structure of J-bimodule via
a · m = m · a := (a + I)m, a ∈ J, m ∈ M . This defines an isomorphism of
the category of Jordan bimodules over J and the category of associative left
modules over U(J). Moreover, by [8] if dimkJ < ∞ then dimkU(J) < ∞,
i.e. U(J) is a finite-dimensional associative algebra (with 1).
Let A be a finite dimensional associative k-algebra with 1, and denote
by k〈x, y〉 the free associative algebra in x, y. Then we say that A has
• a finite type if there are finitely many isomorphism classes of inde-
composable A-modules;
• a tame type if it is not of finite type and for each dimension d there
exists finitely many one-parameter families F1, . . . , FN , such that
every indecomposable module of dimension d is isomorphic to a
module from some Fi;
• a wild type if there exists an A–k〈x, y〉-bimodule M , finitely gen-
erated and free as a k〈x, y〉-module, such that the functor N 7→
M ⊗k〈x,y〉 N from k〈x, y〉-mod to A-mod keeps indecomposability
and isomorphism classes.
The following theorem allows us to classify algebras with respect to
their representation type.
Theorem 2.1. [4] A finite dimensional algebra A has either finite or
tame or wild type.
Having in mind the above isomorphism J-bimod ≃ U(J)-mod, we
will say that a finite dimensional Jordan algebra J is of finite, tame or
wild type if it is true for the finite dimensional associative algebra U(J).
By the Theorem 2.1, every finite dimensional Jordan algebra has either
finite or time or wild type.
Now we recall briefly the notion of a quiver for associative algebra.
For further details and examples we refer to [11].
An associative algebra A is called basic or Morita reduced if
A/RadA ≃ kn for some positive integer n. Two algebras A and B
are called Morita equivalent if the categories A-mod and B-mod are
equivalent. In any class of Morita equivalent algebras there exists a basic
algebra and it is unique up to isomorphism.
Recall that a quiver Q is defined as a set of vertices Q0 and a set of
arrows Q1 together with two mappings s, e : Q1 → Q0 which send an
52 On the representation type of Jordan basic algebras
arrow to its start and end vertex correspondingly. For any basic algebra
Λ we define its quiver Q = Q(Λ) in the following way. Let Λ = S ⊕ RadΛ,
where S ≃ kn is the semi-simple part of Λ and RadΛ is the Jacobson
radical. Choose a set of orthogonal primitive idempotents e1 + · · ·+en = 1
and denote by kij = dimkej(RadΛ/Rad2Λ)ei. Then the vertices of Q are
labelled by the set of idempotents Q0 = {1, . . . , n} and from vertex i to
vertex j lead kij arrows.
Reciprocally, one can construct the path algebra k[Q] starting with
a quiver Q. This algebra has a basis formed by the set of orthogonal
idempotents Q0 and all oriented paths in Q, that is, the sequences x1 . . . xk,
xi ∈ Q1, k > 0 such that s(xi) = e(xi+1), i = 1, . . . , k − 1. The product
of two paths x1 . . . xk and y1 . . . ys is the path x1 . . . xky1 . . . ys if s(xk) =
e(y1) and zero otherwise. If e ∈ Q0 and x ∈ Q1 then ex = x if e(x) = e
and zero otherwise; similarly xe = x if s(x) = e and zero otherwise.
Now we return to original associative algebra A. Let Λ be a basic
algebra which is Morita equivalent to A and Q(Λ) be its quiver, then
there exists an epimorphism
π : k[Q(Λ)] ։ Λ (4)
(see [5]). Let I ⊂ kerπ be the set of generators of the ideal kerπ, then
the pair (Q, I) is called the quiver with relations corresponding to A. It
follows from (4) that k[Q(Λ)]/〈I〉 ≃ Λ. The main use of quivers with
relations is that the category A-mod is equivalent to the category of
representations of the corresponding quiver with relations. Once again,
we reduce the problem of describing modules over associative algebra to
studying modules over its quiver. There are plenty techniques in order to
determine the representation type of a quiver, as well as number of results
for different classes of finite dimensional associative algebras (for example,
for local algebras [15], for algebras with the radical squared zero [6], for
special biserial algebras [7]).
3. Representation type of Jordan basic algebras
Recall that for any a, b ∈ J we write a · b for their product in J ,
while by ab we denote their product in U(J). Any finite-dimensional k-
algebra J may be written as a direct sum of a semi-simple Jordan algebra
S(J) and the radical Rad J of J (i.e. the unique maximal nilpotent
ideal of J), J = S(J) ⊕ Rad J . Denote by QJ(J) := Q(U(J)) and by
QiJ(J) := Qi(U(J)), i = 0, 1.
I . Kashuba, S. Ovsienko, I. Shestakov 53
Theorem 3.1 ([11, Teorem 2.3]). Let J# = J +k1 be the algebra obtained
by the formal adjoining of the identity element to J then the category J-
bimod is isomorphic to the category of unital modules J#-bimod1 of J#.
Thus, without loss of generality we will suppose that J has an identity
element c.
Next we recall the Peirce decomposition of Jordan algebra relative to
idempotent or system of idempotents.
Theorem 3.2 ([9, III.1]). Let e be an idempotent in J then we have the
following decomposition into a direct sum of subspaces
J = J1 ⊕ J 1
2
⊕ J0,
where Ji = {x ∈ J | x · e = ix}, for i = 0, 1
2
, 1.
This decomposition is called the Peirce decomposition of J relative to
idempotent e. The multiplication table for the Peirce components Ji is:
J2
1 ⊆ J1, J1 · J0 = 0, J2
0 ⊆ J0,
J0 · J 1
2
⊆ J 1
2
, J1 · J 1
2
⊆ J 1
2
, J2
1
2
⊆ J0 ⊕ J1. (5)
Furthermore, we have the following generalization: if J is a Jordan
algebra with an identity element c which is a sum of pairwise orthogonal
idempotents ei, i.e. c =
∑n
i=1 ei, we have the refined Peirce decomposition
of J relative to idempotents {e1, . . . , en}:
J =
⊕
16i6j6n
Jij (6)
where Jii = {x ∈ J | x · ei = x} and Jij =
{
x ∈ J | x · ei = x · ej = 1
2
x
}
.
The multiplication table for the Peirce components is:
J2
ii ⊆ Jii, Jij · Jii ⊆ Jij , J2
ij ⊆ Jii ⊕ Jjj ,
Jij · Jjk ⊆ Jik, Jii · Jjj = Jii · Jjk = Jij · Jkl = 0,
(7)
where the indices i, j, k, l are all different.
In the following proposition we recall the Peirce decomposition of
U(J) inherited from the Peirce decomposition of J .
Proposition 3.3 ([12, Prop 5.1]). Let J be a Jordan algebra and U =
U(J) be the universal multiplicative envelope of J and let c be the identity
element in J .
54 On the representation type of Jordan basic algebras
(i) Put C0 = (c − 1)(2c − 1), C1/2 = 4c(1 − c), C1 = c(2c − 1),
where 1 denotes the identity element in U(J). Then the Ci, i = 0, 1
2
, 1 are
central orthogonal idempotents in U , moreover C0 + C 1
2
+ C1 = 1 which
implies the following decomposition of U(J) into the direct sum of ideals
U = UC0 ⊕ UC1 ⊕ UC1/2.
(ii) Let e1, . . . , en be orthogonal idempotents in J such that
∑
ei = c.
Then the elements
Cij = 4eiej , Cii = ei(2ei − 1), C0i = 4ei(1 − c)
are pairwise orthogonal idempotents in U such that
∑
i6j
Cij = C1,
∑
i
C0i = C1/2.
We put also C00 = C0 to have an orthogonal sum 1 =
∑n
i,j=0 Cij.
(iii) If the idempotents ei are central then Cij are central in U and U
is a direct sum of ideals CijU .
The proof follows from (2)-(3), the second relation for ei, ei, ej , i 6= j
provides that ei and ej commute in U(J). They also imply the following
relations between Cij and ek in U(J)
C0iei = 1
2
C0i, Ciiei = Cii, Cijei = 1
2
Cij ,
C0iek = 0, Ciiek = 0, Cijek = 0,
(8)
here all i, j, k are different.
Now suppose that J is a basic Jordan algebra, i.e. is a direct sum of
basic field k. Write S(J) ≃ km = ke1 + · · ·+kem Then by Lemma 5.3 [11]
we know that S(U(J)) = U(S(J)), i.e. the semi-simple part of U(S(J))
is isomorphic to the subalgebra generated by the images of elements of
semi-simple part S(J) of J together with 1 ∈ U(J), thus U(J) is basic
associative algebra and {Cij |0 6 i 6 j 6 m} is the complete set of
primitive orthogonal idempotents. In particular it follows that any basic
Jordan semi-simple algebras is of finite representation type.
Suppose further that (Rad J)2 = 0, then Rad J is completely deter-
mined by S(J)-bimodule structure of Rad J , or equivalently by its Peirce
decomposition relative to {ei|1 6 i 6 m}.
We start with some technical lemma, following two crucial examples.
Lemma 3.4. Suppose that J = ke1 + · · · + kem + Rij, Rij = (Rad J)ij.
(i) Let a ∈ Rij, b ∈ Rkl with (i, j) 6= (k, l), then a and b commute
in U(J).
I . Kashuba, S. Ovsienko, I. Shestakov 55
(ii) Let a ∈ Rii then a and ek, 1 6 k 6 m commute in U(J).
(iii) Let a ∈ Rij then a and ek, k 6= i, j commute in U(J).
(iv) Let a ∈ Rii, then we have the following decomposition of a in U(J)
a = C0ia + Ciia +
∑
1 6 j 6 m,
i 6= j
Cija (9)
(v) Let b ∈ Rij, i 6= j, then we have the following decomposition of b in
U(J)
b = CiibCij +CijbCjj +CjjbCij +CijbCii +C0ibC0j +C0jbC0i (10)
Proof. Since (i, j) 6= (k, l) there exists t such that et acts differently on a
and b. Then equation (3) for et, a, b guarantees that ab = ba. If a ∈ Rii
the same relation for ek, ek, a gives eka − aek + 2δi,kaek − 2δi,keka = 0,
thus we obtain (ii). The argument for (iii) is analogous.
If a ∈ Rii, by (ii) [ek, a] = 0 for any 1 6 k 6 m, therefore CijaCkl = 0
unless i = k, j = l. Relation (2) gives
(2c − 1)(c − 1)a = C00a = 0, (2ek − 1)eka = Ckka = 0,
2ekca − 2eka = −
1
2
C0k = 0, k 6= i.
Thus (9) follows.
Finally suppose that b ∈ Rij , analogously to (iv), C00b = 0. Further by
(iii) we have that [ek, b] = 0 for k 6= i, j, therefore from (8) it follows that
ClkbCst = CstbClk = 0 for any (s, t) 6= (l, k). When k 6= i, j relation (2)
for ek, ek and b provides Ckkb = 0, while for ek, ei and b we obtain
4ekeib + 4beiek − 2bek = Cikb + bCik − 2bek = 0.
It follows that CtkbCtk = 0, 0 6 k 6 m. We continue to eliminate
summands in the Peirce decomposition of b. For ei, b, ej the relation (2)
results in
2eibej + 2ejbei − bei − bej = 0,
then CiibCii = C0ibC0i = CiibC0i = C0ibCii = 0, the same equations hold
if we substitute j instead of i. Moreover we also obtain that CiibCjj =
CjjbCii = 0. Changing the order of the elements in (2) we get
2eiejb + 2bejei − bei − bej = 0
and consequently C0ibCjj = CjjbC0i = C0jbCii = CiibC0j = 0 and (10)
follows.
56 On the representation type of Jordan basic algebras
Now we will define the representation type of basic Jordan algebras
with dim Rad J = 1. There are only two non-isomorphic algebras J ii =
ke1 + · · · + kem + Rii, and J ij = ke1 + · · · + kem + Rij , for i 6= j.
Example 3.5 (Representation type of J ii). Let 〈a〉 = Rad J ii, then from
Lemma 3.4, (iv) it follows that the arrows of Q1J(J ii) are ai := Cija,
b := C0ia and c := Ciia. To calculate the relations, write (2) for a, a, ej
if j 6= i: a2ej = 0, while a2(ei − 1) = 0. Then a2
i = Cija2 = 0 and
b2 = C0ia
2 = 0, while (2) for a, a, a gives a3 = 0 therefore c3 = 0. The
quiver QJ(J ii) is the union of isolated points and the following quivers
corresponding to idempotents Cij , C0i and Cii respectively,
•
ai
��
•
b
��
•
c
�� (11)
with the relations a2
i = b2 = c3 = 0. By Proposition 4.1(vi) Jordan algebra
J ii is of finite representation type.
Example 3.6 (Representation type of J ij). Let 〈b〉 = Rad J ij , then from
Lemma 3.4, (v) it follows that the arrows of Q1J(J ij) are
f := C0ibC0j , g = C0jbC0i, h = CijbCjj ,
k = CjjbCij , l = CiibCij , m = CijbCii.
To calculate the relations, we again write (2) for ei, b, b and obtain b2 =
eib
2 + b2ei, which yields C0ib
2C0i = C0jb2C0j = Ciib
2Cii = Cjjb2Cjj = 0.
Therefore, we have
gf = fg = hk = ml = 0.
Now consider
Cijb2Cij = Cijb(Cii + Cjj)bCij = lm + kh.
On the other hand, (2) for b, ei, b gives 2beib = b2. Substituting (10) for b
and multiplying it by Cij we obtain
Cijb2Cij = 2CijbeibCij = 2Cij(bCjj + bCii)ei(Ciib + Cjjb)Cij
= 2CijbCjjbCij = 2kh.
and we have the relation kh = lm. The quiver QJ(J ij) is the union of
isolated points and the following quivers corresponding to idempotents
C0i, C0j , Cii, Cij and Cjj correspondingly
•
f
)) •
g
ii •
l
)) •
m
ii
h
)) •
k
ii I = 〈fg, gf, hk, ml, kh − lm〉 (12)
I . Kashuba, S. Ovsienko, I. Shestakov 57
By Proposition 4.1 (vi) k[Q]/〈I〉 is of tame representation type and
so is J ij .
Theorem 3.7 (Representation type of basic Jordan algebras J with
(Rad J)2 = 0). Let J = ke1+· · ·+kem+Rad J be basic Jordan algebra with
(Rad J)2 = 0 and let Rad J =
∑
16i6j6m Rij be the Peirce decomposition
of Rad J relative to e1, . . . , em. Then
i) J has finite representation type iff J is semi-simple Jordan algebra.
ii) J has tame representation type iff Rad J 6= 0, for any i, j dim Rij 6 1;
if i 6= j and dim Rij = 1 then dim Rii = dim Rjj = 0 and, finally,
for any j
∑
i6=j
dim Rij 6 2.
Proof. By [9] any semi-simple Jordan algebra has finite representation
type. Both algebras J ii and J ij are of tame representation type, then
from Proposition 4.1, (i) if Rad J 6= 0, algebra J is of wild or tame
representation type.
Next from Remark 5.3, [11], it follows that if J1 = ke1 + · · · + kem +
Rad J1, J2 = ke1 + · · ·+kem +Rad J2 and J = ke1 + · · ·+kem +Rad J1 +
Rad J2 then
Q1J(J) = Q1J(J1) ⊔ Q1J(J2). (13)
First we will prove that if J if of tame type then dim Rad Rij 6 1 for
any i, j ∈ {1, . . . , m}. Suppose dim Rad Rij > 2, then if i 6= j from (13)
it follows that QJ(J) contains the following subquiver Q′ corresponding
to the vertices Cii, Cij and Cii
Q′ : • ,, ��• ,, ��ll`` •ll`` D(Q′) :
•
��''
•
��xx ''��
•
xx��
• • •
By Theorem 4.2 the quotient algebra k[Q′]/ Rad2(k[Q′]) is of wild
representation type thus, by Proposition 4.1(i), J has wild type as
well. If dim Rad Rii > 2, by (13), CiiU(J)Cii contains a wild subalge-
bra k〈x, y, z〉/〈x, y, z〉2, see Proposition 4.1 (iv). As a consequence of
Proposition 4.1 (i) J is wild.
Further, suppose that there exist i 6= j such that dim Rad Rij =
dim Rad Rii = 1. Then by (13) QJ(J) contains the following subquiver
58 On the representation type of Jordan basic algebras
corresponding to the vertices Cii, Cij and Cjj .
Q′′ : • )) • ))
ii
��
•ii
��
D(Q′′) :
•
��
•
���� ��
•
�� ��
• • •
By Theorem 4.2, the quotient algebra k[Q′′]/ Rad2(k[Q′′]) is of wild
representation type, therefore J has wild type as well. This implies
that for any tame algebra J whenever i 6= j and dim Rij = 1 both
dim Rii = dim Rjj = 0.
Finally, suppose there exist i, j, k, l such that dim Rij = dim Rkj =
dim Rlj = 1 then Q(J) contains the following subquiver Q′′′
Cii
++
Cijkk
,,
Cjjkk
,,
Ckjll
,,
Ckkll
Clj
++
KK
Cllll
By the double quiver arguments J has wild representation type.
Next we will show that all the remaining non-semi-simple Jordan
algebras are of tame representation type. By (13)
Q1J(S(J) + ⊕16i,j6mRij) = ⊔16i,j6mQ1J(J ij),
recall that Q1J(J ij) are constructed in Example 3.5 for i = j and in
Example 3.6 for i 6= j. Moreover, if we denote by Iij the relations corre-
sponding to the quiver of algebra J ij then ⊔16i,j6mIij belongs to the set
of generators I of J . The remaining relations we have to check correspond
to the following two cases. First case, suppose there exist i, j, k such
that dim Rij = dim Rkj = 1. Let Rij = 〈b〉 and Rkj = 〈d〉, then denote as
CijbCjj = f , CjjbCij = h, CkjdCjj = g, CjjdCkj = l. Then we have the
following subquiver
Cij
h ,,
Cjj
g
,,
f
kk Ckj
l
ll (14)
and by Example 3.6 hf = 0 and lg = 0. Further from (2) for ei, b, c
it follows that bc = 2eicb + cbei, moreover by Lemma 3.4 (i) b and c
commute, thus fl = gh = 0. The second case to check for new relations
is when there exist i, j such that dim Rii = dim Rjj . Let Rii = 〈a〉 and
I . Kashuba, S. Ovsienko, I. Shestakov 59
Rjj = 〈c〉 and denote as CijaCij = f and CijcCij = g. Then we have the
following subquiver in the vertex corresponding to Cij
•
f
��
g
EE (15)
and by Example 3.5 f2 = g2 = 0. By Lemma 3.4 (i) b, c and ei commute
then (2) for b, c, ei provides that bc = 2eibc, this gives the relation gf = fg.
By Proposition 4.1 (ii) the quiver in (15) has tame representation type.
Further, it remains to prove that if J satisfies the condition ii) of the
theorem it is of tame representation type. Observe that such algebras
may be written and a direct sum of local algebras J ii and algebras
J i1,i2 + J i2,i3 + · · · + J ik−1,ik , then QJ(J) also can be viewed as a disjoint
union of quivers from Example 3.5 and (15) (both are tame) and two
quivers
Ci1i1
h ,,
Ci1i2
g
,,
f
ll Ci2i2
k ,,
m
ll Ci2i3
l
ll Cik−1ik
,,
Cikikmm
I1 = 〈hf − mg, fh, gm, lk, gh, fm, ml, kg〉,
C0i1
a ,,
C0i2
b ,,
c
ll C0i3
d
ll
I2 = 〈ac, ca, bd, db, ba, cd〉.
These quivers correspond to special biserial algebras, check Proposition 4.1,
(vi), thus are of tame representation type.
4. Appendix: quiver type glossary
In the proposition below we summarize some known facts from the
associative theory that we will need to determine the representation type
of algebras.
Proposition 4.1. Let A be a finite dimensional associative k-algebra
and k〈x1, . . . , xn〉 be the free associative algebra in n generators.
(i) If a homomorphic image of an algebra A is of wild type then so is
A. A quotient algebra of tame algebra is either tame or finite. If A
contains subalgebra of wild type then A is wild. In particular if Q(A)
contains subquiver which corresponds to wild algebra then A is wild.
60 On the representation type of Jordan basic algebras
(ii) If A is a local algebra and A ≃ k〈x, y〉/I, where I = 〈yx, xy〉 or
I = 〈x2, y2〉, then A is tame.
(iii) If A is a local algebra which has a homomorphic image isomorphic
to k〈x, y〉/I, where I = 〈x2, xy − yx, y2x, y3〉, then A is wild.
(iv) The algebras k〈x, y, z〉/〈x, y, z〉2 and k〈x, y〉/〈x, y〉3 are wild.
(v) For any positive integer n, the algebra k〈x〉/〈xn〉 is of finite type.
(vi) An algebra A is called special biserial if A is isomorphic to k[Q]/R
for some quiver Q and admissible ideal R, such that
1) any vertex of Q is the starting point of at most two arrows;
2) any vertex of Q is the end point of at most two arrows;
3) if b is an arrow in Q1, then there is at most one arrow a with
ab ∈ R;
4) if b is an arrow in Q1, then there is at most one arrow c with
bc ∈ R.
Any special biserial algebra is of tame or finite representation type.
Proof. For (i) see [5], (ii)–(v) are proved in [15] and (vi) can be found in
[3, Cap. VI].
The following theorem contains the stunning result for an associative
algebra A with Rad2A = 0.
Theorem 4.2 ([6], [14]). An algebra A with Rad2A = 0 is of finite (tame)
representation type if and only if its double quiver D(Q(A)) is a disjoint
finite union of simply laced Dynkin diagrams (correspondingly simply
laced extended Dynkin diagrams).
The simply laced Dynkin diagrams (simple laced extended Dynkin
diagrams) are An, n > 1, Dn, n > 4 and En n = 6, 7, 8 (correspondingly
Ãn, n > 1, D̃n, n > 4 and Ẽn n = 6, 7, 8). Recall that for a quiver
Q = (Q0, Q1) with Q0 = {1, . . . , n}, its double quiver is defined as D(Q) =
(D(Q0), D(Q1)) where D(Q0) = {1′, . . . , n′} ∪ {1′′, . . . , n′′}, D(Q1) has
the same cardinality as Q1, and to every a ∈ Q1 with e(a) = m, s(a) = k
corresponds a unique d(a) ∈ D(Q1) with e(d(a)) = m′′ and s(d(a)) = k′.
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Contact information
Iryna Kashuba,
Ivan Shestakov
Instituto de Matemática e Estatística, Univer-
sidade de São Paulo, R. do Matão 1010, São
Paulo 05311-970, Brazil
E-Mail(s): ikashuba@gmail.com,
ivan.shestakov@gmail.com
Serge Ovsienko Faculty of Mechanics and Mathematics, Kiev
Taras Shevchenko University, Volodymyrska St,
60, Kyiv, 01033 Ukraine
Received by the editors: 28.03.2017.
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