On Classification of Finite-Dimensional Superbialgebras and Hopf Superalgebras

0.43 MB·English

by  Said Aissaoui

Symmetry, Integrability and Geometry: Methods and Applications SIGMA 10 (2014), 001, 24 pages
On Classification of Finite-Dimensional
Superbialgebras and Hopf Superalgebras
Said AISSAOUI † and Abdenacer MAKHLOUF ‡
† Universit´e A-Mira, Laboratoire de Math´ematiques Appliqu´ees,
Targa Ouzemmour 06000 B´ejaia, Algeria
E-mail: [email protected]
‡ Universit´e de Haute Alsace, Laboratoire de Math´ematiques, Informatique et Applications,
4, rue des Fr`eres Lumi`ere F-68093 Mulhouse, France
4 E-mail: [email protected]
1
URL: http://www.algebre.fst.uha.fr/makhlouf/Makhlouf.html
0
2 Received February 08, 2013, in final form December 23, 2013; Published online January 02, 2014

n http://dx.doi.org/10.3842/SIGMA.2014.001
a
J
Abstract. The purpose of this paper is to investigate finite-dimensional superbialgebras
2
and Hopf superalgebras. We study connected superbialgebras and provide a classification

] of non-trivial superbialgebras and Hopf superalgebras in dimension n with n≤4.
A
R Key words: superalgebra; superbialgebra; Hopf superalgebra; classification
.
h 2010 Mathematics Subject Classification: 16T10; 57T05; 17A70
t
a
m
[ Introduction

3
v
Hopf superalgebras are a generalization of the supergroup notion. In physics, infinite-dimen-
8
3 sional Hopf superalgebras based on Lie superalgebras turn out to be related to the integrable
8 S-matrix of the AdS/CFT correspondence. For a large class of finite-dimensional Lie superalge-
0
bras (including the classical simple ones) a Lie supergroup associated to the algebra is defined
.
1 by fixing the Hopf superalgebra of functions on the supergroup [34]. This framework seems to
0
be interesting for studying supermanifolds and supersymmetry. Majid showed that the theo-
3
1 ry of Lie superalgebras and Hopf superalgebras can be reduced to the classical case using the
v: bosonization by K[Z/2Z], where K is an algebraically closed field [24]. Recently some classes
i of Hopf superalgebras were investigated like for example pointed Hopf superalgebras and qu-
X
asitriangular Hopf superalgebras [3, 4, 18], see also [13, 23]. So far very few is known about
r
a general classification of superbialgebras and Hopf superalgebras. Nevertheless, classification of
finite-dimensional Hopf algebras is known for small dimensions and structure of many classes
are deeply studied, see for example [2, 4, 5, 9, 10, 15, 16, 28, 29, 31, 32, 33, 36, 37, 38, 39]. For
general theory of Hopf algebras, we refer to [1, 19, 21, 25, 30, 35], and for applications see for
example [12, 20, 22, 27].
In this paper, we discuss properties of n-dimensional superbialgebras and provide a clas-
sification of non-trivial superbialgebras in dimensions 2, 3 and 4. Moreover we derive a clas-
sification of Hopf superalgebras for these dimensions.
The paper is organized as follows. Section 1 reviews definitions and properties of superbial-
gebras and Hopf superalgebras. Moreover, we provide an outline of the computations leading to
classifications. In Section 2, we focus on trivial and connected superbialgebras. We characterize
connected superbialgebras and classify connected 2-dimensional superbialgebras and Hopf su-
peralgebras. Sections 3 and 4 deal respectively with classification of 3- and 4-dimensional
superbialgebras and Hopf superalgebras. We emphasize on non-trivial superbialgebras, from
2 S. Aissaoui and A. Makhlouf
which we derive the non-trivial Hopf superalgebras. To this end, we establish classification of
3-dimensional superalgebras and use the 4-dimensional classification due to Armour, Chen and
Zhang [7].
Throughout this paper, we work over K, an algebraically closed field of characteristic zero.
Unless otherwise specified, we will denote the multiplication by µ or ·, and the unit element by
η(1) = e = e0 = 1. For simplicity, we will use just product (or multiplication), unit, counit
1 1
and coproduct (or comultiplication) in the sense of even morphism (i.e. we will drop the suffix
‘super’). We also would like to mention that in the last section, most of the results are obtained
by using the computer algebra system Mathematica. The program we have used is available.
1 Definitions of superbialgebras and Hopf superalgebras
In this section, we summarize definitions and properties of superbialgebras and Hopf superalge-
bras. For more details, we refer to [3, 18].
Asuperspace AisaK-vectorspaceendowedwithaZ/2Z-grading, inotherwords, itwritesas
adirectsumoftwovectorspacesA = A ⊕A suchasA istheevenpartandA istheoddpart.
0 1 0 1
Elements of A (resp. A ) are called even homogeneous (resp. odd homogeneous). If a ∈ A ,
0 1 0
we set |a| = deg(a) = 0 and if a ∈ A , |a| = deg(a) = 1. Notice that some authors denote the
1
dimension of A by i|j, such that i = n and j = n , where n = dimA and n = dimA . But
0 1 0 0 1 1
in this paper we preserve this notation for the superalgebra obtained by the ith algebra. Notice
that the following definitions could be described in a symmetric category of superspaces.
Definition 1.1. A superalgebra is a triple (A,µ,η) where A is a superspace, µ : A⊗A → A
(multiplication) and η : K → A (unit) are two superspace morphisms satisfying
µ◦(µ⊗id ) = µ◦(id ⊗µ) (associativity), (1.1)
A A
µ◦η⊗id = µ◦id ⊗η (unity). (1.2)
A A
A superalgebra A is called commutative if the multiplication satisfies µ◦τ = µ where τ is
a superflip. In other words, for all homogeneous elements a,b ∈ A, µ(a⊗b) = (−1)|a||b|µ(b⊗a).
Let (A,µ ,η ), (B,µ ,η ) be two superalgebras. A map f : A → B is a superalgebra
A A B B
morphism if it is a superspace morphism satisfying
f ◦µ = µ ◦f ⊗f and f ◦η = η . (1.3)
A B A B
Definition 1.2. Asupercoalgebra isatriple(C,∆,ε)whereC isasuperspacesuchthat∆ : C →
C ⊗C (comultiplication or coproduct) and ε : C → K (counit) are two superspace morphisms
satisfying
(∆⊗id )◦∆ = (id ⊗∆)◦∆ (coassociativity), (1.4)
C C
(ε⊗id )◦∆ = (id ⊗ε)◦∆ (counity). (1.5)
C C
We use the Sweedleer’s notation for the coproduct, we set for all x ∈ C
(cid:88) (cid:88)
∆(x) = x(1)⊗x(2), (id⊗∆)◦∆(x) = (∆⊗id)◦∆(x) = x(1)⊗x(2)⊗x(3).
(x) (x)
A supercoalgebra C is said to be cocommutative if the comultiplication satisfies ∆ = τ ◦∆,
where τ is the superflip, that is
∀x ∈ C, ∆(x) = (cid:88)x(1)⊗x(2) = (cid:88)(−1)|x(1)||x(2)|x(2)⊗x(1).
(x) (x)
On Classification of Finite-Dimensional Superbialgebras and Hopf Superalgebras 3
Let(A,∆ ,ε ),(B,∆ ,ε )betwosupercoalgebras. Themapf : (A,∆ ,ε ) → (B,∆ ,ε )
A A B B A A B B
is a supercoalgebra morphism if it is a superspace morphism and satisfies
f ⊗f ◦∆ = ∆ ◦f and ε ◦f = ε . (1.6)
A B B A
Now, we consider a superbialgebra structure which is obtained by combining superalgebras
and supercoalgebras. Moreover, if a superbialgebra has an anti-superbialgebra morphism satis-
fying some condition, it is called a Hopf superalgebra.
Definition 1.3. A superbialgebra is a tuple (A,µ,η,∆,ε), where (A,µ,η) is a superalgebra and
(A,∆,ε)isasupercoalgebrasuchthatoneofthesetwoequivalentcompatibilityconditionshold:
1) ∆ : A → A⊗A and ε : A → K are superalgebra morphisms,
2) µ : A⊗A → A and η : K → A are supercoalgebra morphisms.
In other words, ∆ (resp. ε) satisfies the following compatibility condition
∆◦µ = (µ⊗µ)◦(id ⊗τ ⊗id )◦(∆⊗∆), ∆◦η = η⊗η (1.7)
A A
(resp. ε◦µ = µK◦(ε⊗ε), ε◦η = idK),
where µK is the multiplication of K.
The last condition in (1.7) says that the unit 1 = e0 = η(1) of the superbialgebra is a group-
1
like element ∆(1) = 1⊗1.
Remark 1.4. Note that if (A = A ⊕A ,µ,η,∆,ε) is a superbialgebra with η(1) = 1, a unit
0 1
element, then we have
ε(A ) = 0 and ε(1) = 1.
1
Indeed, if ε : A −→ K is a superspace morphism and K is a superspace whose odd part is 0,
then ε sends the odd part of A to odd part of K which is 0. For the second assertion, if A is
a superbialgebra then ε is a superalgebra morphism and it sends the unit element of A to unit
element of K.
If A and B are two superbialgebras over K, we shall call a superspace morphism f : A → B
asuperbialgebramorphism ifitisbothasuperalgebramorphismandasupercoalgebramorphism.
Definition 1.5. A Hopf superalgebra is a superbialgebra admitting an antipode, that is a su-
perspace morphism S : A → A which satisfies the following condition:
µ◦(S ⊗id )◦∆ = µ◦(id ⊗S)◦∆ = η◦ε. (1.8)
A A
A Hopf superalgebra is given by a tuple H = (A,µ,η,∆,ε,S).
Let (A,µ,η,∆,ε,S) be a Hopf superalgebra and S its antipode. Then we have the following
properties:
1) S ◦µ = µ◦τ ◦(S ⊗S);
2) S ◦η = η;
3) ε◦S = ε;
4) τ ◦(S ⊗S)◦∆ = ∆◦S;
5) if A is commutative or cocommutative then S ◦S = id where id : A −→ A is the identity
morphism;
4 S. Aissaoui and A. Makhlouf
6) if the antipode S of the Hopf superalgebra A, with ∆ as comultiplication, is bijective then
A is also a Hopf superalgebra, with the opposite comultiplication ∆(cid:48) = τ ◦ ∆ and the
opposite antipode S(cid:48) = S−1;
7) all finite-dimensional Hopf superalgebras possess bijective antipodes.
Remark 1.6. Some superalgebras do not carry a superbialgebra structure. For example A =
M (K) is a superalgebra with matrix multiplication and such that the even part is A =
2 0
Ke0 ⊕ K e0 and the odd part is A = K e1⊕ K e1, where
1 2 1 1 2
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
1 0 1 0 0 1 0 0
e0 = , e0 = , e1 = , e1 = .
1 0 1 2 0 0 1 0 0 2 1 0
Indeed, thecompatibilityconditionbetweenthecounitεandtheproductε(µ(x⊗y)) = ε(x)ε(y)
isn’t always satisfied, since in one hand we have ε(µ(e1 ⊗ e1)) = ε(e0) = 0, because ε(e1) =
1 2 2 2
ε(e1) = 0, and in the other hand ε(µ(e1⊗e1)) = ε(e0−e0) = 0. Then ε(e0) = 1, which leads to
1 2 1 1 2 2
a contradiction.
In Sections 3 and 4, we aim to classify n-dimensional non-trivial superbialgebras (resp. Hopf
superalgebras), for n = 3 and n = 4. An n-dimensional superbialgebra (resp. a Hopf superalge-
bra) is identified to its structure constants with respect to a fixed basis. It turns out that the
axioms of superbialgebra structure translate to a system of polynomial equations that define
the algebraic variety of n-dimensional superbialgebras which is embedded into K2n30+6n0n21+n0−1.
The classification requires to solve this algebraic system. The calculations are handled using
a computer algebra system. We include in the following an outline of the computation.
1. We provide a list of non-trivial n-dimensional superalgebras:
• For n = 2, the list is taken from [14, 17].
• For n = 3, the list is given in Propositions 3.2 and 3.3.
• For n = 4, the list is taken from [6].
2. Fix a superalgebra A and compute the supercoalgebra’s structure constants {Dk }
(i,s),(j,t)
and {ξ0}, where
i
∆(el) = (cid:88)1 (cid:88)ns (cid:88)nt D(j,s)(k)es⊗et
i (i,l) j k
s=0j=1k=1
with t = (l+s) mod [2] and

ξ0 if l = 0 and i (cid:54)= 1,
 i
ε(el) = 0 if l = 1,
i

1 if l = 0 and i = 1
with respect to bases {e0} and {e1} of A and A , respectively. Supercoal-
i i=1,...,n0 i i=1,...,n1 0 1
gebra structures on A making it a superbialgebra are in one-to-one correspondence with
solutions of the system corresponding to (1.4), (1.5), (1.7).
3. Obtained in (2) is a family {A } of superalgebras whose underlying superalgebra is A.
i i∈I
Wesortthemintoisomorphismclasses. Twosuperbialgebras, givenbytheirstructurecon-
stants,areisomorphicifthereexistmatrices(Tk ,Tk ) definingasuperbialgebramor-
(i,0) (i,1) i,k
phism with respect to the basis, that is satisfying det(Tk )det(Tk ) (cid:54)= 0 and (1.3)–(1.6).
(i,0) (i,1)
4. For each representative of isomorphism classes, determine whether it admits an antipode
or not. It follows from existence or not of structure constants λk , of a morphism S,
(i,s)
S(es) = (cid:80)ns λk es, satisfying (1.8).
i (i,s) k
k=1
On Classification of Finite-Dimensional Superbialgebras and Hopf Superalgebras 5
2 Trivial and connected superbialgebras
We describe in this section some properties of connected superbialgebras, which are superbial-
gebras with 1-dimensional even part and trivial superbialgebras, which are superbialgebras with
trivial odd part.
2.1 Trivial superbialgebras
We have the following obvious result.
Proposition 2.1. Every finite-dimensional bialgebra (resp. Hopf algebra) is a superbialgebra
(resp. Hopf superalgebra), whose odd part is reduced to {0}. This superbialgebra (resp. Hopf
superalgebra) is called trivial superbialgebra (resp. trivial Hopf superalgebra).
ThealgebraicclassificationofHopfalgebrawereinvestigatedbymanyauthors. Hopfalgebras
of prime dimensions were classified in [39]. We refer to [37, 38] for the classification of finite-
dimensional Hopf algebras up to dimension 11, see also [26]. Semisimple Hopf algebras of
dimension 12 were given in [16] and complete classification of dimension 12 provided in [32].
See [8] for dimension 14 and [11] for dimension 20. The minimal dimension where the problem is
unsolvedis24. Fortheclassificationof2and3-dimensionalbialgebras,wereferto[14]. Theyare
based on Gabriel’s result of 2 and 3-dimensional algebras [17]. Classification of 4-dimensional
superbialgebras will be given in a forthcoming paper.
2.2 Connected superbialgebras
In the following we consider connected superbialgebras whose even part are isomorphic to K.
The notions of a connected superalgebra, supercoalgebra and Hopf superalgebra are defined in
the same way.
Lemma 2.2. Let A = A ⊕A be an (n+1)-dimensional connected superspace. If (A,µ,η,∆,ε)
0 1
is a superbialgebra with 1 = η(1), then we have
1) µ(x⊗y) = µ(y⊗x) = 0, ∀x,y ∈ A ,
1
2) ∆(x) = 1⊗x+x⊗1, ∀x ∈ A .
1
n
(cid:80)
Proof. Let (x ,x ,...,x ) be a basis of A , and set µ(x ⊗x ) = α 1, ∆(x ) = (λ x ⊗
1 2 n 1 i j ij i k1 k
k=1
1+λ 1⊗x ), with α ,λ ,λ ∈ K, ∀i,j ∈ {1,...,n}.
1k k ij 1j j1
• The compatibility of the counit and the multiplication ε(µ(x ⊗x )) = ε(x )ε(x ) and the
i j i j
fact that ε(x ) = 0, ∀i ∈ {1,...,n}, show that α = 0, ∀i,j ∈ {1,...,n}. Then we have
i ij
µ(x ⊗x ) = µ(x ⊗x ) = 0, ∀i,j ∈ {1,...,n}.
i j j i
• The compatibility of the counit and the following condition:
(ε⊗id)(∆(x )) = (id⊗ε)(∆(x )) = x , ∀i ∈ {1,...,n},
i i i
n n
(cid:80) (cid:80)
lead to (ε⊗id)(∆(x )) = (λ ε(x )⊗1+λ ε(1)⊗x ) = λ x = x and
i k1 k 1k k 1k k i
k=1 k=1
n n
(cid:88) (cid:88)
(id⊗ε)(∆(x )) = (λ x ⊗ε(1)+λ 1⊗ε(x )) = λ x = x .
i k1 k 1k k k1 k i
k=1 k=1
Then λ = λ = 0, ∀k ∈ {1,...,n}\{i} and λ = λ = 1. So
1k k1 1i i1
∆(x ) = x ⊗1+1⊗x , ∀i ∈ {1,...,n}. (cid:4)
i i i
6 S. Aissaoui and A. Makhlouf
Notice that the first assertion of this lemma is just Proposition 2.11 in [6] for which the proof
is different.
Theorem 2.3. An n-dimensional connected superbialgebra exists only when n < 3.
Proof. Let A = A ⊕A be a superalgebra with multiplication µ and unit 1. Assume A = K1,
0 1 0
dimA = n−1 and (x ,x ,...,x ) be a basis of A .
1 1 2 n−1 1
According to Lemma 2.2, ∆(x ) = 1⊗x +x ⊗1. Compatibility condition leads in one hand
i i i
to ∆(µ(x ⊗x )) = 0, and in the other hand to
i j
µ⊗µ◦τ(∆(x )⊗∆(x )) = µ⊗µ◦τ((1⊗x +x ⊗1)⊗(1⊗x +x ⊗1))
i j i i j j
= µ⊗µ◦τ(x ⊗1⊗x ⊗1+x ⊗1⊗1⊗x +1⊗x ⊗x ⊗1+1⊗x ⊗1⊗x )
i j i j i j i j
= x ⊗x −x ⊗x , ∀i,j ∈ {1,...,n−1}.
i j j i
Since x ⊗x −x ⊗x (cid:54)= 0, ∀i (cid:54)= j, therefore, the compatibility condition is satisfied only if
i j j i
n = 2. (cid:4)
A 2-dimensional superbialgebra is either trivial or connected. For trivial 2-dimensional bial-
gebras, we refer to [14]. In the following, we provide connected 2-dimensional superbialgebras
and Hopf superalgebras classification.
Let (A,µ,η,∆,ε) be a 2-dimensional connected superbialgebra. We set A = K, A =
0 1
span{x} and η(1) = 1 its unit element.
Proposition 2.4. Every 2-dimensional connected superbialgebra is isomorphic to 2-dimensional
connected superbialgebra K[x]/(x2) with deg(x) = 1 and defined by ∆(x) = 1⊗x+x⊗1.
Moreover, it carries a Hopf superalgebra structure with an antipode S defined by S(1) = 1
and S(x) = −x.
Proof. The multiplication, the comultiplication and the counit are defined according to Lem-
ma 2.2 and with this extra product, coproduct and counit x·x = α1, ∆(x) = β1⊗x+γx⊗1,
where α,β,γ ∈ K. Solving the system corresponding to conditions (1.1), (1.2), with respect
to structure constants α, β, γ, leads to the result. For the second assertion, we assume that
the antipode S is defined as, S(1) = 1, S(x) = λx, with λ ∈ K. Applying the identity (1.8)
to x, we obtain only one non trivial 2-dimensional Hopf superalgebra associated to connected 2-
dimensionalsuperbialgebradefined above. Theantipodeis defined asS(1) = 1, S(x) = −x. (cid:4)
Let f : A −→ A be a superalgebra homomorphism, this means that f is an even linear map,
satisfying f(1) = 1 and f ◦µ = µ◦f ⊗f. Let us set f(x) = αx, where α ∈ K\{0}. A direct
calculation shows that f◦µ = µ◦f⊗f is satisfied for any α (cid:54)= 0. Therefore, the automorphism
group of the 2-dimensional superbialgebra is the infinite group
(cid:26)(cid:18) (cid:19) (cid:27)
1 0
, with α ∈ K\{0} .
0 α
3 Classification of 3-dimensional superbialgebras
and Hopf superalgebras
In dimension 3, there are three cases for n = dimA . If n = 1, we have connected superbialge-
0 0 0
bras which are covered by Proposition 2.3. If n = 3, we have trivial superbialgebras for which
0
we refer to [14] for the classification of 3-dimensional bialgebras. It remains to study the case
n = 2.
0
On Classification of Finite-Dimensional Superbialgebras and Hopf Superalgebras 7
3.1 Superalgebras
Let (A,µ,η,∆,ε) be a 3-dimensional superbialgebra such that A = A ⊕A and dimA = 2.
0 1 0
Let {1,x,y} be a basis of A, such that {1,x} generates the even part A and {y} generates the
0
odd part A . Assume that η(1) = 1.
1
We recall here the classification of 2-dimensional algebras.
Proposition 3.1 ([14, 17]). There are, up to isomorphism, two 2-dimensional algebras A =
1
K[x]/(x2) and A = K[x]/(x2−x).
2
Since the even part of a superalgebra is an algebra, we fix the even part multiplication to
be one of the two 2-dimensional algebras, recalled in Proposition 3.1. We denote by µ the
i|j
jth multiplication obtained by extending the multiplication µ of a 2-dimensional algebra to the
i
multiplication of a 3-dimensional superalgebra. We consider first the algebra A .
1
Proposition 3.2. Every non-trivial 3-dimensional superalgebra, where the even part is the
2-dimensional algebra A is isomorphic to one of the following pairwise nonisomorphic 3-di-
1
mensional superalgebras A = K[x,y]/(x2,y2,xy) and A = K[x,y]/(x2,y2 − x,xy), where
1|1 1|2
deg(x) = 0 and deg(y) = 1.
Proof. We set
x·y = αy, y·x = βy, y·y = γe0+σx, with α,β,γ,σ ∈ K.
1
We have x·(y·y) = γx, (x·y)·y = αγe0+ασx, (y·y)·x = γx and y·(y·x) = βγe0+βσx.
1 1
According to associativity and by identification, we obtain αγ = 0, ασ = γ, βγ = 0, βσ = γ.
Then γ = 0 and y·y = σx.
Moreover, x·(x·y) = (x·x)·y = 0 and (y·x)·x = y·(x·x) = 0 lead to α = β = 0. Finally,
it remains to fix σ.
If σ = 0, we obtain the superalgebra defined by A . If σ (cid:54)= 0, then we have the superalgebra
1|1
which is given by the multiplication µ defined as µ (x⊗x) = 0, µ (y⊗x) = 0, µ (x⊗y) = 0,
σ σ σ σ
µ (y⊗y) = σx,σ (cid:54)= 0. It is isomorphic to A (σ = 1). (cid:4)
σ 1|2
For the second algebra A , we obtain three possible extensions as a 3-dimensional superalge-
2
bra.
Proposition 3.3. Every non-trivial 3-dimensional superalgebra where the even part is the
2-dimensional algebra A , is isomorphic to one of the following pairwise nonisomorphic 3-
2
dimensional superalgebras A = K[x,y]/(x2 −x,y2 −x,xy −y), A = K(cid:104)x,y(cid:105)/(x2 −x,y2,
2|1 2|2
xy−y,yx) and A = K[x,y]/(x2−x, y2,xy−y), where deg(x) = 0 and deg(y) = 1.
2|3
Proof. By associativity conditions (x·x)·y = x·(x·y) and y ·(x·x) = (y·x)·x, it follows
that α and β may have only values 0 or 1.
Similarly, conditions x·(y·y) = (x·y)·y and y·(y·x) = (y·y)·x lead to
αγ = 0, ασ = γ +σ, βγ = 0, βσ = γ +σ. (3.1)
We discuss the two cases γ = 0 and γ (cid:54)= 0.
1. If γ = 0, then the system (3.1) reduces to σ(α−1) = 0 and σ(β−1) = 0. So, we deduce
two subcases:
a) If σ (cid:54)= 0, then α = β = 1. We obtain superalgebras which are isomorphic to A (σ = 1).
2|1
b) If σ = 0, we study two cases α = β and α (cid:54)= β,
8 S. Aissaoui and A. Makhlouf
i) If α = β, that means (α,β) ∈ {(0,0),(1,1)}. Then, we obtain two superalgebras
isomorphic to A .
2|3
ii) If α (cid:54)= β, that means (α,β) ∈ {(0,1),(1,0)}. Then, we obtain superalgebras which
are isomorphic to A .
2|2
2. If γ (cid:54)= 0. Then, the system (3.1) reduces to α = β = 0 and σ = −γ. So, we have
superalgebras with multiplications µ defined as
γ
µ (x⊗x) = x, µ (y⊗x) = 0, µ (x⊗y) = 0, µ (y⊗y) = γ(1−x),
γ γ γ γ
which are isomorphic to the superalgebra A . (cid:4)
2|1
3.2 Superbialgebras and Hopf superalgebras
Now, we construct superbialgebra structures associated to the five superalgebras found above.
Proposition 3.4. There is no 3-dimensional superbialgebra, with dimA = 2, associated to
0
superalgebras A and A .
1|1 1|2
Proof. Assume ∆(x) = α1⊗1+β1⊗x+γx⊗1+σx⊗x+δy⊗y where α,β,γ,σ,δ ∈ K. Since
we have x2 = 0, the compatibility condition ε(µ(x⊗x)) = ε(x)ε(x), implies ε(x) = 0.
In one hand, the condition (ε ⊗ id)(∆(x)) = (id⊗ε)(∆(x)) = x implies that α = 0 and
β = γ = 1. In the other hand, the compatibility condition ∆◦µ(x⊗x) = (µ⊗µ)◦(id⊗τ ⊗
id)◦(∆⊗∆)(x⊗x) = 0, leads to βγ = 0. Therefore we have a contradiction. (cid:4)
In the sequel, we consider the superalgebra structures defined by µ , µ and µ . We
2|1 2|2 2|3
denote by Ak the superbialgebra (A,µ ,η,∆k ,εk ). We have Acop = (A,µ,η,∆(cid:48),ε) where
i|j i|j i|j i|j
∆(cid:48) = ∆ ◦ τ and τ is the superflip. By direct calculation, we obtain the following pairwise
non-isomorphic superbialgebras. For all of them we have ∆k (1) = 1⊗1.
i|j
For superalgebra A , we have
2|1
1) A1 with ∆1 (x) = 1⊗x+x⊗1−x⊗x, ∆1 (y) = y⊗1+1⊗y−y⊗x, ε1 (x) = 0;
2|1 2|1 2|1 2|1
2) (A1 )cop.
2|1
For superalgebra A , we have
2|2
1) A1 with ∆1 (x) = 1⊗x+x⊗1−x⊗x, ∆1 (y) = 1⊗y+y⊗1−y⊗x−x⊗y, ε1 (x) = 0;
2|2 2|2 2|2 2|2
2) A2 with ∆2 (x) = 1⊗x+x⊗1−x⊗x+y⊗y, ∆2 (y) = 1⊗y+y⊗1−y⊗x−x⊗y,
2|2 2|2 2|2
ε2 (x) = 0;
2|2
3) A3 with ∆3 (x) = x⊗x, ∆3 (y) = y⊗x+x⊗y, ε3 (x) = 1;
2|2 2|2 2|2 2|2
4) A4 with ∆4 (x) = x⊗x+y⊗y, ∆4 (y) = y⊗x+x⊗y, ε4 (x) = 1.
2|2 2|2 2|2 2|2
For superalgebra A , we have
2|3
1) A1 with ∆1 (x) = 1⊗x+x⊗1−x⊗x, ∆1 (y) = 1⊗y+y⊗1−x⊗y−y⊗x, ε1 (x) = 0;
2|3 2|3 2|3 2|3
2) A2 with ∆2 (x) = 1⊗x+x⊗1−x⊗x, ∆2 (y) = 1⊗y+y⊗1−x⊗y, ε2 (x) = 0;
2|3 2|3 2|3 2|3
3) (A2 )cop;
2|3
4) A4 with ∆4 (x) = 1⊗x+x⊗1−x⊗x, ∆4 (y) = y⊗1+1⊗y,ε4 (x) = 0;
2|3 2|3 2|3 2|3
5) A5 with ∆5 (x) = x⊗x, ∆5 (y) = y⊗x+x⊗y, ε5 (x) = 1.
2|3 2|3 2|3 2|3
On Classification of Finite-Dimensional Superbialgebras and Hopf Superalgebras 9
Theorem 3.5. Every non-trivial 3-dimensional superbialgebra with dimA = 2 is isomorphic to
0
one of the following 3-dimensional pairwise non-isomorphic superbialgebras, which are defined
above,
A1 , (cid:0)A1 (cid:1)cop, Ak (k = 1,...,4), A1 , A2 , (cid:0)A2 (cid:1)cop, A4 , A5 .
2|1 2|1 2|2 2|3 2|3 2|3 2|3 2|3
Theorem 3.6. There is no non-trivial 3-dimensional Hopf superalgebra.
Proof. WecheckthatnooneofthesuperbialgebrasmaycarryastructureofHopfsuperalgebra.
Let S be an antipode of one of the 3-dimensional superbialgebras defined above. Assume that
S(1) = 1, S(x) = λ 1+λ x, S(y) = λ y,
1 2 3
and S satisfies the identity µ◦(S ⊗id)◦∆ = µ◦(id⊗S)◦∆ = η◦ε.
We apply the identity to x and study two cases.
Case 1: ε(x) = 0. For all 3-dimensional superalgebras in this case, we have
∆(x) = 1⊗x+x⊗1−x⊗x+αy⊗y, such that α = 0 or 1.
In one hand, we have,
µ◦(S ⊗id )◦∆(x) = µ◦(1⊗x+x⊗1−S(x)⊗x+αS(y)⊗y)
A
= x+(λ x+λ x)−λ x−λ x = λ 1+(1−λ )x.
1 2 1 2 1 1
In the other hand, we have η◦ε(x) = 0. Therefore, we have a contradiction.
Case 2: ε(x) = 1. In this case, ∆(x) must have the form
∆(x) = x⊗x+αy⊗y, such that α = 0 or 1.
Then, on one hand, we have,
µ◦(S ⊗id )◦∆(x) = µ◦(S(x)⊗x+αS(y)⊗y) = (λ +λ )x.
A 1 2
On the other hand, we have η◦ε(x) = 1. So we have a contradiction. (cid:4)
4 Classification of 4-dimensional superbialgebras
and Hopf superalgebras
Let A = A ⊕A be a 4-dimensional superbialgebra. Then one has to consider four cases for
0 1
n = dimA . If n = 1, the superbialgebras are connected, then it is covered by Proposition 2.3.
0 0 0
If n = 4, then they correspond to trivial superbialgebras. In the sequel, we discuss cases n = 2
0 0
and n = 3.
0
4.1 4-dimensional algebras
Inthefollowing,werecalltheclassificationof4-dimensionalalgebrasgivenbyGabrielin[17]and
the classification of 4-dimensional superalgebras with n = 2 and n = 3 provided by Armour,
0 0
Chen and Zhang in [7].
Theorem 4.1. The following algebras are pairwise non-isomorphic and every 4-dimensional
algebras is isomorphic to one of these 19 algebras:
1) K×K×K×K,
2) K×K×K[x]/x2,
10 S. Aissaoui and A. Makhlouf
3) K[x]/x2×K[y]/y2,
4) K×K[x]/x3,
5) K[x]/x4,
6) K×K[x,y]/(x,y)2,
7) K[x,y]/(x2,y2),
8) K[x,y]/(x3,xy,y2),
9) K[x,y,z]/(x,y,z)2,
(cid:18)K K(cid:19)
10) M = ,
2 K K
  
a 0 0 0
 
 
11) 0 a 0 d/a,b,c,d ∈ K,
c 0 b 0 
 
 0 0 0 b 
12) ΛK2 = exterior algebra of K2,
(cid:18)K K(cid:19)
13) K× ,
0 K
  
a 0 0
 
14) c a 0/a,b,c,d ∈ K ,
 d 0 b 
  
a c d
 
15) 0 a 0/a,b,c,d ∈ K ,
 0 0 b 
16) K[x,y]/(x2,y2,yx),
  
a 0 0
 
17) 0 a 0/a,b,c,d ∈ K ,
 c d b 
18) K(cid:104)x,y(cid:105)/(x2,y2,yx−λxy), for λ (cid:54)= −1,0,1,
19) K(cid:104)x,y(cid:105)/(y2,x2+yx,xy+yx),
where K(cid:104)x,y(cid:105) is the free associative algebra generated by x and y.
4.2 Superalgebras with dim(A ) = 3
0
Let{e0,e0,e0,e1}beabasisofthesuperalgebraA,A = A ⊕A ,suchthatA = span{e0,e0,e0},
1 2 3 1 0 1 0 1 2 3
A = span{e1} and e0 be the unit element of the superalgebra. We recall the multiplication of
1 1 1
all possible graduation obtained from a fixed algebra, we denote by µ the multiplication of
i|j
each 4-dimensional superalgebra in the case dimA = 3. We preserve the notation used in [7],
0
a superalgebra i|j means the jth superalgebra obtained by a fixed ith 4-dimensional algebra.
Proposition4.2([7]). LetKbeanalgebraicallyclosedfieldofcharacteristic0, supposethatAis
a 4-dimensional superalgebra with dimA = 3. Then A is isomorphic to one of the superalgebra
0
in the following pairwise nonisomorphic families:
1|1: K×K×K×K, e0 = (1,1,1,1), e0 = (1,0,0,0), e0 = (0,0,1,1), e1 = (0,0,1,−1);
1 2 3 1
2|1: K×K×K[x]/x2, e0 = (1,1,1), e0 = (1,0,0), e0 = (0,1,0), e1 = (0,0,x);
1 2 3 1
2|2: K×K×K[x]/x2, e0 = (1,1,1), e0 = (1,1,0), e0 = (0,0,x), e1 = (1,−1,0);
1 2 3 1
3|1: K[x]/x2×K[y]/y2, e0 = (1,1), e0 = (1,0), e0 = (x,0), e1 = (0,y);
1 2 3 1
4|1: K×K[x]/x3, e0 = (1,1), e0 = (1,0), e0 = (0,x2), e1 = (0,x);
1 2 3 1
6|1: K×K[x,y]/(x,y)2 : e0 = (1,1), e0 = (1,0), e0 = (0,x), e1 = (0,y);
1 2 3 1
7|1: K[x,y]/(x2,y2), e0 = 1, e0 = x+y, e0 = xy, e1 = x−y;
1 2 3 1