Symmetry, Integrability and Geometry: Methods and Applications
SIGMA 10 (2014), 001, 24 pages
On Classif ication of Finite-Dimensional
Superbialgebras and Hopf Superalgebras
Said AISSAOUI
†
and Abdenacer MAKHLOUF
‡
†
Université A-Mira, Laboratoire de Mathématiques Appliquées,
Targa Ouzemmour 06000 Béjaia, Algeria
E-mail: aissaoui.said@yahoo.fr
‡
Université de Haute Alsace, Laboratoire de Mathématiques, Informatique et Applications,
4, rue des Frères Lumière F-68093 Mulhouse, France
E-mail: Abdenacer.Makhlouf@uha.fr
URL: http://www.algebre.fst.uha.fr/makhlouf/Makhlouf.html
Received February 08, 2013, in final form December 23, 2013; Published online January 02, 2014
http://dx.doi.org/10.3842/SIGMA.2014.001
Abstract. The purpose of this paper is to investigate finite-dimensional superbialgebras
and Hopf superalgebras. We study connected superbialgebras and provide a classification
of non-trivial superbialgebras and Hopf superalgebras in dimension n with n ≤ 4.
Key words: superalgebra; superbialgebra; Hopf superalgebra; classification
2010 Mathematics Subject Classification: 16T10; 57T05; 17A70
Introduction
Hopf superalgebras are a generalization of the supergroup notion. In physics, infinite-dimensional Hopf superalgebras based on Lie superalgebras turn out to be related to the integrable
S-matrix of the AdS/CFT correspondence. For a large class of finite-dimensional Lie superalgebras (including the classical simple ones) a Lie supergroup associated to the algebra is defined
by fixing the Hopf superalgebra of functions on the supergroup [34]. This framework seems to
be interesting for studying supermanifolds and supersymmetry. Majid showed that the theory of Lie superalgebras and Hopf superalgebras can be reduced to the classical case using the
bosonization by K[Z/2Z], where K is an algebraically closed field [24]. Recently some classes
of Hopf superalgebras were investigated like for example pointed Hopf superalgebras and quasitriangular Hopf superalgebras [3, 4, 18], see also [13, 23]. So far very few is known about
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 classification of non-trivial superbialgebras in dimensions 2, 3 and 4. Moreover we derive a classification of Hopf superalgebras for these dimensions.
The paper is organized as follows. Section 1 reviews definitions and properties of superbialgebras 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 superalgebras. 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) = e1 = e01 = 1. For simplicity, we will use just product (or multiplication), unit, counit
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 superalgebras. For more details, we refer to [3, 18].
A superspace A is a K-vector space endowed with a Z/2Z-grading, in other words, it writes as
a direct sum of two vector spaces A = A0 ⊕A1 such as A0 is the even part and A1 is the odd part.
Elements of A0 (resp. A1 ) are called even homogeneous (resp. odd homogeneous). If a ∈ A0 ,
we set |a| = deg(a) = 0 and if a ∈ A1 , |a| = deg(a) = 1. Notice that some authors denote the
dimension of A by i|j, such that i = n0 and j = n1 , where n0 = dim A0 and n1 = dim A1 . But
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 ⊗ idA ) = µ ◦ (idA ⊗µ)
µ ◦ η ⊗ idA = µ ◦ idA ⊗η
(associativity),
(1.1)
(unity).
(1.2)
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, µA , ηA ), (B, µB , ηB ) be two superalgebras. A map f : A → B is a superalgebra
morphism if it is a superspace morphism satisfying
f ◦ µA = µB ◦ f ⊗ f
and
f ◦ ηA = ηB .
(1.3)
Definition 1.2. A supercoalgebra is a triple (C, ∆, ε) where C is a superspace such that ∆ : C →
C ⊗ C (comultiplication or coproduct) and ε : C → K (counit) are two superspace morphisms
satisfying
(∆ ⊗ idC ) ◦ ∆ = (idC ⊗∆) ◦ ∆
(ε ⊗ idC ) ◦ ∆ = (idC ⊗ε) ◦ ∆
(coassociativity),
(1.4)
(counity).
(1.5)
We use the Sweedleer’s notation for the coproduct, we set for all x ∈ C
X
X
∆(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
X
(1)
(2)
∀ x ∈ C,
∆(x) =
x(1) ⊗ x(2) =
(−1)|x ||x | x(2) ⊗ x(1) .
(x)
(x)
On Classification of Finite-Dimensional Superbialgebras and Hopf Superalgebras
3
Let (A, ∆A , εA ), (B, ∆B , εB ) be two supercoalgebras. The map f : (A, ∆A , εA ) → (B, ∆B , εB )
is a supercoalgebra morphism if it is a superspace morphism and satisfies
f ⊗ f ◦ ∆A = ∆B ◦ f
εB ◦ f = εA .
and
(1.6)
Now, we consider a superbialgebra structure which is obtained by combining superalgebras
and supercoalgebras. Moreover, if a superbialgebra has an anti-superbialgebra morphism satisfying 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) is a supercoalgebra such that one of these two equivalent compatibility conditions hold:
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) ◦ (idA ⊗τ ⊗ idA ) ◦ (∆ ⊗ ∆),
(resp.
ε ◦ µ = µK ◦ (ε ⊗ ε),
∆◦η =η⊗η
(1.7)
ε ◦ η = idK ),
where µK is the multiplication of K.
The last condition in (1.7) says that the unit 1 = e01 = η(1) of the superbialgebra is a grouplike element ∆(1) = 1 ⊗ 1.
Remark 1.4. Note that if (A = A0 ⊕ A1 , µ, η, ∆, ε) is a superbialgebra with η(1) = 1, a unit
element, then we have
ε(A1 ) = 0
and
ε(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
a superbialgebra morphism if it is both a superalgebra morphism and a supercoalgebra morphism.
Definition 1.5. A Hopf superalgebra is a superbialgebra admitting an antipode, that is a superspace morphism S : A → A which satisfies the following condition:
µ ◦ (S ⊗ idA ) ◦ ∆ = µ ◦ (idA ⊗S) ◦ ∆ = η ◦ ε.
(1.8)
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
0
A is also a Hopf superalgebra, with the opposite comultiplication ∆ = τ ◦ ∆ and the
0
opposite antipode S = 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 =
M2 (K) is a superalgebra with matrix multiplication and such that the even part is A0 =
Ke01 ⊕ K e02 and the odd part is A1 = K e11 ⊕ K e12 , where
1 0
1 0
0 1
0 0
0
0
1
1
e1 =
,
e2 =
,
e1 =
,
e2 =
.
0 1
0 0
0 0
1 0
Indeed, the compatibility condition between the counit ε and the product ε(µ(x ⊗ y)) = ε(x)ε(y)
isn’t always satisfied, since in one hand we have ε(µ(e11 ⊗ e12 )) = ε(e02 ) = 0, because ε(e12 ) =
ε(e11 ) = 0, and in the other hand ε(µ(e12 ⊗ e11 )) = ε(e01 − e02 ) = 0. Then ε(e02 ) = 1, which leads to
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 superalgebra) 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
3
2
the algebraic variety of n-dimensional superbialgebras which is embedded into K2n0 +6n0 n1 +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].
k
2. Fix a superalgebra A and compute the supercoalgebra’s structure constants {D(i,s),(j,t)
}
0
and {ξi }, where
∆(eli )
=
ns X
nt
1 X
X
(j,s)(k) s
ej
D(i,l)
⊗ etk
s=0 j=1 k=1
with t = (l + s) mod

0

ξi if
ε(eli ) = 0 if


1 if
[2] and
l = 0 and i 6= 1,
l = 1,
l = 0 and i = 1
with respect to bases {e0i }i=1,...,n0 and {e1i }i=1,...,n1 of A0 and A1 , respectively. Supercoalgebra 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 {Ai }i∈I of superalgebras whose underlying superalgebra is A.
We sort them into isomorphism classes. Two superbialgebras, given by their structure conk ,Tk )
stants, are isomorphic if there exist matrices (T(i,0)
(i,1) i,k defining a superbialgebra mork ) det(T k ) 6= 0 and (1.3)–(1.6).
phism with respect to the basis, that is satisfying det(T(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(i,s) , of a morphism S,
ns
P
S(esi ) =
λk(i,s) esk , satisfying (1.8).
k=1
On Classification of Finite-Dimensional Superbialgebras and Hopf Superalgebras
2
5
Trivial and connected superbialgebras
We describe in this section some properties of connected superbialgebras, which are superbialgebras 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).
The algebraic classification of Hopf algebra were investigated by many authors. Hopf algebras
of prime dimensions were classified in [39]. We refer to [37, 38] for the classification of finitedimensional 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
unsolved is 24. For the classification of 2 and 3-dimensional bialgebras, we refer to [14]. They are
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 = A0 ⊕ A1 be an (n + 1)-dimensional connected superspace. If (A, µ, η, ∆, ε)
is a superbialgebra with 1 = η(1), then we have
1) µ(x ⊗ y) = µ(y ⊗ x) = 0, ∀ x, y ∈ A1 ,
2) ∆(x) = 1 ⊗ x + x ⊗ 1, ∀ x ∈ A1 .
Proof . Let (x1 , x2 , . . . , xn ) be a basis of A1 , and set µ(xi ⊗ xj ) = αij 1, ∆(xi ) =
n
P
(λk1 xk ⊗
k=1
1 + λ1k 1 ⊗ xk ), with αij , λ1j , λj1 ∈ K, ∀ i, j ∈ {1, . . . , n}.
• The compatibility of the counit and the multiplication ε(µ(xi ⊗ xj )) = ε(xi )ε(xj ) and the
fact that ε(xi ) = 0, ∀ i ∈ {1, . . . , n}, show that αij = 0, ∀ i, j ∈ {1, . . . , n}. Then we have
µ(xi ⊗ xj ) = µ(xj ⊗ xi ) = 0, ∀ i, j ∈ {1, . . . , n}.
• The compatibility of the counit and the following condition:
(ε ⊗ id)(∆(xi )) = (id ⊗ε)(∆(xi )) = xi ,
lead to (ε ⊗ id)(∆(xi )) =
n
P
∀ i ∈ {1, . . . , n},
(λk1 ε(xk ) ⊗ 1 + λ1k ε(1) ⊗ xk ) =
k=1
n
P
λ1k xk = xi and
k=1
n
n
X
X
(id ⊗ε)(∆(xi )) =
(λk1 xk ⊗ ε(1) + λ1k 1 ⊗ ε(xk )) =
λk1 xk = xi .
k=1
k=1
Then λ1k = λk1 = 0, ∀ k ∈ {1, . . . , n} \ {i} and λ1i = λi1 = 1. So
∆(xi ) = xi ⊗ 1 + 1 ⊗ xi ,
∀ i ∈ {1, . . . , n}.
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 = A0 ⊕ A1 be a superalgebra with multiplication µ and unit 1. Assume A0 = K1,
dim A1 = n − 1 and (x1 , x2 , . . . , xn−1 ) be a basis of A1 .
According to Lemma 2.2, ∆(xi ) = 1 ⊗ xi + xi ⊗ 1. Compatibility condition leads in one hand
to ∆(µ(xi ⊗ xj )) = 0, and in the other hand to
µ ⊗ µ ◦ τ (∆(xi ) ⊗ ∆(xj )) = µ ⊗ µ ◦ τ ((1 ⊗ xi + xi ⊗ 1) ⊗ (1 ⊗ xj + xj ⊗ 1))
= µ ⊗ µ ◦ τ (xi ⊗ 1 ⊗ xj ⊗ 1 + xi ⊗ 1 ⊗ 1 ⊗ xj + 1 ⊗ xi ⊗ xj ⊗ 1 + 1 ⊗ xi ⊗ 1 ⊗ xj )
= xi ⊗ xj − xj ⊗ xi ,
∀ i, j ∈ {1, . . . , n − 1}.
Since xi ⊗ xj − xj ⊗ xi 6= 0, ∀ i 6= j, therefore, the compatibility condition is satisfied only if
n = 2.
A 2-dimensional superbialgebra is either trivial or connected. For trivial 2-dimensional bialgebras, 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 A0 = K, A1 =
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 Lemma 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 2dimensional superbialgebra defined above. The antipode is defined as S(1) = 1, S(x) = −x. 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 6= 0. Therefore, the automorphism
group of the 2-dimensional superbialgebra is the infinite group
1 0
, with α ∈ K\{0} .
0 α
3
Classif ication of 3-dimensional superbialgebras
and Hopf superalgebras
In dimension 3, there are three cases for n0 = dim A0 . If n0 = 1, we have connected superbialgebras which are covered by Proposition 2.3. If n0 = 3, we have trivial superbialgebras for which
we refer to [14] for the classification of 3-dimensional bialgebras. It remains to study the case
n0 = 2.
On Classification of Finite-Dimensional Superbialgebras and Hopf Superalgebras
3.1
7
Superalgebras
Let (A, µ, η, ∆, ε) be a 3-dimensional superbialgebra such that A = A0 ⊕ A1 and dim A0 = 2.
Let {1, x, y} be a basis of A, such that {1, x} generates the even part A0 and {y} generates the
odd part A1 . Assume that η(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 A1 =
K[x]/(x2 ) and A2 = K[x]/(x2 − x).
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 µi|j the
j th multiplication obtained by extending the multiplication µi of a 2-dimensional algebra to the
multiplication of a 3-dimensional superalgebra. We consider first the algebra A1 .
Proposition 3.2. Every non-trivial 3-dimensional superalgebra, where the even part is the
2-dimensional algebra A1 is isomorphic to one of the following pairwise nonisomorphic 3-dimensional superalgebras A1|1 = K[x, y]/(x2 , y 2 , xy) and A1|2 = K[x, y]/(x2 , y 2 − x, xy), where
deg(x) = 0 and deg(y) = 1.
Proof . We set
x · y = αy,
y · x = βy,
y · y = γe01 + σx,
with
α, β, γ, σ ∈ K.
We have x · (y · y) = γx, (x · y) · y = αγe01 + ασx, (y · y) · x = γx and y · (y · x) = βγe01 + βσx.
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 A1|1 . If σ 6= 0, then we have the superalgebra
which is given by the multiplication µσ defined as µσ (x ⊗ x) = 0, µσ (y ⊗ x) = 0, µσ (x ⊗ y) = 0,
µσ (y ⊗ y) = σx, σ 6= 0. It is isomorphic to A1|2 (σ = 1).
For the second algebra A2 , we obtain three possible extensions as a 3-dimensional superalgebra.
Proposition 3.3. Every non-trivial 3-dimensional superalgebra where the even part is the
2-dimensional algebra A2 , is isomorphic to one of the following pairwise nonisomorphic 3dimensional superalgebras A2|1 = K[x, y]/(x2 − x, y 2 − x, xy − y), A2|2 = Khx, yi/(x2 − x, y 2 ,
xy − y, yx) and A2|3 = K[x, y]/(x2 − x, y 2 , xy − y), where deg(x) = 0 and deg(y) = 1.
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 6= 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 6= 0, then α = β = 1. We obtain superalgebras which are isomorphic to A2|1 (σ = 1).
b) If σ = 0, we study two cases α = β and α 6= 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 A2|3 .
ii) If α 6= β, that means (α, β) ∈ {(0, 1), (1, 0)}. Then, we obtain superalgebras which
are isomorphic to A2|2 .
2. If γ 6= 0. Then, the system (3.1) reduces to α = β = 0 and σ = −γ. So, we have
superalgebras with multiplications µγ defined as
µγ (x ⊗ x) = x,
µγ (y ⊗ x) = 0,
µγ (x ⊗ y) = 0,
µγ (y ⊗ y) = γ (1 − x) ,
which are isomorphic to the superalgebra A2|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 dim A0 = 2, associated to
superalgebras A1|1 and A1|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.
In the sequel, we consider the superalgebra structures defined by µ2|1 , µ2|2 and µ2|3 . We
denote by Aki|j the superbialgebra (A, µi|j , η, ∆ki|j , εki|j ). We have Acop = (A, µ, η, ∆0 , ε) where
∆0 = ∆ ◦ τ and τ is the superflip. By direct calculation, we obtain the following pairwise
non-isomorphic superbialgebras. For all of them we have ∆ki|j (1) = 1 ⊗ 1.
For superalgebra A2|1 , we have
1) A12|1 with ∆12|1 (x) = 1 ⊗ x + x ⊗ 1 − x ⊗ x, ∆12|1 (y) = y ⊗ 1 + 1 ⊗ y − y ⊗ x, ε12|1 (x) = 0;
2) (A12|1 )cop .
For superalgebra A2|2 , we have
1) A12|2 with ∆12|2 (x) = 1 ⊗ x + x ⊗ 1 − x ⊗ x, ∆12|2 (y) = 1 ⊗ y + y ⊗ 1 − y ⊗ x − x ⊗ y, ε12|2 (x) = 0;
2) A22|2 with ∆22|2 (x) = 1 ⊗ x + x ⊗ 1 − x ⊗ x + y ⊗ y, ∆22|2 (y) = 1 ⊗ y + y ⊗ 1 − y ⊗ x − x ⊗ y,
ε22|2 (x) = 0;
3) A32|2 with ∆32|2 (x) = x ⊗ x, ∆32|2 (y) = y ⊗ x + x ⊗ y, ε32|2 (x) = 1;
4) A42|2 with ∆42|2 (x) = x ⊗ x + y ⊗ y, ∆42|2 (y) = y ⊗ x + x ⊗ y, ε42|2 (x) = 1.
For superalgebra A2|3 , we have
1) A12|3 with ∆12|3 (x) = 1 ⊗ x + x ⊗ 1 − x ⊗ x, ∆12|3 (y) = 1 ⊗ y + y ⊗ 1 − x ⊗ y − y ⊗ x, ε12|3 (x) = 0;
2) A22|3 with ∆22|3 (x) = 1 ⊗ x + x ⊗ 1 − x ⊗ x, ∆22|3 (y) = 1 ⊗ y + y ⊗ 1 − x ⊗ y, ε22|3 (x) = 0;
3) (A22|3 )cop ;
4) A42|3 with ∆42|3 (x) = 1 ⊗ x + x ⊗ 1 − x ⊗ x, ∆42|3 (y) = y ⊗ 1 + 1 ⊗ y, ε42|3 (x) = 0;
5) A52|3 with ∆52|3 (x) = x ⊗ x, ∆52|3 (y) = y ⊗ x + x ⊗ y, ε52|3 (x) = 1.
On Classification of Finite-Dimensional Superbialgebras and Hopf Superalgebras
9
Theorem 3.5. Every non-trivial 3-dimensional superbialgebra with dim A0 = 2 is isomorphic to
one of the following 3-dimensional pairwise non-isomorphic superbialgebras, which are defined
above,
cop
cop
A22|3
, A42|3 , A52|3 .
A12|1
, Ak2|2 (k = 1, . . . , 4), A12|3 , A22|3 ,
A12|1 ,
Theorem 3.6. There is no non-trivial 3-dimensional Hopf superalgebra.
Proof . We check that no one of the superbialgebras may carry a structure of Hopf superalgebra.
Let S be an antipode of one of the 3-dimensional superbialgebras defined above. Assume that
S(1) = 1,
S(x) = λ1 1 + λ2 x,
S(y) = λ3 y,
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 ⊗ idA ) ◦ ∆(x) = µ ◦ (1 ⊗ x + x ⊗ 1 − S(x) ⊗ x + αS(y) ⊗ y)
= x + (λ1 x + λ2 x) − λ1 x − λ2 x = λ1 1 + (1 − λ1 )x.
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 ⊗ idA ) ◦ ∆(x) = µ ◦ (S(x) ⊗ x + αS(y) ⊗ y) = (λ1 + λ2 )x.
On the other hand, we have η ◦ ε(x) = 1. So we have a contradiction.
4
Classif ication of 4-dimensional superbialgebras
and Hopf superalgebras
Let A = A0 ⊕ A1 be a 4-dimensional superbialgebra. Then one has to consider four cases for
n0 = dim A0 . If n0 = 1, the superbialgebras are connected, then it is covered by Proposition 2.3.
If n0 = 4, then they correspond to trivial superbialgebras. In the sequel, we discuss cases n0 = 2
and n0 = 3.
4.1
4-dimensional algebras
In the following, we recall the classification of 4-dimensional algebras given by Gabriel in [17] and
the classification of 4-dimensional superalgebras with n0 = 2 and n0 = 3 provided by Armour,
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
K[x]/x2 × K[y]/y 2 ,
K × K[x]/x3 ,
K[x]/x4 ,
K × K[x, y]/(x, y)2 ,
K[x, y]/(x2 , y 2 ),
K[x, y]/(x3 , xy, y 2 ),
K[x, y, z]/(x,
y,z)2 ,
K K
10) M2 =
,
K K



a
0
0
0







0
a
0
d


11) 
/a, b, c, d ∈ K ,
c 0 b 0






0 0 0 b
2
12) ΛK2 = exterior
algebra of K ,
K K
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
2 y 2 , yx),
16) K[x,

 y]/(x , 

 a 0 0
17) 0 a 0 /a, b, c, d ∈ K ,


c d b
18) Khx, yi/(x2 , y 2 , yx − λxy), for λ 6= −1, 0, 1,
19) Khx, yi/(y 2 , x2 + yx, xy + yx),
where Khx, yi is the free associative algebra generated by x and y.
3)
4)
5)
6)
7)
8)
9)
4.2
Superalgebras with dim(A0 ) = 3
Let {e01 , e02 , e03 , e11 } be a basis of the superalgebra A, A = A0 ⊕A1 , such that A0 = span{e01 , e02 , e03 },
A1 = span{e11 } and e01 be the unit element of the superalgebra. We recall the multiplication of
all possible graduation obtained from a fixed algebra, we denote by µi|j the multiplication of
each 4-dimensional superalgebra in the case dim A0 = 3. We preserve the notation used in [7],
a superalgebra i|j means the j th superalgebra obtained by a fixed ith 4-dimensional algebra.
Proposition 4.2 ([7]). Let K be an algebraically closed field of characteristic 0, suppose that A is
a 4-dimensional superalgebra with dim A0 = 3. Then A is isomorphic to one of the superalgebra
in the following pairwise nonisomorphic families:
1|1: K × K × K × K, e01 = (1, 1, 1, 1), e02 = (1, 0, 0, 0), e03 = (0, 0, 1, 1), e11 = (0, 0, 1, −1);
2|1: K × K × K[x]/x2 , e01 = (1, 1, 1), e02 = (1, 0, 0), e03 = (0, 1, 0), e11 = (0, 0, x);
2|2: K × K × K[x]/x2 , e01 = (1, 1, 1), e02 = (1, 1, 0), e03 = (0, 0, x), e11 = (1, −1, 0);
3|1: K[x]/x2 × K[y]/y 2 , e01 = (1, 1), e02 = (1, 0), e03 = (x, 0), e11 = (0, y);
4|1: K × K[x]/x3 , e01 = (1, 1), e02 = (1, 0), e03 = (0, x2 ), e11 = (0, x);
6|1: K × K[x, y]/(x, y)2 : e01 = (1, 1), e02 = (1, 0), e03 = (0, x), e11 = (0, y);
7|1: K[x, y]/(x2 , y 2 ), e01 = 1, e02 = x + y, e03 = xy, e11 = x − y;
On Classification of Finite-Dimensional Superbialgebras and Hopf Superalgebras
8|1: K[x, y]/(x3 , xy, y 2 ), e01 = 1, e02 = x, e03 = x2 , e11 = y;
8|2: K[x, y]/(x3 , xy, y 2 ), e01 = 1, e02 = x2 , e03 = y, e11 = x;
9|1: K[x, y, z]/(x, y, z)2 , e01 = 1, e02 = x, e03 = y, e11 = z;






a
0
0
0
1
0
0
0
1 0










0
a
0
d
0
1
0
0
0


 0 0 1
11|1: 
 c 0 b 0/a, b, c, d ∈ K, e1 = 0 0 1 0, e2 = 0 0





0 0 0 b
0 0 0 1
0 0




0 0 0 0
0 0 0 0
0 0 0 1 1 0 0 0 0



e03 = 
0 0 0 0, e1 = 1 0 0 0;
0 0 0 0
0 0 0 0
K K
b c
1 0
13|1: K ×
=
a,
/a, b, c, d ∈ K , e01 = 1,
,
0 K
0 d
0 1
1 0
0 0
0 1
e02 = 0,
, e03 = 0,
, e11 = 0,
;
0 0
0 1
0 0







1 0 0
1 0 0
 a 0 0

14|1:  c a 0/a, b, c, d ∈ K , e01 = 0 1 0, e02 = 0 1 0,


d 0 b
0 0 1
0 0 0




0 0 0
0 0 0
e03 = 0 0 0, e11 = 1 0 0;
0 0 0
1 0 0







1 0 0
1 0 0
 a 0 0

14|2:  c a 0/a, b, c, d ∈ K , e01 = 0 1 0, e02 = 0 1 0,


d 0 b
0 0 1
0 0 0




0 0 0
0 0 0
e03 = 1 0 0, e11 = 0 0 0;
0 0 0
1 0 0







1 0 0
1 0 0
 a c d

15|1: 0 a 0/a, b, c, d ∈ K , e01 = 0 1 0, e02 = 0 1 0,


0 0 b
0 0 1
0 0 0




0 0 1
0 1 0
e03 = 0 0 0, e11 = 0 0 0;
0 0 0
0 0 0







1 0 0
1 0 0
 a c d

15|2: 0 a 0/a, b, c, d ∈ K , e01 = 0 1 0, e02 = 0 1 0,


0 0 b
0 0 1
0 0 0




0 1 0
0 0 1
e03 = 0 0 0, e11 = 0 0 0;
0 0 0
0 0 0







1 0 0
1 0 0
 a 0 0

17|1: 0 a 0/a, b, c, d ∈ K , e01 = 0 1 0, e02 = 0 1 0,


c d b
0 0 1
0 0 0




0 0 0
0 0 0
e03 = 0 0 0, e11 = 0 0 0.
1 0 0
0 1 0
0
0
0
0

0
0
,
0
0
11
12
4.3
S. Aissaoui and A. Makhlouf
Superbialgebras and Hopf superalgebras with dim(A0 ) = 3
We look for all possible superbialgebras which could be obtained from a fixed superalgebra. The
computation are done using a computer algebra system. For the convenience of the presentation,
we only summarize the results in the following proposition. The notations and details are
collected in Appendix A.1.
Proposition 4.3. Let (A, µ, η, ∆, ε) be a 4-dimensional superbialgebra with dim A0 = 3. Then
A is isomorphic to one of the superbialgebra in the following pairwise nonisomorphic families:
superalgebra
1|1
2|1
4|1
6|1
13|1
14|1
14|2
15|1
15|2
17|1
associated superbialgebras
A, µ1|1 , η, ∆k1|1 , εk1|1 , k = 1, . . . , 12
A, µ2|1 , η, ∆k2|1 , εk2|1 , k = 1, . . . , 22
A, µ4|1 , η, ∆k4|1 , εk4|1 , k = 1, 2, 3
A, µ6|1 , η, ∆k6|1 , εk6|1 , k = 1, . . . , 18
A, µ13|1 , η, ∆k13|1 , εk13|1 , k = 1, . . . , 21
A, µ14|1 , η, ∆k14|1 , εk14|1 , k = 1, . . . , 9
A, µ14|2 , η, ∆k14|2 , εk14|2 , k = 1, . . . , 4
A, µ15|1 , η, ∆k15|1 , εk15|1 , k = 1, . . . , 9
A, µ15|2 , η, ∆k15|2 , εk15|2 , k = 1, . . . , 4
A, µ17|1 , η, ∆k17|1 , εk17|1 , k = 1, . . . , 11
Remark 4.4. There is no 4-dimensional superbialgebra with dim A0 = 3 and underlying multiplications µ2|2 , µ3|1 , µ7|1 , µ8|1 , µ8|2 , µ9|1 , µ11|1 .
Now, we look for Hopf superalgebra structures. For a fixed superbialgebra defined above, we
add the antipode’s property. It turns out that there exists only one non-trivial 4-dimensional
Hopf superalgebra in the case dim A0 = 3. It corresponds to the algebra (1|1) and comultiplication ∆21|1 , where A = K × K × K × K. We set x = e01 − 2e02 − e03 = (−1, 1, 0, 0) and
y = e11 = (0, 0, 1, −1) with deg(x) = 0, deg(y) = 1. It leads to basis {1, x, x2 , y} and the algebra
may be written as K[x, y]/(x2 + y 2 − 1, xy) with deg(x) = 0 and deg(y) = 1.
Proposition 4.5. Every non-trivial 4-dimensional Hopf superalgebra where dim A0 = 3 is isomorphic to the 4-dimensional Hopf superalgebra K[x, y]/(x2 + y 2 − 1, xy) with deg(x) = 0,
deg(y) = 1 and such that
∆(x) = x ⊗ x − αy ⊗ y,
∆(y) = x ⊗ y + y ⊗ x,
ε(x) = 1,
ε(y) = 0,
S(x) = x,
S(y) = αy.
where α is a primitive 4th root of unity.
4.4
Superalgebras with dim(A0 ) = 2
Let {e01 , e02 , e11 , e12 } be a basis of the underlying superspace A, such that {e01 , e02 } is a basis of
the even part, {e11 , e12 } a basis of the odd part and e01 is the unit of the superalgebra.
Proposition 4.6 ([7]). Let K be an algebraically closed f ield, suppose that A is a 4-dimensional
superalgebra with dim A0 = 2. Then A is isomorphic to one of the superalgebra in the following
pairwise non-isomorphic families:
1|2: K × K × K × K, e01 = (1, 1, 1, 1), e02 = (1, 1, 0, 0), e11 = (1, −1, 0, 0), e12 = (0, 0, 1, −1);
On Classification of Finite-Dimensional Superbialgebras and Hopf Superalgebras
2|3: K × K × K[x]/x2 , e01 = (1, 1, 1), e02 = (1, 1, 0), e11 = (1, −1, 0), e12 = (0, 0, x);
3|2: K[x]/x2 × K[y]/y 2 , e01 = (1, 1), e02 = (1, 0), e11 = (x, 0), e12 = (0, y);
3|3: K[x]/x2 × K[y]/y 2 , e01 = (1, 1), e02 = (x, y), e11 = (1, −1), e12 = (x, −y);
5|1: K[x]/x4 , e01 = 1, e02 = x2 , e11 = x, e12 = x3 ;
6|2: K × K[x, y]/(x, y)2 , e01 = (1, 1), e02 = (1, 0), e11 = (0, x), e12 = (0, y);
7|2: K[x, y]/(x2 , y 2 ), e01 = 1, e02 = x, e11 = y, e12 = xy;
7|3: K[x, y]/(x2 , y 2 ), e01 = 1, e02 = xy, e11 = x, e12 = y;
8|3: K[x, y]/(x3 , xy, y 2 ), e01 = 1, e02 = x2 , e11 = x, e12 = y;
9|2: K[x, y, z]/(x, y, z)2 , e01 = 1, e02 = x, e11 = y, e12 = z;
K K
a b
1 0
0
10|1: M2 =
=
/a, b, c, d ∈ K , e1 =
,
K K
c d
0 1
0 1
0 0
e11 =
, e12 =
;
0 0
1 0





1 0 0 0
a 0 0 0








0 a 0 d
 /a, b, c, d ∈ K , e0 = 0 1 0 0,
11|2: 
1
0 0 1 0
 c 0 b 0






0 0 0 1
0 0 0 b





0 0 0
0 0 0 0
1 0 0 0





0 1 0 0 1 0 0 0 1 1 0 0 0
e02 = 
0 0 0 0, e1 = 0 0 0 0, e2 = 1 0 0
0 0 0
0 0 0 0
0 0 0 0





a 0 0 0
1 0 0 0








0 a 0 d
 /a, b, c, d ∈ K , e0 = 0 1 0 0,
11|3: 
1



c 0 b 0
0 0 1 0






0 0 0 b
0 0 0 1





0 0 0 0
1 0 0
0
0 0





0
0
0
1
0
1
0
0
 1 
 1 0 0
e02 = 
1 0 0 0, e1 = 0 0 −1 0 , e2 = 1 0
0 0 0 0
0 0 0 −1
0 0
e02
1 0
=
,
0 0

0
0
;
0
0

0 0
0 −1
;
0 0
0 0
12|1: ΛK2 = Khx, yi/(x2 , y 2 , xy + yx), e01 = 1, e02 = x, e11 = y, e12 = xy;
12|2: ΛK2 = Khx, yi/(x2 , y 2 , xy + yx), e01 = 1,




1
 a 0 0

14|3:  c a 0 /a, b, c, d ∈ K , e01 = 0


d 0 b
0




0 0 0
0 0 0
e11 = 1 0 0, e12 = 0 0 0;
0 0 0
1 0 0




1
 a c d

15|3: 0 a 0 /a, b, c, d ∈ K , e01 = 0


0 0 b
0




0 1 0
0 0 1
e11 = 0 0 0, e12 = 0 0 0;
0 0 0
0 0 0
e02 = xy, e11 = x,


0 0
1
1 0, e02 = 0
0 1
0
e12 = y;

0 0
1 0 ,
0 0



0 0
1 0 0
1 0, e02 = 0 1 0,
0 1
0 0 0
16|1: Khx, yi/(x2 , y 2 , yx), e01 = 1, e02 = x, e11 = y, e12 = xy;
13
14
S. Aissaoui and A. Makhlouf
16|2: Khx, yi/(x2 , y 2 , yx), e01 = 1, e02 = y, e11 = x, e12 = xy;
16|3: Khx, yi/(x2 , y 2 , yx), e01 = 1, e02 = xy, e11




1
 a 0 0

17|2: 0 a 0 /a, b, c, d ∈ K , e01 = 0


c d b
0




0 0 0
0 0 0
e11 = 0 0 0, e12 = 0 0 0;
1 0 0
0 1 0
= x, e12 = y;



0 0
1 0 0
1 0, e02 = 0 1 0,
0 1
0 0 0
(18; λ|1): Khx, yi/(x2 , y 2 , yx − λxy), where λ ∈ K with λ 6= −1, 0, 1,
e01 = 1, e02 = x, e11 = y, e12 = xy;
(18; λ|2): Khx, yi/(x2 , y 2 , yx − λxy), where λ ∈ K with λ 6= −1, 0, 1,
e01 = 1, e02 = xy, e11 = x, e12 = y;
19|1: Khx, yi/(y 2 , x2 + yx, xy + yx), e01 = 1, e02 = xy, e11 = x, e12 = y.
4.5
Superbialgebras and Hopf superalgebras with dim(A0 ) = 2
Now, we compute all superbialgebra structures associated to superalgebras described above
where dim A0 = 2. We give fairly basic description of the results, all details are given in
Appendix A.2. We denote by µi|j the multiplication in Ai|j with respect to the basis described
above.
Proposition 4.7. There is no 4-dimensional superbialgebra with dim A0 = 2 endowed with one
of these multiplications µ1|2 , µ3|3 , µ5|1 , µ7|2 , µ7|3 , µ8|3 , µ10|1 , µ11|3 , µ12|1 , µ16|1 , µ16|2 , µ16|3 ,
µ18|1 , µ18|2 , µ19|1 .
Proposition 4.8. Let (A, µ, η, ∆, ε) be a 4-dimensional superbialgebra with dim A0 = 2, then
A is isomorphic to one of the superbialgebra in the following pairwise nonisomorphic families:
superalgebra
2|3
3|2
6|2
11|2
12|2
14|3
15|3
17|2
associated superbialgebras
A, µ2|3 , η, ∆k2|3 , εk2|3 , k = 1, . . . , 4
A, µ3|2 , η, ∆k3|2 , εk3|2 , k = 1, . . . , 9
A, µ6|2 , η, ∆k6|2 , εk6|2 , k = 1, . . . , 11
A, µ11|2 , η, ∆111|2 , ε111|2
A, µ12|2 , η, ∆112|2 , ε112|2
A, µ14|3 , η, ∆114|3 , ε114|3 , k = 1, . . . , 7
A, µ15|3 , η, ∆115|3 , ε115|3 , k = 1, . . . , 7
A, µ17|2 , η, ∆117|2 , ε117|2 , k = 1, 2
It turns out that the only superbialgebras, with dim A0 = 2, which can have a non-trivial
Hopf superalgebra structure are (3|2), (11|2) and (12|2).
Proposition 4.9. Every non-trivial 4-dimensional Hopf superalgebra where dim A0 = 2 is isomorphic to one of the following pairwise non-isomorphic Hopf superalgebras:
1) H1 = Khx, yi/(x2 − x, y 2 , yx) with deg(x) = 0, deg(y) = 1 and such that
∆(x) = 1 ⊗ x + x ⊗ 1 − 2x ⊗ x,
∆(y) = 1 ⊗ y + y ⊗ 1,
ε(x) = ε(y) = 0,
S(y) = −y;
S(x) = x,
On Classification of Finite-Dimensional Superbialgebras and Hopf Superalgebras
15
2) H2 = Khx, yi/(x2 − x, y 2 , xy − yx − y) with deg(x) = 0, deg(y) = 1 and such that
∆(x) = 1 ⊗ x + x ⊗ 1 − 2x ⊗ x,
∆(y) = 1 ⊗ y + y ⊗ 1 − 2x ⊗ xy − 2xy ⊗ x,
ε(x) = ε(y) = 0,
S(y) = y;
S(x) = x,
3) H3 = Khx, yi/(x2 , y 2 , xy + yx) with deg(x) = deg(y) = 1 and such that
∆(x) = 1 ⊗ x + x ⊗ 1,
ε(x) = ε(y) = 0,
∆(y) = 1 ⊗ y + y ⊗ 1,
S(x) = −x,
S(y) = −y;
4) H4 = Khx, yi/(x2 − x, y 2 , yx) with deg(x) = 0, deg(y) = 1 and such that
∆(x) = 1 ⊗ x + x ⊗ 1 − 2x ⊗ x,
∆(y) = 1 ⊗ y + y ⊗ 1 − 2x ⊗ y,
ε(x) = ε(y) = 0,
S(y) = 2xy − y,
S(x) = x,
where Khx, yi stands for noncommutative polynomials.
Proof . The
e11 + e12 .
The Hopf
The Hopf
The Hopf
Hopf superalgebra H1 corresponds to A23|2 (see Appendix A.2) with x = e02 , y =
superalgebra H2 corresponds to A111|2 with x = e02 , y = e11 − e12 .
superalgebra H3 corresponds to A112|2 with x = e11 , y = e12 .
superalgebra H4 corresponds to A13|2 with x = e02 , y = e11 + e12 .
Therefore, gathering the results corresponding to the case dimA0 = 3 and dimA0 = 2 we
obtain five 4-dimensional Hopf superalgebras.
Theorem 4.10. Every non-trivial 4-dimensional Hopf superalgebra is isomorphic to one of the
following Hopf superalgebras
H1 ∼
H2 ∼
= K[Z/2Z] ⊗ ΛK,
= K[Z/2Z] oσ ΛK,
2
∗
H3 ∼
H4 = H2
and
H5 ,
= ΛK ,
where σ : Z/2Z → GL(K) is the non-trivial action of Z/2Z on K, H2∗ is the dual of H2 and H5
is defined as K[x, y]/(x2 + y 2 − 1, xy) (deg(x) = 0, deg(y) = 1) such that
∆(x) = x ⊗ x − αy ⊗ y,
∆(y) = x ⊗ y + y ⊗ x,
ε(x) = 1,
ε(y) = 0,
S(x) = x,
S(y) = αy.
with α4 = 1.
Proof . The first three Hopf superalgebras are cocommutative. We obtain the isomorphisms,
thanks to Kostant’s theorem. However, since H4 is commutative, its dual is cocommutative,
it turns out that we have H4 = H2∗ . The Hopf superalgebra H5 corresponds to A1|1 (Proposition 4.5), it is not cocommutative and is the only one whose even part is 3-dimensional.
In the following table, we collect the results obtained for 4-dimensional superbialgebras and
Hopf superalgebras. Notice that algebras are those classified by Gabriel, see Theorem 4.1. The
superalgebras (i|j) denote j th graduation of the ith algebra, obtained by Armour, Chen and
Zhang, see Proposition 4.2 for the first case (dim A0 = 3) and Proposition 4.6 for the second
case (dim A0 = 2). We quote below the number of corresponding superbialgebras and Hopf
superalgebras.
16
S. Aissaoui and A. Makhlouf
algebra
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
(18; λ)
19
A
superalgebra
1|1
1|2
2|1
2|2
2|3
3|1
3|2
3|3
4|1
5|1
6|1
6|2
7|1
7|2
7|3
8|1
8|2
8|3
9|1
9|2
10|1
11|1
11|2
11|3
12|1
12|2
13|1
14|1
14|2
14|3
15|1
15|2
15|3
16|2
16|1
16|3
17|1
17|2
(18; λ)|1
(18; λ)|2
19|1
] superbialgebras
12
0
22
0
4
0
9
0
3
0
18
11
0
0
0
0
0
0
0
0
0
0
1
0
0
1
21
9
4
7
9
4
7
0
0
0
11
2
0
0
0
] Hopf superalgebras
1
0
0
0
0
0
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Appendix
We list in the following supercoalgebras associated to a given 4-dimensional superalgebra A,
such that A = A0 ⊕ A1 . We denote the comultiplication by ∆ki|j and the counit by εki|j , where i
indicates the item of the 4-dimensional algebra listed in Theorem 4.1, and Ai|j denotes the
superalgebra obtained from the ith algebra, see Proposition 4.2 for the first case (dim A0 = 3)
On Classification of Finite-Dimensional Superbialgebras and Hopf Superalgebras
17
and Proposition 4.6 for the second case (dim A0 = 2). The exponent k indicates the item of
the comultiplication and counit which combined with the multiplication of superalgebra Ai|j
and the unit η provide a 4-dimensional superbialgebra. Recall that for all of them we have
η(1) = 1 (unit element), ∆(1) = 1 ⊗ 1, ε(1) = 1 and ε(A1 ) = 0. In the sequel, we denote
by Aki|j the superbialgebra (A, µi|j , η, ∆ki|j , εki|j ). In order to simplify the superbialgebra in this
appendix, we change the variables for superalgebras A1|1 , A2|1 , A3|2 , A6|1 , A13|1 , A11|2 , A12|2 .
For the superalgebras A1|1 , A3|2 , A11|2 , A12|2 we change the variables as mentioned before
Proposition 4.5 and in the proof of Proposition 4.9. For A2|1 and A13|1 we use x = e02 − e03 ,
y = e11 , and for A6|1 we use x = e02 + e03 , y = e11 . According to the multiplication of each
superalgebra we obtain that A2|1 ∼
= K[x, y]/(x3 − x, xy, y 2 ), A6|1 ∼
= K[x, y]/(x3 − x2 , xy, y 2 ),
A13|1 ∼
= Khx, yi/(x3 − x, xy + yx, xy − y, y 2 ). For the remaining superalgebras we change just the
notation of basis vectors. For dim(A0 ) = 3, we consider the basis {1 = e01 , x = e02 , y = e03 , z = e11 }
and for dim(A0 ) = 2 the basis {1 = e01 , x = e02 , y = e11 , z = e12 }.
A.1
Case dim A0 = 3
Superalgebra A1|1 ∼
= K[x, y]/(x2 + y 2 − 1, xy) with deg(x) = 0 and deg(y) = 1, we have
1) A11|1 with ∆11|1 (x) = 21 (x ⊗ 1 + x2 ⊗ 1 + x2 ⊗ x − x ⊗ x),
∆11|1 (y) = y ⊗ 1 + 12 x2 ⊗ y − 21 x ⊗ y, ε11|1 (x) = −1;
2) A21|1 with ∆21|1 (x) = x ⊗ x − αy ⊗ y, ∆21|1 (y) = x ⊗ y + y ⊗ x, ε21|1 (x) = 1,
where α is a primitive 4th root of unity of K;
3) A31|1 with ∆31|1 (x) = x ⊗ x, ∆31|1 (y) = 1 ⊗ y + y ⊗ x, ε31|1 (x) = 1;
4) A41|1 with ∆41|1 (x) = 21 (x2 ⊗ x2 + x ⊗ x2 + x2 ⊗ x − x ⊗ x),
∆41|1 (y) = 1 ⊗ y + y ⊗ x2 , ε41|1 (x) = −1;
5) A51|1 with ∆51|1 (x) = 12 (x ⊗ 1 + 1 ⊗ x − 1 ⊗ x2 − x2 ⊗ 1 + x2 ⊗ x2 + x ⊗ x),
∆51|1 (y) = y ⊗ 1 + 12 (y ⊗ x + x ⊗ y − y ⊗ x2 + x2 ⊗ y), ε51|1 (x) = 1;
6) A61|1 with ∆61|1 (x) = 12 (x ⊗ x + x ⊗ x2 + x2 ⊗ x − x2 ⊗ x2 ),
∆61|1 (y) = 1 ⊗ y + y ⊗ x2 , ε61|1 (x) = 1;
7) A71|1 with ∆71|1 (x) = −1 ⊗ 1 + 12 (x ⊗ x + x ⊗ 1 + 1 ⊗ x + 1 ⊗ x2 + x2 ⊗ 1 − x2 ⊗ x2 ),
∆71|1 (y) = 21 (x ⊗ y + y ⊗ x + x2 ⊗ y + y ⊗ x2 ), ε71|1 (x) = 1;
8) A81|1 with ∆81|1 (x) = x ⊗ x, ∆81|1 (y) = y ⊗ 1 + x2 ⊗ y, ε81|1 (x) = 1;
1 cop , A11 = (A3 )cop , A12 = (A5 )cop .
9) A91|1 = (A41|1 )cop , A10
1|1 = (A1|1 )
1|1
1|1
1|1
1|1
Superalgebra A2|1 ∼
= K[x, y]/(x3 − x, y 2 , xy) with deg(x) = 0 and deg(y) = 1, εk2|1 (x) = 1 for
k = 1, . . . , 22:
1) Ak2|1 with ∆k2|1 (x) = x ⊗ x for k = 1, . . . , 5:
• ∆12|1 (y) = y ⊗ x + x ⊗ y,
• ∆22|1 (y) = x ⊗ y + y ⊗ x2 ,
• ∆32|1 (y) = y ⊗ x2 + x2 ⊗ y,
• ∆42|1 (y) = 1 ⊗ y + y ⊗ x,
• ∆52|1 (y) = 1 ⊗ y + y ⊗ x2 ;
2) A62|1 with ∆62|1 (x) = 21 (x⊗1+x⊗x+x2 ⊗x−x2 ⊗1), ∆62|1 (y) = 21 (y⊗x+x⊗y+x2 ⊗y+y⊗x2 );
18
S. Aissaoui and A. Makhlouf
3) A72|1 with ∆72|1 (x) = 14 (3x ⊗ x + x ⊗ x2 + x2 ⊗ x − x2 ⊗ x2 ),
∆72|1 (y) = 12 y ⊗ x + 12 x ⊗ y + 12 x2 ⊗ y + 12 y ⊗ x2 ;
4) A82|1 with ∆82|1 (x) = 41 (3x ⊗ x + x ⊗ x2 + x2 ⊗ x − x2 ⊗ x2 ),
∆82|1 (y) = 1 ⊗ y + 12 y ⊗ x + 12 y ⊗ x2 ;
5) A92|1 with ∆92|1 (x) = 21 ( 32 x2 ⊗ x2 − 21 x2 ⊗ x − 12 x ⊗ x2 + 32 x ⊗ x + 1 ⊗ x − x ⊗ 1 − x2 ⊗ 1 − 1 ⊗ x2 ),
∆92|1 (y) = 12 (y ⊗ x + x ⊗ y + x2 ⊗ y + y ⊗ x2 );
1 2
10
2
2
2
6) A10
2|1 with ∆2|1 (x) = 2 (x ⊗ x + x ⊗ x + 1 ⊗ x + x ⊗ 1 − 1 ⊗ x − x ⊗ 1),
1
2
2
∆10
2|1 (y) = 2 (y ⊗ x + x ⊗ y + x ⊗ y + y ⊗ x );
1 2
11
2
2
2
7) A11
2|1 with ∆2|1 (x) = 2 (x ⊗ x + x ⊗ x + 1 ⊗ x + x ⊗ 1 − 1 ⊗ x − x ⊗ 1),
1
2
2
∆11
(2|1) (y) = 1 ⊗ y + 2 (y ⊗ x + x ⊗ y + y ⊗ x − x ⊗ y);
1
12
2
2
8) A12
2|1 with ∆2|1 (x) = 2 (x ⊗ x + x ⊗ x + 1 ⊗ x − 1 ⊗ x ),
1
2
∆12
2|1 (y) = 1 ⊗ y + 2 (y ⊗ x + y ⊗ x);
1
13
2
2
2
2
9) A13
2|1 with ∆2|1 (x) = −1 ⊗ 1 + 2 (x ⊗ x + x ⊗ 1 + 1 ⊗ x + 1 ⊗ x + x ⊗ 1 − x ⊗ x ),
1
2
2
∆13
2|1 (y) = 2 (y ⊗ x + x ⊗ y + x ⊗ y + y ⊗ x );
14
2
2
10) A14
2|1 with ∆2|1 (x) = 1 ⊗ x + x ⊗ 1 − x ⊗ x − x ⊗ x − x ⊗ x ,
2
2
∆14
2|1 (y) = y ⊗ 1 + 1 ⊗ y − x ⊗ y − y ⊗ x ;
15
2
15
2
2
11) A15
2|1 with ∆2|1 (x) = 1 ⊗ x + x ⊗ 1 − x ⊗ x, ∆2|1 (y) = y ⊗ 1 + 1 ⊗ y − x ⊗ y − y ⊗ x ;
1
2
2
2
2
16
12) A16
2|1 with ∆2|1 (x) = 1 ⊗ x + x ⊗ 1 + 2 (−x ⊗ x − x ⊗ x − x ⊗ x + x ⊗ x ),
2
2
∆16
2|1 (y) = y ⊗ 1 + 1 ⊗ y − x ⊗ y − y ⊗ x ;
17
2
2
2
2
13) A17
2|1 with ∆2|1 (x) = 1 ⊗ x + x ⊗ 1 − x ⊗ x − x ⊗ x + x ⊗ x ,
2
2
∆17
2|1 (y) = y ⊗ 1 + 1 ⊗ y − x ⊗ y − y ⊗ x ;
1
18
2
2
2
2
2
2
14) A18
2|1 with ∆2|1 (x) = −1 ⊗ 1 + 1 ⊗ x + x ⊗ 1 + 2 (x ⊗ x + x ⊗ x + x ⊗ x − 3x ⊗ x ),
1
1
1 2
1
2
∆18
2|1 (y) = 2 y ⊗ x + 2 x ⊗ y + 2 x ⊗ y + 2 y ⊗ x ;
1
2
2
2
2
19
2
19
15) A19
2|1 with ∆2|1 (x) = 2 (x ⊗ x + x ⊗ x + x ⊗ x − x ⊗ x ), ∆2|1 (y) = 1 ⊗ y + y ⊗ x ;
15 cop , A21 = (A2 )cop , A22 = (A6 )cop .
16) A20
2|1 = (A2|1 )
2|1
2|1
2|1
2|1
Superalgebra A4|1 , we have ∆k4|1 (x) = x ⊗ x, εk4|1 (x) = 1, εk4|1 (y) = 0 for k = 1, . . . , 3:
1) A14|1 with ∆14|1 (y) = x ⊗ y + y ⊗ x, ∆14|1 (z) = x ⊗ z + z ⊗ x;
2) A24|1 with ∆24|1 (y) = x ⊗ y + y ⊗ 1, ∆24|1 (z) = z ⊗ 1 + x ⊗ z;
3) A34|1 = (A24|1 )cop .
Superalgebra A6|1 ∼
= K[x, y]/(x3 − x2 , y 2 − xy) with deg(x) = 0 and deg(y) = 1, we have
εk6|1 (x) = 1 for k = 1, . . . , 18:
1) A16|1 with ∆16|1 (x) = x ⊗ x2 + x2 ⊗ x − x2 ⊗ x2 , ∆16|1 (y) = y ⊗ x2 + x2 ⊗ y;
2) A26|1 with ∆26|1 (x) = x ⊗ x2 + x2 ⊗ x − x2 ⊗ x2 + y ⊗ y, ∆26|1 (y) = y ⊗ x2 + x2 ⊗ y;
3) Ak6|1 with ∆k6|1 (x) = x ⊗ x for k = 3, . . . , 8 and
• ∆36|1 (y) = x ⊗ y + y ⊗ x,
On Classification of Finite-Dimensional Superbialgebras and Hopf Superalgebras
19
• ∆46|1 (y) = y ⊗ x2 + x ⊗ y,
• ∆56|1 (y) = y ⊗ x2 + x2 ⊗ y,
• ∆66|1 (y) = y ⊗ 1 + x ⊗ y,
• ∆76|1 (y) = y ⊗ 1 + x2 ⊗ y,
• ∆86|1 (y) = 1 ⊗ y + y ⊗ 1;
4) A96|1 with ∆96|1 (x) = x ⊗ x + y ⊗ y, ∆96|1 (y) = x ⊗ y + y ⊗ x;
10
2
2
10
2
2
5) A10
6|1 with ∆6|1 (x) = x ⊗ 1 − x ⊗ 1 + x ⊗ x, ∆6|1 (y) = y ⊗ x + x ⊗ y;
11
2
2
2
2
11
2
6) A11
6|1 with ∆6|1 (x) = x ⊗ x + x ⊗ x − x ⊗ x , ∆6|1 (y) = y ⊗ 1 + x ⊗ y;
12
2
2
12
2
7) A12
6|1 with ∆6|1 (x) = x ⊗ 1 − x ⊗ 1 + x ⊗ x, ∆6|1 (y) = x ⊗ y + y ⊗ 1;
4 cop , A14 = (A11 )cop , A15 = (A6 )cop , A16 = (A5 )cop , A17 = (A12 )cop ,
8) A13
6|1 = (A6|1 )
6|1
6|1
6|1
6|1
6|1
6|1
6|1
6|1
10 )cop .
A18
=
(A
6|1
6|1
Superalgebra A13|1 ∼
= Khx, yi/(x3 − x, y 2 , xy + yx, xy − y) with deg(x) = 0 and deg(y) = 1,
k
we have ε13|1 (x) = 0 for k = 1, . . . , 11 and εk13|1 (x) = 1 for k = 12, . . . , 21:
1) A113|1 with ∆113|1 (x) = 1 ⊗ x + x ⊗ 1 − x2 ⊗ x, ∆113|1 (y) = 1 ⊗ y + y ⊗ 1 − x2 ⊗ y − y ⊗ x2 ;
2) A213|1 with ∆213|1 (x) = 1 ⊗ x + x ⊗ 1 + 12 x ⊗ x − 12 x ⊗ x2 − 12 x2 ⊗ x − 21 x2 ⊗ x2 ,
∆213|1 (y) = 1 ⊗ y + y ⊗ 1 − x2 ⊗ y − y ⊗ x2 ;
3) A313|1 with ∆313|1 (x) = 1 ⊗ x + x ⊗ 1 + 21 x ⊗ x − 12 x ⊗ x2 − 12 x2 ⊗ x − 21 x2 ⊗ x2 ,
∆313|1 (y) = 1 ⊗ y + y ⊗ 1;
4) A413|1 with ∆413|1 (x) = 1 ⊗ x + x ⊗ 1 + −x2 ⊗ x − x2 ⊗ x2 ,
∆413|1 y) = 1 ⊗ y + y ⊗ 1 − y ⊗ x2 − x2 ⊗ y;
5) A513|1 with ∆513|1 (x) = 1 ⊗ x + x ⊗ 1 − 21 x ⊗ x − 12 x ⊗ x2 − 12 x2 ⊗ x + 21 x2 ⊗ x2 ,
∆513|1 (y) = 1 ⊗ y + y ⊗ 1 − 12 x ⊗ y − 12 y ⊗ x − 21 x2 ⊗ y − 12 y ⊗ x2 ;
6) A613|1 with ∆613|1 (x) = 1 ⊗ x + x ⊗ 1 − 21 x ⊗ x − 12 x ⊗ x2 − 21 x2 ⊗ x + 12 x2 ⊗ x2 − 2y ⊗ y,
∆613|1 (y) = 1 ⊗ y + y ⊗ 1 − 12 x ⊗ y − 12 y ⊗ x − 21 x2 ⊗ y − 12 y ⊗ x2 ;
7) A713|1 with ∆713|1 (x) = 1 ⊗ x + x ⊗ 1 − 21 x ⊗ x − 12 x ⊗ x2 + 12 x2 ⊗ x2 − x2 ⊗ x,
∆713|1 (y) = 1 ⊗ y + y ⊗ 1 − 12 x ⊗ y − 12 x2 ⊗ y − y ⊗ x2 ;
8) A813|1 with ∆813|1 (x) = 1 ⊗ x + x ⊗ 1 − x ⊗ x2 , ∆813|1 (y) = 1 ⊗ y + y ⊗ 1 − x2 ⊗ y − y ⊗ x2 ;
9) A913|1 with ∆913|1 (x) = 1 ⊗ x + x ⊗ 1 − x ⊗ x2 , ∆913|1 (y) = y ⊗ 1 + 1 ⊗ y − y ⊗ x2 ;
9
cop ;
10) A10
13|1 = (A13|1 )
11
2
2
11) A11
13|1 with ∆13|1 (x) = 1 ⊗ x + x ⊗ 1 − x ⊗ x − x ⊗ x − x ⊗ x,
2
2
∆11
13|1 (y) = 1 ⊗ y + y ⊗ 1 − x ⊗ y − y ⊗ x ;
1
1
1
1
3
3 2
12
2 1 2
2 1 2
2
12) A12
13|1 with ∆13|1 (x) = 2 1⊗x+ 2 x⊗1− 2 1⊗x − 2 x ⊗1− 4 x⊗x − 4 x ⊗x+ 4 x⊗x+ 4 x ⊗x ,
1
1
1 2
1
2
∆12
13|1 (y) = 2 x ⊗ y + 2 y ⊗ x + 2 x ⊗ y + 2 y ⊗ x ;
13) Ak13|1 with ∆k13|1 (y) = 21 x ⊗ y + 12 y ⊗ x + 12 x2 ⊗ y + 12 y ⊗ x2 for k = 13, . . . , 21:
• ∆13
13|1 (x) = x ⊗ x,
1
1
1 2
1 2
2
2
• ∆14
13|1 (x) = 2 x ⊗ x + 2 x ⊗ x + 2 x ⊗ x − 2 x ⊗ x + 2y ⊗ y,
20
S. Aissaoui and A. Makhlouf
1
1 2
1
1 2
• ∆15
13|1 (x) = 2 x ⊗ x + 2 x ⊗ x + 2 x ⊗ 1 − 2 x ⊗ 1,
1
1
1 2
1
1 2
1
2
2
• ∆16
13|1 (x) = 2 1 ⊗ x + 2 x ⊗ 1 − 2 1 ⊗ x − 2 x ⊗ 1 + 2 x ⊗ x + 2 x ⊗ x ,
3
1
1 2
1 2
2
2
• ∆17
13|1 (x) = 4 x ⊗ x + 4 x ⊗ x + 4 x ⊗ x − 4 x ⊗ x ,
1
1 2
3 2
1
2
2
2
2
• ∆18
13|1 (x) = −1 ⊗ 1 + 2 x ⊗ x + 1 ⊗ x + x ⊗ 1 + 2 x ⊗ x + 2 x ⊗ x − 2 x ⊗ x ,
1
1
1 2
3 2
2
2
2
2
• ∆19
13|1 (x) = −1 ⊗ 1 + 2 x ⊗ x + 1 ⊗ x + x ⊗ 1 + 2 x ⊗ x + 2 x ⊗ x − 2 x ⊗ x + 2y ⊗ y,
1
1 2
1
1 2
3 2
1
2
2
2
• ∆20
13|1 (x) = −1 ⊗ 1 + 2 x ⊗ x + 2 1 ⊗ x + 2 x ⊗ 1 + 2 x ⊗ x + 2 x ⊗ x − 2 x ⊗ x ;
15 cop .
14) A21
13|1 = (A13|1 )
Superalgebra A14|1 , we have εk14|1 (x) = 0, εk14|1 (y) = 0 for k = 1, . . . , 9:
1) Ak14|1 with ∆114|1 (x) = 1 ⊗ x + x ⊗ 1 − x ⊗ x for k = 1, . . . , 6:
• ∆114|1 (y) = 1 ⊗ y + y ⊗ 1 − x ⊗ y − y ⊗ x, ∆114|1 (z) = 1 ⊗ z + z ⊗ 1,
• ∆214|1 (y) = 1 ⊗ y + y ⊗ 1 − x ⊗ y − y ⊗ x, ∆214|1 (z) = 1 ⊗ z + z ⊗ 1 − x ⊗ z,
• ∆314|1 (y) = 1 ⊗ y + y ⊗ 1 − x ⊗ y − y ⊗ x + y ⊗ y, ∆314|1 (z) = 1 ⊗ z + z ⊗ 1 − x ⊗ z,
• ∆414|1 (y) = 1 ⊗ y + y ⊗ 1 − x ⊗ y − y ⊗ x, ∆414|1 (z) = 1 ⊗ z + z ⊗ 1 − x ⊗ z − z ⊗ x,
• ∆514|1 (y) = 1⊗y+y⊗1−x⊗y−y⊗x+y⊗y, ∆514|1 (z) = 1⊗z+z⊗1−x⊗z−z⊗x+z⊗y,
• ∆614|1 (y) = 1 ⊗ y + y ⊗ 1 − x ⊗ z − z ⊗ x + y ⊗ y, ∆614|1 (z) = 1 ⊗ z + z ⊗ 1 − x ⊗ z − z ⊗ x;
2) A714|1 with ∆714|1 (x) = 1 ⊗ x + x ⊗ 1 − x ⊗ x + y ⊗ y,
∆714|1 (y) = 1 ⊗ y + y ⊗ 1 − x ⊗ y − y ⊗ x + y ⊗ y, ∆714|1 (z) = 1 ⊗ z + z ⊗ 1;
3) A814|1 = (A214|1 )cop , A914|1 = (A314|1 )cop .
Superalgebra A14|2 , we have εk14|2 (x) = 0 and εk14|2 (y) = 0 for k = 1, . . . , 4:
1) A114|2 with ∆114|2 (x) = 1 ⊗ x + x ⊗ 1 − x ⊗ x, ∆114|2 (y) = 1 ⊗ y + y ⊗ 1 − y ⊗ z − z ⊗ x,
∆114|2 (z) = 1 ⊗ z + z ⊗ 1 − z ⊗ x − x ⊗ z;
2) A214|2 with ∆214|2 (x) = 1 ⊗ x + x ⊗ 1 − x ⊗ x, ∆214|2 (y) = 1 ⊗ y + y ⊗ 1 − x ⊗ y,
∆214|2 (z) = 1 ⊗ z + z ⊗ 1 − x ⊗ z − z ⊗ x;
3) A314|2 with ∆314|2 (x) = 1 ⊗ x + x ⊗ 1 − x ⊗ x, ∆314|2 (y) = 1 ⊗ y + y ⊗ 1 − x ⊗ y − y ⊗ x + y ⊗ y,
∆314|2 (z) = 1 ⊗ z + z ⊗ 1 − z ⊗ x − x ⊗ z;
4) A414|2 = (A214|2 )cop .
Superalgebra
9
(A14|1 )op , A515|1
(A414|1 )op,cop .
Superalgebra
3
(A14|2 )op .
Superalgebra
A15|1 , we have A115|1 = (A114|1 )op , A215|1 = (A214|1 )op , A315|1 = (A514|1 )op , A415|1 =
= (A614|1 )op , A615|1 = (A314|1 )op , A715|1 = (A414|1 )op , A815|1 = (A714|1 )op , A915|1 =
A15|2 , we have A115|2 = (A114|2 )op , A215|2 = (A414|2 )op , A315|2 = (A214|2 )op , A415|2 =
A17|1 , we have εk17|1 (x) = 1 and εk17|1 (y) = 0 for k = 1, . . . , 11:
1) A117|1 with ∆117|1 (x) = x ⊗ x + y ⊗ y + z ⊗ z, ∆117|1 (y) = x ⊗ y + y ⊗ x + z ⊗ z,
∆117|1 (z) = x ⊗ z + z ⊗ x + z ⊗ y + y ⊗ z;
2) A217|1 with ∆217|1 (x) = x ⊗ x + z ⊗ z, ∆217|1 (y) = x ⊗ y + y ⊗ x, ∆217|1 (z) = z ⊗ x + x ⊗ z;
On Classification of Finite-Dimensional Superbialgebras and Hopf Superalgebras
21
3) A317|1 with ∆317|1 (x) = x ⊗ x + z ⊗ z, ∆317|1 (y) = x ⊗ y + y ⊗ x + z ⊗ z − y ⊗ y,
∆317|1 (z) = z ⊗ x + x ⊗ z;
4) A417|1 with ∆417|1 (x) = x ⊗ x + y ⊗ y, ∆417|1 (y) = x ⊗ y + y ⊗ x,
∆417|1 (z) = z ⊗ x + x ⊗ z − z ⊗ y − y ⊗ z;
5) Ak17|1 , we have ∆k17|1 (x) = x ⊗ x and ∆k17|1 (y) = x ⊗ y + y ⊗ x + y ⊗ y for k = 5, 6, 7:
• ∆517|1 (z) = x ⊗ z + z ⊗ x,
• ∆617|1 (z) = x ⊗ z + z ⊗ x + z ⊗ y − y ⊗ z,
• ∆717|1 (z) = z ⊗ x + x ⊗ z + z ⊗ y;
6) A817|1 = (A717|1 )cop ;
7) Ak17|1 , we have ∆k17|1 (x) = x ⊗ x for k = 9, 10, 11:
• ∆917|1 (y) = x ⊗ y + y ⊗ x − y ⊗ y, ∆917|1 (z) = z ⊗ x + x ⊗ z,
10
• ∆10
17|1 (y) = x ⊗ y + y ⊗ x + z ⊗ z, ∆17|1 (z) = x ⊗ z + z ⊗ x,
11
• ∆11
17|1 (y) = x ⊗ y + y ⊗ x + y ⊗ y + z ⊗ z, ∆17|1 (z) = z ⊗ x + x ⊗ z + z ⊗ y + y ⊗ z.
A.2
Case dim A0 = 2
Superalgebra A2|3 , we have ∆k2|3 (x) = 1 ⊗ x + x ⊗ 1 − x ⊗ x, εk2|3 (x) = 0 for k = 1, . . . , 4:
1) A12|3 with ∆12|3 (y) = 1 ⊗ y + y ⊗ 1 − x ⊗ y, ∆12|3 (z) = 1 ⊗ z + z ⊗ 1 − x ⊗ z − z ⊗ x;
2) A22|3 with ∆22|3 (y) = 1 ⊗ y + y ⊗ 1 − x ⊗ y + z ⊗ x, ∆22|3 (z) = 1 ⊗ z + z ⊗ 1 − x ⊗ z − z ⊗ x;
3) A32|3 = (A12|3 )cop , A42|3 = (A22|3 )cop .
Superalgebra A3|2 ∼
= Khx, yi/(x2 − x, y 2 , yx) with deg(x) = 0 and deg(y) = 1, we have
= 0 for k = 1, . . . , 9:
εk3|2 (x)
1) Ak3|2 with ∆k3|2 (x) = 1 ⊗ x + x ⊗ 1 − 2x ⊗ x for k = 1, 2, and
• ∆13|2 (y) = 1 ⊗ y + y ⊗ 1 − 2x ⊗ y,
• ∆23|2 (y) = 1 ⊗ y + y ⊗ 1;
2) Ak3|2 with ∆k3|2 (x) = 1 ⊗ x + x ⊗ 1 − x ⊗ x, for k = 3, . . . , 9, and
• ∆33|2 (y) = 1 ⊗ y + y ⊗ 1 − x ⊗ y − y ⊗ x + xy ⊗ x + x ⊗ xy,
• ∆43|2 (y) = 1 ⊗ y + y ⊗ 1 − x ⊗ y − xy ⊗ x,
• ∆53|2 (y) = 1 ⊗ y + y ⊗ 1 − x ⊗ y − y ⊗ x,
• ∆63|2 (y) = 1 ⊗ y + y ⊗ 1 − x ⊗ xy − xy ⊗ x,
• ∆73|2 (y) = 1 ⊗ y + y ⊗ 1 − y ⊗ x,
• ∆83|2 (y) = 1 ⊗ y + y ⊗ 1,
• A93|2 = (A43|2 )cop .
Superalgebra A6|2 , we have ∆k6|2 (x) = x ⊗ x and εk6|2 (x) = 1 for k = 1, . . . , 11:
1) Ak6|2 with ∆k6|2 (y) = y ⊗ x + x ⊗ y for k = 1, . . . , 4, and
22
S. Aissaoui and A. Makhlouf
• ∆16|2 (z) = z ⊗ 1 + 1 ⊗ z,
• ∆26|2 (z) = z ⊗ x + x ⊗ z,
• ∆36|2 (z) = z ⊗ x + 1 ⊗ z,
• ∆46|2 (z) = 1 ⊗ z + z ⊗ x + 1 ⊗ y − x ⊗ y;
2) A56|2 with ∆66|2 (y) = y ⊗ 1 + 1 ⊗ y, ∆56|2 (z) = z ⊗ x + x ⊗ z;
3) A66|2 with ∆66|2 (y) = y ⊗ x + 1 ⊗ y, ∆66|2 (z) = z ⊗ x + 1 ⊗ z;
4) A76|2 with ∆76|2 (y) = 1 ⊗ y + y ⊗ x, ∆76|2 (z) = x ⊗ z + z ⊗ x;
5) A86|2 with ∆86|2 (y) = y ⊗ x + x ⊗ y + z ⊗ 1 + 1 ⊗ z − x ⊗ z − z ⊗ x, ∆86|2 (z) = 1 ⊗ z + z ⊗ 1;
6) A96|2 with ∆96|2 (y) = y ⊗ x + 1 ⊗ y + z ⊗ 1 + 1 ⊗ z − z ⊗ x − x ⊗ z, ∆96|2 (z) = z ⊗ x + 1 ⊗ z;
1
1
1
1
10
7) A10
6|2 with ∆6|2 (y) = y ⊗ x + 2 x ⊗ y + 2 1 ⊗ y + 4 1 ⊗ z − 4 x ⊗ z,
1
1
∆10
6|2 (z) = z ⊗ x + 2 1 ⊗ z + 1 ⊗ y − x ⊗ y + 2 x ⊗ z;
6 cop .
8) A11
6|2 = (A3|2 )
Superalgebra A11|2 ∼
= Khx, yi/(x2 − x, y 2 , xy − yx − y) with deg(x) = 0, deg(y) = 1:
1) ∆111|2 (x) = 1 ⊗ x + x ⊗ 1 − 2x ⊗ x, ∆111|2 (y) = 1 ⊗ y + y ⊗ 1 − 2x ⊗ xy − 2xy ⊗ x,
ε111|2 (x) = 1.
Superalgebra A112|2 ∼
= Khx, yi/(x2 , y 2 , xy + yx) with deg(x) = deg(y) = 1:
1) ∆112|2 (x) = 1 ⊗ x + x ⊗ 1, ∆112|2 (y) = 1 ⊗ y + y ⊗ 1.
Superalgebra A14|3 , we have ∆k14|3 (z) = 1 ⊗ z + z ⊗ 1 − x ⊗ z − z ⊗ x and εk14|3 (x) = 0 for
k = 1, . . . , 7:
1) Ak14|3 with ∆k14|3 (x) = 1 ⊗ x + x ⊗ 1 − x ⊗ x for k = 1, . . . , 4:
• ∆114|3 (y) = 1 ⊗ y + y ⊗ 1,
• ∆214|3 (y) = 1 ⊗ y + y ⊗ 1 + x ⊗ z + z ⊗ x,
• ∆314|3 (y) = 1 ⊗ y + y ⊗ 1 − x ⊗ y,
• ∆414|3 (y) = 1 ⊗ y + y ⊗ 1 − x ⊗ y − y ⊗ x;
2) Ak14|3 with ∆k14|3 (x) = 1 ⊗ x + x ⊗ 1 − x ⊗ x + z ⊗ z for k = 5, 6:
• ∆514|3 (y) = 1 ⊗ y + y ⊗ 1,
• ∆614|3 (y) = 1 ⊗ y + y ⊗ 1 + x ⊗ z + z ⊗ x;
3) A714|3 = (A314|3 )cop .
Superalgebra A15|3 , we have A115|3 = (A114|3 )op , A215|3 = (A214|3 )op , A315|3 = (A314|3 )op , A415|3 =
(A414|3 )op , A515|3 = (A514|3 )op , A615|3 = (A614|3 )op , A715|3 = (A514|3 )op,cop .
Superalgebra A17|2 , we have:
1) A117|2 with ∆117|2 (x) = 1 ⊗ x + x ⊗ 1 − x ⊗ x, ∆117|2 (y) = 1 ⊗ y + y ⊗ 1 − y ⊗ x − x ⊗ y,
∆117|2 (z) = 1 ⊗ z + z ⊗ 1 − z ⊗ x − x ⊗ z, ε715|3 (x) = 0;
2) A217|2 with ∆217|2 (x) = x ⊗ x, ∆217|2 (y) = x ⊗ y + y ⊗ x, ∆217|2 (z) = x ⊗ z + z ⊗ x, ε715|3 (x) = 1.
On Classification of Finite-Dimensional Superbialgebras and Hopf Superalgebras
23
Acknowledgements
The authors are grateful to the referees for their valuable remarks and suggestions.
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