On pointed Hopf superalgebras

Abstract

We discuss the relationship between Hopf superalgebras and Hopf algebras. We
list the braided vector spaces of diagonal type with generalized root system of
super type and give the defining relations of the corresponding Nichols
algebras.

Full text

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arXiv:1009.5148v2 [math.QA] 28 Jan 2011
On pointed Hopf superalgebras
Nicol´ as Andruskiewitsch, Iv´ an Angiono, and Hiroyuki Yamane
Dedicado a Jorge Vargas en su sexag´ esimo cumplea˜ nos.
Abstract. We discuss the relationship between Hopf superalgebras and Hopf
algebras. We list the braided vector spaces of diagonal type with generalized
root system of super type and give the defining relations of the corresponding
Nichols algebras.
Introduction
The motivation for this paper is the following: in Heckenberger’s classification
of Nichols algebras of diagonal type with finite root system [He2], there is a large
class of examples that would correspond to contragredient Lie superalgebras. We
want to understand this correspondence. In fact, the explanation is very simple:
Let A be a Hopf algebra with bijective antipode. There is a functor, discovered
by Radford [Ra] and interpreted in categorical terms by Majid [Ma], from the
category of Hopf algebras in the braided category of Yetter-Drinfeld modules over
A to the category of Hopf algebras with split projection to A:
R ? R#A.
See Subsection 1.6 for details; R#A is called the bosonization of R. If A is the
group algebra of Z/2, then the category of super vector spaces fully embeds into
the category of Yetter-Drinfeld modules over A; thus, there is a functor from the
category of Hopf superalgebras to the category of Hopf algebras that we denote
H ? Hσ= H#kZ/2. This functor explains why quantum supergroups appear in
the theory of Hopf algebras, as already noticed by Majid, see [Ma2, Chapter 10.1].
Majid emphasized that the theory of Lie superalgebras and Hopf superalgebras can
be reduced to the classical case using the bosonization by kZ/2. An exposition of
these facts is given in Section 1; we assume that the reader has some familiarity
with the Lifting Method for the classification of Hopf algebras, see [AS3]. However,
2010 Mathematics Subject Classification. Primary 16T05; Secondary 17B37.
Part of this work was done during the visit of the third author to the University of C´ ordoba in
September 2008, supported through Japan’s Grand-in-Aid for Scientific Research (C), 19540027.
The first and second authors were partially supported by ANPCyT-Foncyt, CONICET, Ministerio
de Ciencia y Tecnologa (Crdoba) and Secyt-UNC..
1

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2 ANDRUSKIEWITSCH, ANGIONO AND YAMANE
the main features of this method can be read from the exposition below, dropping
the signs everywhere.
Section 2 is devoted to Nichols algebras of diagonal type that would correspond
to contragredient Lie superalgebras. We list the related diagonal braidings and
discuss the presentation by generators and relations of the corresponding Nichols
algebras applying the method recently presented in [Ang2]. In this way, we recover
results from [Y] and give a partial answer to [And, Question 5.9]. We stress that
Sections 1 and 2 are independent of each other.
Notation. Let k be a field with char k ?= 2; all vector spaces, algebras, tensor
products, etc. are over k except when explicitly stated. For each N > 0, GN
denotes the group of N-th roots of 1 in k.
We use Sweedler’s notation for the comultiplication of a coalgebra D: If x ∈ D,
then ∆(x) = x(1)⊗x(2); if V is a left D-comodule with coaction λ and v ∈ V , then
λ(v) = v(−1)⊗ v(0). If K is a Hopf algebra, thenK
Yetter-Drinfeld modules over K.
KYD denotes the category of
1. Hopf superalgebras
1.1. Super vector spaces. A super vector space is a vector space V graded
by Z/2. We shall write V = V0⊕ V1, to avoid confusion with the coradical
filtration. If v ∈ Vj, then we say that v is homogeneous and write j = |v|. A
super linear map is just a linear map between super vector spaces preserving the
Z/2-grading. The tensor product of two super vector spaces V and W is again
a super vector space, with (V ⊗ W)i= ⊕jVj⊗ Wi+j, i = 0,1. The category of
super vector spaces is symmetric, with super symmetry τsuper: V ⊗ W → W ⊗ V ,
τsuper(v ⊗ w) = (−1)|v||w|w ⊗ v when v and w are homogeneous. A super vector
space V = V0⊕ V1is naturally a Yetter-Drinfeld module over the group algebra
K = kZ/2; namely, if σ denotes the generator of Z/2, then we define σ·v = (−1)|v|v
and δ(v) = σ|v|⊗ v for v homogeneous. The natural embedding of the category of
super vector spaces intoK
KYD preserves the braiding.
Remark 1.1. The categoryK
of the form kχ
and χ the action on the one-dimensional vector space kχ
super vector spaces can be identified with the subcategory ofK
have non-trivial isotypical components only of the types kε
part). Let T be the analogous subcategory whose objects have non-trivial isotypical
components only of the types ksgn
e
or kε
a module category over SV) and T ⊗ T ֒→ SV. In other words,K
tensor category, with even part SV and odd part T .
KYD is semisimple and its irreducible objects are
g, g ∈ Z/2, χ ∈?
Z/2 = {ε,sgn}, meaning that g defines the coaction
g. Then the category SV of
KYD whose objects
e(even part) or ksgn
σ
(odd
σ. Then SV ⊗ T ֒→ T (that is, T becomes
KYD is a super
1.2. Superalgebras. A superalgebra is a Z/2-graded algebra, namely an as-
sociative algebra A with a Z/2-grading A = A0⊕ A1such that AiAj⊆ Ai+j, for
0 ≤ i,j ≤ 1. Given an associative algebra A, a super structure on A is equiva-
lent to an algebra automorphism of order 2, that we call σ by abuse of notation.
Thus σ preserves the Jacobson radical and its powers. Let V = V0⊕ V1be a

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POINTED HOPF SUPERALGEBRAS3
super vector space. Then EndV is a superalgebra, with respect to the grading
EndV = (EndV )0⊕ (EndV )1, where (EndV )i= {T ∈ EndV : T(Vj) ⊆ Vi+j}.
Let A = A0⊕ A1be a superalgebra. Then a super representation on a super
vector space V is a morphism of superalgebras ρ : A → EndV . This amounts to
the same as a module action A ⊗V → V that respects the grading– in words, V is
a (left) supermodule. A superbimodule over A is a bimodule such that both the left
and the right actions are super linear. The corresponding categories are denoted
ASM,ASMA; morphisms are super linear and preserve both structures.
Remark 1.2. Let Aσ= A#k?σ?, the smash product algebra; as a vector space,
this is A ⊗ kZ/2 ≃ A ⊕ Aσ, and the multiplication is given by
(1.1)(a#σk)(b#σl) = (−1)k|b|ab#σk+l,a,b ∈ A,k,l ∈ {0,1}.
Then the categoryASM is naturally equivalent to the categoryAσM of modules
over Aσ. Explicitly, given an A-supermodule V we define the action of Aσas
(a#σk) · v := (−1)|v|ka · v,v ∈ V,a ∈ A,k ∈ Z/2.
Reciprocally, any object ofAσM, with the grading defined by the action of σ and
the action of A given by restriction, is an A-supermodule.
1.3. Supercoalgebras. A supercoalgebra is a Z/2-graded coalgebra, namely
an associative coalgebra C (with comultiplication ∆ and counit ε), provided with
a Z/2-grading C = C0⊕ C1such that ∆(Ci) ⊆?
c ∈ C, then ∆(c) = c(1)⊗ c(2), where c(1)and c(2)are homogeneous.
The dual of a finite-dimensional superalgebra is a supercoalgebra; thus, the
matrix coalgebra EndcV is a supercoalgebra, if V is a finite-dimensional super
vector space.
Given an associative coalgebra C, a super structure on C is equivalent to a
coalgebra automorphism of order 2. In particular, the coradical C0, hence all terms
Cnof the coalgebra filtration, are stable under σ, i. e., are super vector subspaces of
C. Therefore the associated graded coalgebra grC = ⊕n≥0Cn/Cn−1is a Z-graded
supercoalgebra (where C−1= 0).
An element c ∈ C is a group-like if c ?= 0 and ∆(c) = c⊗c; the set of all group-
likes is denoted G(C). Since σ acts by coalgebra automorphisms, it preserves G(C).
We shall say that C is pointed if the coradical C0equals the linear span of G(C). In
the particular case when C is the linear span of G(C), a structure of super coalgebra
is determined by an involution of the set G(C).
Let C = C0⊕ C1be a supercoalgebra. Then a (left) supercomodule is a super
vector space V provided with a coaction δ : V → C ⊗ V that is super; similarly
for right supercomodules and superbicomodules. The corresponding categories are
CSM, SMC,CSMC.
jCj⊗ Ci−j, for 0 ≤ i ≤ 1. We
write the comultiplication of C by the following variation of Sweedler’s notation: If
Remark 1.3. Let Cσ= C#k?σ?, the smash product coalgebra; as a vector
space, this is C ⊗ kZ/2, and the comultiplication is given by
(1.2)∆(c#σk) = c(1)#σ|c(2)|+k⊗ c(2)#σk,c ∈ C,k ∈ Z/2.
Then (grC)σ≃ gr(Cσ).

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4 ANDRUSKIEWITSCH, ANGIONO AND YAMANE
The categoryCSM is naturally equivalent to the categoryCσM of left comod-
ules over Cσ. In this case, given V ∈CSM, the coaction δ : V → Cσ⊗V is defined
by
δ(v) := v(−1)#σ|v(0)|⊗ v(0),v ∈ V.
Reciprocally, any object ofCσM, with the grading and the action of A given by
corestriction, is an A-supercomodule.
1.4. Hopf Superalgebras. A superbialgebra is a bialgebra in the category of
super vector spaces, that is a Z/2-graded algebra and coalgebra B (with respect to
the same grading) such that ∆ and ε are multiplicative, with respect to the product
in B ⊗B twisted by the super symmetry: (a⊗b)(c⊗d) = (−1)|b||c|ac⊗bd. A Hopf
superalgebra is a superbialgebra H such that the identity map has a convolution
inverse S ∈ EndH; S is called the antipode and preserves the super grading.
Example 1.4. When does a usual Hopf algebra H admit a Hopf superalgebra
structure? If so, σ acts by a Hopf algebra automorphism of order 2. However, this
is not enough: for, assume that H = kΓ is a group algebra and let σ0an involution
of the group Γ. If H is a Hopf superalgebra with respect to the automorphism σ
of H defined by σ0, then σ0= id. Indeed, let g ∈ Γ and let xg =1
yg=1
2(eg− eσ0(g)). Then
2(eg+ eσ0(g)),
∆(eg2) = eg2 ⊗ eg2 = egeg⊗ egeg= egeg⊗ xgeg+ egeg⊗ ygeg,
∆(eg)∆(eg) = (eg⊗ (xg+ yg))((xg+ yg) ⊗ eg) = egeg⊗ xgeg+ egeσ0(g)⊗ ygeg.
Hence yg= 0 and σ0(g) = g. The same argument shows that a group-like element in
a Hopf superalgebrais even. In fact, it can be shown a semisimple Hopf superalgebra
over C such that S2= id is purely even [AEG, Cor. 3.1.2].
A Hopf superalgebra H is, in particular, a braided Hopf algebra inK
there is a Hopf algebra Hσ, the Radford-Majid bosonization of H. As a vector
space, Hσ= H ⊗ kZ/2 ≃ H ⊕ Hσ; the multiplication and comultiplication of Hσ
are given by (1.1) and (1.2).
KYD, and
Remark 1.5. Another relation between Hopf algebras and Hopf superalgebras
is given in [AEG, Th. 3.1.1]: There is a one-to-one correspondence between
(1) isomorphisms classes of pairs (H,u), where H is a Hopf algebra and u ∈ H
is a group-like element such that u2= 1, and
(2) isomorphisms classes of pairs (H,g), where H is a Hopf superalgebra and
g ∈ H is a group-like element such that g2= 1 and gxg−1= (−1)|x|x.
Explicitly, given (H,u), H is the algebra H with the grading given by the
adjoint action of u and the comultiplication:
∆super(h) = ∆0(h) + (−1)|h|(u ⊗ id)∆1(h),h ∈ H.

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POINTED HOPF SUPERALGEBRAS5
Here ∆(h) = ∆0(h) + ∆1(h), with ∆k(h) ∈ H ⊗ Hk, k ∈ Z/2. Also, g = u.
Conversely, H = Hσ/(σu−1). Eventually, this correspondence leads to the classifi-
cation of all finite-dimensional triangular Hopf algebras over an algebraically closed
field of characteristic 0 [EG].
Let H = H0⊕H1be a Hopf superalgebra. Then the category of supermodules
HSM is a tensor one (with the underlying tensor product of super vector spaces);
if V,W ∈HSM, v ∈ V is homogeneous, w ∈ W and h ∈ H, then
(1.3)h · (v ⊗ w) = (−1)|h(2)||v|h(1)· v ⊗ h(2)· w.
Moreover, the equivalenceASM ≃AσM in Remark 1.2 is monoidal.
Analogously, the categoryHSM of supercomodules over a Hopf superalgebra
H is a tensor category; here, if V,W ∈HSM, v ∈ V is homogeneous, w ∈ W, then
(1.4)δ(v ⊗ w) = (−1)|v(0)||w(−1)|v(−1)w(−1)⊗ v(0)⊗ w(0).
The equivalenceCSM ≃CσM in Remark 1.3 is also monoidal.
A quasitriangular Hopf superalgebra is a pair (H,R) where H is a Hopf super-
algebra and R ∈ H ⊗H is even, invertible, and satisfies the same axioms as in the
non-super case, see e. g. [AEG]. In particular, it providesHSM with a braiding:
if V,W ∈HSM, v ∈ V and w ∈ W, then cV,W: V ⊗ W → W ⊗ V is given by
(1.5)cV,W(v ⊗ w) = (−1)|v||w|R · (w ⊗ v).
The element R is called a universal R-matrix. Furthermore, any braiding inHSM
arises from a universal R-matrix, cf. [M, 10.4.2]. Also, (H,R) is a triangular Hopf
superalgebra ifHSM is symmetric for the previous braiding.
1.5. Hopf supermodules, Hopf superbimodules and Yetter-Drinfeld
supermodules. Let H be a Hopf superalgebra with bijective antipode. There
is a hierarchy of special modules over H; the proofs of the statements below are
adaptations of the usual proofs for Hopf algebras. We leave to the reader the
pleasant task of checking that the signs match. Analogous results in the more
general context of braided categories have been proved in [B, BD].
• A Hopf supermodule over H is a super vector space V that is simultaneously
a supermodule and a supercomodule, with compatibility saying that the coaction
δ : V → H ⊗V is morphism of H-supermodules. If U is a super vector space, then
H ⊗U is a Hopf supermodule over H with the action and coaction on the left. If V
is a supercomodule, then set VcoH= {v ∈ V : δ(v) = 1⊗v}. There is a Fundamen-
tal theorem for Hopf supermodules: the category of Hopf supermodules over H is
equivalent to the category of super vector spaces, via V ?→ VcoH. Explicitly, if V is
a Hopf supermodule, then the multiplication µ : H ⊗VcoH→ V is an isomorphism
of Hopf supermodules.
• A Hopf superbimodule is a super vector space V that is simultaneously a super-
bimodule and a superbicomodule, with compatibility saying that both coactions
λ : V → H ⊗ V and ρ : V → V ⊗ H are morphisms of H-superbimodules. The
categoryH
Hof Hopf superbimodules is a tensor one, with tensor product ⊗H.
HSMH

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6 ANDRUSKIEWITSCH, ANGIONO AND YAMANE
• A Yetter-Drinfeld supermodule over H is a super vector space V that is simulta-
neously a supermodule and a supercomodule, with compatibility saying that
δ(h · v) = (−1)|v(−1)|(|h(2)|+|h(3)|)+|h(2)||h(3)|h(1)v(−1)S(h(3)) ⊗ h(2)· v(0). (1.6)
The categoryH
HSMH
HYDS of Yetter-Drinfeld supermodules is tensor equivalent to
H. Explicitly,
M ∈H
H
HSMH
H? V = McoH= {m ∈ M : ρ(m) = m ⊗ 1},
with action and coaction h · v = (−1)|v||h(2)|h(1)vS(h(2)), δ = λ;
V ∈H
HYDS ? M = V ⊗ H,
λ(v ⊗ h) = v(−1)h(1)⊗ (v(0)⊗ h(2)),
x · (v ⊗ h)y = (−1)|v||x(2)|x(1)v ⊗ x(2)hy,
ρ(v ⊗ h) = (v ⊗ h(1)) ⊗ h(2)
for v ∈ V , h,x,y ∈ H.
• The tensor categoryH
cM,N: M ⊗HN → N ⊗HM, M,N ∈H
HSMH
Hof Hopf superbimodules is braided, with braiding
HSMH
H, given by
cM,N(m⊗n) = (−1)(|m(0)|+|m(−1)|)(|n(0)|+|n(1)|)m−2n(0)S(n(1))S(m(−1)) ⊗m(0)n(2).
Thus,H
given by
cX,Y(x ⊗ y) = (−1)|x(0)||y|x(−1)· y ⊗ x(0).
HYDS is braided, with braiding cV,W : V ⊗ W → W ⊗ V , V,W ∈H
HYDS,
(1.7)
Remark 1.6. For each Hopf superalgebra H there exists a full embedding of
braided tensor categories i :H
HσYD, given by the restriction of equiva-
lences in Remarks 1.2 and 1.3.
HYDS ֒→Hσ
Example 1.7. Let H be a purely even Hopf superalgebra, that is a usual Hopf
algebra with trivial grading. If V ∈H
HYD and k ∈ Z/2, then V [k] = V with all
elements of degree k, is an object inH
then V [k] is irreducible inH
the form V [k] as above.
HYDS. Moreover, if V is irreducible inH
HYDS. We claim that any irreducible inH
HYD,
HYDS is of
Proof. If W is an irreducible module inH
objects inH
sub-object of WkinH
HYD.
HYDS, then W0and W1are sub-
HYDS, hence W = Wkfor some k. Thus W = U[k], for U an irreducible
?
1.6. Hopf superalgebras with projection and bosonization. In this sub-
section we consider bosonization and Hopf superalgebras with projections; we note
that this construction can be done for general braided categories, see [B, BD].
Let H be a Hopf superalgebra with bijective antipode. If R is a Hopf algebra
in the braided categoryH
HYDS, then we have a Hopf superalgebra R#H: it has
R ⊗ H as underlying super vector space, and its structure is defined by
(a#h)(b#f) := (−1)|h(2)||b|a(h(1)· b)#h(2)f, 1 := 1R#1H,
∆(a#h) := (−1)|(a(2))(0)||h(1)|a(1)#(a(2))(−1)(h(1)⊗ (a(2))(0)#h(2),
ε(a#h) := εR(a)εH(h),
S(a#h) := (−1)|a(0)||h|?1#SH(a(−1)h)??SR(a(0))#1?.

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POINTED HOPF SUPERALGEBRAS7
for each a,b ∈ R and h,f ∈ H. By Remark 1.6, the image of R under the full
embedding i is a Hopf algebra inHσ
HσYD. It is straightforward to prove that
(R#H)σ∼= i(R)#Hσ.(1.8)
Let ι : H ֒→ L and π : L ։ H be morphisms of Hopf superalgebras satisfying
π ◦ ι = idH. Consider the subalgebra of coinvariants
R := LcoH= {x ∈ L : (id⊗π)∆(x) = x ⊗ 1}.
This is a Hopf algebra in the categoryH
of Hopf superalgebras L∼= R#H. Now, ι and π induce Hopf algebra morphisms
ισ: Hσ֒→ Lσand πσ: Lσ։ Hσsuch that πσ◦ ισ= idHσ. Then i(R) coincides
with the subalgebra of coinvariants (Lσ)coHσ, see e. g. [AHS, Lemma 3.1].
HYDS and there exists an isomorphism
1.7. Nichols superalgebras. Let H be a Hopf superalgebra with bijective
antipode. The constructions and results of [S, Section 2] hold in the braided abelian
categoryH
HYDS. We summarize:
Proposition 1.8. [S, Section 2]. Let V be a Yetter-Drinfeld supermodule over
H. Then there is a unique (up to isomorphisms) graded Hopf algebra B(V ) =
⊕n∈N0Bn(V ) inH
• B0(V ) ≃ k,
• V ≃ B1(V ) = P(B(V )) (the space of primitive elements),
• B1(V ) generates the algebra B(V ).
HYDS with the following properties:
Explicitly, B(V ) ≃ T(V )/J(V ), where the ideal J(V ) = ⊕n≥2Jn(V ) has
homogeneous components Jn(V ) that equal the kernel of the quantum symmetrizer
kerSn[W, S]. To be more precise, the braid group in n letters Bnacts on Tn(V )
via the braiding inH
HYDS. Let π : Bn → Sn be the natural projection and let
s : Sn → Bn be the so-called Matsumoto (set-theoretical) section. The element
Sn:=?
conclude from Remark 1.6 that the Nichols algebra functor commutes with the full
embedding i:
σ∈Sns(σ) of the group algebra of Bnis called the quantum symmetrizer;
it acts on Tn(V ) and its kernel is Jn(V ). Because of this explicit description, we
(1.9)B(i(V )) ≃ i(B(V )).
1.8. The lifting method for Hopf superalgebras. Let H be a Hopf super-
algebra with bijective antipode. If the coradical H0of H is a Hopf sub-superalgebra,
then the coradical filtration is also an algebra filtration and the associated graded
coalgebra grH is a graded Hopf superalgebra.
projection grH → H0 splits the inclusion, hence gives rise to a decomposition
grH ≃ R#H0. The graded Hopf algebra R = ⊕n≥0Rn∈H0
properties:
• R0≃ k,
• R1= P(R).
Thus the subalgebra generated by V := R1is isomorphic to the Nichols algebra
B(V ). In this way, the Lifting Method [AS3] can be adapted to the setting of Hopf
superalgebras whose coradical is a Hopf sub-superalgebra. However, there is no
need to start over again since classification problems of Hopf superalgebras reduce
Furthermore, the homogeneous
H0YDS has the following

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8ANDRUSKIEWITSCH, ANGIONO AND YAMANE
to analogous classification problems of Hopf algebras via the functor H ? Hσ.
This principle is illustrated by the following facts:
• The coradical of Hσis H0⊕H0σ and G(Hσ) = G(H) ×?σ?. More generally, the
coradical filtration of Hσis Hσ
n= Hn⊕ Hnσ.
• Hσis pointed if and only if H is pointed.
• The coradical of Hσis a Hopf subalgebra if and only if the coradical of H is a
Hopf sub-superalgebra. If this is the case, then
gr(Hσ) ≃ (grH)σ≃ (R#H0)σ≃ i(R)#(H0)σ.
• The Hopf algebra H is generated (as algebra) by group-like and skew-primitive
elements (generated in degree one, for short) if and only if Hσis generated in degree
one.
Remark 1.9. It was conjectured that a finite-dimensional pointed Hopf algebra
over k is generated in degree one [AS1, 1.4]. This Conjecture was verified in various
cases, see e. g. [AS2, 7.6], [GG], [AG, 2.7], [Ang2, 4.3]. The validity of the
conjecture would imply the validity of the analogous one for Hopf superalgebras.
Because of these considerations, we see that the theory of Hopf superalgebras
is naturally a part of the theory of Hopf algebras.
Example 1.10. There is a full embedding from the category of Lie superalge-
bras to the category of pointed Hopf algebras with group Z/2, given by g ? U(g)σ.
In particular, we see that the classification of finite-dimensional pointed Hopf
superalgebras H with a fixed group of group-like elements Γ reduces to the classi-
fication of finite-dimensional pointed Hopf algebras K such that
◦ G(K) ≃ Γ × Z/2,
◦ there exists a projection of Hopf algebras K → kZ/2 that splits the in-
clusion (from the second factor above).
Example 1.11. Let Γ be a finite abelian group. Assume that k is algebraically
closed. Then any irreducible object inkΓ
kΓYD has dimension one and is of the form
kχ
g, g ∈ Γ, χ ∈?Γ, where g determines the coaction and χ the action. By Example
The corresponding isotypical component of V ∈kΓ
finite-dimensional V ∈kΓ
χj∈?Γ, kj= |xj| ∈ Z/2.
set qij:= χj(gi) and


superalgebra B(V ) has finite dimension if and only if the connected components of
1.7, any irreducible object inkΓ
kΓYD is of the form kχ
g[k], g ∈ Γ, χ ∈?Γ, k ∈ Z/2.
gj[kj], gj ∈ Γ,
kΓYD is denoted Vχ
g[k]. Thus, any
kΓYDS has a basis x1,...,xθ with xj ∈ Vχj
Proposition 1.12. Let gj∈ Γ, χj∈?Γ, kj∈ Z/2, 1 ≤ j ≤ θ. For 1 ≤ i,j ≤ θ,

Let V ∈kΓ
(1.10)
? qij=
qij,i ?= j,
(−1)kiqii,i = j.
kΓYDS with a basis x1,...,xθ, such that xj∈ Vχj
gj[kj]. Then the Nichols

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POINTED HOPF SUPERALGEBRAS9
the generalized Dynkin diagram corresponding to the matrix (? qij)1≤i,j≤θ belong to
Proof. By (1.7) and (1.9), we are reduced to consider the Nichols algebra
of the braided vector space of diagonal type with matrix
Now this matrix and (? qij)1≤i,j≤θare twist-equivalent [AS3, Def. 3.8], hence their
the list in [He2].
?(−1)|xi||xj|qij
?
1≤i,j≤θ.
Nichols algebras have the same dimension [AS3, Prop. 3.9].
?
2. Generalized root systems and Nichols algebras
2.1. Generalized root systems. We recall now the generalization of the
notion of a root system given in [HY].
Fix two non-empty sets X and I, where I is finite, and denote by {αi}i∈I the
canonical basis of ZI.
Definition 2.1. [HY, CH] Assume that for each i ∈ I there exists a map
ri: X → X, and for each X ∈ X a generalized Cartan matrix AX= (aX
the sense of [K] satisfying
ij)i,j∈I in
(1) for all i ∈ I, r2
(2) for all X ∈ X and i,j ∈ I: aX
i= id, and
ij= ari(X)
ij
.
We say that the quadruple C := C(I,X,(ri)i∈I,(AX)X∈C) is a Cartan scheme.
Given i ∈ I and X ∈ X, sX
idenotes the automorphism of ZIsuch that
sX
i(αj) = αj− aX
ijαi,j ∈ I.
The Weyl groupoid of C is the groupoid W(C) for which:
(1) the set of objects is X, and
(2) the morphisms are generated by sX
i ∈ I, X ∈ X.
i, if we consider sX
i∈ Hom(X,ri(X)),
Each morphism w ∈ Hom(W,X1) is a composition sX1
rij−1···ri1(X1), i ≥ 2. We shall write w = idX1si1···simto indicate that w ∈
Hom(W,X1), because the Xj’s are univocally determined by the first one and the
sequence i1,··· ,im.
i1sX2
i2···sXm
im, where Xj =
Definition 2.2. [HY, CH] Given a Cartan scheme C, and for each X ∈ X a set
∆X⊂ ZI, define mX
is a root system of type C if
ij:= |∆X∩(N0αi+N0αj)|. We say that R := R(C,(∆X)X∈X)
(1) for all X ∈ X, ∆X= (∆X∩ NI
(2) for all i ∈ I and all X ∈ X, sX
(3) for all i ∈ I and all X ∈ X, ∆X∩ Zαi= {±αi},
(4) for all i ?= j ∈ I and all X ∈ X, (rirj)mX
0) ∪ −(∆X∩ NI
i(∆X) = ∆ri(X),
0),
ij(X) = X.
We call ∆X
set of negative roots. By simplicity we will write W in place of W(C) when C is
+:= ∆X⊂ NI
0the set of positive roots of X, and ∆X
−:= −∆X
+the

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10ANDRUSKIEWITSCH, ANGIONO AND YAMANE
understood, and for any X ∈ X:
Hom(W,X) := ∪Y ∈XHom(Y,X),
∆X re:= {w(αi) : i ∈ I, w ∈ Hom(W,X)}.
(2.1)
(2.2)
The elements of ∆X reare the real roots of X.
We say that R is finite if ∆Xis finite for all X ∈ X. In such case all the
roots are real, see [CH, Prop. 2.12], and for each pair i ?= j ∈ I and each X ∈ X,
αi+ kαj∈ ∆Xif and only if 0 ≤ k ≤ −aX
ij. Therefore,
(2.3)aX
ij= −max{k ∈ N0: αi+ kαj∈ ∆X}.
Example 2.3. By [HY, Example 3], the root system associated to a finite
dimensional contragradient Lie superalgebra is a generalized root system in this
context. We describe them case by case, considering the irreducible root systems.
We call them super root systems.
Type Aθ: We need to consider a parity of the simple roots p(αi), and extend it to
a group homomorphism p : Zθ→ {±1}. The set X is determined as follows: we
have a symmetry sifrom one point X to a different oneˆ X if p(αi) = −1. The new
parity function ˆ p is determined from p and si:
ˆ p(αk) = p(si(αk)) = p(αk+ mikαi) = p(αk)(−1)mik.
That is, it changes the parity of the vertices k which are connected to i, and keeps
the parity of the non-connected vertices. In this way, X can have more than one
element depending on the parity of the simple roots, but for any X ∈ X we have
the same set of positive roots,
(2.4)∆X
+= {uij:= αi+ αi+1+ ··· + αj: 1 ≤ i ≤ j ≤ θ}.
Type Bθ: As above, X can have more than a point, and again the symmetries that
go from a point to a different one are symmetries of odd vertices, with the same
changes. Anyway, the set of positive roots is the same for any X ∈ X,
(2.5)∆X
+= {uij: 1 ≤ i ≤ j ≤ θ} ∪ {vij:= ui,θ+ uj,θ: 1 ≤ i < j ≤ θ}.
Types Cθ,Dθ: As above we consider a parity function p : Zθ→ Z2. Following
the classical literature, there are sets ∆X(C)
+
described as follows:
of type C and sets ∆X(D)
+
of type D,
∆X(C)
+
= {uij: 1 ≤ i ≤ j ≤ θ} (2.6)
∪ {wij:= ui,θ+ uj,θ−1: 1 ≤ i < j ≤ θ − 1}
∪ {? wi:= ui,θ−1+ ui,θ: 1 ≤ i ≤ θ − 1, p(ui,θ−1) = 1},

Page 11

POINTED HOPF SUPERALGEBRAS 11
∆X(D)
+
= {uij: 1 ≤ i ≤ j ≤ θ,(i,j) ?= (θ − 1,θ)} (2.7)
∪ {αθ−1+ αθ: p(αθ−1) = −1}
∪ {? ui:= ui,θ−2+ αθ: 1 ≤ i ≤ θ − 2}
∪ {? zi:= ui,θ+ ui,θ−2: 1 ≤ i ≤ θ − 2, p(ui,θ−1) = −1}.
+= {α1,α2,α3,α1+ α2,α1+ α3,α2+ α3,α1+ α2+ α3},
∆Xk
+ = {α1,α2,α3,α1+ α2+ α3,α1+ α2+ α3+ αk}
∪ {zij:= ui,θ+ uj,θ−2: 1 ≤ i < j ≤ θ − 2}
Type D(2,1;α): We have four possible sets of roots,
∆X0
(2.8)
(2.9)
∪ {αk+ αj: j ∈ {1,2,3}\ {k}},
where k ∈ {1,2,3}. Here, sk(∆X0) = ∆Xk.
Type F(4): in this case |X| = 6. One of the sets of roots is
∆X
+= {α1, α1+ α2, α1+ α2+ α3, α1+ α2+ 2α3, α1+ 2α2+ α3, (2.10)
α1+ α2+ α3+ α4, α1+ α2+ 2α3+ α4, α1+ 2α2+ 2α3+ α4,
α1+ 2α2+ 3α3+ 2α4, α2+ α3+ α4, α2+ 2α3+ α4, α2+ α3,
α2+ 2α3, α2, α3, α3+ α4, α4}.
The other sets of roots are obtained applying the symmetries si, once one determines
aX
ijas in (2.3).
Type G(3): now, |X| = 4, and one of these sets of positive roots is
∆X
+= {α1, α1+ α2, α1+ α2+ α3, α1+ 2α2+ α3, (2.11)
α1+ 3α2+ α3, α1+ 3α2+ 2α3, α1+ 4α2+ 2α3,
α2, α2+ α3, 2α2+ α3, 3α2+ α3, 3α2+ 2α3, α3}.
We obtain the other sets of positive roots by determining aX
the symmetries si.
ijas in (2.3) and applying
Now we recall the definition of the Weyl groupoid attached to a braided vector
space (V,c) of diagonal type given in [He1], see also [AA]. Fix a basis {x1,...,xθ}
and scalars qij ∈ k×such that c(xi⊗ xj) = qijxj⊗ xi. Let χ : Zθ× Zθ→ k×be
the bilinear form such that χ(αi,αj) = qij. Following [He1], ∆V
degrees of a PBW basis of B(V ), counted with their multiplicities. It is remarked
in [He1] and proved in [AA] that this set does not depend on the PBW basis.
For each n ∈ N we set the following polynomials in q:
?n
q
Let X be the set of ordered bases of Zθ. For each F = {f1,...,fθ} ∈ X, set
qF
ij= χ(fi,fj). Define
?
+denotes the set of
j
?
=
(n)q!
(k)q!(n − k)q!,where (n)q! =
n
?
j=1
(k)q, and (k)q=
k−1
?
j=0
qj.
(2.12)aij(F) := −minn ∈ N0: (n + 1)qF
ii(1 − (qF
ii)nqF
ijqF
ji) = 0
?
,

Page 12

12 ANDRUSKIEWITSCH, ANGIONO AND YAMANE
for each 1 ≤ i ?= j ≤ θ, and set si,F ∈ Aut(Zθ) such that si,F(fj) = fj− aij(F)fi.
Here aii= 2.
Consider for G = Aut(Zθ)×X the groupoid structure given as follows: the set
(g,x)
−→ g(x).
Then the Weyl Groupoid W(χ) of χ is the least subgroupoid of G such that
(id,E) ∈ W(χ), and if (id,F) ∈ W(χ) and si,F is defined, then (si,F,F) ∈ W(χ).
The generalized root system for each object F is ∆VF, where (VF,cF) is the
braided vector space of diagonal type whose braiding matrix is (qF
axioms of a root system by [He1].
of objets is X and the morphisms are x
ij). It satisfies the
Remark 2.4. If aF
ij= 0, then for all k ?= i,j,
qsi(F)
jj
= qF
jj,qsi(F)
jk
qsi(F)
kj
= qF
jkqF
kj.
Remark 2.5. If (qF
matrix of (qF
kj) and then the braiding matrices are twist equivalent. In consequence,
∆Vsi(F)= ∆VF.
ii)aij(F)= qF
ijqF
jifor all j ?= i, then (qsi(F)
kj
) is the transpose
2.2. Diagonal braidings of super type. From now on, k is an algebraically
closed field of characteristic 0.
We shall characterize the Nichols algebras whose root system is one of those
associated to a contragradient finite-dimensional Lie superalgebra.
First we recall some definitions following [He2]. The generalized Dynkin dia-
gram associated to a braided vector space of diagonal type, with braiding matrix
(qij)1≤i,j≤θis a graph with θ vertices, each of them labeled with the corresponding
qii, and an edge between two vertices i,j if qijqji ?= 1, labeled with this scalar.
In this way two braided vector spaces of diagonal type have the same generalized
Dynkin diagram if and only if they are twist equivalent.
A simple chain of length θ is a braided vector space of diagonal type whose
braiding matrix (qij)1≤i,j≤θsatisfies
• (1 + q11)(1 − q11q12q21) = (1 + qθθ)(1 − qθθqθ,θ−1qθ−1,θ) = 0,
• qijqji= 1 if 1 < i,j < θ, |i − j| > 1,
• for any 1 < i < θ, qii= −1, qi−1,iqi,i−1qi+1,iqi,i+1= 1, or qiiqi−1,iqi,i−1=
qiiqi+1,iqi,i+1= 1.
Here C(θ,q;i1,...,ij) denotes a simple chain such that q = q2
and qi−1,iqi,i−1= q if and only if i ∈ {i1,...,ij}.
θθqθ,θ−1qθ−1,θ,
Theorem 2.6. Let (V,c) a braided vector space of diagonal type, with braiding
matrix (qij). Assume that its generalized Dynkin diagram is connected. Then B(V )
has a super root system if and only if its generalized Dynkin diagram is one of the
following ones:
Type Aθ:
(2.13)C(θ,q;i1,...,ij),θ ∈ N,q ∈ k×,q2?= 1,1 ≤ i1< i2< ··· < ik≤ θ.

Page 13

POINTED HOPF SUPERALGEBRAS 13
Type Bθ:
◦q
q−1
◦ζ,ζ ∈ G3, q ∈ k \ {0,1,−1,ζ,ζ2}, (2.14)
?
?
?
?
?
?
C(θ − 1,q2;i1,...,ij)
q−2
◦q,θ ∈ N, q ∈ k×,q ?= ±1, (2.15)
?
?
C(θ − 1,−ζ2;i1,...,ij)
−ζ
◦ζ,θ ∈ N, ζ ∈ G3.(2.16)
Type Cθ, θ ∈ N, q ∈ k×, q4?= 1:
(2.17)
?
?
?
?
C(θ − 1,q;i1,...,ij)
q−2
◦q2.
Type Dθ, θ ∈ N, q ∈ k×, q2?= 1:
?
?
?
?
q
C(θ − 2,q−1;i1,...,ij)
q
◦q−1
◦q−1
?
?
q−1
, (2.18)
?
?
C(θ − 2,q;i1,...,ij)
q−1
◦−1
q2
????????
◦−1
. (2.19)
Type D(2,1;α), q,r,s ∈ C×\ {1}, qrs = 1:
◦q
q−1
◦−1
r−1
◦r, (2.20)
◦−1
?
?
?
?
?
?
?
?
r
?
?
?
??
?
?
?
◦−1
q
s
◦−1
. (2.21)
Type F(4), q ∈ k×, q2,q3?= 1:
◦−1
q−1
◦q
q−2
◦q2
q−2
◦q2, (2.22)
◦−1
q
◦−1
q−2
◦q2
q−2
◦q2, (2.23)

Page 14

14 ANDRUSKIEWITSCH, ANGIONO AND YAMANE
◦−1
q2
q−1
?
?
?
?
?
?
?
?
◦−1
q−2
◦q2
◦q
q−1
, (2.24)
◦q2
q−2
◦−1
q2
◦−1
q−3
????????
◦−1
q
, (2.25)
◦q2
q−2
◦q
q−1
◦−1
q3
◦q−3, (2.26)
◦q2
q−2
◦q2
q−2
◦−1
q3
◦q−3.(2.27)
Type G(3), q ∈ k×, q2,q3?= 1:
◦−1
q−1
◦q
q−3
◦q3, (2.28)
◦−1
q
◦−1
q−3
◦q3, (2.29)
◦−1
q3
?
?
?
??
?
?
?
◦q
q−1
???
??
???
q−2
◦−1
, (2.30)
◦−q−1
q2
◦−1
q−3
◦q3. (2.31)
Proof. When the braiding is of type Aθor Bθit follows by [Ang1, Proposi-
tions 3.9, 3.10]. The proof for the other cases is completely analogous, so we just
show in detail the case Cθ. A first remark is that the submatrix (qij)1≤i,j≤θ−1is of
type Aθ−1, so it is standard and has a generalized Dynkin diagram as (2.13), and
?
qθ,θ−1
qθθ
Also, if there exists 1 ≤ i ≤ θ − 1 such that p(αi) = −1, then the reflection si
changes the set of roots, and by Remark 2.5, qii= −1 ?= qi,i−1qi−1,i,qi,i+1qi+1,i.
If p(αi) = 1 for any i, then the root system is of finite type, and it follows that
the braiding is of Cartan type by [Ang1, Prop. 3.8]. If not, we can assume that
p(αθ−1) = −1 up to applying a suitable sequence of reflections si. Applying sθ−1,
αθbecomes odd for the new parity function, and moreover the reflection sθchanges
the root system, so in the original braiding qθθ?= −1 by Remark 2.5.
We can make also p(αθ−1) = 1 up to applying some reflections. In such case,
aθ−1,θ= −2, aθ−1,θ−2= −1 and applying the reflection sθ−1the vertices θ − 2, θ
are not connected (i.e. αθ−2+ αθ is not a root). In consequence, if (? qij) denotes
the submatrix
qθ−1,θ−1
qθ−1,θ
?
is of type B2.

Page 15

POINTED HOPF SUPERALGEBRAS 15
the braiding matrix after applying the reflection sθ−1, we have
1 = ? qθ−2,θ? qθ,θ−2= χ(αθ−2+ αθ−1,αθ+ 2αθ−1)χ(αθ+ 2αθ−1,αθ−2+ αθ−1)
If we assume q2
θ−1,θ−1?= qθθ, by the possible values of a matrix of type B2in such
conditions we obtain ? qθ−2,θ? qθ,θ−2 = −1, a contradiction. Therefore q2
computation we can see that a braiding as before is of type Cθ.
= q4
θ−1,θ−1q2
θ−2,θ−1q2
θ−1,θ−2qθ,θ−1qθ−1,θ.
θ−1,θ−1=
qθθ, and the braiding has a generalized Dynkin diagram as in (2.17). By direct
?
Remark 2.7. Note that these diagrams correspond with the following ones in
Heckenberger’s classification [He2]:
• row 7 for type G(3), and rows 9, 10 and 11 for type D(2,1;α) in Table 2,
• row 9 for type F(4) in Table 3,
• rows 1 and 2 for type Aθ, rows 3, 4, 5 and 6 for type Bθ, rows 7, 8, 9 and
10 for type Cθ, Dθin Table 4.
2.3. Presentation by generators and relations. In this subsection we will
present the Nichols algebras with super root systems of type A, B, C, D by gener-
ators and relations using the results in [Ang2].
Recall that [x,y]c= xy − χ(α,β)yx if x,y ∈ B(V ) are homogeneous of degree
α,β ∈ Zθ, respectively. In particular, (adcxi)(y) := [xi,y]c.
We will define the hyperletters associated to the root vectors of type A, B, C,
D, following [Ang2, Corollary 2.17].
First of all, xuii= xαi= xi, and recursively,
(2.32)xuij:= [xi,xui+1,j]c, i < j.
Also, xvi,θ= [xui,θ,xθ]c, and recursively,
(2.33)xvij:= [xvi,j+1,xj]c, i < j.
For type C, xwi,θ−1= [xui,θ,xθ−1]c, and then,
xwij:= [xwi,j+1,xj]c,
x? wi:= [xui,θ−1,xui,θ]c.
i < j,(2.34)
(2.35)
For type D, we have x? uθ−2= [xθ−2,xθ], and recursively x? ui= [xi,x? ui+1]. Also,
xzi,θ−2= [xui,θ,xθ−2]c, and
xzij:= [xzi,j+1,xj]c,
x? zi:= [xui,θ−1,x? ui]c.
i < j, (2.36)
(2.37)
In this case, note that xuθ−2,θ= [[xθ−2,xθ]c,xθ−1]c.
For any α ∈ ∆V
+we write qα= χ(α,α), and Nα= ord(qα).
Theorem 2.8. Let (V,c) be a braided vector space of diagonal type, with braid-
ing matrix (qij)1≤i,j≤θ. Assume that the root system of B(V ) is of super type, with
connected components of type A, B, C, D.