Yetter–Drinfel’d Hopf Algebras on Basic Cycle
A class of Yetter–Drinfel’d Hopf algebras on basic cycle is constructed.
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508173753909248 |
|---|---|
| author | Wang, Yanhua Ван, Янхуа |
| author_facet | Wang, Yanhua Ван, Янхуа |
| author_sort | Wang, Yanhua |
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| datestamp_date | 2019-12-05T10:26:31Z |
| description | A class of Yetter–Drinfel’d Hopf algebras on basic cycle is constructed. |
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UDC 512.5
Yanhua Wang (School Math., Shanghai Univ. Finance and Economics, China),
Guohua Liu (Southeast Univ., Nanjing, China)
YETTER – DRINFEL’D HOPF ALGEBRAS ON BASIC CYCLE*
ХОПФОВI АЛГЕБРИ ЄТТЕРА – ДРIНФЕЛЬДА НА БАЗОВОМУ ЦИКЛI
A class of Yetter – Drinfel’d Hopf algebras on basic cycle are constructed.
Побудовано клас хопфових алгебр Єттера – Дрiнфельда на базовому циклi.
1. Introduction. Let H be a Hopf algebra. A Yetter – Drinfel’d module over H is a K-linear space
V such that V is both an H-module and an H-comodule and satisfies a compatibility condition.
Yetter – Drinfel’d Hopf algebras are Hopf algebras in Yetter – Drinfel’d module category. It is a class
of braided Hopf algebras. Nichols algebras [11], (G,χ)-Hopf algebras [12, p. 206] (10.5.11) and
twisted Hopf algebras [10] are important examples of Yetter – Drinfel’d Hopf algebras.
Radford’s projection theorem [13] leads to a decomposition of the given Hopf algebra into a
Radford biproduct of two factors, one is no longer a Hopf algebra, but rather a Yetter – Drinfel’d
Hopf algebra over the other factor. After Radford’s work, some important advances are the followings.
Doi considered Hopf modules in Yetter – Drinfel’d module category in [6]. Scharfschwerdt proved
Nichols – Zoeller theorem for Yetter – Drinfel’d Hopf algebras, see [15]. Schauenburg proved that a
Yetter – Drinfel’d module category is equivalent to a category of the left modules over the Drinfel’d
double, and also to a Hopf bimodule category, see [16]. Sommerhäuser studied Yetter – Drinfel’d Hopf
algebras over groups of prime order in [17]. Andruskiewitsch and Schneider studied Nichols algebras
in [1]. Recently, Grana, Heckenberger and Vendramin classified Nichols algebras of irreducible
Yetter – Drinfel’d module over nonabelian groups in [7].
The quiver methods in the representation theory of algebras were considered by Ringel in [14].
The coalgebra structure on quivers were considered by Chin and Montgomery in [4]. Quivers allow
one to present algebras or coalgebras in a useful way. For example, Cibils and Rosso constructed Hopf
quivers and quiver quantum groups in [3] and [5] respectively. Green and Solberg have investigated
the structure of finite dimensional basic Hopf algebras in [8].
One can get a Hopf algebra or a quantum group via quivers. The constructions of braided Hopf
algebras via quivers are not numerous. In this paper, we provide such an explicit construction via
quivers. Let Cd(n) be a subcoalgebra of the coalgebra KZc
n of paths in the oriented cycle quiver Zc
n of
length n with basis the set of all paths of length strictly less than d. Assume that G = {1, g, . . . , gn−1}
is a group and KG a group Hopf algebra. In this paper, we prove that Cd(n) is a Yetter – Drinfel’d
module over KG. Moreover, Cd(n) is a Yetter – Drinfel’d Hopf algebra over KG, see Theorem 5.
Throughout, K will denote a fixed field. All algebras, coalgebras, (co)modules, ⊗ and Hom
are over K. For basic definitions and facts about coalgebras, Hopf algebras and (co)modules we
refer to Sweedler’s book [18]. In particular, the comultiplication of a coalgebra C is denoted by
* Supported by the Chinese NSF (Grant No. 10901098 and No. 11271239) and Basic Academic Discipline Program,
the 11 th five year plan of 211 Project for Shanghai University of Finance and Economics.
c© YANHUA WANG, GUOHUA LIU, 2014
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 9 1291
1292 YANHUA WANG, GUOHUA LIU
∆(c) =
∑
c1 ⊗ c2 for all c ∈ C, and the structure map of a left C-comodule V is denoted by
ρ(v) =
∑
v−1 ⊗ v0 for all v ∈ V. For quivers we refer to Auslander – Reiten – Smal ’s book [2].
2. Preliminaries. Let (H,m, u,4, ε, S) be a Hopf algebra with antipode S. A left Yetter –
Drinfel’d module over H is a K-vector space V such that V is both a left H-module with action →
and left H-comodule with coaction ρ, and satisfies the compatibility condition:∑
(h→ v)−1 ⊗ (h→ v)0 =
∑
h1v
−1S(h3)⊗ h2 → v0, (1)
for all h ∈ H, v ∈ V. The category of left Yetter – Drinfel’d modules over H is denoted by H
HYD.
The category is a pre-braided category and the pre-braiding is given by
τV,W : V ⊗W −→W ⊗ V, v ⊗ w 7−→
∑
(v−1 → w)⊗ v0.
The above map is a braiding when H has a bijective antipode. Denote by S̄ the inverse of S. The
inverse of τV,W is
τ−1
V,W
: W ⊗ V −→ V ⊗W, w ⊗ v 7−→
∑
v0 ⊗ S̄(v−1)→ w.
Let A be a Yetter – Drinfel’d module. We call the 6-tuple (A,m, u,4, ε, S) a Yetter – Drinfel’d
Hopf algebra (or Hopf algebra in H
HYD) if A satisfies the following conditions:
(a1) (A,m, u) is a left H-module algebra, i.e.,
h→ (ab) =
∑
(h1 → a)(h2 → b), h→ 1A = ε(h)1A.
(a2) (A,m, u) is a left H-comodule algebra, i.e.,
ρ(ab) =
∑
(ab)−1 ⊗ (ab)0 =
∑
a−1b−1 ⊗ a0b0,
ρ(1A) = 1H ⊗ 1A.
(a3) (A,4, ε) is a left H-module coalgebra, i.e.,
4(h→ a) =
∑
(h1 → a1)⊗ (h2 → a2), εA(h→ a) = εH(h)εA(a).
(a4) (A,4, ε) is a left H-comodule coalgebra, i.e.,∑
a−1 ⊗ (a0)1 ⊗ (a0)2 =
∑
a1
−1a2
−1 ⊗ a10 ⊗ a20,∑
a−1εA(a0) = εA(a)1H .
(a5) 4 and ε are algebra maps in H
HYD, i.e.,
4(ab) =
∑
a1(a2
−1 → b1)⊗ a20b2,
4(1) = 1⊗ 1, ε(ab) = ε(a)ε(b), ε(1A) = 1k.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 9
YETTER – DRINFEL’D HOPF ALGEBRAS ON BASIC CYCLE 1293
(a6) There exists a K-linear map S : A −→ A in H
HYD such that it is a convolution inverse of
identity, i.e., S ∗ Id = uε = Id ∗ S.
When the pre-braiding τ is trivial, Yetter – Drinfel’d Hopf algebras are ordinary Hopf algebras,
see [18, p. 8] for details. However, generally, Yetter – Drinfel’d Hopf algebras are not ordinary Hopf
algebras because the bialgebra axiom asserts that they obey (a5).
Let q ∈ K. For nonnegative integer l and 0 ≤ m ≤ l, the Gaussian polynomials is defined to be(
l
m
)
q
:=
(l)!q
m!q(l −m)!q
where
l!q := 1q . . . lq, 0!q := 1, lq := 1 + q + . . .+ ql−1.
Next, we will give several conclusions of Gaussian polynomials. They will be used in next
section. Firstly, we recall the q-Pascal identity, it can be found in [9] (Proposition IV.2.1).(
l
m
)
q
=
(
l − 1
m− 1
)
q
+ qm
(
l − 1
m
)
q
=
(
l − 1
m
)
q
+ ql−m
(
l − 1
m− 1
)
q
. (2)
For any scalar a and a variable element z, for any positive integer l, Kassel proved that
(a− z)(a− qz) . . . (a− ql−1z) =
l∑
k=0
(−1)k
(
l
k
)
q
q
k(k−1)
2 al−kzk
(see [9], IV.2.7). Especially, let a = 1 and z = 1, we have
l∑
k=0
(−1)kq
k(k−1)
2
(
l
k
)
q
= 0. (3)
Moreover, the following equation also holds.
Lemma 1. Let l and k be nonnegative integers. For any integer s, where 0 ≤ s ≤ l + k, we
have ∑
m+p=s
0≤m≤l,0≤p≤k
qm(k−p)
(
l + k − s
l −m
)
q
(
s
m
)
q
=
(
l + k
l
)
q
. (4)
3. Construction. Let Zc
n denote the basic cycle of length n, i.e., an oriented graph with n
vertices e0, . . . , en−1, and a unique arrow ai from ei to ei+1 for each 0 ≤ i ≤ n− 1. The indices are
taken modulo n. Set γmi := ai+m−1 . . . ai+1ai to be the path of length m starting at the vertex ei.
Note that γ0i = ei and γ1i = ai.
Let Cd(n) be the subcoalgebra of KZc
n with basis the set of all paths of length strictly less than d.
Observe that if the order of q is d, then
(
d
l
)
q
= 0 for 1 ≤ l ≤ d− 1. Then Cd(n) is a path coalgebra
with comultiplication 4(γli) =
∑l
k=0 γ
l−k
i+k ⊗ γ
k
i , and counit ε(γli) = δl,0. Here, δl,0 is the Kronecker
symbol.
Define a multiplication on Cd(n) by
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 9
1294 YANHUA WANG, GUOHUA LIU
γliγ
s
j =
(
l + s
l
)
q
γl+s
i+j , (5)
where l+ s < d. Observe that if l+ s ≥ d, then γliγ
s
j = 0 since qd = 1. It is easy to see that the unit
element of Cd(n) is 1 = γ00 .
Definition 1. Let A be a vector space. We call A a pre-bialgebra if A is an algebra and a
coalgebra.
From Definition 1, we know that a pre-bialgebra is a bialgebra if and only if 4 and ε are algebra
morphisms.
The following lemma is routine, we omit the proof.
Lemma 2. Coalgebra Cd(n) is a pre-bialgebra with multiplication (5).
Let G = {1, g, g2, . . . , gn−1} be a group. Then KG is a Hopf algebra, see [12] (1.5.3). It is clear
that Cd(n) becomes a left KG-module with the left module structure
gs → γli = qslγli (6)
and Cd(n) is also a left KG-comodule with comodule structure
ρ(γli) =
∑
gl ⊗ γli. (7)
Then we have the following lemma.
Lemma 3. Coalgebra Cd(n) is a Yetter – Drinfel’d module over KG with module (6) and co-
module (7).
Proof. Take gs ∈ KG and γli ∈ Cd(n). Recall that∑
(gs → γli)
−1 ⊗ (gs → γli)
0 = qslgl ⊗ γli.
Moreover, we have∑
(gs)1(γ
l
i)
−1S((gs)3)⊗ (gs)2 → γli = gsglS(gs)⊗ gs → γli = gl ⊗ qslγli.
This means that (1) holds. Thus Cd(n) is a Yetter – Drinfel’d module over KG.
Next, we will give the main theorem.
Theorem 1. Coalgebra Cd(n) is a Yetter – Drinfel’d Hopf algebra over KG.
Proof. We divide the proof into six steps as the definition of Yetter – Drinfel’d Hopf algebras. In
the following, we take γli, γ
k
j ∈ Cd(n) and gs ∈ G.
It is easy to check that (a1) – (a4) hold. We only need to show (a5) and (a6).
(a5) It is obvious that 4(1) = 1 ⊗ 1, ε(γliγ
k
j ) =
(
l + k
l
)
q
δl+k,0 = δl,0δk,0 = ε(γli)ε(γ
l
j) and
ε(1) = 1. Next, we will prove the comultiplication4 is an algebra map in Yetter – Drinfel’d category.
On one hand, we have
4(γliγ
k
j ) =
(
l + k
l
)
q
4(γl+k
i+j ) =
(
l + k
l
)
q
l+k∑
s=0
γl+k−s
i+j+s ⊗ γ
s
i+j . (8)
On the other hand, we obtain∑
(γli)1((γ
l
i)2
−1 → (γkj )1)⊗ (γli)2
0
(γkj )2 =
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 9
YETTER – DRINFEL’D HOPF ALGEBRAS ON BASIC CYCLE 1295
=
l∑
m=0
k∑
p=0
γl−m
i+m((γmi )−1 → γk−p
j+p )⊗ (γmi )0(γpj ) =
=
l∑
m=0
k∑
p=0
γl−m
i+m(gm → γk−p
j+p )⊗ (γmi γ
p
j ) =
=
l∑
m=0
k∑
p=0
qm(k−p)
(
l −m+ k − p
l −m
)
q
(
m+ p
m
)
q
γl+k−m−p
i+j+m+p ⊗ γ
m+p
i+j . (9)
For s = 0, 1, . . . , l+ k, comparing the coefficient of γl+k−s
i+j+s ⊗ γsi+j in equation (8) and equation (9),
we get (
l + k
l
)
q
γl+k−s
i+j+s ⊗ γ
s
i+j =
∑
m+p=s
0≤m≤l,0≤p≤k
qm(k−p)
(
l + k − s
l −m
)
q
(
s
m
)
q
γl+k−s
i+j+s ⊗ γ
s
i+j
by (4). Thus(
l + k
l
)
q
l+k∑
s=0
γl+k−s
i+j+s ⊗ γ
s
i+j =
l∑
m=0
k∑
p=0
qm(k−p)
(
l −m+ k − p
l −m
)
q
(
m+ p
m
)
q
γl+k−m−p
i+j+m+p ⊗ γ
m+p
i+j .
That means
4(γliγ
k
j ) =
∑
(γli)1((γ
l
i)2
−1 → (γkj )1)⊗ (γli)2
0
(γkj )2.
Hence 4 is an algebra map in Yetter – Drinfel’d category.
(a6) Define S : A −→ A by
S(γli) = (−1)lq
l(l−1)
2 γl−i−l.
Then S is a convolution inverse of identity, since
(S ∗ Id)(γli) =
l∑
m=0
S(γl−m
i+m)γmi =
l∑
m=0
(−1)l−mq
(l−m)(l−m−1)
2 γl−m
−i−lγ
m
i =
=
l∑
m=0
(−1)l−mq
(l−m)(l−m−1)
2
(
l
l −m
)
q
γl−l.
If l = 0,we have (S∗Id)(γ0i ) = γ00 . If l 6= 0,we have
∑l
m=0
(−1)l−mq
(l−m)(l−m−1)
2
(
l
l −m
)
q
γl−l =
= 0 by (3). In a word, (S ∗ Id)(γli) = 0. Similarly, (Id ∗S)(γli) = 0. So S is the convolution inverse
of identity.
Thus Cd(n) is a Yetter – Drinfel’d Hopf algebra over the group algebra KG.
Theorem 1 is proved.
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1296 YANHUA WANG, GUOHUA LIU
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Received 11.09.12,
after revision — 25.11.13
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 9
|
| id | umjimathkievua-article-2222 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:21:00Z |
| publishDate | 2014 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| resource_txt_mv | umjimathkievua/4a/d955825a9ec8f8c85d03a581be53914a.pdf |
| spelling | umjimathkievua-article-22222019-12-05T10:26:31Z Yetter–Drinfel’d Hopf Algebras on Basic Cycle Хопфові алгебри Єттера - Дрінфельда на базовому циклі Wang, Yanhua Ван, Янхуа A class of Yetter–Drinfel’d Hopf algebras on basic cycle is constructed. Побудовано клас хопфових алгебр Єттера-Дрінфельда на базовому циклі. Institute of Mathematics, NAS of Ukraine 2014-09-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2222 Ukrains’kyi Matematychnyi Zhurnal; Vol. 66 No. 9 (2014); 1291–1296 Український математичний журнал; Том 66 № 9 (2014); 1291–1296 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2222/1445 https://umj.imath.kiev.ua/index.php/umj/article/view/2222/1446 Copyright (c) 2014 Wang Yanhua |
| spellingShingle | Wang, Yanhua Ван, Янхуа Yetter–Drinfel’d Hopf Algebras on Basic Cycle |
| title | Yetter–Drinfel’d Hopf Algebras on Basic Cycle |
| title_alt | Хопфові алгебри Єттера - Дрінфельда на базовому циклі |
| title_full | Yetter–Drinfel’d Hopf Algebras on Basic Cycle |
| title_fullStr | Yetter–Drinfel’d Hopf Algebras on Basic Cycle |
| title_full_unstemmed | Yetter–Drinfel’d Hopf Algebras on Basic Cycle |
| title_short | Yetter–Drinfel’d Hopf Algebras on Basic Cycle |
| title_sort | yetter–drinfel’d hopf algebras on basic cycle |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2222 |
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