Weak Hopf Algebras and Singular Solutions of Quantum Yang–Baxter Equation

Mathematical Physics

© Springer-Verlag 2002

Weak Hopf Algebras and Singular Solutions of Quantum Yang–Baxter Equation

Fang Li^1,3,, Steven Duplij²

1 Department of Mathematics, Zhejiang University (Xixi Campus), Hangzhou, Zhejiang 310028, P.R. China.

E-mail: fangli@mail.hz.zj.cn; fli63@yahoo.com

2 Theoretical Physics, Kharkov National University, Kharkov 61077, Ukraine.

E-mail: Steven.A.Duplij@univer.kharkov.ua

3 Institute of Mathematics, Chinese Academy of Sciences, Beijing 100080, P.R. China

Received: 1 May 2001 / Accepted: 1 September 2001

Abstract: We investigate a generalization of Hopf algebrasl_q(2)by weakening the invertibility of the generatorK, i.e. exchanging its invertibilityKK⁻¹=1 to the reg- ularityKKK =K. This leads to a weak Hopf algebrawslq(2)and aJ-weak Hopf algebravslq(2)which are studied in detail. It is shown that the monoids of group-like elements ofwsl_q(2)andvsl_q(2)are regular monoids, which supports the general con- jucture on the connection betweek weak Hopf algebras and regular monoids. Moreover, fromwsl_q(2)a quasi-braided weak Hopf algebraU^w_q is constructed and it is shown that the corresponding quasi-R-matrix is regularR^wRˆ^wR^w =R^w.

1. Introduction

The concept of a weak Hopf algebra as a generalization of a Hopf algebra [29, 1] was introduced in [18] and its characterizations and applications were studied in [20]. A k-bialgebra¹ H = (H, µ, η, , ε)is called a weak Hopf algebra if there existsT ∈ Hom_k(H, H )such thatid ∗T ∗id = id andT ∗id ∗T = T, where T is called a weak antipode ofH. This concept also generalizes the notion of the left and right Hopf algebras [24, 12].

The first aim of this concept is to give a new sub-class of bialgebras which includes all of Hopf algebras such that it is possible to characterize this sub-class through their monoids of all group-like elements [18, 20]. It was known that for every regular monoid S, its semigroup algebrakSoverk is a weak Hopf algebra as the generalization of a group algebra [19].

The second aim is to construct some singular solutions of the quantum Yang-Baxter equation (QYBE) and research QYBE in a larger scope. On this hand, in [20] a quantum quasi-doubleD(H )for a finite dimensional cocommutative perfect weak Hopf algebra

Project (No. 19971074) supported by the National Natural Science Foundation of China.

1 In this paper,kalways denotes a field.

with invertible weak antipode was built and it was verified that its quasi-R-matrix is a regular solution of the QYBE. In particular, the quantum quasi-double of a finite Clifford monoid as a generalization of the quantum double of a finite group was derived [20].

In this paper, we will construct two weak Hopf algebras in the other direction as a generalization of the quantum algebraslq(2)[22, 2]. We show thatwsl2(q)possesses a quasi-R-matrix which becomes a singular (in fact, regular) solution of the QYBE, with a parameterq. In this reason, we want to treat the meaning ofwsl_q(2)and its quasi- R-matrix just assl_q(2)[28, 16]. It is interesting to note thatwsl_q(2)is a natural and non-trivial example of weak Hopf algebras.

2. Weak Quantum Algebras

For completeness and consistency we remind the definition of the enveloping algebra U_q=U_q(sl(2))(see e.g. [16]). Letq∈Candq = ±1,0. The algebraU_qis generated by four variables (Chevalley generators)E,F,K,K⁻¹with the relations

K⁻¹K=KK⁻¹=1, (1)

KEK⁻¹=q²E, (2)

KF K⁻¹=q⁻²F, (3)

EF−FE=K−K⁻¹

q−q⁻¹ . (4)

Now we try to generalize the invertibility condition (1). The first thought is weaken the invertibility to regularity, as it is usually made in semigroup theory [17] (see also [10, 6, 7] for higher regularity). So we will consider such weakening the algebraUq

slq(2) , in which instead of the set

K, K⁻¹

we introduce a pair

Kw, Kw

by means of the regularity relations

K_wK_wK_w =K_w, K_wK_wK_w =K_w. (5) IfK_w satisfying (5) is unique for a givenK_w, then it is called inverse of K_w (see e.g. [27, 11]). The regularity relations (5) imply that one can introduce the variables

J_w =K_wK_w, J_w=K_wK_w. (6) In terms ofJ_w the regularity conditions (5) are

J_wK_w =K_w, K_wJ_w=K_w, (7) JwKw =Kw, KwJw=Kw. (8) Since the noncommutativity of generatorsK_w andK_w very much complexifies the generalized construction², we first consider the commutative case and imply in what follow that

J_w =J_w. (9)

2 This case will be considered elsewhere.

Let us list some useful properties ofJw which will be needed below. First we note that commutativity ofKwandKwleads to idempotency condition

J_w² =J_w, (10)

which means thatJwis a projector (see e.g. [15]).

Conjecture 1. In algebras satisfying the regularity conditions (5) there exists as minimum one zero divisorJw−1.

Remark 1. In addition with unity 1 we have an idempotent analog of unityJ_w which makes the structure of weak algebras more complicated, but simultaneously more inter- esting.

For any variableXwe will define “J-conjugation” as

X_Jw^def= J_wXJ_w (11) and the corresponding mapping will be written as e_w(X) : X → XJw. Note that the mapping e_w(X)is idempotent

e²_w(X)=e_w(X) . (12) Remark 2. In the invertible caseK_w =K, K_w =K⁻¹we haveJ_w =1 and e_w(X)= X=id(X)for anyX, so e_w =id.

It is seen from (5) that the generatorsK_wandK_ware stable under “J_w-conjugation”

K_Jw =J_wK_wJ_w=K_w, K_Jw=J_wK_wJ_w =K_w. (13) Obviously, for anyX

KwXKw =KwXJwKw, (14) and for anyXandY

K_wXK_w=Y ⇒K_wX_JwK_w =Y_Jw. (15) Another definition connected with the idempotent analog of unityJ_w is the “J_w- product” for any two elementsXandY, viz.

X_JwY ^def= XJ_wY. (16) Remark 3. From (7) it follows that the “Jw-product” coincides with the usual product, ifXends with generatorsK_wandK_won right side orY starts with them on left side.

LetJ^(ij)=K_wⁱK^j_wthen we will need a formula J_w^(ij)=K_wⁱ K^j_w=





K_w^i−j, i > j, Jw, i=j, K^j−i_w , i < j,

(17)

which follows from the regularity conditions (7). The variablesJ^(ij)satisfy the regularity conditions

J_w^(ij)J_w^(ji)J_w^(ij)=J_w^(ij) (18) and stable under “J-conjugation” (11)J_wJ^(ij)_w =J_w^(ij).

The regularity conditions (7) lead to the noncancellativity: for any two elementsX andY the following relations hold valid:

X=Y ⇒KwX=KwY, (19)

K_wX=K_wY X=Y, (20)

X=Y ⇒K_wX=K_wY, (21)

K_wX=K_wY X=Y, (22)

X=Y ⇒X_Jw=Y_Jw, (23)

XJw=YJw X=Y. (24)

The generalization ofUq slq(2)

by exploiting regularity (5) instead of invertibility (1) can be done in two different ways.

Definition 1. DefineU_q^w =wsl_q(2)as the algebra generated by the four variablesE_w, F_w,K_w,K_wwith the relations:

K_wK_w =K_wK_w, (25)

K_wK_wK_w =K_w, K_wK_wK_w =K_w, (26) K_wE_w =q²E_wK_w, K_wE_w=q⁻²E_wK_w, (27) KwFw =q⁻²FwKw, KwFw =q²FwKw, (28) EwFw−FwEw =K_w−K_w

q−q⁻¹ . (29)

We callwsl_q(2)a weak quantum algebra.

Definition 2. DefineU_q^v =vsl_q(2)as the algebra generated by the four variablesE_v, Fw,Kv,Kvwith the relations (Jv=KvKv):

K_vK_v=K_vK_v, (30)

KvKvKv=Kv, KvKvKv=Kv, (31) KvEvKv=q²Ev, (32) K_vF_vK_v=q⁻²F_v, (33) E_vJ_vF_v−F_vJ_vE_v= Kv−Kv

q−q⁻¹. (34)

We callvsl_q(2)a J-weak quantum algebra.

In these definitions indeed the first two lines (25)–(26) and (30)–(31) are called to generalize the invertibilityKK⁻¹ =K⁻¹K =1. Each next line (27)–(29) and (32)–

(34) generalizes the corresponding line (2)–(4) in two different ways respectively. In the first almost quantum algebrawsl_q(2)the last relation (29) betweenEandF generators remains unchanged fromsl_q(2), while twoEKandF Krelations are extended to four ones (27)–(28). Invsl_q(2), oppositely, twoEK andF K relations remain unchanged fromsl_q(2)(withK⁻¹→Ksubstitution only), while the last relation (34) betweenE andF generators has the additional multiplierJ_vwhich role will be clear later. Note that theEK andF K relations (32)–(33) can be written in the following form close to (27)–(28):

K_vE_vJ_v=q²J_vE_vK_v, K_vE_vJ_v=q⁻²J_vE_vK_v, (35) KvFvJv=q⁻²JvFvKv, KvFvJv=q²JvFvKv. (36) Using (16) and (7) in the case ofJv we can also present thevslq(2)algebra as an algebra with the “Jv-product”

KvJv Kv=KvJv Kv, (37) K_vJv K_vJvK_v=K_v, K_vJvK_vJvK_v=K_v, (38) KvJv EvJv Kv=q²Ev, (39) K_vJv F_vJv K_v=q⁻²F_v, (40) E_vJv F_v−F_vJvE_v=K_v−K_v

q−q⁻¹ . (41)

Remark 4. Due to (7) the only relation where the “J_w-product” really plays its role is the last relation (41).

From the following proposition, one can find the connection betweenU_q^w =wsl_q(2), U_q^v=vsl_q(2)and the quantum algebrasl_q(2).

Proposition 1.wslq(2)/(Jw−1)∼=slq(2);vslq(2)/(Jv−1)∼=slq(2). Proof. For cancellativeK_wandK_vit is obvious.

Proposition 2. Quantum algebraswsl_q(2)and vsl_q(2)possess zero divisors, one of which is³

J_w,v−1

which annihilates all generators.

Proof. From regularity (26) and (31) it followsK_w,v

J_w,v−1

=0 (see also (1)). Mul- tiplying (27) onJ_wgivesK_wE_wJ_w =q²E_wK_wJ_w ⇒K_w

E_wK_w

K_w =q²E_wK_w. Using the second equation in (27) for the term in the bracket we obtainK_w

q²K_wE_w K_w =q²E_wK_w ⇒(J_w−1) E_wK_w =0.ForF_w similarly, but we use Eq. (28). By analogy, multiplying (32) onJ_vwe haveK_vE_vK_vK_vK_v = q²E_vJ_v ⇒K_vE_vK_v = q²EvJv ⇒ q²Ev = q²EvJv, and soEv(Jv−1)= 0. ForFv similarly, but we use Eq. (33).

Remark 5. Sincesl_q(2)is an algebra without zero divisors, some properties ofsl_q(2) cannot be upgraded towslq(2)andvslq(2), e.g. the standard theorem of Ore extensions and its proof (see Theorem I.7.1 in [16]).

3 We denote byXw,vone of the variablesXworXv.

Remark 6. We conjecture that in U_q^w andU_q^v there are no other than

Jw,v−1 zero divisors which annihilate all generators. In other case thorough analysis of them will be much more complicated and very different from the standard case of non-weak algebras.

We can get some properties ofU_q^wandU_q^vas follows.

Lemma 1. The idempotentJ_w is in the center ofwsl_q(2).

Proof. ForK_w it follows from (13). Multiplying the first equation in (27) onK_w we deriveKw

EwKw

= q²EwJw, and applying the second equation in (27) we obtain EwJw =JwEw. ForFwsimilarly, but we use Eq. (28).

Lemma 2. There are unique algebra automorphismsω_wandω_vofU_q^wandU_q^vrespec- tively such that

ω_w,v(K_w,v)=K_w,v, ω_w,v(K_w,v)=K_w,v,

ωw,v(Ew,v)=Fw,v, ωw,v(Fw,v)=Ew,v. (42) Proof. The proof is obvious, if we note thatω²_w =id andω_v²=id.

As in the case of the automorphismωforsl_q(2)[16], the mappingsω_wandω_vcan be called the weak Cartan automorphisms.

Remark 7. Note thatω_w =ωandω_v=ωin general case.

The connection between the algebras wsl_q(2)andvsl_q(2)can be seen from the following

Proposition 3. There exist the following partial algebra morphism χ : vsl_q(2) → wsl_q(2)such that

χ (X)=e_v(X) (43)

or more exactly: generators X^(v)_w = J_vX_vJ_v = X_vJv for allX_v = K_v, K_v, E_v, F_v satisfy the same relations asX_w (25)–(29).

Proof. Multiplying Eq. (32) onKvwe haveKvEvKvKv=q²EvKv, and using (7) we obtainK_vE_vJ_v=q²E_vJ_vK_v⇒K_vJ_vE_vJ_v=q²J_vE_vJ_vK_v, and so

K_vJvE_vJv =q²E_vJvK_vJv,

which has the shape of the first equation in (27). For F_v similarly using Eq. (33) we obtain

K_vJvF_vJv =q⁻²F_vJvK_vJv.

Equation (34) can be modified using (7) and then applying (11), then we obtain E_vJvF_vJv−F_vJvE_vJv = KvJv −KvJv

q−q⁻¹ which coincides with (29).

For conjugated equations (the second ones in (27)–(28)) after multiplication of (32) on Kv we have KvKvEvKv = q²KvEv ⇒ JvEvJvKv = q²KvJvEvJv or using definition (11) and (7)

KvJvEvJv =q⁻²EvJvKvJv. By analogy from (33) it follows

K_vJvF_vJv =q²F_vJvK_vJv.

Note that the generatorsXw^(v)coincide withXwifJv=1 only. Therefore, some (but not all) properties ofwsl_q(2)can be extended onvsl_q(2)as well, and below we mostly will considerwsl_q(2)in detail.

Lemma 3. Letm≥0 andn∈Z. The following relations hold inU_q^w:

E_w^mK_wⁿ =q^−2mnK_wⁿE^m_w, F_w^mK_wⁿ =q^2mnK_wⁿF_w^m, (44) E_w^mKⁿ_w =q^2mnKⁿ_wE_w^m, F_w^mKⁿ_w=q^−2mnKⁿ_wF_w^m, (45)

[E_w, F_w^m] = [m]F_w^m−1q^−(m−1)K_w−q^m−1K_w

q−q⁻¹ (46)

= [m]q^m−1K_w−q^−(m−1)K_w q−q⁻¹ F_w^m−1, [E^m_w, F_w] = [m]q^−(m−1)K_w−q^m−1K_w

q−q⁻¹ E^m−_w ¹ (47)

= [m]E^m−_w ¹q^m−1K_w−q^−(m−1)K_w q−q⁻¹ .

Proof. The first two relations result easily from Definition 1. The third one follows by induction using Definition 1 and

[E_w, F_w^m] = [E_w, F_w^m−1]F_w+F_w^m−1[E_w, F_w] = [E_w, F_w^m−1]F_w+F_w^m−1K_w−K_w q−q⁻¹ . Applying the automorphismωw(42) to (46), one gets (47).

Note that the commutation relations (44)–(47) coincide with the sl_q(2)case. For vslq(2) the situation is more complicated, because Eqs. (32)–(33) cannot be solved underK_vdue to noncancellativity (see also (19)–(24)). Nevertheless, some analogous relations can be derived. Using the morphism (43) one can conclude that the similar relations (44)–(47) hold forX_w^(v)=J_vX_vJ_v, from which we obtain forvsl_q(2),

J_vE_v^mK_vⁿ=q^−2mnK_vⁿE_v^mJ_v, J_vF_v^mK_vⁿ=q^2mnK_vⁿF_v^mJ_v, (48) J_vE_v^mKⁿ_v=q^2mnKⁿ_vE_v^mJ_v, J_vF_v^mKⁿ_v=q^−2mnKⁿ_vF_v^mJ_v, (49)

JvEvJvF_v^mJv−JvF_v^mJvEvJv= [m]JvF_v^m−1q^−(m−1)K_v−q^m−1K_v

q−q⁻¹ (50)

= [m]q^m−1Kv−q^−(m−1)Kv

q−q⁻¹ F_v^m−1J_v, JvE_v^mJvFvJv−JvFvJvE_v^mJv= [m]q^−(m−1)K_v−q^m−1K_v

q−q⁻¹ E_v^m−1Jv (51)

= [m]J_vE^m−_v ¹q^m−1K_v−q^−(m−1)K_v q−q⁻¹ . It is important to stress that due to noncancellativity of weak algebras we cannot cancel these relations onJv(see (19)–(24)).

In order to discuss the basis ofU_q^w=wsl_q(2), we need to generalize some properties of Ore extensions (see [16]).

3. Weak Ore Extensions

LetRbe an algebra overkandR[t]be the free leftR-module consisting of all polynomials of the formP =_n

i=0aitⁱ with coefficients inR. Ifan =0, define deg(P )=n; say deg(0)= −∞. Letαbe an algebra morphism ofR. Anα-derivation ofRis ak-linear endomorphismδofRsuch thatδ(ab)=α(a)δ(b)+δ(a)bfor alla, b∈R. It follows thatδ(1)=0.

Theorem 1. (i) Assume thatR[t]has an algebra structure such that the natural inclu- sion ofRintoR[t]is a morphism of algebras and deg(P Q) ≤deg(P )+deg(Q) for any pair(P, Q)of elements ofR[t]. Then there exists a unique injective algebra endomorphismαofRand a uniqueα-derivationδofRsuch thatta=α(a)t+δ(a) for alla∈R;

(ii) Conversely, given an algebra endomorphismα ofRand an α-derivationδ ofR, there exists a unique algebra structure onR[t]such that the inclusion ofRintoR[t] is an algebra morphism andta=α(a)t+δ(a)for alla∈R.

Proof. (i) Take any 0 = a ∈ Rand consider the productta. We have deg(ta) ≤ deg(t)+deg(a)=1. By the definition ofR[t], there exists uniquely determined elements α(a)andδ(a)ofRsuch thatta=α(a)t+δ(a). This defines mapsαandδin a unique fashion. The left multiplication byt being linear, so areαandδ. Expanding both sides of the equality(ta)b=t(ab)inR[t]usingta=α(a)t+δ(a)fora, b∈R, we get

α(a)α(b)t+α(a)δ(b)+δ(a)b=α(ab)t+δ(ab).

It follows thatα(ab)=α(a)α(b)andδ(ab)=α(a)δ(b)+δ(a)b, and,α(1)t+δ(1)= t1=t. So,α(1)=1,δ(1)=0. Therefore, we know thatαis an algebra endomorphism andδ is anα-derivation. The uniqueness ofαandδ follows from the freeness ofR[t] overR.

(ii) We need to construct the multiplication onR[t]as an extension of that onRsuch thatta=α(a)t+δ(a). For this, it needs only to determine the multiplicationtafor any a ∈R.

LetM= {(fij)i,j≥1:fij ∈End_k(R)and each row and each column has only finitely manyf_ij =0}andI =



 1

1 ...



is the identity ofM.

Fora∈R, leta :R→Rsatisfyinga(r)=ar. Thena∈End_k(R); and forr∈R, (αa)(r)=α(ar)=α(a)α(r)=(α(a)α)(r) ,(δa)(r)=δ(ar)=α(a)δ(r)+δ(a)r = (α(a)δ +δ(a))(r) , thusαa =α(a)α ,δa=α(a)δ +δ(a) in End_k(R), and, obviously, fora, b∈R,ab =ab;a+b=a+b.

LetT =







 δα δ

α...

...







∈ Mand define/: R[t] →Msatisfying/(_n

i=0a_itⁱ)= _n

i=0(aiI)Tⁱ. It is seen that/is ak-linear map.

Lemma 4. The map/is injective.

Proof. Letp=_n

i=0a_itⁱ. Assume/(p)=0.

Fore_i =











 01

...

0_i−1 1_i 0_i+1

...

0_n













, obviously,{e_i}_i≥1are linear independent. Sinceδ(1)= 0 and

α(1)=1, we haveT e_i =













 0₁

...

0_i−1

δ(1)_i α(1)_i+1

0_i+2 ...

0_n















=e_i+1andTⁱe1=e_i+1for anyi ≥0. Thus,

0 = /(P )e1 =_n

i=0(a_iI)Tⁱe1 =_n

i=0a_ie_i+1. It means thata_i =0 for alli, then ai =ai1=ai1=0. HenceP =0.

Lemma 5. The following relation holdsT (aI)=(α(a)I)T +δ(a)I . Proof. We have

T (aI)=







 α δδ

α...

...











 a

a ...



 =









α(a)δ +δ(a)

α(a)α α(a)δ +δ(a) α(a)α ...

...









=α(a)T +δ(a)I =(α(a)I)T +δ(a)I.

Now, we complete the proof of Theorem 1. LetS denote the subalgebra generated byT andaI (alla ∈R) inM. From Lemma 5, we see that every element ofScan be generated linearly by some elements in the form as(aI)Tⁿ(a ∈R,n≥0).

But/(atⁿ)=(aI)Tⁿ, so/(R[t])=S, i.e./is surjective. Then by Lemma 4,/is bijective. It follows thatR[t]andSare linearly isomorphic.

Defineta=/⁻¹(T (aI)), then we can extend this formula to define the multiplication of R[t]withfg = /⁻¹(xy) for anyf, g ∈ R[t]andx = /(f ),y = /(g). Under this definition,R[t]becomes an algebra and/is an algebra isomorphism fromR[t]to S, and,ta=/⁻¹(T (aI)) =/⁻¹((α(a)I)T +δ(a)I) =α(a)t+δ(a)for alla ∈R.

Obviously, the inclusion ofRintoR[t]is an algebra morphism.

Remark 8. Note that Theorem 1 can be recognized as a generalization of Theorem I.7.1 in [16], sinceRdoes not need to be without zero divisors,αdoes not need to be injective and only deg(P Q)≤deg(P )+deg(Q).

Definition 3. We call the algebra constructed fromαandδa weak Ore extension ofR, denoted asR_w[t, α, δ].

LetS_n,kbe the linear endomorphism ofRdefined as the sum of all n

k

possible com- positions ofkcopies ofδand ofn−kcopies ofα. By inductionn, fromta=α(a)t+δ(a) under the condition of Theorem 1(ii), we gettⁿa =_n

k=0S_n,k(a)t^n−k and moreover,

ni=0a_it^{i m}_i=0b_itⁱ

=_n+m

i=0 c_itⁱ, wherec_i =_i

p=0a_pp

k=0S_p,k(b_i−p+k). Corollary 1. Under the condition of Theorem 1(ii), the following statements hold:

(i) As a leftR-module,R_w[t, α, δ]is free with basis{tⁱ}i≥0;

(ii) Ifαis an automorphism, thenR_w[t, α, δ]is also a right freeR-module with the same basis{tⁱ}_i≥0.

Proof. (i) It follows from the fact thatR_w[t, α, δ]is justR[t]as a leftR-module.

(ii) Firstly, we can show thatR_w[t, α, δ] =

i≥0tⁱR, i.e. for anyp∈R_w[t, α, δ], there area0,a1,· · ·,an∈Rsuch thatp=_n

i=0tⁱai. Equivalently, we show by induction on nthat for anyb∈R,btⁿcan be in the form_n

i=0tⁱai for someai.

Whenn=0, it is obvious. Suppose that forn≤k−1 the result holds. Consider the casen =k. Sinceαis surjective, there isa ∈ Rsuch thatb =αⁿ(a)=S_n,0(a). But tⁿa =_n

k=0Sn,k(a)t^n−k, we getbtⁿ=tⁿa−_n

k=1Sn,k(a)t^n−k =_n

i=0tⁱai by the hypothesis of induction for someaiwithan=a. For anyianda, b∈R,(tⁱa)b=tⁱ(ab) sinceR_w[t, α, δ]is an algebra. ThenR_w[t, α, δ]is a rightR-module.

Supposef (t)=tⁿa_n+ · · · +ta1+A0=0 fora_i ∈Randa_n=0. Thenf (t)can be written as an element ofR[t]by the formulatⁿa =_n

k=0S_n,k(a)t^n−kwhose highest degree term is just that oftⁿa_n =_n

k=0S_n,k(a_n)t^n−k, i.e.αⁿ(a_n)tⁿ. From (i), we get αⁿ(a_n) =0. It impliesa_n = 0. It is a contradiction. HenceR_w[t, α, δ]is a free right R-module.

We will need the following:

Lemma 6. LetRbe an algebra,αbe an algebra automorphism andδbe anα-derivation ofR. IfRis a left (resp. right) Noetherian, then so is the weak Ore extensionR_w[t, α, δ].

The proof can be made similarly as for Theorem I.8.3 in [16].

Theorem 2. The algebrawslq(2)is Noetherian with the basis

P_w= {Eⁱ_wF_w^jK_w^l, E_wⁱ F_w^jK^m_w, E_wⁱ F_w^jJ_w}, (52) wherei, j, lare any non-negative integers,mis any positive integer.

Proof. As is well known, the two-variable polynomial algebrak[K_w, K_w]is Noetherian (see e.g. [15]). ThenA0=k[K_w, K_w]/(J_wK_w−K_w, K_wJ_w−K_w)is also Noetherian.

For anyi, j ≥ 0 anda, b, c ∈ k, if at least one element ofa, b, cdoes not equal 0, aK_wⁱ +bK^j_w+cJ_w is not in the ideal(J_wK_w−K_w, K_wJ_w−K_w)ofk[K_w, K_w]. So, inA0,aK_wⁱ +bK^j_w+cJ_w =0. It follows that{K_wⁱ , K^j_w, J_w:i, j≥0}is a basis ofA0. Letα1satisfyα1(Kw)=q²Kwandα1(Kw)=q⁻²Kw. Thenα1can be extended to an algebra automorphism onA0andA1=A0[F_w, α1,0]is a weak Ore extension ofA0

fromα=α1andδ=0. By Corollary 1,A1is a free leftA0-module with basis{F_w^j}_i≥0. Thus,A1 is ak-algebra with basis{K_w^lF_w^j, K^m_wF_w^j, J_wF_w^j : l andj run respectively over all non-negative integers,mruns over all positive integers}. But, from the definition of the weak Ore extension, we haveK_w^lF_w^j = q^−2ljF_w^jK_w^l,K^m_wF_w^j = q^2mjF_w^jK^m_w, J_wF_w^j =F_w^jJ_w. So, we conclude that{F_w^jK_w^l, F_w^jK^m_w, F_w^jJ_w :landjrun respectively over all non-negative integers,mruns over all positive integers}is a basis ofA1.

Letα2 satisfyα2(F_w^jK_w^l) =q^−2lF_w^jK_w^l,α2(F_w^jK^m_w)= q^2mF_w^jK^m_w,α2(F_w^jJ_w) = F_w^jJ_w. Thenα2can be extended to an algebra automorphism onA1. Letδsatisfy

δ(1)=δ(K_w)=δ(K_w)=0, δ(F_w^jK_w^l)=

j−1

i=0

F_w^j−1q⁻²ⁱKw−q²ⁱKw

q−q⁻¹ K_w^l, δ(F_w^jK^l_w)=

j−1

i=0

F_w^j−1q⁻²ⁱK_w−q²ⁱK_w q−q⁻¹ K^l_w, δ

F_w^jJw

=

j−1

i=0

F_w^j−1q⁻²ⁱK_w−q²ⁱK_w q−q⁻¹ Jw

forj >0 andl≥0. Then just as in the proof of Lemma VI.1.5 in [16], it can be shown thatδcan be extended to anα2-derivation ofA1such thatA2=A1[E_w, α2, δ]is a weak Ore extension ofA1. Then inA2,

E_wK_w=α2(K_w)E_w+δ(K_w)=q⁻²K_wE_w, E_wK_w=q²K_wE_w, E_wF_w=α2(F_w)E_w+δ(F_w)=F_wE_w+K_w−K_w

q−q⁻¹ .

From these, we conclude thatA2 ∼=U_q^w as algebras. Thus, from Lemma 6,U_q^w is Noetherian. By Corollary 1,U_q^wis free with basis{Eⁱ_w}i≥0as a leftA1-module. Thus, as ak-linear space,U_q^w has the basisQ_w = {F_w^jK_w^l E_wⁱ , F_w^jK^m_wE_wⁱ , F_w^jJ_wE_wⁱ : i, j, l run over all non-negative integers,mruns over all positive integers}. By Lemma 3 any x ∈P_w(resp.Q_w)can bek-linearly generated by some elements ofQ_w(resp.P_w), and thereforeP_wandQ_wgenerate the same spaceU_q^w.