Quantizations of the Witt algebra and of simple Lie algebras in characteristic p

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Journal of Algebra 280 (2004) 145–161

www.elsevier.com/locate/jalgebr

Quantizations of the Witt algebra and of simple Lalgebras in characteristicp

Cyril Grunspan

Dipartimento di Matematica, Institute Guido Castelnuovo, Università di Roma La Sapienza,P. le Aldo Moro, 2, 00185 Roma, Italy

Received 25 November 2003

Available online 4 August 2004

Communicated by Corrado de Concini

Abstract

We explicitly quantize the Witt algebra in characteristic 0 equipped with its Lie bialgebra strucdiscovered by Taft. Then, we study the reduction modulop of our formulas. This givesp−1 familiesof polynomial noncocommutative deformations of a restricted enveloping algebra of a simpalgebra in characteristicp (of Cartan type). In particular, this yields new families of noncommutativand noncocommutative Hopf algebras of dimensionpp in charp. 2004 Published by Elsevier Inc.

Keywords:Witt algebra; Lie bialgebra; Finite dimensional Hopf algebra in nonzero characteristic

1. Introduction

Our work extends the class of examples of noncommutative and noncocommfinite dimensional Hopf algebra in finite characteristic (see [6]). We use the general mof quantization by Drinfeld’s twist [1] to quantize explicitly the Lie bialgebra structudiscovered by Taft on the Witt algebra in characteristic zero [6]. Then, we study theof the Witt algebra in characteristicp wherep is a prime number and we show that oformulas lead to quantizations of its restricted enveloping algebra.

E-mail address:[email protected].

0021-8693/$ – see front matter 2004 Published by Elsevier Inc.doi:10.1016/j.jalgebra.2004.04.016

146 C. Grunspan / Journal of Algebra 280 (2004) 145–161

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We start by recalling the definitions of the Witt algebra in characteristic 0 and oclassification of its Lie bialgebra structures, due to Taft. In Section 2, by using thediscovered by Giaquinto and Zhang [3], we quantize this structure explicitly. In Sectwe remark that although the twist used for the quantization involves terms withnegativep-adic valuation, conjugation by this twist preserves the nonnegativep-adic valuation parof the tensor square of the enveloping algebra of the Witt algebra. Moreover, the redmodulop of these formulas is compatible with the structure ofp-Lie algebra on the Witalgebra. We use this fact to equip the restricted enveloping algebra of the Witt awith noncommutative and noncocommutative Hopf algebra structures. We do not ethese Hopf algebras to be triangular since the twist which helps to define it is not doverFp. We thus get new examples of Hopf algebra of dimensionpp in characteristicp.These Hopf algebras contain the Radford algebra as a Hopf-subalgebra [5]. All proofbeen reported in a final section.

1.1. Definition of the Witt algebra

Definition 1. One denotes byW the Q-Lie algebra given by generatorsLr , r ∈ Z, andrelations[Lr,Ls ] = (s − r)Lr+s .

This Lie algebra is isomorphic to the Lie algebra of vector fields on the circle angeneratorsLr , r ∈ Z, correspond to operatorsxr+1 d

dxin Der(Q[x, x−1]). The following

proposition is due to Taft [6].

Proposition 1. There is a triangular Lie bialgebra structure onW given by ther-matrixr − r21 with r := L0 ⊗ Li andi ∈ Z.

In fact, it has been proved by Ng and Taft that these are the only (up to isomorpLie bialgebra structures on the “positive” part of the Witt algebra [2]. The purpose oarticle is to quantize these structures and to investigate the case of finite characteristic.

Notation. In Section 2, one fixesi ∈ Z. In Section 3,p is an odd prime number. We set:

h := 1

iL0, (1)

e := iLi. (2)

For all elementx of a unitaryR-algebra (R a ring) anda ∈ R, we also set:

x(n)a := (x + a)(x + a + 1) · · ·(x + a + n − 1) (3)

andx(n) := x(n)0 .

C. Grunspan / Journal of Algebra 280 (2004) 145–161 147

non-

ra

a

2. Quantization of Taft’s structure

Let us denote by(U(W),m,∆0, S0, ε) the natural Hopf algebra structure onU(W), i.e.,

∆0(Lk) = Lk ⊗ 1+ 1⊗ Lk,

S0(Lk) = −Lk,

ε(Lk) = 0.

We recall the definition of a Drinfeld’s twist.

Definition 2. Let (H,m,∆0, i, ε, S0) be a quantized enveloping algebra withH/tH ∼=U(g) whereg denotes a Lie algebra andt a deformation parameter. An elementJ ∈ H ⊗H

is called a twist if the following conditions are satisfied:

J ≡ 1 (mod t),

(ε ⊗ Id)(J ) = (Id ⊗ ε)(J ) = 1,

(∆0 ⊗ Id)(J ) · (1⊗ J ) = (J ⊗ 1) · (Id ⊗ ∆0)(J ).

In order to quantize the Taft’s structure, we consider the elementF of [3] defined by:

F :=∞∑

r=0

1

r!h(r) ⊗ er tr

wheret denotes a formal variable.The following proposition has been proved in [3] in the context of the unique

abelian Lie algebrab of dimension 2 generated byh ande and applied here forW thanksto the embedding ofb in W given by (1) and (2).

Proposition 2. The elementF is a twist in(U(W)[[t]],m,∆0, S0, ε).

The following general proposition can be found in any book of Hopf algebras.

Proposition 3. Letg be a Lie algebra,(H,m,∆0, i, ε, S0) a quantized enveloping algebwith H/tH ∼= U(g) (t is the formal deformation parameter) andJ ∈ H ⊗H a twist. Then,J is invertible inH ⊗ H , u := m ◦ (S0 ⊗ Id)(J ) is invertible inH and (H,m,∆, i, S, ε)

is a Hopf algebra with∆ := Ad(J−1) ◦ ∆0 and S := Ad(u−1) ◦ S0. Moreover, ifH =U(g)[[t]] equipped with its natural Hopf algebra structure given by∆0(x) = x ⊗ 1 +1 ⊗ x and S0(x) = −x for x ∈ g and if J ≡ 1 + r · t (mod t2) with r ∈ g ⊗ g, then(U(g)[[t]],m,∆, i, S, ε) is a quantization ofg equipped with its triangular Lie bialgebrstructure given by ther-matrix r − r21.


po-e

Hopff

t

We note thatF ≡ 1+ r · t (mod t2). Therefore, the Hopf algebra constructed in Prosition 3 with the help of the twistF on U(W)[[t]] quantizes the Taft’s structure. Thfollowing theorem gives explicitly the quantization.

Theorem 1. There exists a structure of noncommutative and noncocommutativealgebra onU(W)[[t]] denoted by(U(W)[[t]],m,∆,S, ε) which leaves the product oU(W)[[t]] undeformed but with a coalgebra structure defined by:

∆(Lk) := Lk ⊗ (1− et)ki +

∞∑l=0

(−1)lil

∏l−2j=−1(k + j i)

l! h(l) ⊗ (1− et)−lLk+li tl , (4)

S(Lk) := −(1− et)−ki

∞∑l=0

il

∏l−2j=−1(k + j i)

l! Lk+lih(l)1 t l , (5)

ε(Lk) := 0 (6)

with k ∈ Z.

One can simplify formulas of Theorem 4 by introducing the operatord(l) (l ∈ N) onU(W) defined byd(l) := 1

l!ad(e)l . Indeed, it is easy to see that

d(l)(Lk) = il

∏l−2j=−1(k + j i)

l! Lk+li . (7)

We have then:

∆(Lk) := Lk ⊗ (1− et)ki +

∞∑l=0

(−1)lh(l) ⊗ (1− et)−ld(l)(Lk)tl , (8)

S(Lk) := −(1− et)−ki

∞∑l=0

d(l)(Lk) · h(l)1 t l . (9)

We can also give a general formula for the antipode.

Proposition 4. If x ∈ U(W) is homogeneous with respect to the grading given by|Lk| = k,one has:

S(x) = (1− et)−|x|i

∞∑n=0

d(n)(S0(x)

) · h(n)1 tn. (10)

In fact, by applying Lemma 1 below witha = i andk replaced byk − i, we remark thathe structure coefficients are inZ.

Lemma 1. For all integersa, k andl, al∏l−1

j=0(k + ja)/l! is an integer.

This fact allows us to consider the reduction modulop of our formulas.


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3. Quantizations of restricted enveloping algebras

We show that formulas above can be used to define a Hopf algebra structurerestrictedenveloping algebra of the Witt algebra in characteristicp [4].

Definition 3. One denotes byD theFp-Lie algebra given by generatorsDk , k ∈ Fp , andrelations

[Dk,Dl] = (l − k)Dk+l

for k, l ∈ Fp . The algebraD is called the Witt algebra in characteristicp.

The algebraD is isomorphic to Der(Fp[X]/(Xp − 1)) andDk corresponds to the opeatorXk+1 d

dX.

It is well known thatD is a restricted simple Lie algebra [4]. Its structure ofp-Liealgebra is given byD[p]

0 = D0 andD[p]k = 0 for k �= 0. Its restricted enveloping algeb

Uc(D) is isomorphic toU(D)/I whereI is the ideal ofU(D) generated byDp0 − D0

andDp

k with k �= 0. A basis of this algebra is given by monomials∏p−1

k=0 Dαk

k with αk ∈{0, . . . , p − 1} [4]. So dimFp Uc(D) = pp .

Remark 1. The Witt algebra in characteristicp is also sometimes defined as the Lie algewitt given by generators:ek , k ∈ {−1, . . . , p − 2}, and relations:

[ek, el] ={

(l − k)ek+l if k + l � p − 2,

0 otherwise.(11)

The Lie algebrawitt is isomorphic to Der(Fp[Y ]/(Yp)) and ek stands for the operatoY k+1 d

dY. This Lie algebra is also isomorphic toD as it can be seen by settingX = Y + 1.

The corresponding Lie algebra morphism mapsek to∑k

l=−1(−1)l(k+1l+1

)Dl .

The twistF does not make sense overFp because of the coefficient 1/n! in the defin-ition of F . However, as we have seen at the end of Section 2, formulas of Theoremdefined overZ. Moreover, Lemma 2 below shows that these coefficients are compwith the reduction modulop.

Lemma 2. Leta, k, l be integers. Then, the residue class ofal∏l−1

j=0(k + ja)/l! modulop

depends only onl and on the residue class ofa andk.

By definition, we will denote byN(a, k, l) the common residue modulop of all integersof the formαl

∏l−1j=0(κ + jα)/l! with cl(α) = a and cl(κ) = k.

Theorem 2. Let i be an element ofFp −{0}. We define a Hopf algebra(Uc(D),m,∆,S, ε)

overFp[t] as follows: the algebra structure is undeformed and the coalgebra structugiven by:


atione,

ted

∆(Dk) := Dk ⊗ (1− et)ki−1 +p−1∑l=0

(−1)lN(i, k − i, l)h(l) ⊗ (1− et)−lDk+li tl , (12)

S(Dk) := −(1− et)−ki−1p−1∑l=0

N(i, k − i, l)Dk+lih(l)1 t l , (13)

ε(Dk) := 0 (14)

with, k ∈ Fp, h := 1/i ande := iDi .

The sums above are finite. It follows that this structure is a polynomial deform(Uc(D)[t],m,∆,S, ε) of the restricted enveloping algebra ofD. The above sums are finitwhich explains why they define a polynomial deformation. Thus, we can specializet to anyelement ofFp . Sincei is an arbitrary element ofFp −{0}, this givesp − 1 new families ofnoncommutative and noncocommutative Hopf algebra of dimensionpp in characteristicp.

Note 1. If we setα := (1− et)−1, then we have:[h,α] = α2 −α; hp = h; αp = 1; ∆(h) =h⊗α + 1⊗h, α is “group-like”,S(h) = hα−1 andε(h) = 0. So, the subalgebra generaby h ande is a subHopf algebra ofUc(D) isomorphic to the Radford algebra [5].

4. Proofs

4.1. Some relations inW

Notations. For every elementx of a unitaryR-algebra (R a ring) anda ∈ R, we set:

x[n]a := (x + a)(x + a − 1) · · · (x + a − n + 1)

andx[n] := x[n]0 .

Lemma 3. For all elementx of an unitaryQ-algebra,a, b ∈ Q andr, s, t ∈ Z, one has:

x(s+t )a = x(s)

a x(t)a+s, (15)

x[s+t ]a = x[s]

a x[t ]a−s, (16)

x[s]a = x

(s)a−s+1, (17)

∑s+t=r

(−1)t

s!t ! x[s]a x

(t)b =

(a − b

r

)= (a − b) · · · (a − b − r + 1)

r! , (18)

∑s+t=r

(−1)t

s!t ! x[s]a x

[t ]b−s =

(a − b + r − 1

r

)= (a − b) · · ·(a − b + r − 1)

r! . (19)


(19)

,has

Proof. All these identities come from polynomial identities inQ[X]. It is enough to provethem forx ∈ N. The left-hand side of (18) is equal to the coefficient ofXr in

( +∞∑s=0

1

s!x[s]a Xs

)·( +∞∑

t=0

(−1)t

t ! x(t)a Xt

).

But

+∞∑s=0

1

s!x[s]a Xs = (1+ X)x+a and

+∞∑t=0

(−1)t

t ! x(t)a Xt = (1+ X)−(x+b).

Therefore, the left-hand side of (18) is equal to the coefficient ofXr in (1+ X)x+a · (1+X)−(x+b) = (1+X)a−b. Hence, we get (18). Identities (15), (16), (17) are obvious andcomes from (18) and (17).�Lemma 4. For a ∈ Q, k ∈ Z andm,n ∈ Z, the following equalities are satisfied inU(W):

Lkh[n]a = h

[n]a− k

i

Lk, (20)

Lkh(n)a = h

(n)

a− ki

Lk, (21)

enh[m]a = h

[m]a−ne

n, (22)

enh(m)a = h

(m)a−ne

n, (23)

LrLms =

m∑l=0

(−1)l

(l−2∏

j=−1

(r + js)

)(m

l

)Lm−l

s Lr+ls . (24)

Proof. One hasLkL0 = (L0 − k)Lk and, by induction,LkLn0 = (L0 − k)nLk . So,

Lkhn = (h − k/i)nLk and also,LkP(h) = P(h − k/i)Lk for all P ∈ Q[X]. Hence,

we get (20) and (21). In particular, one haseP (h) = P(h − 1)e and, by inductionenP (h) = P(h − n)en. Hence, we get also (22) and (23). On the other hand, oneLrLs = LsLr − (r − s)Lr+s . So (24) is true form = 1. If it is true up to rankm, then

LrLm+1s =

m∑l=0

(−1)l

(l−2∏

j=−1

(r + js)

)(m

l

)Lm−l

s Lr+lsLs

=m∑

l=0

(−1)l

(l−2∏

j=−1

(r + js)

)(m

l

)Lm−l

s

(LsLr+ls − (

r + (l − 1)s))

Lr+(l+1)s

=m∑

(−1)l

(l−2∏

(r + js)

)(m

l

)Lm+1−l

s Lr+ls

l=0 j=−1


+m∑

l=0

(−1)l+1

(l−1∏

j=−1

(r + js)

)(m

l

)Lm−l

s Lr+(l+1)s

=m∑

l=0

(−1)l

(l−2∏

j=−1

(r + js)

)(m

l

)Lm+1−l

s Lr+ls

+m+1∑l=1

(−1)l

(l−2∏

j=−1

(r + js)

)(m

l − 1

)Lm+1−l

s Lr+ls

=m+1∑l=0

(−1)l

(l−2∏

j=−1

(r + js)

)((m

l − 1

)+

(m

l

))Lm+1−l

s Lr+ls

=m+1∑l=0

(−1)l

(l−2∏

j=−1

(r + js)

)(m + 1

l

)Lm+1−l

s Lr+ls .

Therefore, the result is proved by induction.�4.2. Quantization in characteristic 0

Definition 4. For aa ∈ Q, we set:

Fa :=∞∑

r=0

(−1)r

r! h[r]a ⊗ er tr ,

Fa :=∞∑

r=0

1

r!h(r)a ⊗ er tr ,

ua := m ◦ (S0 ⊗ Id)(Fa),

va := m ◦ (Id ⊗ S0)(Fa).

We set alsoF :=F0, F := F0, u := u0 andv := v0.

Lemma 5. For a, b ∈ Q, one has

FaFb = 1⊗ (1− et)a−b, (25)

vaub = (1− et)−(a+b). (26)

Proof. One has

FaFb =∞∑

r,s=0

(−1)r

r!s! h[r]a h

(s)b ⊗ erestr ts =

∞∑m=0

(−1)m

( ∑r+s=m

(−1)s

r!s! h[r]a h

(s)b

)⊗ emtm

(18)=∞∑

(−1)m(

a − b

m

)⊗ emtm = 1⊗ (1− et)a−b.

m=0


On the other hand, one hasS0(h(n)a ) = (−1)nh

[n]−a andS0(e

n) = (−1)nen. So, by definition,one has:

ua =∞∑

n=0

(−1)n

n! h[n]−ae

ntn, vb =∞∑

m=0

1

m!h[n]b entn.

We deduce that:

vaub =∞∑

m,n=0

(−1)m

m!n! h[n]a enh

[m]−b emtm+n (22)=

∞∑r=0

∑m+n=r

(−1)m

m!n! h[n]a h

[m]−b−ne

r tr

(19)=∞∑

r=0

(a + b + r − 1

r

)er tr = (1− et)−(a+b). �

Corollary 1. For a ∈ Q, Fa andua are invertibleF−1a =Fa andu−1

a = v−a . In particular,F−1 =F andu−1 = v.

Lemma 6. For a ∈ Q andk ∈ Z, one has:

(Lk ⊗ 1) · Fa = Fa− ki· (Lk ⊗ 1), (27)

(1⊗ Lk) · Fa =∞∑l=0

(−1)lil

l!

(l−2∏

j=−1

(k + j i)

)Fa+l · (h(l)

a ⊗ Lk+li

)t l , (28)

Lkua = ua+ k

i·

∞∑l=0

il

l!

(l−2∏

j=−1

(k + j i)

)Lk+lih

(l)−a+1t

l . (29)

Proof. One has

(Lk ⊗ 1) · Fa =∞∑

m=0

1

m!Lkh(m)a ⊗ emt

(21)=∞∑

m=0

1

m!h(m)

a− ki

Lk ⊗ emt = Fa− ki· (Lk ⊗ 1).

Hence, we get (27). One has also:

(1⊗ Lk) · Fa =∞∑

m=0

1

m!h(m)a ⊗ Lke

mt

(24)=∞∑

m=0

1

m!h(m)a ⊗

m∑l=0

(−1)lil

(l−2∏

j=−1

(k + j i)

)(m

l

)em−lLk+li t

m

=∞∑ ∞∑ 1

(m + l)! (−1)lil(

m + l

l

)(l−2∏

(k + j i)

)h(m+l)

a ⊗ emLk+li tm+l

l=0 m=0 j=−1


(15)=∞∑l=0

∞∑m=0

(−1)lil

m!l!

(l−2∏

j=−1

(k + j i)

)h

(m)a+lh

(l)a ⊗ emLk+li t

m+l

=∞∑l=0

(−1)lil

l!

(l−2∏

j=−1

(k + j i)

)( ∞∑m=0

1

m!h(m)a+l ⊗ emtm

)·h(l)

a ⊗ Lk+li tl

=∞∑l=0

(−1)lil

l!

(l−2∏

j=−1

(k + j i)

)Fa+l · h(l)

a ⊗ Lk+li tl .

Hence, we get (28). On the other hand, one has:

Lkua =∞∑

m=0

(−1)m

m! Lkh[m]−a emtm

(20)=∞∑

m=0

(−1)m

m! h[m]−a− k

i

Lkemtm

(24)=∞∑

m=0

m∑l=0

(−1)lil

(l−2∏

j=−1

(k + j i)

)(m

l

)(−1)m

m! h[m]−a− k

i

em−lLk+li tm

=∞∑l=0

∞∑m=0

1

(m + l)! (−1)mil(

m + l

l

)(l−2∏

j=−1

(k + j i)

)h

[m+l]−a− k

i

emLk+li tm+l

(16)=∞∑l=0

il

l!

(l−2∏

j=−1

(k + j i)

) ∞∑m=0

(−1)m

m! h[m]−a− k

i

h[l]−a− k

i−m

emLk+li tm+l

=∞∑l=0

il

l!

(l−2∏

j=−1

(k + j i)

) ∞∑m=0

(−1)m

m! h[m]−a− k

i

emtmh[l]−a− k

i

Lk+li tl

=∞∑l=0

il

l!

(l−2∏

j=−1

(k + j i)

)ua+ k

ih

[l]−a− k

i

Lk+li tl

(20)= ua+ ki

∞∑l=0

il

l!

(l−2∏

j=−1

(k + j i)

)Lk+lih

[l]−a+l t

l

(17)= ua+ k

i

∞∑l=0

il

l!

(l−2∏

j=−1

(k + j i)

)Lk+lih

(l)−a+l t

l . �


Hopff

by

Theorem 1. There exists a structure of noncommutative and noncocommutativealgebra onU(W)[[t]] denoted by(U(W)[[t]],m,∆,S, ε) which leaves the product oU(W)[[t]] undeformed but with a deformed comultiplication defined by:

∆(Lk) := Lk ⊗ (1− et)ki +

∞∑l=0

(−1)lil

∏l−2j=−1(k + j i)

l! h(l) ⊗ (1− et)−lLk+li tl , (30)

S(Lk) := −(1− et)−ki

∞∑l=0

il

∏l−2j=−1(k + j i)

l! Lk+lih(l)1 t l , (31)

ε(Lk) := 0 (32)

with k ∈ Z.

Proof. It follows from (27) and (28) that:

∆(Lk) =F0 · (Lk ⊗ 1+ 1⊗ Lk) · F =F0 · (Lk ⊗ 1) · F +F0 · (1⊗ Lk) · F

=F0 · F− ki· (Lk ⊗ 1) +

∞∑l=0

(−1)lil

l!

(l−2∏

j=−1

(k + j i)

)(F0 · Fl) · (h(l) ⊗ Lk+li

)t l

= (1⊗ (1− et)

ki) · (Lk ⊗ 1)

+∞∑l=0

(−1)lil

l!

(l−2∏

j=−1

(k + j i)

)h(l) ⊗ (1− et)−lLk+li t

l

and, from (29) and (26), one gets:

S(Lk) = −v0 · Lk · u0 = −v0 · uki·

∞∑l=0

il

l!

(l−2∏

j=−1

(k + j i)

)Lk+lih

(l)1

= −(1− et)−ki ·

∞∑l=0

il

l!

(l−2∏

j=−1

(k + j i)

)Lk+lih

(l)1 .

Hence we get the result.�Proposition 4. If x ∈ U(W) is homogeneous with respect to the grading defined|Lk| = k, one has:

S(x) = (l − et)−|x|i

∞∑n=0

d(n)(S0(x)

) · h(n)1 tn. (33)


ing

Proof. By induction onr using (29), we first show that

Lkr · · ·Lk1u

= ukr +k2+···+k1i

∑d(lr)(Lkr )h

(lr )

− k1+···+kr−1i

· · ·d(l2)(Lk2)h(l2)

− k1i +1

d(l1)(Lk1)h(l1)1 t l1+···+lr .

Then, with the help of relation (21) betweenLu andh(l)a , we show that

Lkr · · ·Lk1u

= ukr+k2+···+k1i

∑d(lr )(Lkr ) · · ·d(l2)(Lk2)d

(l1)(Lk1)h(l1)1 h

(l2)l1+l2+1 · · ·hlr

l1+···+lr+1tl1+···+lr

= ukr+k2+···+k1i

∑d(lr )(Lkr ) · · ·d(l2)(Lk2)d

(l1)(Lk1)h(l1+···+lr )1 t l1+···+lr

= ukr+k2+···+k1i

∞∑n=0

∑l1+···+lr=n

d(lr)(Lkr ) · · ·d(l2)(Lk2)d(l1)(Lk1)h

(n)1 t l1+···+lr

= ukr+k2+···+k1i

∞∑n=0

d(n)(Lkr · · ·Lk1)h(n)1 tn.

Therefore, one has:

S(Lk1 · · ·Lkr ) = v0S0(Lk1 · · ·Lkr )u = (−1)rv0Lkr · · ·Lk1u

= (−1)rv0 · ukr+k2+···+k1i

∞∑n=0

d(n)(Lkr · · ·Lk1)h(n)1 tn

(26)= (−1)r(1− et)−k1+···+kr

i

∞∑n=0

d(n)(Lkr · · ·Lk1)h(n)1 tn

= (1− et)−k1+···+kr

i

∞∑n=0

d(n)(S0(Lk1 · · ·Lkr )

)h

(n)1 tn. �

In the same way, one can also prove the following lemma.

Lemma 7. Leta(x) andb(x) be the functions defined inU(W) bya(x) := F−1 · (x ⊗1) ·Fandb(x) := F−1 ·(1⊗x) ·F . Then,a(x) = x ⊗(1−et)|x|/i andb(x) = ∑∞

n=0(−1)nh(n) ⊗(1− et)−nd(n)(x)tn for any elementx in U(W) homogeneous with respect to the graddefined by|Lk| = k.

4.3. Reduction to characteristicp

Lemma 1. For all integersa, k andl, al∏l−1 (k + ja)/l! is an integer.
j=0


eat

Proof. First, suppose thatp does not dividea. Replacingk by k − (l − 1)a anda by −a

if necessary, one can assume that a is positive. LetS be the set{k, k + a, . . . , k + (l − 1)a}and for j ∈ N, Sj := {n ∈ S/valp(n) � j } (respectivelySj := {n ∈ S/valp(n) = j })where valp(n) denotes thep-adic valuation of any integern. Setnj := card(Sj ). Clearly,card(Sj ) = nj − nj+1 and

valp

(l−1∏j=0

(k + ja)

)=

∑j�1

j card(Sj

) =∑j�1

j (nj − nj+1) =∑j�1

nj .

Fix j ∈ N. Set s := Inf Sj with s = k + s0a and 0� s0 � l − 1. If, s0 � pj , thenk +(s0 − pj )a ∈ Sj andk + (s0 − pj )a < s. So, s0 ∈ {0, . . . , pj − 1}. Set alsoTj := {r ∈{0, . . . , l −1} | r = s0 (mod pj )}. If t ∈ Sj with t = k + ra ∈ S then,pj | t − s = (r − s0)a.So,pj | r − s0 sincep anda are relatively prime. So,r ∈ Tj . Conversely, ifr ∈ Tj , thenk + ra ∈ Sj . It follows that,nj = cardTj . So,Tj = {s0, s0 + pj , . . . , s0 + (nj − 1)pj } andnj − 1 is the maximal integer such thats0 + (nj − 1)pj � l − 1. In particular, in the caswhenk = a = 1, one hass = pj , s0 = pj − 1 andnj is the maximal integer such thnjp

j � l. Hence, in this case, one hasnj = [l/pj ] and

valp(l!) =∑j�1

[l

pj

]. (34)

In the general case, from the property satisfied bynj − 1, we deduce thats0 + njpj � l.

So,

nj � l − s0

pj� 1

pj�

[1

pj

].

Hence, we get

valp

(l−1∏j=0

(k + ja)

)=

∑j�1

nj �∑j�1

[1

pj

]= valp(l!).

Therefore, valp(al∏l−1

j=0(k + ja)/l!) � 0.Suppose now thatp dividesa. Then,

valp

(al

∏l−1j=0(k + ja)

l!

)� valp

(al

)valp(l!) � l −

∑j�1

[l

pj

]� l −

∑j�1

l

pj= p − 2

p − 1l � 0.

Therefore in any cases, we have

valp

(al

l−1∏ (k + ja)

l!

)� 0.

j=0


c-

re is

-

Sincep is an arbitrary prime number, this implies thatal∏l−1

j=0(k + ja)/l! is an inte-ger. �

By definition,Zp denotes the ring ofp-adic integers.

Definition 5. We denote byWZp (respectivelyWFp ) the Lie algebraW seen as aZp-Liealgebra (respectivelyFp-Lie algebra) andJ the ideal ofU(WFp ) generated byLk+p − Lk

with k ∈ Z, Lpr − Lr with r divisible by p (r ∈ Z) and L

ps with s not divisible byp

(s ∈ Z). We denote also byπ theZ-algebra morphism ofU(WZp ) ontoU(WFp ) obtainedby reduction of scalar modulop and� the projection ofU(WFp ) ontoU(WFp )/J .

Formulas (30) and (31) giving the comultiplication and the antipode inU(W) stillmake sense inU(WZp ) (respectivelyU(WFp )) and define an Hopf algebra struture onU(WZp ) (respectivelyU(WZp )) denoted by(U(WZp ),m,∆,ε,S) (respectively(U(WZp ),m,∆,ε,S)) (one denotes by the same letters∆, S andε the comultiplication,the antipode and the co-unity in all the Hopf algebras we meet). Moreover,π is aZ-Hopfalgebra morphism fromU(WZp ) ontoU(WFp ).

Theorem 2. Let i be an element ofFp −{0}. We define a Hopf algebra(Uc(D),m,∆,S, ε)

overFp[t] as follows: the algebra structure is undeformed and the coalgebra structugiven by:

∆(Dk) := Dk ⊗ (1− et)ki−1 +p−1∑l=0

(−1)lN(i, k − i, l)h(l) ⊗ (1− et)−lDk+li tl , (35)

S(Dk) := −(1− et)−ki−1p−1∑l=0

N(i, k − i, l)Dk+lih(l)1 t l , (36)

ε(Dk) := 0 (37)

with, k ∈ Fp, h := −1iD0 ande := iDi .

Proof. Clearly, Uc(D) and U(WFp )/J are isomorphic. Moreover, inU(WFp ), one hash(l) ≡ 0 (mod J ) for l � p. So, to prove the theorem, it is enough to prove thatJ is a Hopfideal inU(WFp ), i.e.,f (J ) ≡ 0 (mod J ) for f = ∆, S, ε.

To avoid confusions, we denote byLk , k ∈ Z, the generators ofWZp andlk , k ∈ Z, thegenerators ofWFp . So, one hasπ(Lk) = lk . According to formulas giving comultiplications and antipods inU(WZp ) andU(WFp ), one has fori, k ∈ Z,

∆(lk+p) = lk+p ⊗ (1− et)ki−1(1− et)pi−1

+∞∑

(−1)lil

∏l−2j=−1(k + p + j i)

l! h(l) ⊗ (1− et)−l lk+p+li tl . (38)

l=0


he

l to 0

So, ∆(lk+p) ≡ ∆(lk) (mod J ) since ep ≡ 0 (mod J ). We show also thatS(lk+p) ≡S(lk) (mod J ) and ε(lk+p) ≡ ε(lk) (mod J ). On the other hand, by Proposition 4, tfollowing identities are satisfied inU(WZp ):

S(L

pk

) = −(1− et)−kpi−1∞∑l=0

d(l)(L

pk

) · h(l)1 t l

= −(1− et)−kpi−1 · Lpk −

∞∑l=1

d(l)(L

pk

) · h(l)1 t l .

Moreover, from (24), we see that

[Li,L

pk

] =p∑

l=1

(−1)l

(l−2∏

j=−1

(i + jk)

)(p

l

)L

p−lk Li+kl .

If p does not dividek, then∏p−2

j=−1 = (i + jk) is a product ofp terms all distinctmodulo p. So, one of them is 0. Hence, any term of the last sum above is equamodulop. Hence, ad(e)(Lp

k ) ∈ p · U(WZp ) and alsod(l)(Lpk ) ∈ p · U(WZp ) for 1 � l �

p − 1. As a result, by using the projectionπ of U(WZp ) onU(WFp ), we see that:

S(lpk

) = −(1− et)−kpi−1lpk −

∞∑l=p

d(l)(lpk

) · h(l)1 t l .

But lpk andh(l)1 for l � p are equal to 0 modulop. So,S(l

p

k ) ≡ 0 (mod J ).

On the other hand, ifp dividesk, then∏p−2

j=−1(i + jk) ≡ ∏p−2j=−1 i = ip ≡ i (mod p).

So,

ad(e)(L

pk

) = i[Li,L

pk

] = (−1)pi2Li+kp + x

with x ∈ p · U(WZp ). So,

d(2)(L

pk

) = 1

2

[iLi, (−1)pi2Li+kp + x

] = (−1)pi3 · kp

2L2i+kp + 1

2[iLi, x]

∈ p · U(ZZp ).

By induction, we see thatd(l)(Lpk )−kpi−1 ∈ p ·U(WZp ) for 2 � l � p − 1. Hence, usingπ ,

one obtains:

S(lp

k

) = −(1− et) · lpk − (−1)pi2li+kp · h1 −∞∑

d(l)(lp

k

) · h(l)1 t l .

l=p


re,

n ofand E.si for

.

. Pure

We use now the fact thatlpk − lk , ep andh(l)1 with l � p are equal to 0 modulop to deduce

that

S(lp

k

) = −(lk − i2li+kph1

) ≡ −(lk − i2li+kh1

) ≡ S(lk) (modJ ).

In the same way, one has:

∆(L

p

k

) = F−1 · ∆0(Lk)p · F = F−1 · (Lk ⊗ 1+ 1⊗ Lk)

p · F

= F−1 ·(

Lpk ⊗ 1+ 1⊗ L

pk +

p−1∑l=1

(p

l

)Ll

k ⊗ Lp−lk

)· F

= a(L

pk

) + b(L

pk

) +p−1∑l=1

(p

l

)a(Lk)

l · b(Lk)p−l .

But from Lemma 7, we know that

a(L

p

k

) = Lp

k ⊗ (1− et)kpi−1,

b(L

pk

) =∞∑l=0

(−1)lh(l) ⊗ (1− et)−ld(l)(L

pk

)t l

= 1⊗ Lp

k +∞∑l=1

(−1)lh(l) ⊗ (1− et)−ld(l)(L

p

k

)t l .

This lemma also shows thata(Lk), b(Lk) ∈ U(WZp ). By the same argument as befowe see that∆(l

p

k ) ≡ ∆(lk) (mod J ) if p dividesk and∆(lp

k ) ≡ 0 (mod J ) if p does notdivide k. Moreover, it is obvious thatε(lpk ) ≡ 0 (mod J ) for all k. Therefore,J is a Hopfideal inU(WFp ). �

Acknowledgments

I am very grateful to B. Enriquez for his help and support during the preparatiothis article. I am also grateful to P. Baumann, J. Bichon, F. Gavarini, Laura StevensTaft for comments and S.-H. Ng who kindly e-mailed me his thesis. I thank C. Procehospitality at Castelnuovo Institute and C. Kassel for hospitality at IRMA (Strasbourg).

References

[1] V. Drinfeld, Constant quasiclassical solutionsof the Yang–Baxter quantumequation, Soviet MathDokl. 28 (3) (1983) 667–671.

[2] S.-H. Ng, E. Taft, Classification of the Lie bialgebra structures on the Witt and Virasoro algebras, JAppl. Algebra 151 (1) (2000) 67–88.


e-
[3] A. Giaquinto, J. Zhang, Bialgebra action, twists and universal deformation formulas, J. Pure Appl. Algbra 128 (2) (1998) 133–151.
[4] N. Jacobson, Lie Algebras, Dover, 1962.[5] D.E. Radford, Operators on Hopf algebras, Amer. J. Math. 99 (1977) 139–158.[6] E. Taft, Witt and Virasoro algebras as bialgebras, J. Pure Appl. Algebra 87 (3) (1993) 301–312.

Quantizations of the Witt algebra and of simple Lie algebras in characteristic p

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