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GRADED RADICAL W TYPE LIE ALGEBRAS I
KI-BONG NAM
Received 24 August 2001 and in revised form 28 January 2002
We get a newZ-graded Witt type simple Lie algebra using a generalized polynomial ring which is the radical extension of the polynomial ringF[x]with the exponential functionex. 2000 Mathematics Subject Classification: 17B20.
1. Introduction. LetFbe a field of characteristic zero (not necessarily algebraically closed). Throughout this paper,Z+ and Zdenote the nonnegative integers and the integers, respectively. LetF[x]be the polynomial ring in indeterminatex. LetF(x)= {f (x)/g(x)|f (x), g(x)∈F [x], g(x)≠0}be the field of rational functions in one variable. We define theF-algebraV√m,espanned by
edxf1a1/b1···fmam/bmxt|d, a1, . . . , am, t∈Z, fi≠x, a1, b1
=1, . . . , am, bm
=1, 1≤i≤m ,
(1.1)
whereb1, . . . , bmare fixed nonnegative integers, and(ai, bi)=1,1≤i≤m, means that aiandbiare relatively primes, andf1, . . . , fnare the fixed relatively prime polynomials inF[x]. TheF-subalgebraV√+m, eofV√m,eis spanned by
edxf1a1/b1···fmam/bmxt|d, a1, . . . , am∈Z, t∈Z+, fi≠x, a1, b1
=1, . . . , am, bm
=1, 1≤i≤m .
(1.2)
LetW√m,e(∂)be the vector space over Fwith elements{f ∂|f ∈V√m,e}and the standard basis{edxf1a1/b1···fmam/bmxt∂|edxf1a1/b1···fmam/bmxt∂∈W√m,e}. Define a Lie bracket onW√m,e(∂)as follows:
[f ∂, g∂]=f
∂(g)
∂−g
∂(f )
∂, f , g∈V√m,e. (1.3) It is easy to check that (1.3) defines a Lie algebraW√m,e(∂)with the underlying vector spaceW√m,e(∂)(see also [1,3,5]). Similarly, we define the Lie subalgebraW√+m,e(∂)of W√m,e(∂)using theF-algebraV√+m,einstead ofV√m,e.
The Lie algebraW√m,e(∂)has a naturalZ-gradation as follows:
W√m,e(∂)=
d∈Z
W√dm,e, (1.4)
whereW√dm,eis the subspace of the Lie algebraW√m,e(∂)generated by elements of the form{edxf1a1/b1···fmam/bmxt∂|f1, . . . , fn∈F[x], a1, . . . , am, t∈Z, m∈Z+}. We call the subspaceW√dm,ethed-homogeneous component ofW√m,e(∂).
We decompose thed-homogeneous componentW√dm,eas follows:
W√dm, e=
s1,...,sm∈Z
W(d,s1,...,sm), (1.5)
whereW(d,s1,...,sm)is the subspace ofW√dm,espanned by edxf1s1/b1···fmsm/bmxq∂|q∈Z
. (1.6)
Note thatW(0,0,...,0)is the Witt algebraW (1)as defined in [3].
The two radical-homogeneous componentsW(d,a1,...,am)andW(d,r1,...,rm)are equiva- lent ifa1−r1, . . . , am−rm∈Z. This defines an equivalence relation onW√dm,e. Thus we note that the equivalent class ofW(d,a1,...,am) depends only ona1, . . . , am. From now onW(d,a1,...,am)will represent the radical homogeneous equivalent class ofW(d,a1,...,am)
without ambiguity. It is possible to choose the minimal positive integersa1, . . . , amfor the radical homogeneous equivalent componentW(d,a1,...,am).
We give the lexicographic order on all the radical homogeneous equivalent compo- nentsW(d,a1,...,am)usingZ×Zm+.
The radical equivalent homogeneous componentW√dm,ecan be written as follows:
W√dm,e=
(a1,...,am)∈Zm+
W(d,a1,...,am). (1.7)
Thus for any elementl∈W√m,e(∂),lcan be written uniquely as follows:
l=
(d,a1,...,am)∈Z×Zm+
l(d,a1,...,am). (1.8)
For any such elementl∈W√m,e(∂),H(l)is defined as the number of different homoge- neous components oflas in (1.4), andLd(l)as the number of nonequivalent radicald- homogeneous components oflin (1.8). For each basis elementedxf1a1/b1···fmam/bmxt∂ ofW√m,e(∂)(orW√+m,e(∂)), define degLie(edxf1a1/b1···fmam/bmxt∂)=t. Since every ele- mentlofW√m,e(∂)is the sum of the standard basis element, we may define degLie(l) as the highest power of each basis element ofl. Note that the Lie algebraW√m,e(∂)is self-centralized, that is, the centralizerCl(W√m,e(∂))of every elementlinW√m,e(∂)is one dimensional [1]. We find the solution of
11/3=y (1.9)
inZ7. Equation (1.9) implies that
1≡y3mod 7. (1.10)
The solutions of (1.10) are 1, 2, or 4. Thus 11/3=1, 2, or 4 mod 7. Thus the radical number inZpis not uniquely determined generally. So we may not consider the Lie algebras in this paper over a field of characteristicpdifferently from the Lie algebras in [2,3,4]. It is easy to prove that the Lie algebraW(0,...,0)is simple [3].
2. Main results. We need several lemmas forTheorem 2.5.
Lemma2.1. For any elementlin the(d, a1, . . . , am)-radical-homogeneous compo- nent ofW√m(∂), and for any elementl1∈W(0,0,...,0),[l, l1]is an element in the(d, a1, . . . , am)-radical homogeneous equivalent component.
The proof ofLemma 2.1is straightforward.
Lemma2.2. A Lie idealIof W√m,e(∂)which contains∂isW√m,e(∂).
Proof. LetIbe the ideal in the lemma. The Lie subalgebra which has the standard basis{xi∂|i∈Z+}is simple. LetI be any ideal ofW√m,e(∂)which contains∂. Then for anyf ∂∈W√m,e(∂),
[x∂, f ∂]=x∂(f )∂−f ∂∈I. (2.1)
On the other hand,
[∂, xf ∂]=f ∂+x∂(f )∂∈I. (2.2)
Thus by subtracting (2.2) from (2.1) we get 2f ∂∈I. Therefore, we have proven the lemma, sinceI∩W(0,0,...,0)contains nonzero elements and soI⊃W(0,0,...,0).
Lemma 2.3. A Lie ideal I of W√m,e(∂) which contains a nonzero element in W(d,a1,...,am)isW√m,e(∂), for a fixed(d, a1, . . . , am)∈Z×Z+.
Proof. LetI be a Lie ideal ofW√m,e(∂) and la nonzero element in the idealI.
Then we take an elementl1=e−dxf1−a1/b1···fm−am/bmxp∂withpa sufficiently large positive integer such that[l, l1]≠0. Then[f ∂, [l, l1]]is a nonzero element inW(0,0,...,0)
by taking an elementf1t1···fmtm∈F[x], wheret1, . . . , tmare sufficiently large integers.
ThusI∩W(0,0,...,0)contains nonzero elements, and hence,∂∈I∩W(0,0,...,0)by simplicity ofW(0,0,...,0). Then the lemma follows fromLemma 2.2.
Throughout this paper,abmeans thatais a number sufficiently larger thanb.
Lemma2.4. LetI be any nonzero Lie ideal ofW√m,e(∂). For any nonzero element l∈I, there is an elementxs∂,s0, such that[xs∂, l]is the sum of elements inW√m,e(∂) withdegLie([xs∂, l]) >0.
Proof. It is straightforward by choosing a sufficiently large positive integers.
Theorem2.5. The Lie algebraW√m,e(∂)is simple.
Proof. LetIbe a nonzero Lie ideal ofW√m,e(∂). Letlbe a nonzero element ofI.
ByLemma 2.4, we may assume thatlhas polynomial terms with positive powers for each basis element ofl. We prove this theorem in several steps.
Step1. Iflis in the 0-homogeneous component, then the theorem holds. We prove this step, by induction on the numberL0(l)of nonequivalent radical-homogeneous components of the element lofI. If L0(l)is 1 and l∈W(0,0,...,0), then the theorem holds by Lemmas2.2,2.3, and the fact thatW(0,0,...,0)is simple.
Assume that l∈ W(0,0,...,0,ar,...,am) with ar ≠ 0. If we take an element f1hr/kr···
fnhm/kmxhm+1∂such thathrkr, . . . , hnkr and(hr+kr)/kr∈Z+, . . . , (hm+km)/
km∈Z+, then we havel1=[f1hr/kr···fmhm/kmxhm+1∂, l]≠0. This implies thatl1is in W (0,0, . . . ,0). Thus we have proven the theorem byLemma 2.2.
By induction, we may assume that the theorem holds forl∈Isuch thatL0(l)=k, for some fixed nonnegative integerk >1. Assume thatL0(l)=k+1. Iflhas aW(0,0,...,0)
radical-homogeneous equivalent component, we takel2∈W(0,0,...,0)such that[l, l2]can be written as follows:[l, l2]=l3+l4wherel3is a sum of nonzero radical-homogeneous components, andl4=f ∂withf∈F[x]. Thus we have the nonzero element
∂, ···, [∂, l]···
=l2∈I (2.3)
which has no terms in the homogeneous equivalent componentW(0,0,...,0), where we applied Lie brackets untill2has no terms in the radical homogeneous equivalent com- ponentW(0,0,...,0). Thenl2∈Isuch thatH(l2)≤k. Therefore, we have proven the theo- rem by Lemmas2.2,2.3, and induction. Iflhas no terms in the radical homogeneous equivalent component(0,0, . . . ,0), thenlhas a term in the radical homogeneous equiv- alent componentW(0,a1,...,an). Take an elementl3=f1c1/p1···fmcm/pmxcm+1∂such that c1, . . . , cm+1are sufficiently large positive integers such thatc1+a1∈Z···cm+am∈Z, and which is in a radical homogeneous equivalent componentW(0,a1,...,am). Then[l3, l]
is nonzero and which has a term in the radical homogeneous equivalent component W(0,0,...,0). So in this case we have proven the theorem by induction.
Step2. Assume thatlis in thed-homogeneous component such that 0≠dand L0(l)=1, then the theorem holds. By taking e−dxxt∂, we have 0≠[e−dxxt∂, l]∈ W(0,0,...,0) by taking a sufficiently large positive integer t. Thus we have proven the theorem byStep 1.
Step 3. If l is the sum of (k−1) nonzero homogeneous components and 0- homogeneous component, then the theorem holds. We prove the theorem by induc- tion on the number of distinct homogeneous components by Steps1and2. Assume that we have proven the theorem whenl has(k−1)radical-homogeneous compo- nents. Assume thatlhas terms inW(0,0,...,0). ByStep 1, we have an elementl1∈I, such thatl1=l2+f ∂, wherel2has(k−1)homogeneous components andf∈F[x]. Then 0≠∂, [···, [∂, l1]···]∈Ihas(k−1)homogeneous components, where we applied the Lie bracket until it has no terms inW(0,0,...,0). Therefore, we have proven the theorem by induction.
Assume thatlhas a(k)homogeneous equivalent components. We may assumel has the terms which is in 0≠d-homogeneous component. By taking a sufficiently large positive integerr, we have[e−dxxr∂, l]≠0 and it has(k)homogeneous com- ponents with a term in the radical-homogeneous componentW(0,0,...,0). Therefore, we have proven the theorem byStep 3.
Corollary2.6. The Lie algebraW√+m,e(∂)is simple.
Proof. It is straightforward fromTheorem 2.5without usingLemma 2.4.
Corollary2.7. The Lie subalgebraW√0m,eof W√m,e(∂)is simple.
Proof. It is straightforward fromStep 1ofTheorem 2.5.
Proposition 2.8. For any nonzero Lie automorphism θ of W√+m,e(∂), θ(∂)= ∂ holds.
Proof. It is straightforward from the relationθ([∂, x∂])=θ(∂)and the fact that W√+m,e(∂)is self-centralized andZ-graded.
Acknowledgments. The author thanks the referee for the valuable suggestions and comments on this paper. The author also thanks Professor Kawamoto for his comments on radical numbers ofZp.
References
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[3] K.-B. Nam,GeneralizedW andHtype Lie algebras, Algebra Colloq.6(1999), no. 3, 329–
340.
[4] ,ModularWandHtype Lie algebras, Bulletin of South Eastern Asian Mathematics, Springer-Verlag, 2002, to appear.
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Ki-Bong Nam: Department of Mathematics and Computer Science, University of Wisconsin-Whitewater, Whitewater, WI53190, USA
E-mail address:[email protected]
