LIE STRUCTURES ON f[x,,•••,x.,y]1(y3-3py-q)By Fujio KuBo

Bull. Kyushu Inst. Tech.

(Math. Natur. Sci.) No. 35, 1988, pp. 1-6

LIE STRUCTURES ON f[x,,•••,x.,y]1(y3-3py-q)

By Fujio KuBo

(Received November 27, 1987)

1. Introduction

The Poisson Lie structures have been investigated in many papers (for example Berezin [2]). Recently Kubo and Mimura [4] extended these Lie structures to

commutative associative algebras containing a finite-dimensional Lie algebra and studied Lie structures on f[x,,..., x., y]1(y2-2ay+b) (a,bEf[x,,..., jx]). In this paper we

shall investigate Lie structures on factor algebras f[xi,..., x., y]/(y3-3py-g).

2. Notationsandpreliminaries

Let L be a finite-dimensional Lie algebra over a field f of characteristic zero with a basis {xi,..., x.}. Let A be a commutative associative algebra over { with an identity 1 and have deriations di,..., d. satisfying the following conditions, for i, 1'=1,..., n,

LcA, d,(xj)=6ij, didj=djdi.

ThenaLie structure on A is defined as follow ([4]): For any elements a, bEA, the product [a, b] in A is given by

[a, b] =2i,j[xi, xj]di(a)dj(b) = 2 i,j,k ck' xkdi(a)dj(b)

where eV are the structure constants with respect to a basis {xi,..., x.} ofL. Then A is a Lie algebra with this product and this Lie algebra is denoted by

L(L; A, {d,}).

ExAMpLE (Tensor products ofLie algebras constructed above). Let L, G be finite- dimensional Lie algebras over f with basis {xi,..., x.}, {yi,..., y.}, respectively. Let A,

B be commutative associative algebras with identities such that LcA, GcB and there exist derivations ui,..., u. of A and vi,..., v. of B satisfying the conditions; uiu,• ---

ujui, v,vt=vtv,, ui(xj)=6iJ•, v.(yt) ==6,t (i, 1''-- 1,..., n; s, t=1,..., m). Consider the

Author is partially supported by Grand-in-Aid for Encouragement of Young Scientist of Ministry of Education of Japan.

commutative associative algebra C =A(g)fBand a Lie algebra M==L(g)ff+f(g),G with [a(g)1, a'(g) 1] -= [a,a'] (g)1, [1(g)b, 1Qb'] =1(21) [b, b'].

[L (g) ,f, f(gÅr ,G] == O, (a, deA, b, b'EB).

We define derivations Ui,..., U., Vi,..., V. of C by Ui (a Qb) - ui(a)Q b, V, (a (gÅr b) = a (g) v,(b)

(aE4 bE B, i:- 1,..., n, s= 1,..., m). Then for a basis {xi (g) 1, 1 (g) y,: i== 1,..., n, s=1,..., m}, we have U,(xj(g)1)=6ij, Ui(1(El)y,) =O, V,(xj)=O, V,(y,)=6,,(i,j=1,..., n: s, t=1,..., m). Further theses derivations are commutable each other. Then a Lie structure ofL(M; C, {Ui, V,}) is given by, for a, a'EA, b, b'EB,

[a (g) b, a' (g)b'] = 2,,j[x-i(g) 1, xj (g) 1] Ui(a (8)b) Uj(d (El)b')

+ ]2 ,,t[1(2i)y,, 1 (Ei) .vt] V.(a(2)b) V,(a'(g)b') = [a, a'] (Ei)bb'+aa'(g) [b, b'].

To express the structures of Lie algebras we employ some notations. Let I, K be Lie algebras with a homomorphism (S of K into the derivation algebra Der(I). Then the vector space L=I+K is made into a Lie algebra by [a+b, c+d] =([a, c] +6(d)(a)

-j(b)(c))+ [b, d] (a, cEI, h, dEK). This Lie algebra L is called a split extension of I by Kand denoted by I+,K([1: p. 22]).

Let A be a commutative associative algebra over f. Assume that A has a Lie structure whose product [,] satisfies the condition; [ab, c] = [a, c]b+a[b, c](a, b, cE A). For an element wEA, we define the product [a, b].(a, bE A) by

[a, b]. =a[b, w] -h[a, w].

Then by a simple computation this product [ , ]. makes A into a Lie algebra.

3. Lie structures on f[x,,..,,x., y]1(y3-3py-q)

Let L be a finite-dimensional Lie algebra over f with a basis {xi,..., x.} and R the polynomial algebra f[x,,..., x.]. Consider the polynomial algebra R [y] over R and take a polynomial y3-3py-q (p, qER). Let us denote by

A(p, q) -= R[y]/(y3-3py- q)

acommutative associative factor algebra, where (y3-3py-q) is the ideal of R[y]

generated by the polynomial y3 - 3py - q.

Let T(y)=y3+ay2+by+c(a,b, cER), A a factor algebra R[y]/((T(y)) and Di,..., D. derivations of A such that for i, 1'=1,,.., n, DiDj=DjDi, DilR==O/axi.

Lie structures on f[xi,,,,, x., y] 1(y3-3py-q) 3

Then we put p = (a2 - 3b)19, q = ( - 2a3 + 9ab - 27c)127. For an associative isomorphism ip: A(p, q)-ÅrA with di(y) =y+(a!3), we set d,=ip-'Diip(i=1,..., n). Then d,,..., d.

are derivations of A(p, q) such that didj=djdi, ipdi == Diip, dilR=O/Oxi(i, 1'=1,..., n), in other words di is a differential automorphism([3]). Therefore by [4, Lemma 5]

we have the following

PRoposiTioN 1. Let A, p, q, di, Di (l=1,..., n) be given above. Then a Lie algebra L(L; A, {Di}) is isomorphic to a Lie algebra L(L; A(p, q),{d,}).

Hence we shall only investigate Lie structures on A(p, q),

We first extend a derivation Olaxi on R to A(p, q). Let di,..., d. be derivations of A(p, q) such that

(*) d,l.=:DIOx,, d,dj=djd,,

di(y) = ai + biy + c,y2

(ai, bi, cieR: i, 7'=1,..., n). Since di(y3-3py-q) =(-3pai+3qb,-di(q))+3(2pb,+

qci-di(p))y+3(ai+2pci)y2 =O in A(p, q), we have the following conditions; for i, 7'=1,... n,

(1) -3pai+3qbi-di(q), (2) 2pbi+qci=di(p),

(3) ai+2pci--O.

By the condition that didj(y)=d,•di(y) and didj(y)=:(di(a,•)+aibj+2qcicj)+

(di(bj) +bibj + 2aicj + 6pc icj)y + (di(cj) + bjci + 2bicj)y2, we have the following

conditions; for i, 7' -- 1,..., n,

(4) di(aj)+aibj=dj(ai)+ajbi,

(5) di (bj)+2aic,•=dj(bi)+2ajci, (6) di(cj)+bic,•=dj(ci)+bjci.

LEMMA 2. if there exist derivations di,..., d. of A(p,q) satisfying the condition (*), then 4p3-q2Ef. Further if 4p3 =q2, then p, qE{.

PRooF. By (1), (2), (3) we have 3(4p3-g2)b, == d,(4p3-q2)/2 (i=1,...,n).

Therefore 4p3-g2Ef. Assume that 4p3 =q2 and pq#O, and takeanon-zero element hER such that p== h2, q=2h3. By(2), h(bi+cih)= di(h) for i=1,..., n, we have hef.

QE. D.

From novv on, the Lie product [f, g] (f, geR) will be calculated in L(L;R,

{o/o.xi})•

3.l. Lie structures on A(p, q); 4p3#g2

In this section we assume that 4p3lq2. We shall show that di(y)-O (i---=1,,.., n).

By the proofof Lemma 2, we have bi=O (i=1,..., n). If p=O or q==O, then by (1), (2), (3) we have ai=c,=O. If pqlO, then comparing the degrees in x, of the both sides in (2) we have ci ==O and so ai=O.

We now compute a Lie product ff, y] (fGR) of L(L;A(p, q), {di}) as follow:

[f, y] =2i,j[xi xj]di(f)d,(y) =o.

Therefore we have the following

THEoREM 3. Assume that 4p37Eq2. Then the Lie structure of L(L;A(p, q),

{di}) is given by

[f, gy]=[f, g], [f, gy2] .. [f, g]y2,

[fy,gy]= [f, g]y2, [fy,gy2]-g[f, g]+3p[f, g]y, [fy2, gy2] -q[f, g]y+3p[f, g]y2 (f, gER).

CoRoLLARy 4. (1) Lie algebras L(L;A(O, q), {d,})(q 7EO) are isomorphic to a Z3-graded Lie algebra Ro+Ri+R2 such that

[fo, gi] == [f, g]i (i-O, 1, 2), [fi,gi]-[f, g]o,

[fi, g2] =q[f, g]o, [f2, g2] -g[f, g]i•

(2) Lie algebrasL(L;A(p,O),{d,})(p7EO) are isomorphic to the split extension (Ro -F Ri) -F iR ofa Z2-graded Lie algebra Ro+R, by R, where, for f, g, hER, [fo, gi] =3p[f, g]i(i--O, 1), [fi, gi]=[f, g]o,

and the Lie homomorphism 6 of R into Der(Ro+Ri) is given by 6(h)(fo+gi)- [f, h]o+ [g, h]i•

3.2. Lie structures on A(O, O)

In this section we shall discuss about Lie structures on A(O, O) =R[y]/(y3), Let di,..., d. be derivations ofA(O, O) satisfying the condition (*). Then by (3), (4), (5), (6) we have ai=:O, di(bj)=dj(bi) for i, 1'=1,..., n. Therefore there exists an element b of R such that

Lie structures on f[xi,..., x., y] /(y3-3py-q) 5

bi=OblOxi fori=1,...,n.

We restrict the condition (6) to that di(c,•) == dj(ci) (i, .f -- 1,..., n). For example this condition holds if the degrees of ci are less than that of b. Then we can choose an element c ofR such that ci=Oc!Oxi. Therefore we have, for i, j-- 1,..., n,

di(y) = di(b)y+ d,(c)y2 d,(b)d,• (c) == d,• (h)di(c).

THEoREM 5. Let b, cER satisf.y the condition (Oh/Ox,)(Oc!Oxj) == (ab/Oxj)

(OclOxi) (i, J'=1,..., n), and deLfine derivations d,,..., d. of' A(O, O) by d,(y) == d,(b)y +d,(c)y2. Then the Lie structure Qf L(L; A(O, O),{di'i) is given by

[f, gy] == ([f, g] +g[f, b] )y +g[f, c] y2, [,L gy2] -([f, g] +2g [f, h] )y2, [fy, gy] -=([f, g] -f[g, b] +g[f, h]).v2, [fy,gy2] ==O.

[fy2, gy2] =-O (f, gER).

CoRoLLARy6. The Lie algebra L(L;A(O,O),{di})gtven in Theorem 5 is

isomorph ic to the split extension (Ro + R i ) + jR of a Z2-g ra ded Lie atg ebra Ro + Ri by R, where forf, g, hG R,

[Ro, Ro] ==O, [fi,gi] -=([f, g] -f[g, b] +g[f, b])o, [R,, R,] ==O,

and the Lie homomorphism 6 of R into Der(Ro+Ri) is given by 6(h) (f,+g,)== ([f, h]+2f[b, h] +g[c, h]),+(g[b, h]+[g, h]),.

3.3. Lie structures on A (p, g); 4p3=g2, pg40

In this section we assume that 4p3==q2, pg ifO. By Lemma 2 and the conditions (1), (2), (3) we have a,-(4p2!q2)b,, ci=(-2p/q)b, (i-1,..., n). Since di(bj)=dj(bi)

(i, J' -- 1,..., n) by (3), (4), there exists an element b of R such that bi= di(b)(i=1,..., n).

Hence we have, for i=1,..., n,

di(y) -= (4p21q)d,(b) + di(b)y - (2plq)d,(b)y2.

We now compute a Lie product [.L y](feR) of L(L; A(p, q),{d,}) as follow;

[f, y] = Z ,,i [x ,, x,•]d,( f) dj (y)

= (4p21q) [f, b] + [f, b] y - (2pl q) [f, b] y2.

Therefore we have the following

THEoRE"f7.,., Assume that 4p3=e2, pq7EO. rhen the Lie strueture ojf'

L(L; A(p, e), {di-}) is given by

U, gy] = (4p2/q)g[L b] +Åq[L g] +g[A b])År, -- Åq2piq) g[f, b]y2, '[f, gy2] == - 4pg [f, b] - (4p2/q)g [f, b] y+ ( V, g] + 2g V, b] )y2, Efy, gy] ==2p[f, g]'+(2p2tq)[f. g]".v+([f, g] - [.L g]')y2,

[fy, g.v 2] == q( [L g] -f[g, bl + 2g [L b] ) + p(3 [L g] -f [g, b] + 2g [f, b] )N +{2p2/qÅr (f[g, b] ---2g[f, bl)y2,

[fy2, gy2] =4p2[L g]"+e([.L g]+[f, g]')y+p(3[L g] -2Y, gl")y2 ,vhereL gER and [f, g]'= [L g]h (Section 2).

References

[ 1 ] R.K. Amayo and -1.N. Stewart, Infinite-dimensional Lie algebrfis, Noordhoff, Leyden, 1974, [21 F.A. Berezin, Some remarks on the associative envelope of a Lie algebra (in Russian), Funkclon. Anal. Priloz. 1 (1967), 1--14.

r3] 1. Kaplansky, An introduction to differential algebras, Actualites Sci. Ind,, Hermann. 1957.

[4] F. Kubo and F. Mimura, Lie structures on differential algebras, to appea:, Hiroshima Malh, J.

Department of Mathematics K.vushu Institute of Technelogy