DERIVATIONS SATISFYING AN IDENTITY ON SYMMETRIC ALGEBRAS OF LIE ALGEBRAS
By
Fujio KuBo
(Received November 30, 1992)
'
Introduction
Let L be a finite-dimensional Lie algebra over an algebraically closed field of characteristic zero with a Lie product [,] and S(L) the symmetric algebra of L.
The Lie product of L is canonically extended to that of S(L) and S(L) turns a Pois- son algebra (see g 1).
Kubo and Mimura [3] gave a bracket Åq,År on the Poisson algebra (S(L), [,]) relative to an associative derivation D of S(L) by Åqa, bÅr=[a, b]+D(a)b- aD(b) (a,bES(L)). Then it is shown in [3] that S(L) isa Lie algebra under the bracket Åq,År,in other words the product Åq,År is a deformation of the Lie product
[,] (cf. [5]), if and only if D is a Lie derivation of S(L) and satisfies the identity (*) [a, b]D(c)+[b, c]D(a)+[c, a]D (b)=O for any a, b, cE S(L).
In succession, Kubo [4] described the structures of all nonabelian Lie algebras each of which has a nonzero derivation D satisfying the identity (*). It has been left to study such derivations D. In this paper we shall express the explicit forms of matrices for D relative to the appropriate ordered bases.
1. Notations and preliminary results
An associative commutative algebra A over a field of characteristic zero contain- ' ing an identity, equipped with a Lie bracket [,] such that [ab, c]=a[b, c]+b[a,
c] (a, b, c E A), is called a Poisson algebra (K. H. Bhaskara and K. Viswanath [2]).
Let D be an associative derivation of A, and define a bracket Åq , År on A by Åqa, b År= [a, b] +D (a) b- aD (b),
(a, b, cEA). We call the algebra A with this bracket Åq , År the D-extension of the Poisson algebra (A, [,]) ([3]). Then the bracket Åq , År satisfies the Jacobi identi- ty if and only if D satisfies the conditions: D([a, b])= [D(a), b]+[a, D(b)] and [a, b]D(c)+[b, c]D(a)+[c, a]D(b)=O for any a, b, cEA ([3, Theorem 8]).
Throughout this paper let k be an algebraically closed field of chracteristic zero, L a finite-dimensional nonabelian Lie algebra over k with a Lie product [,] and
S(L) the symmetric algebra of L. Let {xi,"',xn} be a basis of L. We regard S(L) a Poisson algebra with a Lie product
Ocr Ob
[a, b]= li.li,j [jci, •Xj] o,z, oxj '
Let Der(L) be the set of all derivations of L. For DE Der(L) we write ID(a, b, c) = [a, b]D(c) +[b, c]D(a)+ [c, a]D(b)
(a, b, c E! S(L)), where we extend D to the associative derivation of S(L). Then we denote by
D*(L) == {D E Der(L)11D =O on L}
As we see in the above paragragh the bracket Åq,År on S(L), defined by Åqa, bÅr =[a, b]+D(a)bmaD (b)
(a, b E S(L)), satisfies the Jacobi identity if and only if ID =O on L.
Let L K be Lie algebras over k, cr a Lie homomorphism of K into Der(I). We put
L =I+K
the direct sum of vector spaces I and K. We define a Lie product on L making it into a Lie algebra by
[i+k, 1'+l]=[i, 7']+cr (k) (1')-a(l) (i)+ [k, l]
(i, 7' E I, k, lE K). We denote this Lie algebra by L=I+aK and call the split ex- tension of I by Kralative to a ([1, p 22]). In case that Iis an ideal of a Lie algebra L with I nK=:O, the notion I+K means the split extension relative to the ajoint rep- resentation of K on L
We employ the following notations:C(L) is the center of L. Let U be an ideal of L. ForxEL we write adux(u)=[u, x] (uE U). For subspaces M, Nof L, MN means the subspace { 2:iminilmi E M, ni E N} of S(L). Triangular brakets Åq , År denote the Lie subalgebra generated by their contents. We write L2 =[L, L]. Letf be a linear operator on a vector space V. We express a matrix F of frelative to an ordered basis {vi, "', vn} for Vby the rule: (f(vi)"'f(vn)) == (vi"'vn)F.
Now we shall list the Lie algebras on which we work in this paper. Let L be a finite-dimensional nonabelian Lie algebra such that D*(L) tO. The structures of such Lie algebras are described in [4] as follows : There exists an abelian ideal U of L such that
(S) L=UOS, where S is simple of dimension 3;
(Nl) L = U+H where H is the Lie algebra generated by x, y, z with multiplica-
tion [x, y]=z, [x,2], [y, 2] E 4(L);
(N2) L=U(DH, His the Lie algebra with a basis { vi, v2, v3, x, y, z} and a mul- tiplication table for this basis
[x, y] =z, [y, z] = rv2, [z, x] == rvl [X, V3]=Vl, [Y, V3]=V2
for some rE k, the product is zero if it is not in the table;
(N3) L= U-i- ÅqiÅr, 6 is a nilpotent linear operator on U;
(Sl) L == UOH, H is the 2dimensional nonabelian Lie algebra ; (S2) L == U+Åq6 År, 6 is a nonnilpotent linear operator on U.
}
We note that the Lie algebra L of the types (Nl), (N2) are nilpotent and those of the types (Sl), (S2) are nonnilpotent solvable. The readers might find a Lie algebra of [4, Theorem 5.1 (2)] ruled out in our list above. This is the Lie algebra H with a basis {vi, v2, v3, x, y} and multiplication [x, y]=x, [vi,x]=v2, [vi, y]=
v3. While preparing this manuscript, we found that D*(H)=O.
LEMMA 1.1. ([4, Lemma 1.1) LetDE D*(L), M, N be subspaces ofL and U an ideal ofL.
(1) If [M, M] 4 O and [M, N] =O then D(N) =O. In particu lar, D(C (L)) == O for any nonabelian Lie alegebra L.
(2) IfL/U is nonabelian thenD(U) g U.
(3) lfU is abelian then forany u, vE U, xE L, [u, x]D (v)= [v, x]D (u).
2. Nonsolvable case (Type (S) in gl)
Let L be a nonsolvable Lie algebra such that D*(L) 7E O. In this case we have a little to do. Since L is of the type (S) in gl, we write L=U(DS, where S is split
" simple of dim 3 with a basis {x, y, h} and multiplication [x, y] =h, [x, h] =2x, [y, h]='2y. Since D(S)gS, D(U) =O by [4, Lemma 2.1] and Lemma 1.1 (1), we can find an sES such that D=adLs. Conversely it is easy to see that JD(t, y,
• h) ==Ofor anyD=:adLs (sES). Hence we have
THEoREM 2.1. IfL is a nonsolvable Lie algebra such that D*(L) 7EO, then D"(L)
=adLL.
3. Split extended case (Types (N3), (Sl), (S2) in gl)
In this section let L be a nonabelian split extension U+ Åq6 År of an abelian ideal
U of L by one dimensional subalgebra Åq6År. Since 6E Der(U) =Endk(U), 6 can be of the form 6(u)= [u, 6] for uG U. Note that 6 is not a zero linear operator on
U, otherwise L would be abelian.
Let us begin with showing
THEoREM 3.1. LetL be a nonabelian split ez;tension U-i- Åq6 År and DE D*(L).
Ifrank 6 2 2 then
D1u=p6 forsomePEk andD(6) E U.
PRooF. We choose wi, w2 E U such that [wi,x] and [w2,y] are linearly inde- pendent. We employ Lemma 1.1 (3) two times. Since [wi, 6]D(w2)=[w2, 6]D(wi), we can write D(wi)=P[wi, 6] for somePE k. Hence by [u, a]D(wi)= [wi, 6]D(u) for any uE U, D(u) =P[u, 6]=Pi(u). This shows the first assertion. By the derivation property of D, [D(6), U]=O, this forces D(6) E U. 1
Let us consider the case that rank 6= 1. Taking abasis {ui, "', un} of U rela- tive to which the matrix of 6 is a Jordan normal form, we have the following multi- plication tables for the bases
(A) [ui, 6] =Pui for some nonzeroPE k, [u,, 6]=O for i År- 2, (B) [ui, 6]=u2, [ui, 6] ==O for i -År 2.
THEoREM 3.2. Let L be a nonabelian split extension U+ Åq6 År andD E D*(L).
Assume that rank 6 =1 and 6 is a nonnilpotent linear operator on U. Then D1u=s6
for somesE k andD(a) E U.
PRooF. This is the case (A) given above. For i År- 2, D(ui)=O by [ui, 6]D(ui)
== [ui, i]D(ui)=O. Write D(ui)=Zaiui+aO, D(6)=Zdiui+d6 (ai, a, di, dE k).
By the derivation property of D, a2 ="'=an= a=b=O. Thus D(ui) =aiui and D(6) E U. 1
Now let So, Si, S2 be the "step matrices" of size m Å~2 of the forms
So= oo
ooo
, Si= 1*
ooo
, S3= 10 Olo
.
THEoREM 3.3. Let L be a nonabelian split extension U+ Åq6 År and D E D*(L).
Assume that rank 6 =1 and 6 is a nilpotent linear operator on U. Then there exists an ordered basis {ui', 6', a2t "', un' } forL zvith a multiplication table
[ui', 6']=u2', [ui', 6'] =:O (i År- 2), [ui', uj'] =O for 1 S i, 1' Åq- n,
such that the matrix forD relatiwe to this ordered basis is one offollozving forms :
ooooO
pos, o '
oo10O
ops, o
'
qOo -q O
ooSi o
(O S i Åq- 2), zuhere Si 's, are the step matrices ofsize (n-2) Å~2 given just above and p, qE k zvith q; O.
PRooF. In this case, we choose a basis {ui, 6, u2, "', un} for L with the multi- plication table (B) given before: [ui, 6]=u2, [ut, 6]=O for i -År 2, [ui, uj] ==O for 1 S i, ]' .Åq n. Here, a matrix of linear operator on L is represented in this ordered basis.
We first consider a Lie automorphism of L whose matrix F is of the following form:
F=
A 8 o
a3i a32 1 O'''O
o
O :• F,
o
' F-i =
A-i 8 o
b-a 1 O-•O
o
O i F,-i
o
where A = [Zll Zl: ] , det A= 1, a =det [ aa 3'1 aa ;:] , b = det [ Zil aa!; ] and Fi is an
invertible matrix of size n-2.
As in the proof of Theorem 3.2, D(ui)=O for i År- 2. Write D(ui)=aiui+cr6+
' cr2u2+wi,D(6)=Biui+B6+B2u2+w2 (wiE U). By the derivation property, cri+B
=O Hence the matrix D for our derivation D is of the form
•
D=
cr:D!IBI2 OOo O, where Di -= (a.i -B.',l,
and D2 is a matrix of size (n-2) Å~2. Aquick calculation shows
F-iDF =
A-iD,A O
' (a3i, a32)A-'DiA+(cr2, B2)A O • • • oF,'iD,A O
We know that there exists a nonsingular matrix Fi such that Fi-iD2A turns one of the step matrices So, Si, S2.
We now take a nonsingular matrix such that A-iDiA is the Jordan normal form for Di, if necessary, devided by det A, so that A-'DiA is one of the following forms :
(a) [g81• (b) (?8]• (c) [g P,]
q#O, because the eigenvalues for Di equal Å} cri2+crBi. Let us express the form of the matrix F"DF for each case. Let B=[bi, b2] be a 1Å~2 matrix such that bi =
(3, i) -entry of F-'DF for i =1, 2.
(a) : Di ==O and any 2Å~2 matarix of determinant 1 is chosen as A. Since B=
[cr2, B2]A, we can take A satisfying B=[P, O] for somePE k. Hence F-'DF is the first form in the statement of this theorem, by our choice of A.
(b) : Since B = -[a32, O] +[cr2, B2]A, we chose a32 so that B vanishes. By this choice of A, F-iDF is dressed in the second form.
(c) : In this case, B=[-qa3i, qa32] +[cr2, B2]A, we choose a3i, a32 such that B vanishes, thereby we are in the last, form. 1
We note that the Lie algebra of the type (Sl) is one of the Lie algebras discus- sed in Theorem 3.2.
4. Nilpotent case (Types (Nl), (N2) in g1)
4. 1. Type (Nl). In this subsection, let L be the split extension U+Hof an abelian ideal U of L and a subalgebra H of L generated by x, y, z with multiplication [x, y]
==2, vi :[x, z], v2=[y, z] supplied with vi, v2 E Åq(L).
If vi = v2 = O, th en L= Åq U, y, zÅr + Åqx År. So L is in the form of Theorem 3.2. If vi and v2 are linearly dependent, we may write vitO, v2=Pvi for someP
E k. Replacing yby y' =ymPx, we have [x, y']=z, [x, z]= vi, [y', z]=O. Regarded as L=Åq U, u', vi, zÅr+ÅqxÅr, L is already discussed in Theorem 3.1.
It remains to prove the following
THEoREM 4.1. Let L be the Lie algebra given in the first paragrmph in this sub- section, andDED*(L). Assume that vi and v2 are linearly independent. Then D(U)=O, D(H)gH and there exists an ordered basis {x', y'i z', vi', vi} forH with a multiplication table
[x ', y' ] == z', [ar ', z' ] = vl', [y', 2t ] = v2',
the product is zero zf not in the table, such that the matrix of DiH relative to this basis is one of the following forms :
ooooo ooooo ooooosoooo osooo
'
oo ooo oo ooo kO OOO oo ooo
OO -k OO
.
zvhere, s, k E k.
PRooF. We shall first express the form of D, so that we will see D(H) g H.
Let M be the subspace spanned by x, y, z and put C=C(L). Then L=C+M.
Write D(w) == wM+wc (wM E M, wc E C) for wE M. Since JD(x, y, 2) is equivalent to 22M (mod CL), zzc+xMv2-yMvi (mod (MM+CC)), xcv2-ycvi(mod ML), we have
the three equations zzM = O, zzc = yMvi - JvM v2, xcv2 = yc vi. Observing that vi and v2 are linearly independent, we can write
D (x) = c.2+svi, D (y) = c,2+ sv2, D (z) = c,vi - c.v2 for some cx, cy, sE k.
It is enough for us to work on H by the paragraph above. We represent a ma- trix for every linear operator in the ordered basis {x, y, z, vi, v2} for H.
We take an automorphism of H whose matrix F is of the form
F= a31 a32 1 OO
O Z A
' F-i =
A-i O
, O
b-a 1 OO
O -a32 A-1
a31
whereA == [aa,'l aa;i] with det A=:1, [Z]=A[.Zli]. Then F-'DF is of the form
.
,
(*)
oo ooo oo ooo pq ooo
eO qOO oe-poo
,
where [P, q] ==[cx, cy]A and e=s-a32P+a3iq. (We label this matrix "(*)" to use this for the next subsection.) If [cx, cy] 4[O, O], we choose A such taht [cx, cg]A=
[k, O], if necessary devided by det A, and put a32= s/k to kill e. In case that [cx, cy]
= [O, O], we haveP =q= O, and e =s. 1
4. 2. Type (N2). In this subsection, let Lbe the Lie algebra U(l)Hof an abel- ian ideal Uof L and an ideal H of L such that Hhas a basis {x, u, z, vi, v2, v3} for
which the multiplication table is that of the type (N2) in gl. If rtO, we replace v3 by z+rv3 GC(L) and rvi, rv2, by vi, v2 respectively. Then L is decomposed to (U(D
Åqv3År)(DÅqx, y, 2, vi, v2År, thereby we are in the same situation of Theorem 4.1, so nothing to do for this case.
Suppose that r = O, so that our muliplication table is as follow
[X, Y] =Z, [Y, Z]=O, [2, X]=O, [X, V3]=Vl, [Y, V3]=V2.
Let D E D*(L). Similarly as in the proof of Theorem 4.1, by ID (x, y, z)=ID(v3, x, y)
=O, we can write
D (x) = cx2+ svi, D (y) = cyz+zv2,
D(z)=O, D(v3)=c,vi-c.v,
for some cx, cu,sEk. Hence D(H)gH. We present a matrix for every linear
operator in the ordered basis {x, y, v3, z, vi, v2} for H.
We take an automorphism of H whose matrix F is of the form
F=
A 88 o
a3i a32 1 O OO
OO O1 OO
o 8ZA
, F-i=
ooA-i
oo O
b-a 1o oo
OO O1 OO
O -a32
O o a,, A-i
where A, a and b are as in the proof of Theorem 4.1. Then F-'DF is of the form
o •- o
:
G' o
where G is the same form labeled "(*)" in the proof of Theorem 4.1. Thus we have
THEoREM 4.2. Let L be the Lie algebra UeH of an abelian ideal U ofL and an idealH ofL such that H has a basis {x, y, z, vi, v2, v3} for zvhich the multiPlica- tion table
[x, y] =z, [y, z]=O, [z, x]=O, [X, V3]=Vl, [Y, V3]=V2.
LetDED*(L). Then D(U)=O, D(H) gH and there exists an automorphism' of
H such that the matrix for DIH relative to the ordered basis {x', y', z', vi', v2', v3'} is
one of the follozving forms :
oosO
osO
oo' oo o
O -k
zvhere s, k E k.
5. Multiplication tables of (S(L), Åq , År)
Finally we shall list the typical types of Lie algebras L such that D*(L) 7E O, their bases (1), multiplication tables (2) with respect to the bracket [,] for the bases
(1), the forms (3) of DE D*(L) and the multiplication tables (4) for the deformed Lie algebras (S (L), Åq , År).
SL (2, k) (Theorem 2. 1) : (1) {x, y, z}
(2) [x, y]=h, [x, h]=2x, [y, h]=-2y (3) D=adLw for some wE SL(2, k) Åqf, gÅr =[f, g]+[f, w]g' [g, w]f(4)
for f, gE SL(2, k)
Split extended case1 (Theorem 3.1):
Put U=Åqui, "', unÅr.
(1) {ui, "', un, 6}
(2) [U, U]=O,rankOÅr-2
i (3) Dlu=sOforsomesEk,D(6)=vEU.
(4)
Åqui, ujÅr=s([ui, 6]uj-[uj, 6]ui), Åqui, 6År =(1+s6) [ui, CS]-uiv
. for any i.
Split extended case 2 (Theorem 3.2) : (1) {u,,u,,O}
(2) [ui, 6]=Pui for some nonzeroPE k, [u2, 6] =O.
(3) D(ui) ==Psui for some sE k, D(u2)=O, D(6)=vE U.
(4)
Åqui, 6År= ip+Ps6-v)u,, Åqu,, SÅr=-u,v, Åqul, u,År=Psuiu2.
Split extended case 3 (Theorem 3.3) : (1) {u,, u,, O}
(2) [u,, 6]=u,, [u,, 6]=O, [u,, u,]=O
(3) The matrix of D relative to the ordered basis {ui, 6, u2} is one of the follow- ing forms:
D,-
l:• g 081, D,== [g• ;• g•], D,-[i -Z• :Ol,
where q 40.
(4)
Åq,År ÅqUl,U2År
D, PUIU2
D, u,S
D, qulu2
Åqul,u2År Åqul,aÅr Åqu2,iÅr
Puiu, u,+Pu,6 O
u2S u,-Puiu2+62 -pu22
quiu2 u2+2qui6 qu26
Nilpotent case1 (Theorem 4.1):
(1) {x, y, z, v,, v,}
(2) [X, y]==Z, [X, 2]=Vi, [Y, 2]=V2 (3) D, : D, (x) = sv,, D, (y) =sv,
D, : D,(x)=s2, D,(z) =-sv, (s E k)
(4)
D,: z
Åq
'
År r y z Vl
x o 2+S(ViY-V2X) (1+s2)v,
y o (1+s2)v, SVIV2
z o o
Vl o
V2 o
D,: 2
Åq
' År x y z Vl
,zr o (1+sy)z v,+s(z2+xv,) SVI
y o (1+sy)v, o
2 o -sv
Vl o
V2 o
Nilpotent case 2 (Theorem 4.2) (1) {x, y, 2, Vi, V2, V3}
(2) [x, y]=Z, [X, Z]==O, [Y, 2]==O, [X, V3]=Vi, [Y, V3]=V2
(3) D, : D, (x) =sv,, D, (y)=sv, D, : D, (x) == sz, D, (v,) = -sv,
(4)
Åq,ÅrX Y Z VI V2 V3
x u z
Di: vi
V2 V3 Åq,År x y z
D2: vi
V2 V3
O z+s(v,y-v,x) sv,2 sv,2 sv,v, (1+sv,)v,
O sv2 sviV2 SV2 (1+SV3)V2
ooo o
oo o
oo
o
O (1+sy)z sz2 sv,z sv2z v,+S(V2X+V3Z)
O O O O (1+sy)v,
0 O O SV2
O O SVIV2 O sv22
o
References
[1] R. K Amayo and I. N. Stewart, Infinite-dimensional Lie algebras, Noordhoff, Leyden, 1974.
[ 2 ] K. Bhaskara and K. Vismanath, Poisson algebras and Poisson manifolds, Pittman, Lon- don, 1988.
[ 3 ] E Kubo and F. Mimura, Extensions of Poisson algebras by derivations, Hiroshima Math.
J. 20 (1990), 37-46.
, [4] E Kubo, An identity on symmetric algebras of Lie algebras, Nonassociative algebras and related topics, (Hiroshima, 1990) , 129-137, World Sci. Publishing, River Edge, NJ, 1991.
[5] A, Nijenhuis and R. W. Richardson, Cohomology and deformations in graded Lie alge- bras, Bull. Amer. Math. Soc. 72 (1966), 1-29.
s
Department of Mathematics KNushu Institute of Technology
