504 ARCH. MATH.

Finitely presented infinite-dimensional simple Lie algebras

By

IFAi~ STEWART

A Lie algebra (or a group) is finitely presented if it can be defined by a finite set of generators subject to finitely many relations. Higman [5] has recently constructed finitely presented infinite simple groups. Here we show that some Lie algebras which have already occurred in several interesting situations afford analogous examples for Lie algebras over fields of characteristic zero.

Consider the Lie algebra W with basis {w-l, w0, wl, w2 . . . . } and multiplication

(1) [w~w~] = (i -- 1) w~+t .

This plays a role in .4mayo's investigation [1] of Lie algebras with subalgebras of codimension 1, where it is shown to be simple (as is easily verified). I t is a subalgebra of the generalized Wit t a l g e b r a ~ Z of [3], p. 206. In [2] Amayo proves that it satisfies the maximal condition for subalgebras. I t is the only known instance (apart from trivial variations) of an infiuite-dimensional Lie algebra with this property. Further, it is closely related to the infinite-dimensional Lie algebras of Cartan type (Guillemin [4]).

In what follows we shall write

[u, nv] ---- [... [[u v] v]. . . v] (n factors v)

for elements u, v of a Lie algebra.

Theorem 1. Over any field o] characteristic zero the inlinite.dimensional simple Lie algebra W has a finite presentation by generators {x, y} subject to the relations:

(2) [y, ax] ---- 6x ,

(3) [[xy] [y, ~x]] -- 6 [xy],

(4) [[y, 2x] y] = -- 12 y ,

(5) [y, 3[x y]] = -- 54 [x, 8Y], "

(6) [y, 5[xy]] = 60 [[y, a[xy]] y].

Proof . We put the relations in more suggestive form. Define x-1 ---- x, x0 = ~ [y, 2x], xx = -- �89 [x y], x2 = y, and inductively

Vol. XXVI, 1975 Finitely presented simple Lie algebras 505

1 (7) X~+l - - i - - 1 [x~xi] (i ~ 2) .

Then the relations (2)- - (6) , toge ther wi th the definitions of x0 and xi , t ake the form

[ X - - 1 X 0 ] = - - x - i , [ x - i x i ] = - 2 x 0 ,

[ x - i x ~ ] = - 3 x i ,

(S) Ix0 x i ] = - x i ,

[xo x2] = - 2 x 2 ,

Ix2 x3] = - x s ,

Ix2 x s ] = - 3 xT.

These are all relat ions which hold in W, if w's are subs t i tu ted for x's. Our content ion will be t h a t the remaining relat ions

(9) [x~xl] = (i -- 2.) xi+j

follow f rom (8). We prove this by induction.

(a) ] ---- 0: We know (9) for i < 2. For i > 2 we have

1 [x~+i xo] - i - 1 [[x~ x i ] x0] =

1 - - i - 1 ( [ [z~x0]Xl ] + [ x d x i x o ] ] ) =

= (i + 1) xi+i

using induct ion on i and (7). Hence (9) holds for ] = 0.

(b) j = 1 : The definition (7) implies (9).

(c) 2.---- - - 1 : Again we know the result for i--_<2. For i > 2 we have

1 [z~+ix-i] - i - 1 [[z~ xi] z-i] =

1 ---- i - - i ([[X~X-1] Xl] J r [x~[xix-1]]) :

= (i + 2) x~

using (7) and p a r t (a). (d) The final case, where i, 2" > 2, we prove b y induct ion on i A- 2.- I f n = i § ] =< 5

the resul t is t rue b y (8). I f i => 3 we have

1 [ x ~ + i x r i - 1 [ [x~xi ] zr =

1 (10) - i - 1 ( [ [z~zj] x i ] + [ x ~ [ x i x j ] ] ) =

( i - - j ) ( i + i - - i) 1 - - J - - i - - 1 x i + j + i q- ~ [x~ x / + i ] �9

506 I. STEWART ARCH. MATH.

Hence if we can determine [Xn-lX2], all other [~i+lXj] for which i -{- 1 ~- j---- n-l- 1 are uniquely determined, using (10) and induction on ].

Since n :> 5 we have

(11) (n -- 7) [xn-lx2] = [[xn-4x3] x2] ---- = [ [zn-4 x2] z3] + [xn-4 [x3 z2]] = ---- (n -- 6) [Xn-2 xs] + [xn-4 xs].

From (10) we also have

(12) (n - - 3) [xn-zx2] ~- (n - - 1) (n -- 4)xn+l - - [xn-2xa],

(13) (n -- 4) [xn-2x3] ~-- (n - - 1) (n -- 6) xn+l - - 2[xn-ax4] ,

(14) (n -- 5) [Xn-3 x4] = (n - - 1) (n -- 8)X,+l -- 3 [xn-4x5].

We can regard (11)-- (14) as a system of equations for the elements [xn-4 xs], [xn-s x4], [xn-2x3], [xn-lx2]. The determinant of the system is -- (n - -1 ) (n~-1) (n - -6 ) which is not zero unless n---- 6 (since we know n _~ 5). However, when n ---- 6 we already have [xs, x~] --~ 3x7 from (8). Hence for all n ~ 5 we obtain, either by solving (11)--(14) or appealing to (8),

[x~ - i x2] = (n - 3) z ~ + l .

Using induction on ] and (10) it now follows that (9) holds for i ~ ] = n ~ 1, and the induction step is complete. Now W satisfies the relations (9) on substituting w's for x's, and is generated by w-1 and w2. Hence we have a finite presentation for W. This proves the theorem.

The "generalized Wit t algebras" of [3], p. 206 are defined over any field ~ as follows. Let G be a subgroup of the additive group l+, take a vector space over ~ with basis {wg : g e G}, and define

[wg w~] ---- (g - - h) wg+~ (g, h e G) .

The resulting Lie algebra $4/'a is simple (or 1-dimensional) except when ~ has char- acteristic 2, in which case r 2 is simple. I t is easy to see t h a t ~ a is finitely generated if and only ff G is a finitely generated abelian group. For ~ of prime characteristic this implies $~a finite-dimensional. An argument similar to, and in part based upon, tha t above, will show tha t for ~ of characteristic zero r is finitely presented ff and only ff G is finitely generated. Thus we obtain families of finitely presented infinite- dimensional simple Lie algebras.

I t is natural to ask how large these families are, up to isomorphism. I t is shown in [3], p. 211, t ha t if G and H are subgroups of ~+ then ~ is isomorphic to Sf'A ff and only if G and H are pro]ectively equivalent: that is, there exists 0 ~= ~ e ~ such that

I f ~ contains an element 0 such that 1, 0, and 03 are linearly independent over Q, it is easy to check that the additive subgroups generated by {1, ~0} and {1, riO} for 0 ~= ~, fl e Q are projeetively equivalent ff and only if :r • ft. Hence we obtain (at least) a countably infinite family. Otherwise ~ is either Q or a quadratic extension Q (VD) where D is a squarefree integer. I f ~ = ~ there is a single projective equiv-

Vol. XXVI, 1975 Finitely presented simple Lie algebras 507

alence class of nontrivial finitely generated additive subgroups, because Q+ is locally cyclic, and r acts transitively by multiplication. In the case t = r the question reduces to a purely group-theoretical one. Let U be the group of unimodular 2 • 2 integer matrices, and R the group of matrices

for 0 4 ~ q ~ . Then there is a bijeetion between the set of double cosets Ux_R in GL2 (II~) and the set of projective equivalence classes of 2-dimensional lattices in II~ ( ~ ) . Since all lattices in iI~ (VD) have dimension ~ 2, and those of dimension 1 are pro- jectively equivalent, we obtain an infinite family in this ease if and only if the set of double cosets is infinite. This seems likely, but we are unable to verify it. What we have proved may be stated as:

Theorem 2. Over every field of characteristic zero, other than Q or a quadratic extension o/ 4, there exists at least a countable infinity of non-isomorphic finitely presented infinite-dimensional simple Lie algebras.

I t should also be noted that the algebra W is not isomorphic to any r For the results of [3], p. 209, show tha t W has precisely one 3-dimensional subalgebra, namely <w-l, w0, wl>, whereas 3r has infinitely many of the form <w_a, w0, wg> provided I GI # 1.

Acknowledgement. This work was supported by a Forschungsstipendinm from the Alexander yon Humboldt-Stiftung, Bonn-Bad Godesberg, W. Germany.

References

[1] R. K. A~AYO, Quasi-ideals of Lie algebras II. Preprint, University of Bonn 1974. [2] R. K. A~c~Yo, Lie algebras with max. Preprint, University of Bonn 1974. [3] R. K. _h.~ro and I. N. ST]~WA:aT, Infinite-dimensional Lie algebras. Leyden 1974. [4] V. Gvr~.T.v~nr Infinite-dimensional primitive Lie algebras. J. Differential Geometry 4,

257--282 (1970). [5] G. H I G ~ , (unpublished).

Eingegangen am 3. 11. 1974

Anschrift des Autors:

Ian Stewart Mathematics Institute University of Warwick Coventry England