- HOMOGENEOUS EXPANDING SOLITONS 33
Many noncompact homogeneous spaces admit no Einstein metrics what-
soever. For instance, there is the following result of Milnor [266, Theorem
2.4].
THEOREM 1.46. If the Lie algebra of a Lie group N is nilpotent but not
commutative, then the Ricci curvature of any left-invariant metric on N has
mixed sign.
The Ricci soliton structure equation
(1.64) Re= ->..g ..:...cxg
illustrates the sense in which a Ricci soliton may be regarded as a gener-
alization of an Einstein metric. Therefore, it is natural to look for Ricci
solitons on homogeneous spaces, such as nonabelian nilpotent Lie groups,
that do not admit any Einstein metric. Existence and uniqueness of Ricci
solitons on such spaces has been investigated by Lauret [244].
To describe his results, we must recall some notation from the coho-
mology of a Lie algebra g, relative to its adjoint representation. (See [232]
and [369].) A k-cochain is a skew-symmetric k-linear map g x · · · x g---+ g.
Denote the vector space of all k-cochains by Ck, noting the natural identi-
fications c^0 = g, C^1 = End(g) = g* 0 g, and C^2 = A^2 (g*) 0 g. The Lie
algebra cohomology of g relative to its Chevalley complex,
Hk(g) = ker(8k)/ im(8k-^1 ),
is derived from the coboundary operators 8k : ck ___, ck+l defined by
8k(A)(X1,.. .,xk+I) = '"" Lt (-1)^1 "+k [Xj,A(X1, ... ,Xj, A ... ,xk+l)]
I::;j::;k+I
+
l::;i<j::;k+I
In particular, one has 8o(X)(Y) = -[Y, X] = adx(Y) for all X, Y E g, and
81 (A) (X, Y) = [X, A(Y)] - [Y, A(X)] - A([X, Y])
for all A E End(g) and X, YE g. A derivation of g is an element of ker(81).
Let g be a left-invariant metric on a simply-connected Lie group G.
Regarding the Ricci curvature Re of g as an endomorphism, Lauret considers
the condition
(1.65) 81(Rc)(·, ·) = ->..[-, ·] (>.. E IR),
which is easily seen to be equivalent to
(1.66). Re= ->..I +D (>.. E IR, DE ker(81)).
Notice that equations (1.65) and (1.66) relate the geometry of (G,g) to
algebraic data of g. This turns out to be a productive point of view. We
now survey some of the results it generates, referring the reader to [244] for
the detailed proofs.
