By Nomizu K., Sasaki T.
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Extra info for Affine differential geometry. Geometry of affine immersions
Let p be a finite-dimensional representation of g. A symmetric bilinear form B. on g defined by B,(x,y) = tr(p(x)p(y)), is called associated with p. This bilinear form is invariant. The bilinear form associated with the adjoint representation of g is called the Killing form of the Lie algebra g. We will denote the Killing form by K. If Ij is an ideal of g, then the Killing form of h coincides with the restriction of the Killing form of g to f . If a Lie algebra g can be endowed with a nondegenerate bilinear form then the adjoint and coadjoint representations of g are equivalent.
Similarly, for an arbitrary Lie algebra g we define the adjoint representation ad : g --+ 91(g) with the help of the relation ad(x)y - (x, y). Note that the operators ad(x), x E g are, in fact, the inner derivations of g. The adjoint representation equips a Lie algebra g with the structure of a g-module. It is this structure of a g-module that we will have in mind in saying that we consider a Lie algebra g to be a g-module. A Lie algebra g is called reductive if it is a semisimple g-module. Let V be a module over a Lie algebra g.
It is clear that JJ _ -1; hence, JJ is a complex structure called the canonical complex structure on V V. The corresponding complex linear space is called the complexification of the real vector space V and is denoted by Vc. The initial real vector space V can be identified with the subset of Vc, formed by the vectors of the form 22 Introductory data on Lie algebras (v, 0). Note that (v, 0) = J0(v, 0) = (0, v); hence, an arbitrary vector w of Vc can be represented as w=v+u, where v, u E V; and such a representation is unique.
Affine differential geometry. Geometry of affine immersions by Nomizu K., Sasaki T.