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V is non-degenerate for some (and hence for any) injective f . If G is semisimple then its center is finite and the adjoint representation Ad : G -+ GL(g) gives an embedding ad : g c+ gI(g). The image of g in gI(g) is a Lie algebra of derivations D E gI(g) such that The corresponding standard invariant form is called the Killing form, The Killing form is non-degenerate if and only if g is semisimple. 22 2 Actions with Finitely Many Orbits Algebraic Tori A connected algebraic group is called an algebraic torus if all its elements are semisimple.

X * is a hypersurface and Ax is the determinant a symmetric matrix. 15 The short grading of E6 gives the following action. G = C* x Spinlo and 11 is the half-spinor representation. Except for the zero orbit and 44 2 Actions with Finitely Many Orbits the dense orbit there is only one orbit O. In particular, the projectivization of O is smooth and self-dual. It is a spinor variety S5. It could also be described via Cayley numbers. Let Ca be the algebra of split Cayley numbers (therefore @a= 0@ C, where O is the real division algebra of octonions).

12 Consider the short grading of I = so,+z that corresponds to ,6 = a l . Then G = C* x SO, and ll = Cn with the simplest action. There are two non-zero orbits: the dense one and the self-dual quadric hypersurface Q c Cn preserved by G. 13 Consider the short grading of I = 502, = D, that corresponds to ,6 = a, or ,6 = an-1. Then G = C* x SL, acts naturally on 11 = A2Cn. There are r = [n/2] non-zero orbits Oi, i = 1,. . ,r. Oi is the variety of skewsymmetric matrices of rank 2r. The projectivization of O1 is identified with X = Gr(2, Cn) in the Pliicker embedding.