Pairs of semisimple Lie algebras and their maximal reductive subalgebras (CROSBI ID 140828)
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Podaci o odgovornosti
Širola, Boris
engleski
Pairs of semisimple Lie algebras and their maximal reductive subalgebras
Let g be a semisimple Lie algebra over a field K, char(K)=0, and g_1 a subalgebra reductive in g. Suppose that the restriction of the Killing form B of g to g_1xg_1 is nondegenerate. Consider the following statements: (1) For any Cartan subalgebra h_1 of g_1 there is a unique Cartan subalgebra h of g containing h_1 ; (2) g_1 is self-normalizing in g ; (3) The B-orthogonal p of g_1 in g is simple as a g_1-module for the adjoint representation. We give some answers to this natural question: For which pairs (g, g_1) do (1), (2) or (3) hold? We also study how p in general decomposes as a g_1-module, and when g_1 is a maximal subalgebra of g. In particular suppose (g, sigma) is a pair with g as above and sigma its automorphism of order m. Assume that K contains a primitive m-th root of unity. Define g_1:=g^sigma, the fixed point algebra for sigma. We prove the following generalization of a well known result for symmetric Lie algebras, i.e., for m=2: (a) (g, g_1) satisfies (1) ; (b) For m prime, (g, g_1) satisfies (2).
Pair of Lie algebras; Semisimple Lie algebra; Reductive subalgebra; Cartan subalgebra; Self-normalizing subalgebra; Finite-order automorphism; Fixed point algebra
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Podaci o izdanju
11 (3)
2008.
233-250
objavljeno
1386-923X
10.1007/s10468-007-9068-z