2-step Nilpotent Lie Groups Arising from Semisimple Modules
نویسنده
چکیده
Let G0 denote a compact semisimple Lie algebra and U a finite dimensional real G0 module. The vector space N0 = U ⊕ G0 admits a canonical 2-step nilpotent Lie algebra structure with [N0,N0] = G0 and an inner product 〈, 〉, unique up to scaling, for which the elements of G0 are skew symmetric derivations of N0. Let N0 denote the corresponding simply connected 2-step nilpotent Lie group with Lie algebra N0, and let 〈, 〉 also denote the left invariant metric on N0 determined by the inner product 〈, 〉 on N0. In this article we investigate the basic differential geometric properties of N0 by using elementary representation theory to study the complexification N = N0 = V ⊕G, where V = U and G = G0 . The weight space decomposition for V and its real analogue for U describe the bracket structures for N = N0 and N0. The Weyl group W of G acts on the real Lie algebra N0 by automorphisms and isometries. The Lie algebra N0 admits a Chevalley rational structure for which the the weight spaces of U are rational. We use the roots of G and the weights of V to construct totally geodesic, rational subalgebras of N0 = U ⊕ G0. Mathematics 2000 Subject Classification Primary : 53C30 Secondary : 22E25, 22E46
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