Appendix B The Basics of Lie Algebra, Lie Groups, and Their Representations

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Z , the connected nilpotent subgroups generated by {E | } The interest in these subgroups is that almost all elements of G c admit a Gauss decomposition: g = z + hz = b + z = z + b , z Z , h Hc , b B (B21)

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It follows that the quotients X + = G c /B and X = B + \G c are compact complex homogeneous manifolds, on which G c acts by holomorphic transformations

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B31 Extensions of Lie algebras and Lie groups

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It is useful to have a method for constructing a group G from two smaller ones, one of them at least becoming a closed subgroup of G Several possibilities are available Here, we describe the two simplest ones

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(1) Direct product

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This is the most trivial solution, which consists in glueing the two groups together, without interaction Given two (topological or Lie) groups G 1 , G 2 , their direct product G = G 1 ~ G 2 is simply their Cartesian product, endowed with the group law: ( g 1 , g 2 )( g 1 , g 2 ) = ( g 1 g 1 , g 2 g 2 ) , g 1 , g 1 G 1 , g 2, g 2 G 2 (B22)

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With the obvious identi cations g 1 ~ ( g 1 , e 2 ), g 2 ~ (e 1 , g 2 ), where e j denotes the neutral element of G j , j = 1, 2, it is clear that both G 1 and G 2 are invariant subgroups of G 1 ~ G 2 In the case of Lie groups, the notion of direct product corresponds to that of direct sum of the corresponding Lie algebras, g = g1 g2 , and again both g1 and g2 are ideals of g

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(2) Semidirect product

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A more interesting construction arises when one of the groups, say, G 2 , acts on the other one, G 1 , by automorphisms More precisely, there is given a homomorphism from G 2 into the group Aut G 1 of automorphisms of G 1 Although the general de nition may be given as in the rst case, we consider only the case where G 1 = V is Abelian, in fact a vector space (hence group operations are noted additively), and G 2 = S is a subgroup of Aut V Then we de ne the semidirect product G = V S as the Cartesian product, endowed with the group law: (v , s)(v , s ) = (v + s (v ), ss ) , v , v V , s, s S (B23)

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The law (B23) entails that the neutral element of G is (0, e S ) and the inverse of (v , s) is (v , s) 1 = ( 1 (v ), s 1 ) = ( s 1 (v ), s 1 ) It is easy to check that V is an invariant s subgroup of G, while S is not in general As a matter of fact, S is invariant if and

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B3 Lie Groups

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only if the automorphism is trivial, that is, the product is direct Indeed one has readily (v , s)(v , e S )(v , s) 1 = ( s (v ) , e S ) V , and (v , s)(0, s )(v , s) 1 = (v , ss )( 1 (v ), s 1 ) = (v ss s 1 (v ), ss s 1 ) s In addition to the Weyl Heisenberg group G WH = R R2 that we discussed in 3, the following groups are examples of semidirect products of this type are: The Euclidean group E (n) = Rn The Poincar group the Lorentz group;

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P+ (1, 3)

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SO(n);

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