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B3 Lie Groups
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N 0 of the origin 0 g and an open neighborhood N e of the identity element e of G Each X (t) de nes a one-parameter subgroup of G, with in nitesimal generator X, X (t) = exp(tX ) , and every one-parameter subgroup is obtained in this way If G is a matrix group, the elements X of the Lie algebra are also matrices and the exponential map comes out in terms of matrix exponentials Using this tool, we may sketch the fundamental theorems of Lie as follows: Every Lie group G has a unique Lie algebra g, obtained as the vector space of in nitesimal generators of all one-parameter subgroups, in other words, the tangent space T e G, at the identity element of G Given a Lie algebra g, one may associate with it, by the exponential map X exp X , a unique connected and simply connected Lie group G, with Lie algebra g (G is called the universal covering of G) Any other connected Lie group G with the same Lie algebra g is of the form G = G/D , where D is an invariant discrete subgroup of G Furthermore, a Lie group G is simple, or semisimple, if and only if its Lie algebra g is simple, or semisimple Here again, one may start with a real Lie group G, build its complexi cation G c , and nd all real forms of G c One, and only one, of them is compact (it corresponds, of course, to the compact real form of the Lie algebra) For instance, the complex Lie group SL(2, C) has two real forms, SU (2) and SU (1, 1), the former being compact A Lie group has natural actions on its Lie algebra and its dual These are the adjoint and coadjoint actions, respectively, and may be understood in terms of the exponential map For g G, g g g g 1 de nes a differentiable map from G to itself The derivative of this map at g = e is an invertible linear transformation, Ad g , of T e G (or equivalently, of g) onto itself, giving the adjoint action Thus, for t ( , ), for some > 0, such that exp(tX ) G and X g, Y = d [ g exp(tX ) g 1 ]|t=0 := Ad g (X ) dt (B11)
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is a tangent vector in T e G = g If G is a matrix group, the adjoint action is simply Ad g (X ) = g X g 1 (B12)
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Now considering g Ad g as a function on G with values in End g, its derivative at the identity, g = e, de nes a linear map ad : g End g Thus, if g = exp X , then Ad g = exp(adX ), and it can be veri ed that (adX )(Y ) = [X , Y ] The corresponding coadjoint actions are now obtained by dualization: the coadjoint action Ad# of g G on the dual g , of the Lie algebra, is given by g Ad# (X ); X = X ; Ad g 1 (X ) , g X g , X g, (B13)
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where ; = ; g ,g denotes the dual pairing between g and g Once again, the (negative of the) derivative of the map g Ad# at g = e is a linear transformation g
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Appendix B The Basics of Lie Algebra, Lie Groups, and Their Representations
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ad# : g End g , such that for any X g, (ad# )(X ) is the map (ad# )(X )(X ); Y = X ; (ad)(X )(Y ) , X g , Y g (B14)
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Clearly, Ad# = exp( ad# X ) If we introduce a basis in g and represent Ad g by a mag trix in this basis, then, in terms of the dual basis in g , Ad# is represented by the g transposed inverse of this matrix Under the coadjoint action, the vector space g splits into a union of disjoint coadjoint orbits OX = {Ad# (X )| g G} g (B15)
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According to the Kirillov Souriau Kostant theory (see [226]), each coadjoint orbit carries a natural symplectic structure In addition, is G-invariant This implies in particular that the orbit is of even dimension and carries a natural G-invariant (Liouville) measure Therefore, a coadjoint orbit is a natural candidate for realizing the phase space of a classical system and hence a starting point for a quantization procedure Semisimple Lie groups have several interesting decompositions In the sequel, we present three of them, the Cartan, the Iwasawa, and the Gauss decompositions
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