By Martin Schottenloher (auth.)
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Additional info for A Mathematical Introduction to Conformal Field Theory: Based on a Series of Lectures given at the Mathematisches Institut der Universität Hamburg
The homomorphism fl is called a splitting map. A central extension which splits is called a trivial extension, since it is equivalent to the exact sequence of Lie-algebra homomorphisms 0 >a ~a@g ~g >0. (Equivalence is defined in analogy to the group case, cf. the discussion after 3. 1 49 Central Extensions and Equivalence If, in the preceding examples, the sequence of Lie groups splits in the sense of Sect. 3 (with a differentiable homomorphism S : G ---. 2 with splitting map S. In general, the reverse implication holds for connected and simply connected Lie groups G only.
This implies [[vl(P)gj - ulgjll < IIZ~(u)ugj - z ~ ( u ) v i g j l l + I I @ ( U ) - 1)Vlgjll < 6 ~+~ E for j = 1 , . . e. vf(P) 6 B. Hence, the image vf(D) of the neighborhood D of V1 is contained in B. 2 Bargmann 's Theorem 55 is a differentiable principal fibre bundle. The difficulty in defining a Lie-group structure on the unitary group lies in the fact that the corresponding Lie algebra should contain the (bounded and unbounded) self-adjoint operators on ]HI. E is by construction the fibre product of ~ and T.
Note that ~" U(H) ---. Aut(IP) is a homomorphism of groups. 2 (Wigner [Wig31], Chap. 20, Appendix). For every transformation T E Aut(]P) there is a unitary or an anti-unitary operator U with T = ~(U). The elementary proof of Wigner has been simplified by Bargmann [Bar64]. Let U(IP) "= ~ (U(EI)) C Aut(lP). Then U(IP) is a subgroup of Aut(IP). 3 The sequence A 1 ~U(1) ~, U(H) ~,U(P) ,1 with L()~) :-- )rids, )~ e U(1), is an exact sequence of homomorphism and, hence, defines a central extension of U(P) by U(1).