By Martin Schottenloher (auth.)

ISBN-10: 3540617531

ISBN-13: 9783540617532

ISBN-10: 3540706909

ISBN-13: 9783540706908

**Read Online or Download A Mathematical Introduction to Conformal Field Theory: Based on a Series of Lectures given at the Mathematisches Institut der Universität Hamburg PDF**

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**Additional resources for A Mathematical Introduction to Conformal Field Theory: Based on a Series of Lectures given at the Mathematisches Institut der Universität Hamburg**

**Sample text**

7) and T is a continuous homomorphism. 8: For every A E U(P) there is an open neighborhood W C U(IP) and a continuous map u : W ---, U(H) with o u = idw. Let now V "= T-i(W). e. (u o T(g), g) E E for g e V. # is continuous because u and T are continuous. This implies that ¢ " U(1) x V ---, r - l ( v ) C E, is a bijective map (A, g) ~ (Au o T(g), g), with a continuous inverse map ~)-l(v~ g) = ()~(U), g), where A(U) e U(1) for U e ~ - I ( W ) is given by the equation U = A(U)u o ~(U). Hence, the continuity of ¢-1 is a consequence of the continuity of the multiplication U(1) x U(H)---, U(H), (A, U)~-, AU.

7 (Bargmann [Bar54]) Let G be a connected and simply connected, finite-dimensional Lie group with H 2 (Lie G, R) = 0. e. for every continuous homomorphism T : G ---, U(IP) there is a continuous homomorphism S" G ---, U(H) with T = ~ o S. U(1) u(~) T ^ , u(~) ,1 4. Central Extensions of Lie Algebras 52 A Here, E = { (U, g) e U(]HI) x G[ ~(U) = Tg}, r = pr 2 and T = pr:. E is a topological group as a subgroup of the topological group U(]E) x G (cf. 7) and T is a continuous homomorphism. 8: For every A E U(P) there is an open neighborhood W C U(IP) and a continuous map u : W ---, U(H) with o u = idw.

In this case, the sequence of Lie groups splits if and only if the associated sequence of Lie algebras splits. All this follows immediately from the properties stated at the beginning of this section. 3 For every central extension of Lie algebras 0 >a ">~ 7r ~ g >0 there is a linear homomorphism ~ : g ~ [) with lr o ~ = idg. = [9 (x), Z (Y)] - Z (IX, Y]) for x, Y e g. Then ~ is a splitting map if and only if 0 = O. It can easily be checked that the map 0 : g × g --"* a (depending on /3) always has the following three properties: 1 o O: g × 9 ~ a is bilinear and alternating.

### A Mathematical Introduction to Conformal Field Theory: Based on a Series of Lectures given at the Mathematisches Institut der Universität Hamburg by Martin Schottenloher (auth.)

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