By Martin Schottenloher (auth.)

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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

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1 ( B ) C U(][-]I) open. 2 Quantization of Symmetries 41 The strong topology can be defined on any subset M C B(]HI)"= { B" ]HI ---, IHIIB is R-linear and bounded} of JR-linear continuous operators, hence in particular on M~ = { U" ][-]I---, IHI[U unitary or anti-unitary}. ) is induced by ~" M~ Aut(IP) (cf. 2). This topology on Aut(IP) (or U(IP)) is called the strong topology as well. When quantizing a classical symmetry group the following question arises: given a continuous homomorphism T : G ~ U(~*), does there exist a continuous homomorphism S : G ~ U(]HI), such that the following diagram commutes?

E is by construction the fibre product of ~ and T. 8 with general fibre U(1), this must also hold for E --, G. 7, to show that E actually is a Lie group. 9 For every finite-dimensional semi-simple Lie algebra g over IK one can show: H 2 (g, IK) = 0 (cf. [HN91]). As a consequence of the above discussion we thus have the following result which can be applied to the quantization of certain important symmetries: if G is a connected and simply connected Lie group with simple Lie algebra Lie(G) = it, then every continuous representation T • G --, U(IP) has a lift to a unitary representation.

It follows that O (X, Y) = # ([X, Y]). Conversely, if O has this form, it clearly satisfies 1 o and 2 °. The linear map a : g --+ ll = g @ a defined by a ( X ) : = X + #(X), X E g, turns out to be a Lie-algebra homomorphism: o([x,z]) = [X, Y]g + # ([X, Y]) = [X,Y]g+e(X,Y) = = [X + # (X), Y + # (Y)]~ [a(X),a(Z)]~. Hence, a is a splitting map. 5 Alt2(g, a ) : = { O ' g x g ~ a[ t3 satisfies condition 1°} Z 2 (g, a) : = { O e Alt2(g, a ) l e satisfies condition 2 ° } B 2 (g, a) := { e t t x tt --, a[ 3# e HomK (g, a)" O = t5} H2 (it, a) := z ~ ( g .

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