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Example text

12) is also associative. 4), (1,e)(A,g) = (A~z(e,g),g)= ( A , g ) = (A,g)(1, e). 13) This shows that (1,e) is the unit element of Gex. We leave it to the reader show that the inverse of an element (~, g) is given by , ~o(g,g_~),g . 14). Verify also that the subset of G~x given by (C*,e) := {()~,e) I A E C*) is an abelian subgroup of Gex. 15) Projective representations and central extensions 47 From the considerations given above follows that G~x is indeed a group. We now show that this group has a non-trivial center.

I) otr = td o ~ and A' o (I) = id o A. 8). 1" Commuting diagram of a universal central extension. 46). In this expression we use the notion of the residue of a Laurent polynomial P(A), denoted ResP(A). Although this may be well known we give its definition. e. M P(A) -- E l-N alAt (at e C; M , N E Z ; M > N). 42), | | y) _ y). 53) A moment of reflection shows that g(fl+m,o is precisely the residue of (d)~l/d)~) Am. Indeed, Res (\ ddAAl ~] Am - Res(eA l+m-1) -eSl+m,o. 54) Using this result one easily verifies the following lemma.

9) gives r ~) - K (d(~), y). 4)of the 2-cocycle r and the properties of the Cartan-Killing form K (in particular its associativity) we show that the linear map d is a derivation of M. To this end we consider K(d([~, V]), z) - r = r [y, z]) + r V], z) - -r z], ~) - r ~], V) [z, x]) - K(d(x), [y, z]) + K(d(y), [z, x]) = K([d(~), y], z) + K([~, d(y)], z) = K([d(x), y] + Ix, d(y)], z) (x, y, z E M). 13) Invoking the fact that K is non-degenerate on M we obtain d([x, y]) = [d(x), y] + [x, d(y)] (x, y E M).

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Algebra of PD operators with constant analytic symbols


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