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The COPROMAPH convention sequence has now developed right into a major foreign area the place primary strategies in mathematical and theoretical physics and their physics purposes should be conceived, constructed and disseminated. simple rules for addressing a number of modern difficulties in mathematical and theoretical physics are offered in a nonintimidating surroundings. specialists give you the reader the basics to foretell new probabilities in physics and different fields.

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Examples: the sphere 5 " in Rn+1, a (one- or two-sheeted) hyperboloid in E n + 1 , a torus in ffi3, the Mobius strip in E 3 . 21. A Lie group is a group G which is at the same time a smooth manifold such that the map G x G -» G, (g,h) H-> gh~x is (infinitely often) differentiable. Examples: . Abelian Lie groups: finite dimensional vector spaces (with vector addition) , S 1 (with multiplication of complex numbers) . Matrix groups: SO(2), SO(3), SU(2), SO(l,3), GL(n,E) = GL(E"). Note that, if G is a Lie group and H a closed subgroup of G, then the quotient G/H is a smooth manifold.

It is the set of fixed points of a. The commutation relations of g are the following: [1,6] C 6, (thus 6 is a subalgebra) [e,p] = p, [p,p]Ct. Then the Lie algebra g* = t ©p*, where p* = ip, is another real form of g c , and it is noncompact (this construction is called Weyl's unitary trick). The commutation relations of g* read: [t, 6] C E, [e,p*] = p*. [p*,p*]c-e. (6 is still a subalgebra) 27 Thus classifying the real forms amounts to classify the involutive automorphisms, and again the result was obtained by Cartan.

Then one can define the unitary quasi-regular representation: [UqL(g)f}(x) = J^^f(g-lx),feSj = L2(X,dvL). (19) In these expressions, the square root factor is a Radon-Nikodym derivative, which compensates for the non-invariance of the measure. Examples: . G = SO(3), H = SO(2), X = S2. G = S0 0 (l,3), H = SO(3), X = S0 0 (l,3)/SO(3) = two-sheeted hyperboloid. 3. Two representations, T\ and Ti in the Hilbert spaces f)i,i}2, respectively, are equivalent if there exists an invertible operator S : Sji -> f)2 such that T2(g) = ST1(g)S-\ for a l l 3 € G.