# Download Algebre lineaire by Seymour Lipschutz PDF

By Seymour Lipschutz

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We will ﬁnd that their representation theory has a lot in common with that of the ﬁnite groups, in particular the fact that all irreducible representations are ﬁnite-dimensional and contained in the regular representation (the famous Theorem of Peter and Weyl). But there is an important diﬀerence: the number of equivalence classes of irreducible representations may be inﬁnite. We do not prove all the modiﬁcations in the general theorems when linear representations are specialized to continuos representations, but concentrate in the next chapters on an explicit description of the representation theory for SU(2) resp.

53]). 1 to the central fact that the characters χ1 , . . , χh belonging to a complete set of equivalence classes of irreducible representations of G form an orthonormal basis of the space of class functions H0 . We recall that a function f deﬁned on G is a class function if one has f (g) = f (tgt−1 ) for all g, t ∈ G. 6: Let f be a class function on G and (π, V ) a representation of G. Let πf be the endomorphism of V deﬁned by πf := f (t)π(t). e. πf = λ idV , with χ(t)f (t) = (m/n) < χ, ¯ f >.

B) The map G −→ GL(H), g −→ π(g) is continuous. 10). 5: A representation (π, V ) is bounded iﬀ we have sup g∈G π(g) < ∞. 2 The Continuity Condition 35 In all examples in this section and in most cases of the following sections all these conditions are fulﬁlled. 134. But we will show for some examples how the continuity of a representation comes out. Later we will not examine continuity and leave this to the reader’s diligence. 1: The linear representation π = χk , k ∈ Z of G = SO(2) in V = C given by cos ϑ sin ϑ ), ϑ ∈ R, χk (r(θ)) := eikϑ for all g = r(ϑ) = ( − sin ϑ cos ϑ is obviously continuous: The map (g, z) −→ eikϑ z is continuous in g = r(ϑ) for each z ∈ C or even contiuous in both variables g and z, because the exponential function is (in particular) continuous and an open neighborhood of e ∈ SO(2) is homeomorphic to an open interval in R containing 0.