Elements de Mathematique. Algebre. Chapitre 10 by N. Bourbaki

By N. Bourbaki

Les ?‰l?©ments de math?©matique de Nicolas Bourbaki ont pour objet une pr?©sentation rigoureuse, syst?©matique et sans pr?©requis des math?©matiques depuis leurs fondements.

Ce dixi??me chapitre du Livre d Alg??bre, deuxi??me Livre du trait?©, pose les bases du calcul homologique.

Ce quantity est a ?©t?© publi?© en 1980.

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Example text

13) Equivalently, we can describe U(1) as the group of transformations U(1) = {z −→ wz : w, z ∈ C, |w| = 1}. 14) π 2, As already noted, a special case occurs when θ = in which case w = i. 2.

10) In the remainder of this chapter, we set the stage for later developments by discussing the properties of several important unitary groups. 2. 4 that complex multiplication can be interpreted geometrically as a rescaling and a rotation. A pure rotation is therefore obtained by multiplying by a unit complex number. In other words, if |w| = 1, then |wz| = |z|, that is, the length of z is preserved under multiplication by w. What do unit-normed elements w ∈ C look like? 12) for some θ. Thus, which can also be written as U(1) = {w ∈ C : |w| = 1}.

8) and it is straightforward to generalize this construction to higher dimensions. 10) for the m × n matrices whose where we have introduced the notation K elements lie in K; these expressions are often taken as the definitions of the orthogonal and special orthogonal groups, respectively. In the remainder of this chapter, we set the stage for later developments by discussing the properties of several important orthogonal groups. 1. 2 The Geometry of SO(2) From the definition in the previous section, we have SO(2) = {M ∈ R2×2 : M T M = I, det M = 1}.

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