Classifying space for U(n): Difference between revisions

Revision as of 18:45, 26 April 2009

Revision as of 18:46, 26 April 2009

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In [[mathematics]], the '''[[classifying space]] for [[U(n)unitary group|U(''n'')]]''' may be constructed as either

# the [[Grassmannian]] of ''n''-planes in an infinite-dimensional complex [[Hilbert space]]; or,

# the direct limit, with the induced topology, of [[Grassmannian|Grassmannians]] of ''n'' planes.

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The [[total space]] <math>EU(n)</math> of the [[universal bundle]] is given by

:<math>EU(n)=\{e_1,\ldots,e_n : (e_i,e_j)=\delta_{ij}, e_i\in \mathcal{H} \}. \, </math>

Here, ''H'' is an infinite-dimensional complex Hilbert space, the <math>e_i</math> are vectors in ''H'', and <math>\delta_{ij}</math> is the [[Kronecker delta]]. The symbol <math>(\cdot,\cdot)</math> is the [[inner product]] on ''H''. Thus, we have that ''EU(n)'' is the space of [[orthonormal]] ''n''-frames in ''H''.

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The [[group action]] of ''U''(''n'') on this space is the natural one. The [[base space]] is then

:<math>BU(n)=EU(n)/U(n) \, </math>

and is the set of [[Grassmannian]] ''n''-dimensional subspaces (or ''n''-planes) in ''H''. That is,

:<math>BU(n) = \{ V \subset \mathcal{H} : \dim V = n \} \, </math>

so that ''V'' is an ''n''-dimensional vector space.