Entropy (classical thermodynamics)

2013-11-05T17:29:56Z

132.216.24.103: /* Refrigerators */

:{{For|a lemma on Lie algebras|Whitehead's lemma (Lie algebras)}}
'''Whitehead's lemma''' is a technical result in [[abstract algebra]] used in [[algebraic K-theory]]. It states that a [[matrix (mathematics)|matrix]] of the form

:<math>
\begin{bmatrix}
u & 0 \\
0 & u^{-1} \end{bmatrix}</math>

is equivalent to the [[identity matrix]] by [[elementary matrices|elementary transformations]] (that is, transvections):

:<math>
\begin{bmatrix}
u & 0 \\
0 & u^{-1} \end{bmatrix} = e_{21}(u^{-1}) e_{12}(1-u) e_{21}(-1) e_{12}(1-u^{-1}). </math>

Here, <math>e_{ij}(s)</math> indicates a matrix whose diagonal block is <math>1</math> and <math>ij^{th}</math> entry is <math>s</math>.

The name "Whitehead's lemma" also refers to the closely related result that the [[derived group]] of the [[stable general linear group]] is the group generated by [[elementary matrices]].<ref name=Mil31>{{cite book | last1=Milnor | first1=John Willard | author1-link= John Milnor | title=Introduction to algebraic K-theory | publisher=[[Princeton University Press]] | location=Princeton, NJ | mr=0349811 | year=1971 | zbl=0237.18005 | series=Annals of Mathematics Studies | volume=72 | at=Section 3.1 }}</ref><ref name=Sn164>{{cite book | title=Explicit Brauer Induction: With Applications to Algebra and Number Theory | volume=40 | series=Cambridge Studies in Advanced Mathematics | first=V. P. | last=Snaith | authorlink= | publisher=[[Cambridge University Press]] | year=1994 | isbn=0-521-46015-8 | zbl=0991.20005 | page=164 }}</ref> In symbols,
:<math>\operatorname{E}(A) = [\operatorname{GL}(A),\operatorname{GL}(A)]</math>.

This holds for the stable group (the [[direct limit]] of matrices of finite size) over any ring, but not in general for the unstable groups, even over a field. For instance for
:<math>\operatorname{GL}(2,\mathbb{Z}/2\mathbb{Z})</math>

one has:
:<math>\operatorname{Alt}(3) \cong [\operatorname{GL}_2(\mathbb{Z}/2\mathbb{Z}),\operatorname{GL}_2(\mathbb{Z}/2\mathbb{Z})] < \operatorname{E}_2(\mathbb{Z}/2\mathbb{Z}) = \operatorname{SL}_2(\mathbb{Z}/2\mathbb{Z}) = \operatorname{GL}_2(\mathbb{Z}/2\mathbb{Z}) \cong \operatorname{Sym}(3),</math>
where Alt(3) and Sym(3) denote the [[alternating group|alternating]] resp. [[symmetric group]] on 3 letters.

==See also==
*[[Special linear group#Relations to other subgroups of GL(n,A)]]

==References==
<references/>

[[Category:Matrix theory]]
[[Category:Lemmas]]
[[Category:K-theory]]
[[Category:Theorems in abstract algebra]]

{{Abstract-algebra-stub}}

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Entropy (classical thermodynamics)