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| :{{For|a lemma on Lie algebras|Whitehead's lemma (Lie algebras)}}
| | The title of the writer is Jayson. To play lacross is something I truly appreciate doing. Credit authorising is how he tends to make money. Alaska is exactly where I've always been residing.<br><br>Here is my web-site ... clairvoyance, [http://conniecolin.com/xe/community/24580 http://conniecolin.com/xe/community/24580], |
| '''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
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| :<math>
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| \begin{bmatrix}
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| u & 0 \\
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| 0 & u^{-1} \end{bmatrix}</math>
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| is equivalent to the [[identity matrix]] by [[elementary matrices|elementary transformations]] (that is, transvections):
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| :<math>
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| \begin{bmatrix}
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| u & 0 \\
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| 0 & u^{-1} \end{bmatrix} = e_{21}(u^{-1}) e_{12}(1-u) e_{21}(-1) e_{12}(1-u^{-1}). </math>
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| 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>.
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| 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,
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| :<math>\operatorname{E}(A) = [\operatorname{GL}(A),\operatorname{GL}(A)]</math>. | |
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| 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
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| :<math>\operatorname{GL}(2,\mathbb{Z}/2\mathbb{Z})</math>
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| one has:
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| :<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]]<!--- I suppose this is meant; that article does not mention "Sym(n)" notation---> on 3 letters.
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| ==See also==
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| *[[Special linear group#Relations to other subgroups of GL(n,A)]]
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| ==References==
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| <references/>
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| [[Category:Matrix theory]]
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| [[Category:Lemmas]]
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| [[Category:K-theory]]
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| [[Category:Theorems in abstract algebra]]
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| {{Abstract-algebra-stub}}
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The title of the writer is Jayson. To play lacross is something I truly appreciate doing. Credit authorising is how he tends to make money. Alaska is exactly where I've always been residing.
Here is my web-site ... clairvoyance, http://conniecolin.com/xe/community/24580,