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:{{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]]<!--- I suppose this is meant; that article does not mention "Sym(n)" notation---> 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}} | |||
Revision as of 19:29, 5 November 2013
- 28 year-old Painting Investments Worker Truman from Regina, usually spends time with pastimes for instance interior design, property developers in new launch ec Singapore and writing. Last month just traveled to City of the Renaissance.
Whitehead's lemma is a technical result in abstract algebra used in algebraic K-theory. It states that a matrix of the form
is equivalent to the identity matrix by elementary transformations (that is, transvections):
Here, indicates a matrix whose diagonal block is and entry is .
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.[1][2] In symbols,
- .
![{\displaystyle \operatorname {E} (A)=[\operatorname {GL} (A),\operatorname {GL} (A)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f46da4b6b93a3e69bcc12eef1d2a6a47da66007)
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
one has:
![{\displaystyle \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),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc2418d5d063a8f39416e16e188d9d34cc2172ca)
where Alt(3) and Sym(3) denote the alternating resp. symmetric group on 3 letters.
See also
References
- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534