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:{{For|A lemma in Lie algebras|Whitehead's lemma (Lie algebras)}} |
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'''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 |
'''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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Revision as of 01:01, 8 May 2013
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[1] that the derived group of the stable general linear group is the group generated by elementary matrices. In symbols,
- .
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:
where Alt(3) and Sym(3) denote the alternating resp. symmetric group on 3 letters.
See also
References