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:{{For|A lemma in Lie algebras|Whitehead's lemma (Lie algebras)}}
:{{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
'''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



Revision as of 01:02, 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

  1. ^ J. Milnor, Introduction to algebraic K -theory, Annals of Mathematics Studies 72, Princeton University Press, 1971. Section 3.1.