Entropy (classical thermodynamics)

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

${\begin{bmatrix}u&0\\0&u^{-1}\end{bmatrix}}$

is equivalent to the identity matrix by elementary transformations (that is, transvections):

${\begin{bmatrix}u&0\\0&u^{-1}\end{bmatrix}}=e_{21}(u^{-1})e_{12}(1-u)e_{21}(-1)e_{12}(1-u^{-1}).$

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

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. In symbols,

$\operatorname {E} (A)=[\operatorname {GL} (A),\operatorname {GL} (A)]$.

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

$\operatorname {GL} (2,\mathbb {Z} /2\mathbb {Z} )$

one has:

$\operatorname {Alt} (3)\cong [\operatorname {GL} _{2}(\mathbb {Z} /2\mathbb {Z} ),\operatorname {GL} _{2}(\mathbb {Z} /2\mathbb {Z} )]<\operatorname {E} _{2}(\mathbb {Z} /2\mathbb {Z} )=\operatorname {SL} _{2}(\mathbb {Z} /2\mathbb {Z} )=\operatorname {GL} _{2}(\mathbb {Z} /2\mathbb {Z} )\cong \operatorname {Sym} (3),$

where Alt(3) and Sym(3) denote the alternating resp. symmetric group on 3 letters.

See also

• Special linear group#Relations to other subgroups of GL(n,A)

References

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