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WP:CHECKWIKI error fix. Section heading problem. Violates WP:MOSHEAD.

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In [[mathematics]], a '''restricted Lie algebra''' is a [[Lie algebra]] together with an additional "''p'' operation."

==Definition==
Let ''L'' be a Lie algebra over a field ''k'' of characteristic ''p>0''. A ''p'' '''operation''' on ''L'' is a map <math>X \mapsto X^{[p]}</math> satisfying
* <math>\mathrm{ad}(X^{[p]}) = \mathrm{ad}(X)^p</math> for all <math>X \in L</math>,
* <math>(tX)^{[p]} = t^pX^{[p]}</math> for all <math>t \in k, X \in L</math>,
* <math>(X+Y)^{[p]} = X^{[p]} + Y^{[p]} + \sum_{i=1}^{p-1} \frac{s_i(X,Y)}{i}</math>, for all <math>X,Y \in L</math>, where <math>s_i(X,Y)</math> is the coefficient of <math>t^{i-1}</math> in the formal expression <math>\mathrm{ad}(tX+Y)^{p-1}(X)</math>.

If the characteristic of ''k'' is 0, then ''L'' is a restricted Lie algebra where the ''p'' operation is the identity map.

==Examples==
For any associative algebra ''A'' defined over a field of characteristic ''p'', the bracket operation <math>[X,Y] := XY-YX</math> and ''p'' operation <math>X^{[p]} := X^p</math> make ''A'' into a restricted Lie algebra <math>\mathrm{Lie}(A)</math>.

Let ''G'' be an algebraic group over a field k of characteristic ''p'', and <math>\mathrm{Lie}(G)</math> be the [[Zariski tangent space]] at the identity element of ''G''. Each element of <math>\mathrm{Lie}(G)</math> uniquely defines a left-invariant vector field on ''G'', and the commutator of vector fields defines a Lie algebra structure on <math>\mathrm{Lie}(G)</math> just as in the [[Lie group]] case. If ''p>0'', the [[Frobenius map]] <math>x \mapsto x^p</math> defines a ''p'' operation on <math>\mathrm{Lie}(G)</math>.

==Restricted universal enveloping algebra==
The functor <math>A \mapsto \mathrm{Lie}(A)</math> has a [[left adjoint]] <math>L \mapsto U^{[p]}(L)</math> called the '''restricted universal enveloping algebra'''. To construct this, let <math>U(L)</math> be the [[universal enveloping algebra]] of ''L'' forgetting the ''p'' operation. Letting ''I'' be the two-sided ideal generated by elements of the form <math>x^p - x^{[p]}</math>, we set <math>U^{[p]}(L) = U(L) / I</math>.

==See also==
Restricted Lie algebras are used in [[Nathan Jacobson|Jacobson]]'s Galois correspondence for [[purely inseparable extension]]s of fields of exponent 1.

==References==
* [[Armand Borel]], ''Linear Algebraic Groups'' 2nd edition, [[Graduate Texts in Mathematics]] '''126''', Springer-Verlag.
*{{Citation | last1=Block | first1=Richard E. | last2=Wilson | first2=Robert Lee | title=Classification of the restricted simple Lie algebras | doi=10.1016/0021-8693(88)90216-5 | mr=931904 | year=1988 | journal=[[Journal of Algebra]] | issn=0021-8693 | volume=114 | issue=1 | pages=115–259}}

[[Category:Algebraic groups]]
[[Category:Lie algebras]]

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en>BG19bot: WP:CHECKWIKI error fix. Section heading problem. Violates WP:MOSHEAD.