Restricted Lie algebra

In mathematics, a restricted Lie algebra is a Lie algebra together with an additional 'p operation.' In mathematics, a restricted Lie algebra is a Lie algebra together with an additional 'p operation.' Let L be a Lie algebra over a field k of characteristic p>0. A p operation on L is a map X ↦ X [ p ] {displaystyle Xmapsto X^{}} satisfying If the characteristic of k is 0, then L is a restricted Lie algebra where the p operation is the identity map. For any associative algebra A defined over a field of characteristic p, the bracket operation [ X , Y ] := X Y − Y X {displaystyle :=XY-YX} and p operation X [ p ] := X p {displaystyle X^{}:=X^{p}} make A into a restricted Lie algebra L i e ( A ) {displaystyle mathrm {Lie} (A)} . Let G be an algebraic group over a field k of characteristic p, and L i e ( G ) {displaystyle mathrm {Lie} (G)} be the Zariski tangent space at the identity element of G. Each element of L i e ( G ) {displaystyle mathrm {Lie} (G)} uniquely defines a left-invariant vector field on G, and the commutator of vector fields defines a Lie algebra structure on L i e ( G ) {displaystyle mathrm {Lie} (G)} just as in the Lie group case. If p>0, the Frobenius map x ↦ x p {displaystyle xmapsto x^{p}} defines a p operation on L i e ( G ) {displaystyle mathrm {Lie} (G)} . The functor A ↦ L i e ( A ) {displaystyle Amapsto mathrm {Lie} (A)} has a left adjoint L ↦ U [ p ] ( L ) {displaystyle Lmapsto U^{}(L)} called the restricted universal enveloping algebra. To construct this, let U ( L ) {displaystyle U(L)} be the universal enveloping algebra of L forgetting the p operation. Letting I be the two-sided ideal generated by elements of the form x p − x [ p ] {displaystyle x^{p}-x^{}} , we set U [ p ] ( L ) = U ( L ) / I {displaystyle U^{}(L)=U(L)/I} . It satisfies a form of the PBW theorem. Restricted Lie algebras are used in Jacobson's Galois correspondence for purely inseparable extensions of fields of exponent 1.