Restricted Lie algebra

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

Definition[edit]

Let L be a Lie algebra over a field k of characteristic p>0. A p operation on L is a map satisfying

  • for all ,
  • for all ,
  • , for all , where is the coefficient of in the formal expression .

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

Examples[edit]

For any associative algebra A defined over a field of characteristic p, the bracket operation and p operation make A into a restricted Lie algebra .

Let G be an algebraic group over a field k of characteristic p, and be the Zariski tangent space at the identity element of G. Each element of uniquely defines a left-invariant vector field on G, and the commutator of vector fields defines a Lie algebra structure on just as in the Lie group case. If p>0, the Frobenius map defines a p operation on .

Restricted universal enveloping algebra[edit]

The functor has a left adjoint called the restricted universal enveloping algebra. To construct this, let be the universal enveloping algebra of L forgetting the p operation. Letting I be the two-sided ideal generated by elements of the form , we set . It satisfies a form of the PBW theorem.

See also[edit]

Restricted Lie algebras are used in Jacobson's Galois correspondence for purely inseparable extensions of fields of exponent 1.

References[edit]