Restricted Lie algebra

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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 X\mapsto X^{{[p]}} satisfying

  • {\mathrm {ad}}(X^{{[p]}})={\mathrm {ad}}(X)^{p} for all X\in L,
  • (tX)^{{[p]}}=t^{p}X^{{[p]}} for all t\in k,X\in L,
  • (X+Y)^{{[p]}}=X^{{[p]}}+Y^{{[p]}}+\sum _{{i=1}}^{{p-1}}{\frac {s_{i}(X,Y)}{i}}, for all X,Y\in L, where s_{i}(X,Y) is the coefficient of t^{{i-1}} in the formal expression {\mathrm {ad}}(tX+Y)^{{p-1}}(X).

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 [X,Y]:=XY-YX and p operation X^{{[p]}}:=X^{p} make A into a restricted Lie algebra {\mathrm {Lie}}(A).

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

Restricted universal enveloping algebra

The functor A\mapsto {\mathrm {Lie}}(A) has a left adjoint L\mapsto U^{{[p]}}(L) called the restricted universal enveloping algebra. To construct this, let 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]}}, we set U^{{[p]}}(L)=U(L)/I.

See also

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

References

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