Restricted Lie algebra

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

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

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

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

Restricted universal enveloping algebra

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

See also

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

References

• Borel, Armand (1991), Linear Algebraic Groups, Graduate Texts in Mathematics, 126 (2nd ed.), Springer-Verlag, Zbl 0726.20030.
• Block, Richard E.; Wilson, Robert Lee (1988), "Classification of the restricted simple Lie algebras", Journal of Algebra, 114 (1): 115–259, doi:10.1016/0021-8693(88)90216-5, ISSN 0021-8693, MR 0931904.
• Montgomery, Susan (1993), Hopf algebras and their actions on rings. Expanded version of ten lectures given at the CBMS Conference on Hopf algebras and their actions on rings, which took place at DePaul University in Chicago, USA, August 10-14, 1992, Regional Conference Series in Mathematics, 82, Providence, RI: American Mathematical Society, p. 23, ISBN 978-0-8218-0738-5, Zbl 0793.16029.