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# Commutative Restricted Lie Algebras

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Part of the Contemporary Mathematicians book series (CM)

## Abstract

A Lie algebra of characteristic p ≠ 0 is called restricted if in addition to the usual compositions one has defined a unary operation aa [p] such that
$$\begin{array}{*{20}c} {\left( {\alpha a} \right)^{\left[ p \right]} = \alpha ^p a^{\left[ p \right]} ,} \\ {\left[ {a,b^{\left[ p \right]} } \right] = \left[ { \cdots \left[ {ab} \right] \cdots b} \right],\left( {pb's} \right)} \\ {\left( {a + b} \right)^{\left[ p \right]} = a^{\left[ p \right]} + b^{\left[ p \right]} + \sum\limits_1^{p - 1} {S_i \left( {a,b} \right),} } \\ \end{array}$$
where is i (a, b) is the coefficient of λ i −1 in
$$\left[ { \cdots \left[ {\left[ {a,\lambda a + b} \right]\lambda a + b} \right] \cdots \lambda a + b} \right],^2 {\text{ }}\left( {p - 1\left( {\lambda a + b} \right)'s} \right).$$
Examples of such algebras are subspaces of associative algebras of characteristic p ≠ 0 which are closed under the Lie multiplication [ab] = abba and under pth powers. Then one may take a [p] = a p . It is known that every restricted Lie algebra is isomorphic to one of this type. For this reason we may simplify our notation in the sequel and write a p for a [p]. We call the mapping aa p the p-operator in $$\mathfrak{L}$$.

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## Bibliography

1. 1.
A. S. Amitsur, A generalization of a theorem on linear differential equations, Bull. Amer. Math. Soc. vol. 54 (1948) pp. 937–942.
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N. Jacobson, Theory of rings. New York, 1943.Google Scholar
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N. Jacobson, Restricted Lie algebras of characteristic p, Trans. Amer. Math. Soc. vol. 50 (1941) pp. 15–25.Google Scholar
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O. Ore, On a special class of polynomials, Trans. Amer. Math. Soc. vol. 35 (1933) pp. 559–584.

## Copyright information

© Birkhäuser Boston 1989

## Authors and Affiliations

1. 1.Yale UniversityUSA