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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.CrossRefGoogle Scholar
  2. 2.
    N. Jacobson, Theory of rings. New York, 1943.Google Scholar
  3. 3.
    N. Jacobson, Restricted Lie algebras of characteristic p, Trans. Amer. Math. Soc. vol. 50 (1941) pp. 15–25.Google Scholar
  4. 4.
    O. Ore, On a special class of polynomials, Trans. Amer. Math. Soc. vol. 35 (1933) pp. 559–584.CrossRefGoogle Scholar

Copyright information

© Birkhäuser Boston 1989

Authors and Affiliations

  1. 1.Yale UniversityUSA

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