Commutative Restricted Lie Algebras

Part of the Contemporary Mathematicians book series (CM)


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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© Birkhäuser Boston 1989

Authors and Affiliations

  1. 1.Yale UniversityUSA

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