A new class of simple Lie algebras. (English) Zbl 0055.26503

The author defines a class of simple Lie algebras of finite characteristic as follows. Let \(\mathfrak F\) be a field of characteristic \(p\), and let \(\mathfrak B_n\) be the algebra (with unit-element) of all commutative polynomials in \(x_1,\ldots, x_n\) with the defining relations \(x_1^p = \cdots = x_n^p = 0\). The algebra \(\mathfrak W_n\) of derivations over of \(\mathfrak F\) may be considered as a Lie algebra in a well known way. For each element \(A = \sum_i a_i\frac{\partial}{\partial x_i}\) of \(\mathfrak W_n\) the divergence of \(A\) is defined to be \(\delta A = \sum_i \frac{\partial a_i}{\partial x_i}\). The derivations whose divergence is zero, form a subalgebra \(\mathfrak M_n\) (of dimension \((n - 1) p^n+ 1\) over \(\mathfrak F\) of \(\mathfrak W_n\), and for \(n > 2\) the algebra \(\mathfrak T_n= \mathfrak M_n^2\) is simple. Its dimension over \(\mathfrak F\) is \((n -1) (p^n - 1)\), and this shows \(\mathfrak T_n\) to be in general distinct from the known simple Lie algebras of characteristic \(p\). The elements of \(\mathfrak T_n\) are also characterized in a different way as “truncated derivations”.


17B50 Modular Lie (super)algebras
17B20 Simple, semisimple, reductive (super)algebras
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