## 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”.

### MSC:

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

### Keywords:

simple Lie algebras; prime characteristic
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