Zassenhaus algebra

A Lie algebra of special derivations of the divided power algebra

over a field of characteristic . It is usually denoted by , is -dimensional and has a basis with commutator

It also has another basis, (if ), with commutator , where is a finite field of order . Zassenhaus algebras appeared first in this form in 1939 [a8] (see also Witt algebra). is simple if (cf. also Simple algebra), has an ideal of codimension if , , and is -dimensional non-Abelian if , . It is a Lie -algebra if and only if . The -structure on can be given by , , . By changing to other additive subgroup of , or by changing the multiplication, one can get different algebras. For example, the multiplication

where is an additive homomorphism of finite fields, gives rise to the Albert–Zassenhaus algebra.

Suppose that all algebras and modules are finite-dimensional and that the ground field is an algebraically closed field of characteristic .

Let be the -module defined for on the vector space by

For example, and are isomorphic to the adjoint and co-adjoint modules, has an irreducible submodule , and is irreducible if . Any irreducible restricted -module is isomorphic to one of the following modules: the -dimensional trivial module ; the -dimensional module ; or the -dimensional module , , . The maximal dimension of irreducible -modules is , but there may be infinitely many non-isomorphic irreducible modules of given dimension. The minimal dimension of irreducible non-trivial -modules is , and any irreducible module of dimension is isomorphic to . Any irreducible -module with a non-trivial action of is irreducible as a module over the maximal subalgebra . In any case, any non-restricted irreducible -module is induced by some irreducible submodule of [a3].

Any simple Lie algebra of dimension with a subalgebra of codimension is isomorphic to for some [a4]. Albert–Zassenhaus algebras have subalgebras of codimension . Any automorphism of is induced by an admissible automorphism of i.e., by an automorphism such that is a linear combination of , where , , . There are infinitely many non-conjugate Cartan subalgebras (cf. also Cartan subalgebra) of dimension if , and exactly two non-conjugate Cartan subalgebras of dimension if [a2]. The algebra of outer derivations of is -dimensional and generated by the derivations . The algebra has a -parametric deformation [a6]. A non-split central extension of was constructed first by R. Block in 1968 [a1]. The characteristic- infinite-dimensional analogue of this extension is well known as the Virasoro algebra. The list of irreducible -modules that have non-split extensions is the following: . All bilinear invariant forms of are trivial, but it has a generalized Casimir element . The centre of the universal enveloping algebra is generated by the -centre and the generalized Casimir elements [a7].

References

[a1]	R.E. Block, "On the extension of Lie algebras" Canad. J. Math. , 20 (1968) pp. 1439–1450
[a2]	G. Brown, "Cartan subalgebras of Zassenhaus algebras" Canad. J. Math. , 27 : 5 (1975) pp. 1011–1021
[a3]	Ho-Jui Chang, "Über Wittsche Lie-Ringe" Abb. Math. Sem. Univ. Hamburg , 14 (1941) pp. 151–184
[a4]	A.S. Dzhumadil'daev, "Simple Lie algebras with a subalgebra of codimension 1" Russian Math. Surveys , 40 : 1 (1985) pp. 215–216 (In Russian)
[a5]	A.S. Dzhumadil'daev, "Cohomology and nonsplit extensions of modular Lie algebras" Contemp. Math. , 131:2 (1992) pp. 31–43
[a6]	A.S. Dzhumadil'daev, A.I. Kostrikin, "Deformations of the Lie algebra " Proc. Steklov Inst. Math. , 148 (1980) pp. 143–158 (In Russian)
[a7]	Y.B. Ermolaev, "On structure of the center of the universal enveloping algebra of a Zassenhaus algebra" Soviet Math. (Iz.VUZ) , 20 (1978) Izv. VUZ Mat. , 12(199) (1978) pp. 46–59
[a8]	H. Zassenhaus, "Über Lie'she Ringe mit Primzahlcharacteristik" Abh. Math. Sem. Univ. Hamburg , 13 (1939) pp. 1–100