Zassenhaus algebra
A Lie algebra of special derivations of the divided power algebra
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over a field of characteristic
. It is usually denoted by
, is
-dimensional and has a basis
with commutator
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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
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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
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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 ![]() |
[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 |
Zassenhaus algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Zassenhaus_algebra&oldid=50210