Lie p-algebra
restricted Lie algebra
An algebra over a field
of characteristic
(or, more generally, over a ring of prime characteristic
), endowed with a
-mapping
such that the following relations are satisfied:
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Here is the inner derivation of
defined by the element
(the adjoint transformation) and
is a certain element of
that is a linear combination of Lie monomials
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with or
for all
.
A typical example of a Lie -algebra is obtained if one considers an arbitrary associative algebra
over
(cf. Associative rings and algebras) as a universal algebra, with the following two derivation operations:
i) ,
ii) .
In particular, the property is a direct consequence of the identity
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for , in which case
for
. Since any Lie algebra can be imbedded in a suitably chosen associative algebra
with the operations i) and ii) (the Poincaré–Birkhoff–Witt theorem), one often replaces
, with some risk of ambiguity, by
.
As in every structure theory, the structure-preserving mappings are of particular relevance.
For any Lie -algebra
there is a
-universal (restricted universal) enveloping associative algebra
. If
, then
. This remark shows that for an arbitrary Lie algebra it makes sense to talk about its smallest
-envelope, or about its
-closure.
An ordinary Lie subalgebra (Lie ideal) of
is called a
-subalgebra (
-ideal) if
for all
. A homomorphism
of Lie
-algebras is called a
-homomorphism if
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If is a linear Lie
-algebra over
, one also calls this a
-representation
of
.
The specification of a -structure
on a Lie algebra
with basis
and zero centre
is uniquely and completely determined by specifying the images
of the basis elements
. On the other hand, a commutative Lie algebra
, for which one always has
, is endowed with a
-structure by considering the pair
, where
is an arbitrary
-semi-linear mapping,
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Over an algebraically closed field , every finite-dimensional commutative Lie
-algebra splits into the direct sum
of a torus
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and a nilpotent subalgebra (cf. Nilpotent algebra) , where the identity
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holds for sufficiently large (see [1]).
Important sources of Lie -algebras are the theory of algebraic groups, the theory of formal groups and the theory of inseparable fields (see [2]). The Lie algebra
of all derivations of an arbitrary algebra
is a
-subalgebra of
.
The class of simple Lie -algebras (restricted simple Lie algebras) is especially interesting for several reasons. For each finite-dimensional Lie algebra
over the complex numbers
, let
be the
-span of a Chevalley basis of
, and extend scalars to
:
. The quotient algebra
is simple and restricted. The simple Lie algebras obtained in this way are known as algebras of classical type:
(
),
(
),
(
),
(
),
,
,
,
,
. Besides the classical algebras, there are four other classes of simple Lie
-algebras: general algebras
,
(
); special algebras
,
(
); Hamiltonian algebras
,
(
); contact algebras
,
(
, where
for
(
) and
for
(
)). The simple Lie
-algebras just described are called algebras of Cartan type. They are obtained by replacing the ring of power series
in the Lie–Cartan construction (see Lie algebra, 3)) by that of the
-truncated polynomials
,
, or
. In the symbols
the index
has an invariant meaning; namely, it is the dimension of a maximal toroidal subalgebra. The main Block–Wilson classification theorem [5]: Let
be a finite-dimensional simple Lie
-algebra over an algebraically closed field
of characteristic
; then
is of classical or Cartan type. This result was conjectured by A.I. Kostrikin and I.R. Shafarevich (see [3]). It is not known whether the statement above will be true for
(presumably so), but for
, however, the situation is necessarily more complicated. For example, for
the classical Lie algebra
is included in a parametric family of
-dimensional simple Lie
-algebras
,
.
The theory of modular Lie algebras, i.e. Lie algebras over fields of characteristic , was created in the last half-century. It is symbolically said that its source is the discovery of E. Witt (1937) of the simple non-classical Lie algebra
. Here it should be noted that there is a much more involved construction of the simple Lie algebras of Cartan type that are not
-algebras. By dropping the requirement of being restricted, additional difficulties arise also in the study of representations, cohomology, deformations, and other problems in the theory of modular Lie algebras. The study of interrelations between constructions taking these into account and not taking into account the restrictedness condition, forms an important part of the theory (cf. [6]).
References
[1] | N. Jacobson, "Lie algebras" , Interscience (1962) ((also: Dover, reprint, 1979)) |
[2] | G.B. Seligman, "Modular Lie algebras" , Springer (1967) |
[3] | A.I. Kostrikin, I.R. Shafarevich, "Cartan pseudogroups and Lie ![]() |
[4] | H. Zassenhaus, "Ueber Liesche Ringe mit Primzahlcharacteristik" Abh. Math. Sem. Hansische Univ. , 13 (1939) pp. 1–100 |
[5] | R.E. Block, R.L. Wilson, "Classification of the restricted simple Lie algebras" J. of Algebra , 114 (1988) pp. 115–259 |
[6] | H. Strade, R. Farnsteiner, "Modular Lie algebras and their representations" , M. Dekker (1988) |
Comments
In characteristic 2 and 3 there exist infinitely many simple Lie -algebras, of dimension 31 and 10, respectively (cf. [a1]).
References
[a1] | V.G. Kac, B.Yu. Veisfeiler, "Exponentials in Lie algebras of characteristic ![]() |
[a2] | A. Borel, "Linear algebraic groups" , Benjamin (1969) |
Lie p-algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lie_p-algebra&oldid=17551