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A restricted Lie algebra is a Lie algebra over a field of
characteristic p>0, equipped with a restriction map x to xp,
satisfying the axioms given in
[Jac62].
A restrictable Lie algebra is a Lie algebra which can be equipped with a
restriction map.
A Lie algebra is restrictable if and only if
(ad) L is closed under the pth power map.
Hence restrictable Lie algebras have a standard restriction map induced by the
adjoint representation.
For many purposes, it suffices to know that a Lie algebra is restrictable,
without needing to know a restriction.
By convention, a Lie algebra over a field of characteristic zero is always
considered restrictable, and the restriction map is the identity map.
In Magma, we do not make a distinction between the concepts of restricted and
restrictable.
Note however that a Lie algebra can have a nonstandard restriction map.
IsRestricted(L) : AlgLie -> BoolElt, Map
IspLieAlgebra(L) : AlgLie -> BoolElt, Map
Returns true if, and only if, the Lie algebra L is restrictable.
If L is restrictable, the restriction map is returned as a second value.
pMap(L) : AlgLie -> Map
The restriction map of the Lie algebra L.
If L is not restrictable, an error is signalled.
> L:= LieAlgebra( "A2", GF(5) );
> IsRestrictable( L );
true Mapping from: AlgLie: L to AlgLie: L given by a rule [no inverse]
> pmap:= pMap( L );
> pmap( 2*L.3 + L.4);
(0 0 0 1 0 0 0 0)
pSubalgebra(Q) : SetEnum[AlgLieElt] -> AlgLie
Given a sequence Q of elements from the Lie algebra L, the function
returns the restricted subalgebra generated by the elements of Q, i.e.,
the smallest subalgebra containing Q which is also closed under the
restriction map. If the parent of Q is not restrictable, an error is
signalled.
Given Lie algebras L and M such that L≤M, this function returns
the closure of L under the restriction map of M. If L is not a
subalgebra of M or M is not restrictable, an error is signalled.
IspSubalgebra(L, M) : AlgLie, AlgLie -> AlgLie
Return true if and only if the Lie algebra L is a restricted Lie
subalgebra of M with the same restriction map. Note that if L is
constructed using the pClosure intrinsic, this will always be true.
However if L is constructed as a subalgebra, this may be false even
if L is restrictable, since the restriction map of L will be the
standard map rather than the restriction map of M.
Given Lie algebras L and M such that L≤M, this function returns
the quotient of L by the p-closure of the Lie algebra M, with
respect to the inherited restriction map.
Let G be a p-group. Then the quotients of the successive terms of the
Jennings series of G can be viewed as vector spaces over the field of
p elements. The direct sum of these vector spaces carries the structure
of a Lie algebra (coming from the commutator of G). This function returns
two values. Firstly, the Lie algebra constructed from G by this process.
This Lie algebra is graded. The second returned value is a sequence of
sequences of two elements. The first element is the degree of a homogeneous
component while the second element is its dimension. The basis elements
of the Lie algebra are ordered according to increasing degree. This means
that from the dimensions of the homogeneous components it is possible to
derive the degree of each basis element.
Lie algebras constructed in this way are naturally restricted. Moreover,
if x is a homogeneous element of degree d, then the p-th power
image of x is homogeneous of degree pd.
> G:= SmallGroup( 3^6, 196 );
> L, gr:= JenningsLieAlgebra( G );
> L;
Lie Algebra of dimension 6 with base ring GF(3)
> gr;
[
[ 1, 3 ],
[ 2, 1 ],
[ 3, 2 ]
]
> // So the first three basis elements are of degree 1,
> // the fourth basis element is of degree 2, and so on.
> pmap:= pMap( L );
> pmap( L.1 );
(0 0 0 0 1 1)
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