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.
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.
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)
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.
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)