1. Penvelope

1.1 The functions

1.1-1 PEnvelopingAlgebra

> PEnvelopingAlgebra( L )( operation )

Here L is a restricted Lie algebra. This function returns the p-enveloping algebra of L.

It is possible to add the option cut, as in PEnvelopingAlgebra( L : cut );. In this case the quotient of the p-enveloping algebra by the c+1-st power of the Jacobson radical will be returned. In this case the Lie algebra has to be graded.


gap> L:= JenningsLieAlgebra(H1);
<Lie algebra of dimension 6 over GF(3)>
gap> U:= PEnvelopingAlgebra( L : cut );
<algebra of dimension 75 over GF(3)>

1.1-2 QuotientByMaximalIdeal

> QuotientByMaximalIdeal( U )( operation )

Here U must be a p-enveloping algebra. This function returns a list of two elements. The first is an associative algebra isomorphic to a quotient of U (the ideal is constructed by taking consecutive complements to the Lie algebra inside subspaces consisting of u such that Uu=uU=0). The second element of the list is a set of basis elements that form a basis of the Lie algebra inside the algebra.


gap> U:= PEnvelopingAlgebra( L : cut );
<algebra of dimension 26 over GF(3)>
gap> QuotientByMaximalIdeal( U );
[ <algebra of dimension 9 over GF(3)>, [ v.2, v.3, v.4, v.5, v.6, v.7 ] ]

1.1-3 UElement

> UElement( a )( operation )

Here a is a nilpotent element of a ring, for example a p-enveloping algebra. This function returns the invertible element 1+a.


gap> L:= JenningsLieAlgebra( H2 );
<Lie algebra of dimension 6 over GF(3)>
gap> U:= PEnvelopingAlgebra( L : cut );
<algebra of dimension 23 over GF(3)>
gap> r:= QuotientByMaximalIdeal( U );
[ <algebra of dimension 10 over GF(3)>, [ v.2, v.3, v.4, v.5, v.6, v.7 ] ]
gap> UElement( r[2][1] );
1 + v.2

1.1-4 PCGroupByUElements

> PCGroupByUElements( list )( operation )

Here list is a list of elements made by UElement. This function returns a pc group isomorphic to the group generated by the elements in list.


gap> h:= List( r[2], UElement );
[ 1 + v.2, 1 + v.3, 1 + v.4, 1 + v.5, 1 + v.6, 1 + v.7 ]
gap> PCGroupByUElements( h{[1,2]} );
#I  default `IsGeneratorsOfMagmaWithInverses' method returns `true' for 
[ 1 + v.2, 1 + v.3 ]
The size of G is 9
Group([ f1, f2 ])




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