Finitely presented Lie algebras can be constructed from free Lie algebras
by using the / constructor, i.e., FL/[r1...rk] is the quotient of
the free Lie algebra FL by the ideal generated by the elements
r1...rk of FL. If the finitely presented Lie algebra K happens to be
finite dimensional then an isomorphic structure constants Lie algebra
can be constructed by NiceAlgebraMonomorphism(K), which returns a surjective
homomorphism. The structure constants Lie algebra can then be accessed by
calling Range for this map. Also limited computations with elements of the
finitely presented Lie algebra are possible.
gap> L:= FreeLieAlgebra( Rationals, "s", "t" ); <Lie algebra over Rationals, with 2 generators> gap> gL:= GeneratorsOfAlgebra( L );; s:= gL[1];; t:= gL[2];; gap> K:= L/[ s*(s*t), t*(t*(s*t)), s*(t*(s*t))-t*(s*t) ]; <Lie algebra over Rationals, with 2 generators> gap> h:= NiceAlgebraMonomorphism( K ); [ [(1)*s], [(1)*t] ] -> [ v.1, v.2 ] gap> U:= Range( h ); <Lie algebra of dimension 3 over Rationals> gap> IsLieNilpotent( U ); true gap> gK:= GeneratorsOfAlgebra( K ); [ [(1)*s], [(1)*t] ] gap> gK[1]*(gK[2]*gK[1]) = Zero( K ); true
FpLieAlgebraByCartanMatrix( C ) F
Here C must be a Cartan matrix. The function returns the finitely-presented Lie algebra over the field of rational numbers defined by this Cartan matrix. By Serre's theorem, this Lie algebra is a semisimple Lie algebra, and its root system has Cartan matrix C.
gap> C:= [ [ 2, -1 ], [ -3, 2 ] ];; gap> K:= FpLieAlgebraByCartanMatrix( C ); <Lie algebra over Rationals, with 6 generators> gap> h:= NiceAlgebraMonomorphism( K ); [ [(1)*x1], [(1)*x2], [(1)*x3], [(1)*x4], [(1)*x5], [(1)*x6] ] -> [ v.1, v.2, v.3, v.4, v.5, v.6 ] gap> SemiSimpleType( Range( h ) ); "G2"
NilpotentQuotientOfFpLieAlgebra( FpL, max ) F
NilpotentQuotientOfFpLieAlgebra( FpL, max, weights ) F
Here FpL is a finitely presented Lie algebra. Let K be the quotient of FpL by the max+1-th term of its lower central series. This function calculates a surjective homomorphism of FpL onto K. When called with the third argument weights, the k-th generator of FpL gets assigned the k-th element of the list weights. In that case a quotient is calculated of FpL by the ideal generated by all elements of weight max+1. If the list weights only consists of 1's then the two calls are equivalent. The default value of weights is a list (of length equal to the number of generators of FpL) consisting of 1's.par If the relators of FpL are homogeneous, then the resulting algebra is naturally graded.
gap> L:= FreeLieAlgebra( Rationals, "x", "y" );; gap> g:= GeneratorsOfAlgebra(L);; x:= g[1]; y:= g[2]; (1)*x (1)*y gap> rr:=[((y*x)*x)*x-6*(y*x)*y, 3*((((y*x)*x)*x)*x)*x-20*(((y*x)*x)*x)*y ]; [ (-1)*(x*(x*(x*y)))+(6)*((x*y)*y), (-3)*(x*(x*(x*(x*(x*y)))))+(20)*(x*(x*((x*y)*y)))+(-20)*((x*(x*y))*(x*y)) ] gap> K:= L/rr; <Lie algebra over Rationals, with 2 generators> gap> h:=NilpotentQuotientOfFpLieAlgebra(K, 50, [1,2] ); [ [(1)*x], [(1)*y] ] -> [ v.1, v.2 ] gap> L:= Range( h ); <Lie algebra of dimension 50 over Rationals> gap> Grading( L ); rec( min_degree := 1, max_degree := 50, source := Integers, hom_components := function( d ) ... end )
[Top] [Previous] [Up] [Next] [Index]
GAP 4 manual