doc /// Key "Quotient Lie algebras and subspaces" Description Text The most common situation for a Lie algebra L (in this package) is that it is finitely presented, i.e., $L$ is given by a finite number of generators, yielding a free Lie algebra $F$, modulo a finite list of homogeneous elements in $F$. The ambient Lie algebra of $L$, see @TO "ambient(LieAlgebra)"@, is equal to $F$. Example F = lieAlgebra{a,b,c} M = F/{a a b, a a c} L1 = M/{a b c} describe M describe L1 Text There is also the possibility to build quotients by Lie ideals. A Lie ideal is of type {\tt LieIdeal}, and may be constructed in different ways, e.g., as the kernel of a homomorphism. In general, Lie ideals are not finitely generated (or not known to be, as $J$ below), but a finitely generated Lie ideal may be formed using the constructor @TO lieIdeal@. Building a quotient by a finitely generated ideal is the same as above, taking the Lie algebra modulo the generators of the ideal. Example F = lieAlgebra{a,b,c} I = lieIdeal{a a b,a a c,a b c} L2=F/I describe L2 L1==L2 Text The Lie algebra $L3$ below is a quotient of the finitely presented Lie algebra $M$ by the ideal $J$, which is not known to be finitely generated. The ambient Lie algebra of $L3$ is $M$ and {\tt ideal(L3)} is $J$. The Lie algebras $L2$ and $L3$ are isomorphic, but are presented in different ways. Example f = map(L1,M) J = kernel f L3 = M/J describe L3 dims(1,6,L2) dims(1,6,L3) Text If two quotients by Lie ideals are performed successively, then the program converts the final result to a quotient of the first Lie algebra by a single ideal. In the example below, $L5=(M/J)/K$ and this is transformed to $M/P$, where $P$ is the inverse image of $K$ under the natural map $M \ \to\ M/J$. Example L4 = L3/{a b,a c} g = map(L4,L3) K = kernel g L5 = L3/K ambient L5 ideal L5===inverse(map(L3,M),K) Text If a quotient by a Lie ideal that is not known to be finitely generated is followed by a quotient with finitely many generators, then the programs converts it by changing the order of the operations. In the example below, {\tt L6=(M/J)/\{a b\}} and this is transformed to {\tt (M/\{a b\})/Q}, where $Q$ is the image of $J$ under the natural map $M \ \to\ M/\{a b\}$\ (this in fact is an ideal since the map is surjective). Example L6 = (M/J)/{a b} L7 = ambient L6 use M L7 == M/{a b} Q = image(map(L7,M),J) ideal L6===new LieIdeal from Q Text It may also happen that L has a non-zero differential, see @TO differentialLieAlgebra@. The differential is given as the list {\tt diff(L)} of elements in $F$ that consists of the values of the differential on the generators of $F$, see @TO "diff(LieAlgebra)"@. Note that {\tt ideal(D)} (shown below) has been produced by the program to get the square of the differential to be zero. The extra {\tt - (b b a)} in {\tt ideal(L)} below is added by the program to ensure that the ideal generated by {\tt b c2} is invariant under the differential. Example F = lieAlgebra({a,b,c2,c3,c4},Signs=>{0,0,1,0,1}, Weights => {{1,0},{1,0},{2,1},{3,2},{5,3}}, LastWeightHomological=>true) D=differentialLieAlgebra{0_F,0_F,a b,a c2,a b c3} describe D L=D/{b c2} describe L Text In addition to the constructor @TO lieIdeal@ there are also the constructors @TO lieSubAlgebra@ and @TO lieSubSpace@ yielding finitely generated Lie subalgebras and finitely generated subspaces respectively. Example L = lieAlgebra{a,b,c} A = lieSubAlgebra{a,b c} basis(4,A) S=lieSubSpace{a,b c} dims(1,4,S) Text Ideals, subalgebras and subspaces are both inputs and possible outputs of several methods. The methods @TO "image(LieAlgebraMap,LieSubSpace)"@ and @TO "inverse(LieAlgebraMap,LieSubSpace)"@, which are used above, have image and kernel of a Lie algebra map or derivation as special cases. The method @TO "quotient(LieIdeal,FGLieSubAlgebra)"@ has @TO "annihilator(FGLieSubAlgebra)"@ and @TO center@ as special cases. Example L = lieAlgebra{a,b,c} I = lieIdeal{a a c+b a c-a b a,c c a-b b a } M = L/I J=lieIdeal{a b} A = quotient(J,lieSubAlgebra{a c}) dims(1,3,A) basis(2,A) member((c b) (a c),J) Text One may also form the sum, @TO (symbol +,LieSubSpace,LieSubSpace)@, and intersection, @TO (symbol \@,LieSubSpace,LieSubSpace)@, of two Lie subspaces (in particular subalgebras or ideals). Example L = lieAlgebra{a,b,c} I = lieIdeal{a b} J = lieIdeal{b c} T = I+J U = I@J dims(1,5,T) dims(1,5,U) 2*dims(1,5,I) Text Finally, the methods @TO boundaries@ and @TO cycles@ give the subalgebras {\tt image(d)} and {\tt kernel(d)} respectively, where $d$ is the differential, while @TO lieHomology@ gives the homology as a vector space. SeeAlso "Second Lie algebra tutorial" "Differential Lie algebra tutorial" "Homomorphisms and derivations" /// end

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