So far our algorithm has generated the elements of a pseudo-Hall-basis of a Lie algebraG inductively with a loop using the monomial degree as its counter. This is straightforward and convenient in some applications, but impractical in others:
• For instance, G might already be equipped with a different gradation, and we would want our output to be exactly all elements of the Pseudo-Hall-basis up to a given degree, sorted by degree (regarding that gradation).
• We might want to drop the restiction that #X, the number of generators, is finite. But then, the number of brackets that have to be considered in the very first step (i.e. l=2) is infinite and the algorithm will never terminate. A strategy we could pursue here would be to equipXwith a gradation such thatXis degreewise finite, so that there are only finitely many generators of each degree that have to be considered within each iteration of the loop corresponding to a degree.
Taking this into account, for this section let G be Lie algebra that is graded with a N+ -gradation degeand generated by a degreewise finite setX. We will call degeexternal gradation in the sequel to distinguish it from the monomial degree degH. As before, let<S be a total order onM(X) and<Hbe a Hall order onM(X).
Note that this implies that that G is degreewise finite dimensional, i.e. dim Gl <∞for all l∈N+, because by theorem 4.2.4,Gis a quotient algebra of the free Lie algebraL(X), which has to be degreewise finite dimensional with respect to dege.
To modify the algorithm accordingly, two aspects have to be taken into consideration.
Remark 4.4.1. Since elements ofX can have any degree now, instead of being considered only once during the initialization (in step 1 ofPHall), they need to be included inSin every step of the iteration overl(which is done in step 4).
Remark 4.4.2. The requirement that degeis strictly positive implies
dege(m,n)=dege(m)+dege(n)>max(degem+degen). (4.25) This implies that in stepl, the arguments of the Lie brackets considered forSare of strictly lower degree thanl, and, as can be shown by induction overl, are elements of ¯Hl−1. If this condition is dropped, further modifications to the algorithm will have to be made as lined out in the next section.
Considering this, we obtain the following modified algorithm. The input are the setX of generators of G and a maximal (external) degreelmax to compute up to. The output are ordered tuplesHandB.
start
H:=(), B:=(), l:=1
l=lmax+1?
S:={h∈X| dege(h)=l} ∪ (h,g)∈H×H|dege(h)+
dege(g) = land IsHallElement(h,g)=True
SortSby<Ss.t.
s1 <S s2 <S . . . <S sr
i:=1
silin. dep.
ofH?
B:=B_ {si+P
jci jhi j} withci j ∈ C,hi j ∈ H
s.t.si+P
jci jhi j = 0
H := H_{si}
i:=i+1
i=r+1?
l:=l+1
return (H,B) no
yes
yes no
yes no
Figure 4.4: Flow chart of algorithmPHallPosExtGrad
Algorithm 4PHallPosExtGrad
1: H:=()
2: B:=()
3: forl=1, . . . ,lmaxdo
4: S := {h ∈ X|dege(h) = l} ∪
(h,g) ∈ H × H| dege(h) + dege(g) = land IsHallElement(h,g)=True
5: S :=Sort(S, >S) .such thats1 <Ss2<S. . . <Ssr
6: fori=1, ...,rdo
7: ifsiis linearly dependent ofH(in G)then
8: B:=B_{si+P
jci jhi j}withci j∈C, hi j∈Hsuch thatsi+P
jci jhi j=0
9: else
10: H:=H_{si}
11: return(H, B)
Remark 4.4.3. Theorem 4.3.3 and its proof still hold (with deg = dege), so (H, B)=PHallPosExtGrad(X,lmax) is a pseudo-Hall-basis of G up to external degreelmax. Remark 4.4.4. If the external degree used is the monomial degree, then G0 = {}, and the algorithm coincides with the algorithmPHallforl>1.
Example 4.4.5. We compute the first few steps of PHallPosExtGrad for the (N+-graded) subalgebra of the Witt algebra w(def. 4.3.16) that is generated byL1 andL2, using<H=<S
DegLex withL1<L2(cf. example 4.2.14 and 4.3.17).
l i si(Hall element) si(standard basis) lin. dep. ofH
1 1 L1 L1 no
2 1 L2 L2 no
3 1 [L1,L2] −L3 no
4 1 [L1,[L1,L2]] 2L4 no
5 1 [L2,[L1,L2]] L5 no
5 2 [L1,[L1,[L1,L2]]] −6L5 yes 6 1 [L2,[L1,[L1,L2]]] −4L6 no 7 1 [L2,[L2,[L1,L2]]] −3L7 no 7 2 [[L1,L2],[L1,[L1,L2]]] 2L7 yes For contrast, if we use another sorting order,<S=Lex, we find
l i si(Hall element) si(standard basis) lin. dep. ofH
1 1 L1 L1 no
2 1 L2 L2 no
3 1 [L1,L2] −L3 no
4 1 [L1,[L1,L2]] 2L4 no
5 1 [L1,[L1,[L1,L2]]] −6L5 no
5 2 [L2,[L1,L2]] L5 yes
6 1 [L1,[L1,[L1,[L1,L2]]]] 24L6 no 6 2 [L2,[L1,[L1,L2]]] −4L6 yes 7 1 [L1,[L1,[L1,[L1,[L1,L2]]]]] −120L7 no 7 2 [[L1,L2],[L1,[L1,L2]]] 2L7 yes 7 3 [L2,[L1,[L1,[L1,L2]]]] 18L7 yes
Comparing the two results, it is apparent that not only the output depends on <S, but also the number of computations required (cf. remark 4.3.7).
Remark 4.4.6. If G is a degreewise finite dimensional graded Lie algebra, any basisbG of G satisfies the requirements forXin this chapter. In conjunction with a slight modification of the algorithm given above, this can be used the following way (that emphasizes the practical importance of the choice of <S) to iteratively expand a tupleY of elements of G so that it generates ¯Glwith increasingl:
Algorithm 5PHallFindGenerators
1: X:=bG 2: H:=()
3: Y:=()
4: B:=()
5: forl=1, . . . ,lmaxdo
6: S := {h ∈ X|dege(h) = l} ∪
(h,g) ∈ H × H| dege(h) + dege(g) = land IsHallElement(h,g)=True
7: S :=Sort(S, >S) .such thats1 <Ss2<S. . . <Ssr 8: fori=1, ...,rdo
9: ifsiis linearly dependent ofH(in G)then
10: B:=B_{si+P
jci jhi j}withci j∈C, hi j∈Hsuch thatsi+P
jci jhi j=0
11: else
12: H:=H_{si}
13: ifdegHsi = 1then
14: Y:=Y_{si}
15: return(H, Y, B)
By theorem 4.3.3, if we run the algorithm, we obtain a basisHof ¯Llmax (the fact that bG may be infinite is not a problem because the algorithm only ever considers one stratum at a time which is finite dimensional). Because every element ofHthat has monomial degree 1 is added toY, the tupleYgeneratesH, and with it ¯Llmax.
NowHdepends on the sorting order<Sused in the algorithm. If<Ssatisfiesg<Shfor all g,h ∈G with degHg<degHh, thenYwill be kept minimal in the sense that elements ofbG will only be added toYif they are not in the subalgebra generated by the previous elements ofY.
If, on the contrary, we choose a sorting order<Sthat reverses the inequalities given above, HandY will reproducebG andB will be a list of the structure constants of G (both up to degreelmax).
4.4.1 Application: Observations about the multiplet1g1
In section 2.4, we had seen that the multiplet1g1 might be of relevance to the structure ofg ford=3. Therefore, giving a little attention to the subalgebra generated by1g1is warranted.
Call this subalgebraF. When we use the algorithmPHallPosExtGradto explore the structure ofF, we find that for eachlin{2, . . . ,8}– this is the limit of computation on the machine used –Hlconsists of one element of magnetic quantum numbers 0,±1 each, and because of lemma 2.4.1, they must form a multiplet of spin 1. This can be generalized to a conjecture:
Conjecture 4.4.7. For alll∈N,
F∩gl 1V. (4.26)
We also find thatBlconsists of monomials in the elements of1g1of a particular form, as opposed to linear combinations of them, which is the general case. We can generalize this to the “⊂” part of another conjecture, using the notion of thecentralizer Z(S) of a subsetSof a Lie algebraG, defined as
These equalities have been verified using the Mathematica program CenterTest (cf.
B.6.8) up to l=6, the limit of computation.
Remark 4.4.9. While for all 3-dimensional sl2-modules 1V, we have [1V,1V] Φ1V with the epimophism Φ as in lemma 2.4.1, the analogue to conjecture 4.4.7 is not necessarily true for every 3-dimensionalsl2-module, not even for every suchsl2-module that occurs in the weight space decomposition ofg. Take for instance the multipletW 1Voccurring in [2g1, 1g1]3V⊕2V⊕1V. A basis ofWis given by the elements
Remark 4.4.10. If conjecture 4.4.7 is true, then the “⊃” part of the equations (4.28) and (4.30) can be proven for arbitraryl:
Proof. • The fact that the right hand sides are subsets ofgl follows from the fact that the right hand sides are multiple brackets oflelements ofg1.
• Bothv:=1v1andw:=adl1−v10 1v1have magnetic quantum number 1, so [v,w]∈g2l. Since conjecture 4.4.7 is supposed to be true, [v,w] is an element in a multiplet of spin 1. The only way to satisfy these conditions is [v,w]=0, which proves (4.31).
• Analogous for equation (4.32).