Iterative generation and subsequent extension to a basis

The approach given in 4.5.1 has the disadvantage that the sequence in which elements ofS were considered for extension to a basis of G_kcould not be choosen freely. If we can iteratively generate all candidates for extension to a basis first and only then extend to a basis, we can avoid this problem. We will see that imposing the following additional conditions on the Lie algebra G allows to avoid cases where the algorithm never terminates.

Definition 4.5.8(pseudo-Hall-exhaustibility). A Lie algebra G equipped with aN0-gradation deg_ewith the properties

1. the dimensions dim G_l are finite for alll(G is degreewise finite dimensional), 2. for alllexistsn_l ∈N0such that

ad_gnl ad_gnl−1. . .ad_g1g=0 ∀g_i∈G₀,g∈G_l (4.37) is calledpseudo-Hall-exhaustible.

Remark 4.5.9. In particular, this obviously implies that G₀is a finite-dimensional subalgebra.

Example 4.5.10. The following Lie algebras are pseudo-Hall-exhaustible:

1. finite-dimensional nilpotent Lie algebras with the trivial gradation; here G=G₀, 2. finitely generated Lie algebras that are graded by the monomial degree; here G0 = 0,

G₁is spanned by the generators,

3. he(0+),g_l>0i ⊂g(the Pohlmeyer-Rehren Lie algebra ind=3).

The last example is a special case of the following more general proposition.

Proposition 4.5.11. LetGbe a degreewise finite-dimensionalN0-graded Lie algebra with semisimple stratumG0.

Recall that in this situation (cf. subsection 2.2.3)G₀ has a root space decomposition with base {α1, . . . , αb}and set of positive roots R⁺, and (cf. subsection 2.3.1) each stratumG_lcan be decomposed into weight spaces relative toG0.

Then the Lie subalgebraG˜ (ofG) generated by

{x∈G^β₀ |β∈R⁺} ∪G≥1 (4.38) is pseudo-Hall-exhaustible.

Proof. Consider that forl>1, ˜G_l =G_lis a G₀-module, so the same weight space decomposi-tion applies. Without loss of generality (otherwise, decomposegandg_iaccordingly)g∈G^λ_l for some weightλ∈Γandg_i ∈ g^β₀ⁱ for some positive rootβi∈R⁺for alli. By theorem 2.3.3:2, there existki ∈N0,i∈ {1, . . . ,b}such that

λ = σl−

b

X

i=1

k_iαi, (4.39)

whereσlis the highest weight of G_l. In particular, K:=

n

X

i=1

k_i ≥0. (4.40)

Similarly, by the definition ofR⁺, there exist coefficientsm_i^j∈N0such that for allβ^j ∈R⁺: β^j =

n

X

i=1

m_i^jαi (4.41)

withPn

i=1mⁱ_j>0 (otherwise,βj=0<R⁺). Now, because of theorem 2.3.2:2

adgpadg_p−1. . .adg₁g∈G˜^λ_l^p (4.42)

with

λp := λ−

=:k_i^j

z }| {



k_i−

p

X

j=1

m_i^j



αi. (4.43)

But forp>Kwe have

K^p :=

n

X

i=1

k^p_i <0, (4.44)

so G^λ_l^p = 0 (λp is not a weight) because otherwise, there would be a contradiction to the analogue to equation (4.40) forλp(instead ofλ) that would hold ifλpwere a weight.

Remark 4.5.12. Pseudo-Hall-exhaustibility is a stronger property for graded Lie algebras than being degreewise finite dimensional. For instance, the Pohlmeyer-Rehren Lie algebra ford =3 is degreewise finite dimensional but not pseudo-Hall-exhaustible, because G can be decomposed into eigenspaces of ad_h, an inner automorphism that is not identically zero.

Let now G be a pseudo-Hall-exhaustible Lie algebra generated by the degreewise finite setX. In this context, we can formulate the following variation ofPHall:

Algorithm 8PHallExh

1: (H,B) :=PHall(X₀,l_max) with large enoughl_max .see remark 4.3.13

2: fork=1, . . . ,l_maxdo

3: S˜ := {h ∈ X|deg_e(h) = k} ∪

(h,g) ∈ H × H| deg_e(h) + deg_e(g) = kand IsHallElement(h,g)=True

4: while# ˜S>0do

5: Remove thosesi∈S˜with vanishing image in G and append them toB

6: S:=S_S˜

7: S˜ :=

(h,g) ∈ S˜ ×H| deg_e(g) = 0 and IsHallElement(h,g) = True _ (g,h) ∈ H×S,˜ | deg_e(g)=0 and IsHallElement(g,h)=True

8: S :=Sort(S, >S) .such thats₁ <Ss₂<S. . . <Ss_r

9: fori=1, ...,rdo

10: ifs_iis linearly dependent ofH(in G)then

11: B:=B_{s_i+P

jc_{i j}h_{i j}}withc_{i j}∈C, h_{i j}∈Hsuch thats_i+P

jc_{i j}h_{i j}=0

12: else

13: H:=H_{s_i}

14: return(H,B)

Remark 4.5.13. Since each iteration of thewhileloop in step 4 consists of adjoining elements of external degree zero, condition 3 of pseudo-Hall-exhaustability implies that after then_k +1-th iteration of +1-the loop, ˜S={}, so the exit condition of thewhileloop is met, the loop is finite and the algorithm terminates.

Outlook

5.1 Interplay of the Hall algorithm and representation theory

Throughout the entire chapter 4, we have always used Hall orders<H and sorting orders<S

that were given a priori. It might be worthwhile to go the other way around andconstruct Hall and sorting orders that are tailored to the problem at hand.

Of particular interest might be Hall and sorting orders that give us a shortest pseudo-Hall-basis by minimizing #H_k.

In situations where some of the representation theory of the Lie subalgebraG₀is already known (for instance,g₀ so(d,C) in the case of the Pohlmeyer-Rehren Lie algebra that will be considered in the sequel), in the spirit of remark 4.5.5 we might want to find a setHthat consists – as far as possible given the constraints, for instance imposed by the axioms for Hall sets – of lowest weight vectors of irreducibleg₀-modules (expressed in terms of brackets of elements of lower degree) and their multiple adjoints of elements of g₀satisfying equation (4.34). This would allow us to see theg₀-modules more clearly in the pseudo-Hall-basis.

Unfortunately, it isn’t immediately clear how to approach this problem, and it also isn’t clear how successful we can hope to be.

Example 5.1.1. Consider the situation of remark 4.5.6 for d = 3 and further suppose that we have found some Hall elementv0 =[a0,v₁] ∈ g^m_l . (v0 ∈g^m_l with somem,lautomatically follows if the elements ofXhave definite degree and magnetic quantum number.) Now, the condition

v∈H ⇒ adⁿ_xv∈H for alln∈N0 (5.1) would be desirable because this aligns the Hall set neatly with the irreducibleg₀-modules.

But then, in particular [x,v₀] ∈ H. By axiom 2 (b) of a Hall set (4.2.6), this impliesv₀ ∈Xor a₀≤_H x. In the former case,v₀∈g_l∩X. We consider the latter case.

Now suppose the conditions from 4.5.6 on <H are met, then this implies a0 = x, and therefore furtherv₁ ∈ g^m−1

l ). Since we havea₀ =x, we can inductively apply the reasoning applied tov0tov₁and inductively further tov_i(that have the propertyv0=adⁱ_xv_i). Because we cannot continue the iteration more thanm+l+1 times before the latter case is excluded due to corollary 2.3.12:3, it follows that

v₀ =ad^r_xv_r

with somev_r∈Xwithr≤m+l+1 and deg_ev_r=deg_ev₀, in particularv_r∈g^m_l ^−r.

We have shown that wecan’t express the elements vr as brackets of elements of lower degree, unlike our intention. Since this can be done for arbitrarily highl, the setXmust be infinite.

If we want to uphold condition (5.1), we therefore have to choose a Hall order<H that violates the conditions from 4.5.6. But unfortunately, it is not obvious how to do this.

Im Dokument Structure Analysis of the Pohlmeyer-Rehren Lie Algebra and Adaptations of the Hall Algorithm to Non-Free Graded Lie Algebras (Seite 115-120)