Root subsystems of rank 2 hyperbolic root systems
Lisa Carbone, Matt Kownacki, Scott H. Murray, Sowmya Srinivasan
RROOT SUBSYSTEMS OF RANK 2 HYPERBOLIC ROOT SYSTEMS
LISA CARBONE, MATT KOWNACKI, SCOTT H. MURRAY AND SOWMYA SRINIVASANA
BSTRACT . Let ∆ be a rank 2 hyperbolic root system. Then ∆ has generalized Cartan matrix H ( a, b ) = (cid:16) − b − a (cid:17) indexed by a, b ∈ Z with ab ≥ . If a (cid:54) = b , then ∆ is non-symmetric and is generated by onelong simple root and one short simple root; whereas if a = b , ∆ is symmetric and is generated by two longsimple roots. We prove that if a (cid:54) = b , then ∆ contains an infinite family of symmetric rank 2 hyperbolicroot subsystems H ( k, k ) for certain k ≥ , generated by either two short or two long simple roots. We alsoprove that ∆ contains non-symmetric rank 2 hyperbolic root subsystems H ( a (cid:48) , b (cid:48) ) , for certain a (cid:48) , b (cid:48) ∈ Z with a (cid:48) b (cid:48) ≥ . One of our tools is a characterization of the types of root subsystems that are generated by a subsetof roots. We classify these types of subsystems in rank 2 hyperbolic root systems.
1. I
NTRODUCTION
Let ∆ be the root system of a rank 2 Kac–Moody algebra g . Then ∆ has generalized Cartan matrix H ( a, b ) := (cid:0) − b − a (cid:1) for some a, b ∈ Z with ab ≥ . Let S = { α , α } denote a basis of simple roots of ∆ . If a (cid:54) = b , then ∆ is non-symmetric and its base consists of a long simple root and a short simple root,whereas if a = b , ∆ is symmetric and its base consists of two long simple roots.The root system ∆ contains two types of roots; real and imaginary. The real roots of ∆ are of the form wα i for some w ∈ W , where W is the Weyl group of the root system. In the rank 2 hyperbolic case, W ∼ = D ∞ , is the infinite dihedral group. The additional imaginary roots will not play a significantrole in this work. The real roots are supported on the branches of a hyperbola in R (1 , , with a pair ofbranches for each root length (Figure 1).We are also motivated by the following question. If α and β are real roots of a Kac-Moody group, thenthe commutator [ χ α , χ β ] involves the root group corresponding to α + β . The commutator [ χ α , χ β ] isthen trivial if the sum α + β is not a root. In order to determine the non–trivial commutators in the Kac–Moody group associated to H ( a, b ) , we may therefore reduce to the study of rank 2 root subsystems ina general Kac–Moody root system. This observation provides the setting for the current work.As observed by Morita ([Mor], [Mor2]), to determine the group commutators, it is also necessary tocharacterize the set of positive Z –linear combinations ( Z > α + Z > β ) ∩ ∆ re ( H ( a, b )) for real roots α, β whose sum is a real root. A similar question was answered in arbitrary Kac–Moody root systems byBillig and Pianzola ([BP]). We obtain an explicit description of this set for all rank 2 hyperbolic rootsystems.In future work, we will use these results to determine the non–trivial group commutators and theirstructure constants ([CMW]).We obtain proofs of the results stated by Morita ([Mor]) that if a and b are both greater than one, thenno sum of real roots can be a real root. It follows that the prounipotent subgroup corresponding to thepositive real roots on a single branch of the hyperbola is commutative. When a or b = 1 we prove,as stated in [Mor], that the prounipotent subgroup generated by all the positive real short root groups This research made extensive use of the Magma computer algebra system. a r X i v : . [ m a t h - ph ] N ov s metabelian and the prounipotent subgroup generated by all the positive real long root groups iscommutative. Our results in Sections 3 and 4 also cover the affine cases H (2 , and H (4 , .In order to make our results precise, we use two different concepts of a subsystem generated by a subset Γ of real roots: namely a subsystem Φ(Γ) , corresponding to a reflection subgroup of the Weyl groupand consisting entirely of real roots; and ∆(Γ) consisting of all roots that can be written as an integrallinear combination of elements of Γ . Such a ∆(Γ) subsystem contains both real and imaginary roots andcorresponds to a certain subalgebra of the Kac–Moody algebra.We have classified both kinds of subsystem inside a rank 2 infinite root system, and found that the twoconcepts of subsystem are equivalent in almost all cases: Theorem 1.1.
Let ∆ be a rank 2 infinite root system and let Γ be a set of real roots which generate ∆ , that is, ∆(Γ) = ∆ . Then either Φ(Γ) is the set of all real roots in ∆ or it is the set of all short real roots in ∆ . The secondcase occurs only if a = 1 or b = 1 and Φ(Γ) consists of short roots.
Our classification also gives us the following result, which holds for either concept of subsystem:
Theorem 1.2. If ∆ is a rank 2 hyperbolic root system, then ∆ contains symmetric rank 2 hyperbolic root sub-systems of type H ( k, k ) for infinitely many distinct k ≥ . If ∆ is non-symmetric of type H ( a, b ) , then it alsocontains non-symmetric rank 2 hyperbolic root subsystems of type H ( a(cid:96), b(cid:96) ) for infinitely many distinct (cid:96) ≥ . We also classify the rank 2 Φ -subsystems as finite, affine or hyperbolic systems: Theorem 1.3.
Let ∆ be a rank 2 root system and let Γ be a nonempty set of real roots in ∆ . (i) If ∆ is finite, then Φ(Γ) is finite. (ii) If ∆ is affine of type (cid:101) A , then Φ(Γ) has finite type A or affine type (cid:101) A . (iii) If ∆ is affine of type (cid:101) A (2)2 , then Φ(Γ) has finite type A , or affine type (cid:101) A or (cid:101) A (2)2 . (iv) If ∆ is hyperbolic, then Φ(Γ) has finite type A or hyperbolic type. We mention the following related works: This work was inspired by the papers [Mor] and [Mor2]where the results of interest were stated without proof. Feingold and Nicolai ([FN], Theorem 3.1) gavea method for generating a subalgebra corresponding to a ∆(Γ) –type root subsystem for a certain choiceof real roots in any Kac–Moody algebra.As in Section 4 of this paper, Casselman ([C]) reduced the study of structure constants for Kac–Moodyalgebras to rank 2 subsystems. Some of our results in Section 4 overlap with Section 4 of [C].Tumarkin ([T]) gave a classification the sublattices of hyperbolic root lattices of the same rank. However,he requires conditions on the possible angles between roots that exclude all but a finite number of rank2 hyperbolic root systems. In contrast, for our intended application to Kac–Moody groups, we requirethe explicit construction of the embedding of the simple roots of a subsystems into the ambient system,rather than just describing its root lattice.The authors are very grateful to Chuck Weibel for his careful reading of the MSc thesis ([Sr]) of thefourth author. This research was greatly facilitated by experiments carried out in the computationalalgebra systems Magma ([BCFS]) and Maple ([M]). . R EAL R OOTS
Let A = H ( a, b ) be the × generalized Cartan matrix A = H ( a, b ) = ( a ij ) i,j =1 , = (cid:18) − b − a (cid:19) for positive integers a, b , with Kac–Moody algebra g = g ( A ) , root system ∆ = ∆( A ) , and Weyl group W = W ( A ) . When ab < , A is positive definite and so ∆ is finite. When ab = 4 , A is positive semi-definite but not positive definite and so ∆ is affine. When ab > , A is indefinite but every propergeneralized Cartan submatrix is positive definite, and so A is hyperbolic. Without loss of generality, weassume that a ≥ b .Let S = { α , α } denote a basis of simple roots of ∆ . We have the simple root reflections w j ( α i ) = α i − a ij α j for i = 1 , with matrices with respect to S [ w ] S = (cid:18) − b (cid:19) , [ w ] S = (cid:18) a − (cid:19) . The Weyl group W = W ( A ) is the group generated by the simple root reflections w and w .Let B = B ( a, b ) = (cid:18) a/b − a − a (cid:19) be a symmetrization of A . This defines the symmetric bilinear form ( u, v ) = [ u ] TS B [ v ] S and quadraticform || u || = ( u, u ) , which are preserved under the action of W . So the set of real roots is ∆ re = W α ∪ W α . Since ∆ is finite, affine, or hyperbolic, the set of imaginary roots is ∆ im = { α ∈ Z α + Z α | α (cid:54) = 0 and || α || ≤ } . A diagram of the hyperbolic root system H (5 , is given in Figure 1.Every root α ∈ ∆ has an expression of the form α = k α + k α where the k i are either all ≥ , in whichcase α is called positive , or all ≤ , in which case α is called negative . The positive roots are denoted ∆ + ,the negative roots ∆ − .Now || α || = 2 a/b and || α || = 2 . So all real roots xα + yα in the orbit W α satisfy ax − abxy + by = b, and all real roots xα + yα in the orbit W α satisfy ax − abxy + by = a. These curves are displayed in Figures 1–3 as blue (resp. red) dotted lines. These curves are elliptical forfinite systems, straight lines for affine systems, and hyperbolas for hyperbolic systems. If ∆ is nonsym-metric ( a > b ) the roots in W α are called long and the roots in W α are called short . If ∆ is symmetric( a = b ) then all roots are considered to be long . Note that (with the exception of A ), the real roots fallinto two distinct orbits under the action of W . The diagrams use red for the orbit of α , blue for the orbitof α , and black for the imaginary roots. The horizontal lines indicate the action of w while the verticallines indicate the action of w . This is the transpose of the generalized Cartan matrix A in [ACP]. IGURE
1. Root system of type H (5 , For j ∈ Z , we define α LLj := ( w w ) j α , α LUj := ( w w ) j w α α SUj := ( w w ) j α , α SLj := ( w w ) j w α . All real roots are given by these four sequences. If ab ≥ , then these are all distinct, and a root is positiveif and only if j ≥ . The roots for j = − , , are given in Table 1. j − α LLj − α − aα α ( ab − α + aα α LUj − α α + aα ( ab − α + a ( ab − α α SUj − bα − α α bα + ( ab − α α SLj − α bα + α b ( ab − α + ( ab − α T ABLE
1. Examples of real rootsThe following lemma characterizes the real roots in terms of recursive sequences η j and γ j . Values ofthese sequences for small j are given in Table 2. Lemma 2.1. ([ACP], Lemmas 3.2 and 3.3) For all integers j , α LLj = η j α + aγ j α , α LUj = η j α + aγ j +1 α ,α SUj = bγ j α + η j α , α SLj = bγ j +1 α + η j α , γ j η j ab − ab − a b − ab + 13 a b − ab + 3 a b − a b + 6 ab − a b − a b + 10 ab − a b − a b + 15 a b − ab + 15 a b − a b + 21 a b − ab + 5 a b − a b + 28 a b − a b + 15 ab − a b − a b + 36 a b − a b + 35 ab − a b − a b + 45 a b − a b + 70 a b − ab + 17 a b − a b + 55 a b − a b + 126 a b − ab + 7 a b − a b + 66 a b − a b + 210 a b − a b + 28 ab − a b − a b + 78 a b − a b + 330 a b − a b + 84 ab − a b − a b + 91 a b − a b + 495 a b − a b + 210 a b − ab + 1 T ABLE
2. Values of η j and γ j for small j where (i) γ = 0 , γ = 1 , η = 1 , η = ab − ; (ii) η j = abγ j − η j − ; (iii) γ j = η j − − γ j − ; (iv) both sequences X j = η j and γ j satisfy the recurrence relation X j = ( ab − X j − − X j − . Note that these are both generalized Fibonacci sequences provided that ab > . In particular, γ j is theLucas sequence with parameters P = ab − , Q = 1 .The following is a useful lemma giving negatives of roots: Lemma 2.2.
For all j ∈ Z , γ − j = − γ j and η − j = − η j − . Also − α LLj = α LU − j − , − α LUj = α LL − j − , − α SUj = α SL − j − , − α SLj = α SU − j − .
3. S
UMS OF REAL ROOTS
Let ∆ be an infinite rank 2 root system of type H ( a, b ) with a ≥ b and ab ≥ . In this section we determineall real roots α, β ∈ ∆ for which α + β is also a real root.We will split our analysis into two cases: that in which a ≥ b > , and that in which a > b = 1 . We findin the first case that the sum of two real roots is never a real root, and in the second case that there arecertain β ∈ ∆ re so that β ± α i ∈ ∆ re .3.1. The case a ≥ b > .Lemma 3.1. If a ≥ b > , then bγ < η < bγ < η < bγ < · · · , aγ < η < aγ < η < aγ < · · · . In fact the gaps between sequence elements are nondecreasing, that is, for j ≥ , η j +1 − bγ j +1 ≥ bγ j +1 − η j ≥ η j − bγ j ,η j +1 − aγ j +1 ≥ aγ j +1 − η j ≥ η j − aγ j . Proof.
To see that the gaps in the sequences are nondecreasing, we apply Lemma 2.1 as follows: η j +1 − bγ j +1 = ( a − bγ j +1 − η j ≥ bγ j +1 − η m = ( b − η j − bγ j ≥ η j − bγ j . The other result is similar. (cid:3) he inequalities in Lemma 3.1 show that the real roots have the ”staircase pattern” shown in Figure 2.F IGURE
2. The positive real roots for H ( a, b ) with a ≥ b > Proposition 3.2. If a ≥ b > and α, β ∈ ∆ re , then α + β / ∈ ∆ re .Proof. We can write α = wα i for i = 1 or 2, and some w ∈ W . We may also replace β by w − β . Thus wewish to determine the β ∈ ∆ re for which α i + β ∈ ∆ re .Replacing β by − β if β ∈ ∆ re − , we wish to determine the β ∈ ∆ re + for which α i ± β ∈ ∆ re . Thus we maytake α = ± α i and β ∈ ∆ re + .From Figure 2, it is clear that β ± α i ∈ ∆ re only when one of the differences η j +1 − η j or γ j +1 − γ j equals1. By Lemma 3.1, this cannot occur. (cid:3) The case a > b = 1 . This case is considerably more intricate.We define real functions Ψ ± ( x ) := (cid:16) ( x − ± (cid:112) x ( x − (cid:17) , for which ψ ± = Ψ ± ( ab ) are the character-istic roots of the recurrence equation in Lemma 2.1(iv).The following lemma gives a bound on these parameters. Lemma 3.3. If ab > , then ψ + > . and < ψ − < . .Proof. We can find Ψ (cid:48)± ( x ) = 12 (cid:32) ∓ x − (cid:112) x ( x − (cid:33) and so Ψ + is increasing for x ≥ and Ψ − is positive and decreasing for x ≥ . Hence ψ + ≥ Ψ + (5) > . and < ψ − ≤ Ψ − (5) < . . (cid:3) efine λ := ψ + ψ + − , µ := 1 (cid:112) ab ( ab − . The following lemma is an easy consequence of Lemma 3.3 and shows that the sequences η j and γ j areeach within a small constant of being exponential with base ψ + . Lemma 3.4. If ab > , then, for j ≥ , λψ j + − . < η j < λψ j + ,µψ j + − . < γ j < µψ j + , where ψ + > . , < λ < . , and < µ < . . The following lemma shows that the roots have the ”staircase pattern” shown in Figure 3.6.
Lemma 3.5. If a > and b = 1 , then γ < η = γ < γ < η < γ < η < · · · , aγ < η < η < aγ < η < aγ < η < aγ < · · · . Proof.
For the first two inequalities, the previous lemmas show that < µ < . and < λ < . .Also, we have: λψ + − aµ = ψ ψ + − − a (cid:112) a ( a − ≥ . − . > and µψ − λ = ψ (cid:112) a ( a − − ψ + ψ + − > . ψ − > . For j ≥ we have γ j +1 = η j − γ j < η j . Similarly for j ≥ we have η j +1 = aγ j +1 − η j < aγ j . Now η j − aγ j − > λψ j + − . − aµψ j − = ( λψ + − aµ ) ψ j − − . ≥ ψ j − − . > and so aγ j < η j +1 . And γ j +1 − η j − > µψ j +1+ − . − λψ j − = ψ j − ( µψ − λ ) − . ≥ ψ j − − . > and so γ j +1 > η j − . (cid:3) We now use the above results to determine the β ∈ ∆ re for which β ± α i ∈ ∆ re , for i = 1 , . IGURE
3. The positive real roots for H ( a, with a > Theorem 3.6. If a ≥ , b = 1 and β ∈ ∆ re + then (i) β + α ∈ ∆ re if and only if β = α ; (ii) β − α ∈ ∆ re if and only if β = α + α ; (iii) β + α ∈ ∆ re if and only if β = α or α + ( a − α ; (iv) β − α ∈ ∆ re if and only if β = α + α or α + aα .Proof. For (i), note that if β = α , then β + α = α + α = α SL , by Table 1. Thus β + α is a real root.Conversely, let β ∈ ∆ re such that β + α ∈ ∆ re . First suppose that β = α SUj for some j . By Lemma 2.1,we have β j = γ j α + η j α . Then β + α = ( γ j + 1) α + η j α . If β + α is a long root on a lower branch, then again by Lemma 2.1 we have β + α = η k α + aγ k α for some k . Then we must have γ j + 1 = η k and η j = aγ k , but by Lemma 3.1 there are no j, k suchthat η j = aγ k . Similarly, there are no j, k such that γ j + 1 = η k and η j = aγ k +1 , so β + α cannot be along root on an upper branch. If β + α is a short root on an upper branch, then Lemma 2.1 implies that γ j + 1 = γ k and η j = η k , but again, no such j, k can exist: if β + α is a short root on an upper branch,then we must have j, k so that γ j + 1 = γ k +1 and η j = η k . The second of these conditions implies that j = k . Then by the first of these conditions we have γ j +1 − γ j = 1 . Then j must be . So β + α = α SL = α + α , o β is α . Similarly we may check that if β = α LLj , β = α LUj , or β = α SLj for some j > , then β + α (cid:54)∈ ∆ re .Following similar reasoning, we can check that the second, third and fourth claims hold. (cid:3) By a straightforward case-by-case argument, we can now prove the following result about lengths ofsums of roots.
Theorem 3.7.
Let ∆ be an infinite rank 2 root system. (i) If α, β, α + β ∈ ∆ re with α and β short, then α + β is long. (ii) If α, β, α + β ∈ ∆ re with α short and β long, then α + β is short. (iii) If α, β ∈ ∆ re with α and β long, then α + β / ∈ ∆ re . We note that (i) and (iii) are not true in finite root systems of type A or G . However there is a slightlyweaker result that holds in any symmetrizable system: Theorem 3.8.
Let ∆ be a symmetrizable root system and suppose α, β, α + β ∈ ∆ re . (i) If || α || = || β || , then || α + β || = a || α || for some positive integer a . (ii) If || α || (cid:54) = || β || , then || α + β || = min( || α || , || β || ) .Proof. We only need to consider the rank 2 subsystem Z { α, β } ∩ ∆ . These results are easily shown to betrue if the subsystem has finite type A , B , or G , and they follow from the previous proposition if thesubsystem is infinite. (cid:3)
4. S
UBSYSTEMS
Root systems can be used to describe three different structures: Coxeter groups, Kac–Moody algebrasand Kac–Moody groups. These three structures lead to two different concepts of subsystem, since theLie correspondence ensures that Kac–Moody algebras and groups give the same subsystems.In this section, we describe two distinct types of root subsystem and show that the two concepts usuallycoincide, but not always. We also classify all subsystems of infinite rank 2 root systems.Suppose ∆ is a symmetrizable root system with simple roots Π = { α , . . . , α (cid:96) } . Let W = W (∆) be theWeyl group and let ∆ re = W Π denote the real roots.For Γ ⊆ ∆ re , the reflection subgroup generated by Γ is defined as W Γ = (cid:104) w α : α ∈ Γ (cid:105) . Then W Γ is also a Coxeter group. We define Φ(Γ) = W Γ Γ , that is, the closure of Γ under the action of W Γ . We call this a Φ -subsystem (also noted in [C], Proposition7). Note that a Φ -subsystem consists entirely of real roots.Let α be any real root. Then there is a corresponding pair of root vectors x α and x − α in g = g (∆) whichgenerate a subalgebra isomorphic to sl . We denote this subalgebra by sl ( α ) . Now let Γ ⊆ ∆ re . We maydefine the fundamental Kac–Moody subalgebra corresponding to Γ to be g Γ = (cid:104) h , sl ( α ) : α ∈ Γ (cid:105) . Then g Γ is a Kac–Moody algebra and its root system is ∆(Γ) = Z Γ ∩ ∆ , hat is the set of all roots in ∆ that can be written as an integer linear combination of elements of Γ . Wecall this a ∆ -subsystem . The Kac–Moody subalgebra of [FN], Theorem 3.1 is of this type. We also define ∆ re (Γ) = Z Γ ∩ ∆ re .4.1. Subsystems corresponding to submatrices.
In this section we discuss one of the easiest ways toconstruct subsystems, and show that ∆ - and Φ -subsystems coincide in this case. Let g be a Kac–Moodyalgebra with generalized Cartan matrix A = ( a ij ) i,j ∈ I , I = { , , . . . , (cid:96) } , Cartan subalgebra h of dimen-sion (cid:96) − rank( A ) , simple roots Π = { α , . . . , α (cid:96) } ⊆ h ∗ and simple coroots Π ∨ = { α ∨ , . . . , α ∨ (cid:96) } ⊆ h . Let Q denote the root lattice of g and let g = h ⊕ (cid:16)(cid:76) α ∈ Q \{ } g α (cid:17) denote the root space decomposition. Let ∆ denote the set of all roots.Let B = ( a ij ) i,j ∈ K be a submatrix of A for some K ⊆ I with | K | = (cid:96) . Let h ( B ) be a subspace of h ofdimension (cid:96) − rank( A ) containing Π( B ) ∨ = { α ∨ i | i ∈ K } , and such that Π( B ) = { α i | h ( B ) ∗ | i ∈ K } islinearly independent. Set Q = (cid:76) i ∈ K Z α i . Then g ∼ = h ( B ) ⊕ (cid:77) α ∈ Q \{ } g α is the Kac–Moody algebra of B with Cartan subalgebra h ( B ) , simple roots Π( B ) and simple coroots Π( B ) ∨ ([K], Exercise 1.2). We identify Q ⊆ Q ⊂ h ∗ with Z Π( B ) ⊂ h ( B ) ∗ in the obvious way. Proposition 4.1. (Proposition 6, [Mo]) Let A = ( A ij ) i,j ∈ I be a generalized Cartan matrix with I = { , , . . . , (cid:96) } .Let K ⊂ I , K (cid:54) = ∅ . Let Π( A ) = { α , . . . , α (cid:96) } be the simple roots of the root system ∆( A ) . Let B = ( A ij ) i,j ∈ K .Let ∆( B ) denote the root system corresponding to B . Let Π( B ) = Π( A ) ∩ ∆( B ) . Then Π( B ) (cid:54) = ∅ and ∆( B ) has the properties: ∆( B ) = ∆ ∩ Z Π( B ) where Z Π( B ) denotes all integral linear combinations of Π( B ) and ∆( B ) re = ∆ re ∩ Z Π( B ) , ∆( B ) im = ∆ im ∩ Z Π( B ) . Proposition 4.2.
Using the notation above, define Φ( B ) = W Π( B ) (Π( B )) . Then ∆( B ) re = Φ( B ) . That is, Φ( B ) and ∆( B ) re subsets coincide.Proof. Our claim is that ∆ re ∩ Z Π( B ) = W Π( B ) (Π( B )) . The inclusion ∆( B ) re ⊆ Φ( B ) is clear. To prove the reverse inclusion, let α ∈ Φ( B ) . Then α ∈ ∆ re and α ∈ Z Π( B ) by definition. Hence Φ( B ) ⊆ ∆( B ) re . (cid:3) The following lemma establishes a useful property of ∆( B ) subsystems. Lemma 4.3. ∆( B ) re subsets are closed under taking integral sums of the simple roots corresponding to B .Proof. We have ∆( B ) re = ∆ re ∩ Z Π( B ) , but since B is a subsystem arising from a submatrix of thegeneralized Cartan matrix, B has an associated root lattice Q = (cid:76) i ∈ K Z α i which is closed under takingintegral sums. (See also Section 4, in particular Lemma 6, of [C]). (cid:3) Since Φ( B ) and ∆( B ) re subsets coincide, we claim that Φ( B ) subsets have this property as well. roposition 4.4. Using the notation above, Φ( B ) subsets are closed under taking integral sums of the simpleroots corresponding to B .Proof. Let α, β ∈ Φ( B ) . We recall that − pα + β, . . . , β − α, β, α + β, . . . , qα + β is the α –string through β . We claim that for s, t ∈ Z , sα + tβ ∈ Z Π( B ) . Let α = (cid:88) i ∈ K a i α i and β = (cid:88) i ∈ K b i α i . Writing elements of the root string in terms of their coordinates on the root lattice Q , it is clear that theyare all elements of Z Π( B ) . For example − pα + β = (cid:80) i ∈ K ( − pa i + b i ) α i . (cid:3) Classification of Φ -subsystems in rank 2. Our next step is to classify the Φ -subsystems in anyinfinite rank 2 root system using explicit formulas for the Weyl group reflections. Let ∆ be a root systemof type H ( a, b ) for a ≥ b and ab ≥ . Let Γ ⊆ ∆ re be nonempty.First we note that Φ(Γ) is closed under negation, since w α α = − α . So, using the formulas of Lemma 2.2, Φ(Γ) = { α LLj , α LU − j − , α SUk , α SL − k − | j ∈ I L , k ∈ I S } , for some index sets I L , I S ⊆ Z . Every real root has the form α = wα i for i = 1 , and w ∈ W , so thereflection in α is w α = ww i w − . We obtain the following formulas for the reflections corresponding toeach real root: w LLj = w LU − j − = ( w w ) j w , w SUj = w SL − j − = ( w w ) j w . We can use this to easily prove formulas for the action of a reflection on a real root:
Lemma 4.5.
For all j, k ∈ Z , w LLk α LLj = − α LL k − j , w SUk α SUj = − α SU k − j ,w LLk α SUj = − α SU − k − j − , w SUk α LLj = − α LL − k − j − . Lemma 4.6.
Given integers j and k : (i) If j, k ∈ I L , then j + ( k − i ) Z ⊆ I L . (ii) If j, k ∈ I S , then j + ( k − j ) Z ⊆ I S . (iii) If j ∈ I L , k ∈ I S , then j + (2 j + 2 k + 1) Z ⊆ I L and k + (2 j + 2 k + 1) Z ⊆ I S .Proof. Suppose I S contains (cid:96) := j + ( n − k − j ) and m := j + n ( k − j ) . Then Lemma 4.5 shows that j + ( n + 1)( k − j ) = 2 (cid:96) − m ∈ I S and j + ( n − k − j ) = 2 m − (cid:96) ∈ I S . Part (i) now follows by bidirectionalinduction. Part (ii) is similar.Let d := 2 j +2 k +1 . Now suppose j, j + nd ∈ I L and k, k + nd ∈ I S . Then j − ( n +1) d = − k − ( j + nd ) − ∈ I L and so j + ( n + 1) d = 2 j − ( j − ( n + 1) d ) ∈ I S . Similar arguments show that j + ( n − d ∈ I S and k + ( n ± d ∈ I L . Part (iii) now follows by bidirectional induction. (cid:3) We can now classify the Φ -subsystems in terms of their index sets: Proposition 4.7. (i) If I S is empty, then I L = r + d Z for some r, d ∈ Z with d ≥ and ≤ r < d . (ii) If I L is empty, then I S = r + d Z for some r, d ∈ Z with d ≥ and ≤ r < d . (iii) Otherwise, I L = r + (2 d + 1) Z and I S = d − r + (2 d + 1) Z for some d ≥ and d ≤ r ≤ d . ype Integer conditions Simple roots Cartan Matrix Inner product matrixI L r arbitrary α LLr A ab A I S r arbitrary α SUr A A II L d > , ≤ r < d α LLr , α
LUd − r − H ( δ d , δ d ) ab H ( δ d , δ d ) II S d > , ≤ r < d α SUr , α
SLd − r − H ( δ d , δ d ) H ( δ d , δ d ) II LS d ≥ , − d ≤ r ≤ d α LLr , α
SUd − r H ( a(cid:15) d , b(cid:15) d ) B ( a(cid:15) d , b(cid:15) d ) T ABLE Φ -subsystems of rank 2 root systems Proof. (i) Suppose I S empty and let J = { j ∈ Z | α LLj ∈ Γ or α LU − j − ∈ Γ } , so Φ(Γ) = Φ( { α LLj | j ∈ J } ) .If J contains a single element, then take r to be that element and d = 0 . Otherwise, let d be the greatestcommon divisor of all the integers j − k for j, k ∈ J with j (cid:54) = k . Let r be the remainder of j ∈ J divided by d , which is the same for all j ∈ J . Then standard properties of integer lattices together withLemma 4.6(i) show that J ⊆ r + d Z ⊆ I L . It now suffices to show that { α LLj , α LU − j − | j ∈ r + d Z } is a Φ -subsystem, but this follows immediatelyfrom Lemmas 2.2 and 4.5. The proof of (ii) is similar to (i).(iii) The orbits of W on ∆ re are W α and W α , so Φ(Γ) ∩ W α = { α LLj , α LU − j − | i ∈ I L } and Φ(Γ) ∩ W α = { α SUj , α SL − j − | j ∈ I S } are both Φ -subsystems in their own rights. By (i) and (ii), I L = r + d Z and I S = r + d Z for some d i ≥ , ≤ r i < d i , for i = 1 , . For every m ∈ Z , we have r ∈ I L and r + md ∈ I S , so Lemma 4.6(iii) implies that r + (2 r + 2 r + 2 md + 1) m Z ⊆ r + d Z . Hence d | (2 r + 2 r + 1) + 2 md , for all m ∈ Z . So d | r + 2 r + 1 and hence d is odd, say d = 2 d + 1 . Also d | d and hence d | d . Reversing theroles of I L and I S we also get d | d , so d = d = 2 d + 1 . We can choose r such that r ≡ r (mod 2 d + 1) and − d ≤ r ≤ d , so that I L = r + (2 d + 1) Z . Finally r + 2 r + 1 ≡ d + 1) , so r ≡ r + 2 dr + 2 dr + d ≡ r − r − r + d ≡ d − r (mod 2 d + 1) , and hence I S = d − r + (2 d + 1) Z . (cid:3) Theorem 4.8.
Let ∆ be an infinite rank 2 root system of type H ( a, b ) with a ≥ b and ab ≥ . Every nonempty Φ -subsystem of ∆ has simple roots, Cartan matrix, and inner product matrix given by one of the rows in Table 3where δ d := η d − η d − and (cid:15) d := γ d +1 − γ d . In particular all Φ -subsystems of ∆ have rank at most .Proof. Let Φ (cid:48) be a Φ -subsystem of ∆ . First suppose that Φ (cid:48) ⊆ W α . Then Proposition 4.7(i) implies that Φ (cid:48) = { α LLj , α LU − j − | j ∈ r + d Z } , for some d ≥ and ≤ r < d . If d = 0 , this gives us type I L . Otherwiseit is easily shown that every positive root in Φ (cid:48) is a positive linear combination of α LLr and α LUd − r − , sothis forms a base. The Cartan matrix and inner product matrix can be computed directly from the base.For example, if the Cartan matrix is ( c ij ) then c = 2( α LLr , α
LUd − r − )( α LLr , α
LLr ) = ba ( α LLr , α
LUd − r − ) = ba (cid:0) ( w w ) r α , ( w w ) r α LUd − (cid:1) = ba ( α , α LUd − )= ba (cid:0) (cid:1) (cid:18) a/b − a − a (cid:19) (cid:18) η d − aγ d (cid:19) = ba (cid:16) ab η d − − a γ d (cid:17) = 2 η d − − abγ d = η d − − η d = − δ d , where the second last equality follows from Lemma 2.1(ii). This gives type II L .Similarly we get types I S and II S from Proposition 4.7(ii), and type II LS from Proposition 4.7(iii). (cid:3) δ d = η d − η d − (cid:15) d = γ d +1 − γ d ab − ab − a b − ab + 2 a b − ab + 53 a b − a b + 9 ab − a b − a b + 14 ab − a b − a b + 20 a b − ab + 2 a b − a b + 27 a b − ab + 95 a b − a b + 35 a b − a b + 25 ab − a b − a b + 44 a b − a b + 55 ab − a b − a b + 54 a b − a b + 105 a b − ab + 2 a b − a b + 65 a b − a b + 182 a b − ab + 13 T ABLE
4. Values of δ d and (cid:15) d for small d Values of δ d and (cid:15) d for small d are given in Table 4.We can also classify of Φ -subsystems as finite, affine or hyperbolic systems: Theorem 4.9.
Let ∆ be a rank 2 root system and let Γ be a nonempty set of real roots in ∆ . (i) If ∆ is finite, then Φ(Γ) is finite. (ii) If ∆ is affine of type (cid:101) A , then Φ(Γ) has finite type A or affine type (cid:101) A . (iii) If ∆ is affine of type (cid:101) A (2)2 , then Φ(Γ) has finite type A , or affine type (cid:101) A or (cid:101) A (2)2 . (iv) If ∆ is hyperbolic, then Φ(Γ) has finite type A or hyperbolic type.Proof. Part (i) is clear. The finite type A occurs exactly when Γ ⊆ {± α } , so we will assume from nowon that this is not the case.If ∆ is affine, then ab = 4 and it is easy to show from the recursion formulas in Lemma 2.1 that δ d = η d − η d − = 2 and (cid:15) d = γ d +1 − γ d = 1 . Parts (ii) and (iii) now follow.If ∆ hyperbolic, then ab > , and so for d > δ d = η d − η d − = ( ab − η d − − η d − − η d − > η d − − η d − − η d − = η d − − η d − = δ d − . By induction we get δ d ≥ δ = ( ab − − ab − > for all d > . It now follows that H ( δ d , , δ d ) ishyperbolic since δ d > .A similar argument shows that (cid:15) d > (cid:15) d − for d > , and so (cid:15) d ≥ (cid:15) = 1 for d ≥ . and so H ( a(cid:15) d , b(cid:15) d ) ishyperbolic. (cid:3) As part of the last proof we showed that the sequences δ d and (cid:15) d are strictly increasing when ∆ ishyperbolic, so Theorem 1.2 is now proved for Φ -subsystems.We now consider the classification of ∆ -subsystems of ∆ . Let Γ ⊆ ∆ re nonempty and recall that ∆(Γ) = Z Γ ∩ ∆ , ∆ re (Γ) = Z Γ ∩ ∆ re . Since the imaginary roots of an affine or hyperbolic root system are justthe linear combinations of real roots with nonpositive norm, it will suffice to describe ∆ re (Γ) . From thedefinition of a reflection, we can see that w α ∆ re (Γ) ⊆ ∆ re (Γ) for all α ∈ Γ , and so Φ(Γ) ⊆ ∆ re (Γ) . We also have
Φ(∆ re (Γ)) = ∆ re (Γ) , so the real roots of a ∆ -subsystem always form a Φ -subsystem, butpossibly for a dif and only iferent set of generators. The classification of ∆ subsystems reduces to divis-ibility properties for the sequences η j and γ j . emma 4.10. Let a ≥ b ≥ with ab ≥ , and let d ≥ , i ∈ Z . Then γ d δ j − d = γ j − γ j − d , (1) η d (cid:15) j − d − = γ j − γ j − d − , (2) η d δ j − d = η j − η j − d − , (3) abγ d (cid:15) j − d = η j − η j − d . (4) Proof.
The equations are easy to prove for d = 0 . Note that δ j − = η j − − η j − = ( abγ j − η j ) − ( abγ j − − η j − ) = ab(cid:15) j − − δ j , and (cid:15) j − = γ j − γ j − = ( η j − γ j +1 ) − ( η j − − γ j ) = δ j − (cid:15) j . Assume all of the equations hold for d ≤ e . First we prove (1) and (4) for d = e + 1 : γ e +1 δ j − e − = ( η e − γ e )( ab(cid:15) j − e − − δ j − e )= abη e (cid:15) j − e − − abγ e (cid:15) j − e − − η e δ j − e + γ e δ j − e = ab ( γ j − γ j − e − ) − ( η j − − η j − e ) − ( η j − η j − e − ) + ( γ j − γ j − e )= γ j + ( abγ j − η j − − η j ) + ( η j − e − − abγ j − e − ) + ( η j − e − γ j − e )= γ j + 0 − η j − e − − γ j − e − = γ j − γ j − e − ,abγ e +1 (cid:15) j − e − = ab ( η e − γ e )( δ j − e − (cid:15) j − e )= abη e δ j − e − abγ e δ j − e − abη e (cid:15) j − e + abγ e (cid:15) j − e = ab ( η j − η j − e − ) − ab ( γ j − γ j − e ) − ab ( γ j +1 − γ j − e ) + ( η j − η j − e )= η j + ab ( η j − γ j − γ j +1 ) − ab ( η j − e − − γ j − e ) + ( abγ j − e − η j − e )= η j + 0 − abγ j − e − + η j − e − = η j − η j − e − . The proof of (2) and (3) for d = e + 1 is similar. Equations (1)-(4) follow by induction for d ≥ . (cid:3) Lemma 4.11.
Let a ≥ b ≥ with ab ≥ , and let d ≥ , j ∈ Z . (i) gcd( a, η j ) = gcd( b, η j ) = 1 . (ii) γ d | γ j if and only if j ∈ d Z . (iii) η d | γ j if and only if j ∈ (2 d + 1) Z . (iv) η d | η j if and only if j ∈ d + (2 d + 1) Z . (v) γ d | η j if and only if d = 1 , when ab > . (vi) γ d | η j if and only if d = 2 e + 1 is odd and j ∈ e + (2 e + 1) Z , when ab = 4 .Proof. (i) This follows from the fact that η j ≡ ( − j (mod ab ) , which is easily proved by induction.Cases (ii)-(v) proceed by repeated application of Lemma 4.10. Part (v) also uses Lemma 2.1. Part (vi) isimmediate. (cid:3) heorem 4.12. Let ∆ be a rank 2 root system of type H ( a, b ) with a ≥ b and ab ≥ . Let Γ ⊆ ∆ re be nonempty. (i) If a > , b = 1 and Φ(Γ) is the subsystem consisting of all short roots in ∆ re , then ∆ re (Γ) = ∆ re (cid:54) = Φ(Γ) . (ii) If a = 4 , b = 1 and Φ(Γ) is a subsystem of type II S with base α SUr , α
SLd − r − for some odd d = 2 e + 1 and ≤ r < d , then ∆ re (Γ) (cid:54) = Φ(Γ) is a subsystem of type II LS with base α LLs , α
SUe − s where s ≡ e − r (mod d ) and − e ≤ s ≤ e . (iii) In all other cases, ∆ re (Γ) = Φ(Γ) .Proof. If Φ(Γ) has type I L or I S , then it is clear that ∆ re (Γ) = Φ(Γ) .Suppose Φ(Γ) has type II L . Since Φ(Γ) = ( w w ) r Φ( { α LL , α LUd − } ) , it suffices to consider r = 0 .Now α LL = α and α LUd − = η d − α + aγ d α , so ∆ re (Γ) = Z { α LL , α LUd − } ∩ ∆ re = Z { α , aγ d α } ∩ ∆ re . The root α LLj = η j α + aγ j α is in ∆ re (Γ) if and only if γ d | γ j if and only if j ∈ d Z by Lemma 4.11(ii).Also α SUj = bγ j α + η i α is in ∆ re (Γ) if and only if aγ d | η j which is not possible by Lemma 4.11(i) since a > . Hence ∆ re (Γ) = Φ(Γ) .Suppose Φ(Γ) has type II LS . Since Φ(Γ) = ( w w ) d − r Φ( { α LLd , α SU } ) , it suffices to consider r = d .We have α LLd = η d α + aη d α and α SU = α , so ∆ re (Γ) = Z { α LLd , α SU } ∩ ∆ re = Z { η d α , α } ∩ ∆ re . Now α LLj = η j α + aγ j α is in ∆ re (Γ) if and only if η d | η j if and only if j ∈ d + (2 d + 1) Z byLemma 4.11(iv). And α SUj = bγ j α + η j α is in ∆ re (Γ) if and only if η d | bγ i if and only if j ∈ (2 d + 1) Z by Lemma 4.11(iii). Hence ∆ re (Γ) = Φ(Γ) .Finally suppose Φ( A ) has type II S . Since Φ(Γ) = ( w w ) r Φ( { α SU , α SLd − } ) , it suffices to consider r = 0 .Now α SU = α and α SLd − = bγ d α + η d − α , so ∆ re (Γ) = Z { α LL , α LUd − } ∩ ∆ re = Z { bγ d α , α } ∩ ∆ re . We have α SUj = bγ i α + η j α ∈ ∆ re (Γ) if and only if γ d | γ j if and only if j ∈ d Z . And α LLj = η j α + aγ j α is in ∆ re (Γ) if and only if bγ d | η j . By Lemma 4.11(i), (v), and (vi), this can only happen if a > , b = 1 and d = 1 ; or a = 4 , b = 1 and d odd.If a > , b = 1 , and d = 1 , then Φ(Γ) is the set of all short real roots, and bγ d | η j for all j so ∆ re (Γ) = ∆ re .If a = 4 , b = 1 , and d = 2 e + 1 , then bγ d | η i if and only if j ∈ e + (2 e + 1) Z , so I S = (2 e + 1) Z , I L = e + (2 e + 1) Z , and hence ∆ re (Γ) has type II LS with the given basis. In all other cases ∆ re (Γ) = Φ(Γ) . (cid:3) Theorem 1.1 and Theorem 1.2 for ∆ -subsystems now follow. EFERENCES[ACP] Andersen, K. K. S., Carbone, L. and Penta, D.
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