Let G be a split real Kac–Moody group of arbitrary type and let K be its maximal compact subgroup, i.e. the subgroup of elements fixed by a Cartan–Chevalley involution of G. We construct non-trivial spin covers of K, thus confirming a conjecture by Damour and Hillmann. For irreducible simply-laced diagrams and for all spherical diagrams these spin covers are two-fold central extensions of K. For more complicated irreducible diagrams these spin covers are central extensions by a finite 2-group of possibly larger cardinality. Our construction is amalgam-theoretic and makes use of the generalised spin representations of maximal compact subalgebras of split real Kac–Moody algebras studied by Hainke, Levy and the third author. Our spin covers contain what we call spin-extended Weyl groups which admit a presentation by generators and relations obtained from the one for extended Weyl groups by relaxing the condition on the generators so that only their eighth powers are required to be trivial.
In [9, Section 3.5] it turned out that the existence of a spin-extended Weyl group would be very useful for the study of fermionic billards. Lacking a concrete mathematical model of that group , Damour and Hillmann in their article instead use images of afforded by various generalised spin representations as described in [10, 11], which can be realised as matrix groups.
In [9, Section 3.5, footnote 18, p. 24], Damour and Hillmann conjecture that the spin-extended Weyl group can be constructed as a discrete subgroup of a double spin cover of the subgroup of elements fixed by the Cartan–Chevalley involution of the split real Kac–Moody group of type . The purpose of this article is to confirm this conjecture, and to generalise it to arbitrary diagrams resp. arbitrary generalised Cartan matrices
In the simply-laced case our result is as follows.
Let Π be an irreducible simply-laced Dynkin diagram, i.e., a Dynkin diagram affording only single edges, let be a set of labels of the vertices of Π, and let be the subgroup of elements fixed by the Cartan–Chevalley involution of the split real Kac–Moody group of type Π. For each let and for each let
Moreover, for with , let be the standard embedding as “upper-left diagonal block” and let be the standard embedding as “lower-right diagonal block”.
Then up to isomorphism there exists a uniquely determined group, denoted , whose multiplication table extends the partial multiplication provided by , where is the equivalence relation determined by
for all and . Furthermore, there exists a canonical two-to-one central extension .
The system is called an amalgam of groups, the pair consisting of the group and the set of canonical embeddings , a universal enveloping group; the canonical embeddings are called enveloping homomorphisms. Formal definitions and background information concerning amalgams can be found in Section 3. Since all are isomorphic to one another, it in fact suffices to fix one group instead with connecting homomorphisms .
The formalization of the concept of standard embedding as “upper-left/lower right diagonal block” can be found in Section 10. Note that, since the are only given up to isomorphism, these standard embeddings are only well-defined up to automorphism of , which leads to some ambiguity. Since by  the group (and therefore each of its central extensions by a finite group) is a topological group, one may assume the to be continuous, thus restricting oneself to the ambiguity stemming from the two continuous automorphisms of , the identity and the inversion homomorphisms. This ambiguity is resolved in Section 10.
Theorem A provides us with the means of characterizing .
Let be an irreducible simply-laced Dynkin diagram, a set of labels of the vertices of Π, and for each let
be the canonical enveloping homomorphisms,
be elements of order eight whose polar coordinates involve the angle ,
Then satisfies the defining relations
To be a set of defining relations means that any product of the that in represents the identity can be written as a product of conjugates of ways of representing the identity via (R1), (R2), (R3).
As a by-product of our proof of Theorem A we show in Section 19 that for non-spherical diagrams Π the groups and are never simple; instead they always admit a non-trivial compact Lie group as a quotient via the generalised spin representation described in . The generalised spin representation of is continuous, so that the obtained normal subgroups are closed. Similar non-simplicity phenomena as abstract groups have been observed in . Furthermore, we observe that for arbitrary simply-laced diagrams the image of under the generalised spin representation is finite, generalizing [9, Lemma 2, p. 49].
Sections 3, 4, 5, 6 and 8 are introductory in nature; we revise the notions of amalgams, Cartan matrices and Dynkin diagrams and fix our notation for orthogonal and spin groups. Sections 9 and 10 deal with the classification theory of amalgams and, as a blueprint for Theorem A, identify and as universal enveloping groups of -, resp. -amalgams of type . In Section 11 we prove Theorem A.
Sections 13, 14 and 15 provide us with the necessary tools for generalizing our findings to arbitrary diagrams; they deal with equivariant coverings of the real projective plane by the split Cayley hexagon and the symplectic quadrangle and with coverings of the real projective plane and the symplectic quadrangle by trees. In Section 16 we study - and -amalgams for this larger class of diagrams. Section 17 deals with the general version of Theorem A. Section 18 deals with the proof of Theorem B and its generalization. In Section 19 we observe that our findings provide epimorphisms from and onto non-trivial compact Lie groups.
denotes the set of positive integers.
Throughout this article we use the convention if the set is used as an index. For example, if is a group, then is the same group. Note that this does not apply to superscripts, so and may differ.
For any group G, consider the following maps:
Both maps commute with any group homomorphism.
For any group G, we denote by the centre of G.
Let U be a group, and a set. A U-amalgam over I is a set
such that is a group and is a monomorphism for all . The maps are called connecting homomorphisms. The amalgam is continuous if U and are topological groups, and is continuous for all .
Let and be U-amalgams over I. An epimorphism, resp. an isomorphism of U-amalgams is a system
consisting of a permutation and group epimorphisms, resp. isomorphisms
such that for all
that is, the following diagram commutes:
More generally, let
be a U-amalgam and let
be a V-amalgam. An epimorphism is a system
consisting of a permutation , group epimorphisms and group epimorphisms
such that for all ,
that is, the following diagram commutes:
If (and only if) in the epimorphism each is an isomorphism, then one obtains an epimorphism of U-amalgams by defining and via
If this turns out to be an isomorphism of U-amalgams, by slight abuse of terminology we also call the epimorphism α an isomorphism of amalgams.
More generally, an amalgam can be defined as a collection of groups and a collection of groups with connecting homomorphisms
Since in our situation for all i there exist isomorphisms , it suffices to consider the connecting homomorphisms .
In the more general setting, an isomorphism of amalgams consists of a permutation π of the index set I and isomorphisms
A routine calculation shows that U-amalgams and isomorphisms of U-amalgams are special cases of amalgams and isomorphisms of amalgams as found in the literature.
Given a U-amalgam , an enveloping group of is a pair consisting of a group G and a set
of enveloping homomorphisms such that
that is, for the following diagram commutes:
We write and call τ an enveloping morphism. An enveloping group and the corresponding enveloping morphism are faithful if is a monomorphism for all .
Given a U-amalgam , an enveloping group is called a universal enveloping group if, given an enveloping group of , there is a unique epimorphism such that for all with one has . We write and call τ a universal enveloping morphism. By universality, two universal enveloping groups and of a U-amalgam are (uniquely) isomorphic.
The canonical universal enveloping group of the U-amalgam is the pair , where is the group given by the presentation
and where with the canonical homomorphism for all . The canonical universal enveloping group of a U-amalgam is a universal enveloping group (cf. [25, Lemma 1.3.2]).
Let U and V be groups and I an index set. Suppose
is a U-amalgam over I,
is a V-amalgam over I,
is an amalgam epimorphism ,
with is an enveloping group of .
Then the following hold:
There is a unique enveloping group , , of such that the following diagram commutes for all :
Suppose , , is a universal enveloping group of . Then there is a unique epimorphism such that the following diagram commutes for all :
If α is an isomorphism and is a universal enveloping group, then is also an isomorphism.
(a) Let . Since is an epimorphism, we must have
for the diagrams to commute; the claimed uniqueness follows. The fact that is an epimorphism also implies
Moreover, for we find
Hence is indeed an enveloping group of .
(b) On the one hand, by (a) the lower left triangle in the following diagram commutes:
On the other hand, by the definition of universal enveloping group there is a unique epimorphism making the upper right triangle commute. The claim follows.
(c) This follows from part (b) by interchanging the roles of , G and , . ∎
The following proposition will be crucial throughout this article. The typical situation in our applications will be , , .
Let U, and be groups and I an index set. Suppose
is a -amalgam over I such that
is a U-amalgam over I,
is an amalgam epimorphism ,
with is a universal enveloping group of ,
with is a universal enveloping group of ,
For , with define and , as well as . Then if
it follows that is a central extension of G by .
In this situation the epimorphism is called an -fold central extension of amalgams.
We proceed by proving the following two assertions:
Consider the following commutative diagram:
The hypothesis implies
whence centralises for all . Since is generated by the , one has
i.e., (a) holds.
Commutativity of the diagram implies and so the homomorphism theorem yields an epimorphism , . We construct an inverse map by exploiting that G and are universal enveloping groups of , resp. , in order to show that this epimorphism actually is an isomorphism. Indeed, for , let . Then
Thus one obtains a well-defined homomorphism
Then is an enveloping group for . In particular, for and one has
Since is a universal enveloping group of , there exists a unique epimorphism such that for one has
By the definition of and one finds
But and are universal enveloping groups; their uniqueness property implies that and and hence as claimed . We have shown assertion (b). ∎
Let I be a non-empty set. A generalised Cartan matrix over I is a matrix such that for all ,
is a non-positive integer,
if then .
The matrix A is of two-spherical type if for all with .
A Dynkin diagram (or short: diagram) is a graph Π with vertex set and edge set such that each edge has an edge valency of 1, 2, 3 or and, in addition, edges with valency 2 or 3 are directed. If is directed from v to w, we write . Let , and let , , , resp. be the subsets of of edges of valency 1, 2, 3, resp. . The elements of , , are called edges of type , resp. . The diagram Π is irreducible if it is connected as a graph, it is simply laced if all edges have valency 1, it is doubly laced if all edges have valency 1 or 2, and it is two-spherical if no edge has valency . If is finite, then a labelling of Π is a bijection , where .
Throughout this text, we assume all diagrams to have finite vertex set.
Let I be a non-empty set and a two-spherical generalised Cartan matrix. Then this induces a two-spherical Dynkin diagram with vertex set as follows: For with , there is an edge between i and j if and only if . The valency of the edge then is . If , then the edge is directed if and only if .
Conversely, given a two-spherical Dynkin diagram Π with vertex set V, we obtain a two-spherical generalised Cartan matrix over by setting for ,
These two operations are inverse to each other, i.e., we have and .
Note that in [4, Definition VI.§1.3, p. 167] the opposite convention for the entries of the Cartan matrix is used.
If the generalised Cartan matrix A is not of two-spherical type, it is nevertheless possible to associate a Dynkin diagram to it by labelling the edge between i and j with whenever . In this case it is, of course, not possible to reconstruct the values of and from the diagram Π.
Therefore, by convention, in this article for each edge between i and j with label we consider the values of and as part of the augmented Dynkin diagram: write between the vertex i and the label and between the vertex j and the label. In addition, an edge with label such that and have different parity shall be directed , if is even, and , if is odd. See Figure 1 for an example.
In this section we fix notation concerning the compact real orthogonal groups.
Given a quadratic space with , we set
Given , let , be the standard quadratic form on , and
Since an element of has determinant 1 or -1, we have .
Let and let be the standard basis of . Given a subset , we set
There are canonical isomorphisms
that map an endomorphism into its transformation matrix with respect to the standard basis , resp. the basis . Moreover, there is a canonical embedding
inducing a canonical embedding
which, by slight abuse of notation, we also denote by . We will furthermore use the same symbol for the (co)restriction of to . The most important application of this map in this article is for with providing the map
Let be a quadratic space and let be the tensor algebra of V. The identity provides a ring monomorphism , and the identity a vector space monomorphism ; these allow one to identify , V with their respective images in . For
define the Clifford algebra of as
The transposition map is the involution
which yields a -grading of , i.e.,
Furthermore, following [13, Section 3.1], we define the Clifford conjugation
and the spinor norm
In the following, is an anisotropic quadratic space such that .
Given , the map
is the twisted conjugation with respect to . Using the canonical identification of V with its image in , we define
which is the twisted adjoint representation.
Given and , we set
Recall that is defined to be the standard quadratic form on (cf. Definition 1) and note that in the literature one can also find the opposite sign convention.
(a) Let and let be the standard basis of . Then the following hold in for ,
The first identity is immediate from the definition. The second identity follows from polarization, as in the tensor algebra one has
where denotes the bilinear form associated to . The third identity is immediate from the first two.
(b) One has , where denotes the quaternions. Indeed, given a basis , , of , a basis of , considered as an -vector space, is given by 1, , , . By (a) the latter three basis elements square to -1 and anticommute with one another. Note, furthermore, that under this isomorphism the Clifford conjugation is transformed into the standard involution of the quaternions, i.e., the conjugation obtained from the Cayley–Dickson construction. Consequently, the spinor norm is transformed into the norm of the quaternions.
The map induces a homomorphism
Cf. [13, Proposition 1.9]. ∎
is the pin group with respect to , and
is the spin group with respect to . By Lemma 6 and the -grading of , the sets and are indeed subgroups of . Given , define
The following hold:
One has and .
The twisted adjoint representation induces an epimorphism . In particular, given , we obtain epimorphisms
with in both cases.
The group is a double cover of the group .
See [13, Theorem 1.11]. ∎
(a) By slight abuse of notation, suppressing the choice of basis, we will also sometimes denote the map by ρ.
(b) Let and be such that and , respectively, and let . Then we have . We will explicitly determine these groups for some canonical subgroups of and .
Let , let , and let be the natural embedding of algebras afforded by the inclusion
of bases of , resp. V. Then restricts and corestricts to an embedding
of groups such that the following diagram commutes:
In analogy to Notation 2 we will use the same symbol for the (co)restriction of to . The most important application of this map in this article is for with providing the map
Let . By definition,
Since for all by Remark 5 (a), for each -basis vector of and all one has
and thus for all ,
As is generated as an -algebra by the set , we in particular have
Therefore . Finally,
Since , one has
Let and . Then
Let , let and let . Then there exists an isomorphism such that the following diagram commutes:
As in Notation 2 we slightly abuse notation and also write for the map
and for the map
Consequently, we obtain the following commutative diagram:
According to [13, Corollary 1.12], the group is generated by the set and each element of the group can be written as a product of an even number of elements from this set. That is, each element is of the form
The requirement is equivalent to
Certainly, contains all elements of the form with , i.e., one obtains
i.e., the map
is a group homomorphism from the real numbers onto the circle group. The twisted adjoint representation maps the element to the transformation
i.e., the rotation of the Euclidean plane by the angle , corresponding to the matrix
In other words, is the double cover of the circle group by itself, cf. Theorem 8 (b).
Similarly, each element is of the form
and each element of the form
the real quaternions, identify with the centre of via , , let
be the standard involution, and let
be the group of unit quaternions.
By [13, Section 1.4] one has
The isomorphism in fact is an immediate consequence of the isomorphism from Remark 5 (b) plus the observation that this isomorphism transforms the spinor norm into the norm of the quaternions.
where denotes the Clifford conjugate of , cf. Definition 1. We conclude that for every there exist uniquely determined , such that
Hence, for , one has
That is, the map
is a well-defined bijection and, since
it is in fact an isomorphism of groups.
Consequently, there exists a group epimorphism
Using this isomorphism
there exists a natural homomorphism
Note that the restrictions and are both injections of into , in fact into , as the norm is multiplicative. Since the kernel of this action has order two, the homomorphism must be onto by Proposition 8. We conclude that the group is isomorphic to the group consisting of the maps
for an alternative proof see [37, Lemma 11.22].
A similar argument (or a direct computation using the twisted adjoint representation) shows that the natural homomorphism
is an epimorphism and, thus, that the group is isomorphic to the group consisting of the maps
see also [37, Lemma 11.24].
There also exists a group epimorphism
induced by the map
Altogether, one obtains the following commutative diagram:
Similarly, for let
be the canonical projection. By the homomorphism theorem of groups the map factors through and induces the following commutative diagram:
Then and and there is a continuous group isomorphism
Given an automorphism , there is a unique automorphism such that . Moreover, γ is continuous if and only if is continuous.
Define . Then
Uniqueness follows as , is an isomorphism. ∎
Given an automorphism , there is a unique automorphism such that
Let , where
and observe that . The claim now follows as in the proof of Proposition 2. ∎
Let . Given an automorphism , there is a unique automorphism such that
For , both and are perfect, cf. [24, Corollary 6.56]. By Theorem 8 (b) the group is a central extension of . Since is simply connected (see, e.g., [13, Section 1.8], it is in fact the universal central extension of .
The universal property of universal central extensions (cf. for example [19, Section 1.4C]) yields the claim: Indeed, there are unique homomorphisms
The universal property therefore implies , i.e., is an automorphism.
In fact, all automorphisms are continuous by van der Waerden’s Continuity Theorem, cf. [24, Theorem 5.64]. ∎
Let be a homomorphism such that
for some . Then .
In this section we discuss amalgamation results for compact real orthogonal groups. The results and exposition are similar to [3, 17]. The key difference is that the amalgams in the present article are constructed starting with the circle group instead of the perfect group . This leads to some subtle complications that we will need to address below.
Let Π be a simply-laced diagram with labelling . An -amalgam with respect to Π and σ is an amalgam
and for ,
The standard -amalgam with respect to Π and σ is the -amalgam
with respect to Π and σ with
for all .
The key difference between the standard -amalgam and an arbitrary -amalgam with respect to Π and σ is that, for instance, can be an arbitrary automorphism of . Automatic continuity (like van der Waerden’s Continuity Theorem, cf. [24, Theorem 5.64 and Corollary 6.56]) fails for automorphisms of the circle group whereas it does hold for the group . Hence, obviously, not every automorphism of is induced by an automorphism of and so it is generally not possible to undo the automorphism inside . Therefore Goldschmidt’s Lemma (see [15, Lemma 2.7], also [25, Proposition 8.3.2], [14, Lemma 6.16]) implies that for each diagram Π there exist plenty of pairwise non-isomorphic abstract -amalgams.
However, by [27, Section 4.G], [21, Corollary 7.16], a split real Kac–Moody group and its maximal compact subgroup (i.e., the group of elements fixed by the Cartan–Chevalley involution) both carry natural group topologies that induce the Lie group topology on their respective fundamental subgroups of ranks one and two and make the respective embeddings continuous.
It is therefore meaningful to use continuous -amalgams for studying these maximal compact subgroups. Such continuous amalgams are uniquely determined by the underlying diagram Π, as we will see in Theorem 8 below.
For each group isomorphic to one of , , , we fix a matrix representation that allows us to identify the respective groups accordingly. Our study of amalgams by Goldschmidt’s Lemma [15, Lemma 2.7] then reduces to the study of automorphisms of these groups.
Then the map , is an automorphism of such that
Given , we have
The second assertion follows analogously. ∎
The only influence of the labelling σ of an amalgam is the choice of which of the vertices , corresponds to which subgroup of . We now show that this choice does not affect the isomorphism type of the amalgam.
Let be a simply-laced diagram with labellings . Then