On Quaternionic Tori and their Moduli Spaces
Cinzia Bisi, Graziano Gentili
TL;DR
This paper classifies quaternionic tori, which are quotients of quaternions by lattices, by constructing a moduli space using slice regular functions and analyzing their automorphism groups.
Contribution
It introduces a moduli space for quaternionic tori, classifies them via special lattice bases, and identifies automorphism groups and boundary subsets.
Findings
Constructed a 12-dimensional moduli space for quaternionic tori.
Identified all tori with non-trivial automorphism groups.
Classified boundary points corresponding to automorphism groups.
Abstract
Quaternionic tori are defined as quotients of the skew field of quaternions by rank-4 lattices. Using slice regular functions, these tori are endowed with natural structures of quaternionic manifolds (in fact quaternionic curves), and a fundamental region in a -dimensional real subspace is then constructed to classify them up to biregular diffeomorphisms. The points of the moduli space correspond to suitable \emph{special} bases of rank-4 lattices, which are studied with respect to the action of the group , and up to biregular diffeomeorphisms. All tori with a non trivial group of biregular automorphisms - and all possible groups of their biregular automorphisms - are then identified, and recognized to correspond to five different subsets of boundary points of the moduli space.
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On Quaternionic Tori and their Moduli Space
Cinzia Bisi ∗, Graziano Gentili ∗
First author: Dipartimento di Matematica ed Informatica, Universitá di Ferrara, Via Machiavelli n.35, Ferrara, 44121 Italy
Second author: Dipartimento di Matematica e Informatica “U. Dini”, Universitá di Firenze, Viale Morgagni 67/A, 50134 Firenze, Italy
Abstract.
Quaternionic tori are defined as quotients of the skew field of quaternions by rank-4 lattices. Using slice regular functions, these tori are endowed with natural structures of quaternionic manifolds (in fact quaternionic curves), and a fundamental region in a -dimensional real subspace is then constructed to classify them up to biregular diffeomorphisms. The points of the moduli space correspond to suitable special bases of rank-4 lattices, which are studied with respect to the action of the group , and up to biregular diffeomeorphisms. All tori with a non trivial group of biregular automorphisms - and all possible groups of their biregular automorphisms - are then identified, and recognized to correspond to five different subsets of boundary points of the moduli space.
Key words and phrases:
Regular functions over quaternions, Quaternonic tori and their moduli space
2010 Mathematics Subject Classification:
Primary: 30G35 Secondary: 32G15, 14K10
∗ Both authors are supported by Progetto MIUR di Rilevante Interesse Nazionale *“Varietà reali e complesse: geometria, topologia e analisi armonica” * and by GNSAGA of INdAM. The first author is also supported by Progetto FIRB Geometria differenziale e teoria geometrica delle funzioni
1. Introduction
A new notion of regularity for quaternion-valued functions of a quaternionic variable was introduced in 2006, in [19, 20]. The newly defined class of (slice) regular functions has already proved to be interesting as a quaternionic counterpart of complex holomorphic functions. In this quaternionic setting, a Casorati-Weierstrass Theorem was proved in [30] and it allowed the study of the group of all biregular transformations of the space of quaternions . This group turned out to coincide with the group of all affine transformations of of the form , with and . As we can see, notwithstanding the fact that the algebraic, abstract structure of the group of biholomorphic transformations of is still unknown, that of biregular transformations of inherits the simplicity of the group .
The fact that all quaternionic regular affine transformations of form a group under composition, the group , permits the direct construction of a class of natural quaternionic manifolds (actually quaternionic curves): the quaternionic tori. These tori are studied in the present paper. Together with the quaternionic projective spaces, [28], and the Hopf surfaces, [1], they are among the few directly constructed quaternionic manifolds, and bear with them the genuine interest that accompanies any analog of elliptic complex Riemann surfaces.
In this paper we construct quaternionic tori, realized as quotients of with respect to rank- lattices, and endow them with natural structures of quaternionic -dimensional manifolds. We then use the basic features of quaternionic regular maps to characterize biregularly diffeomorphic tori by means of properties of their generating lattices; this approach introduces into the scenery the group , that plays a fundamental role in this context. In fact the use of classical results on the reduction of Gram matrices, based on the Minkowski-Siegel Reduction Algorithm, allows us to express any “normalized” rank- lattice of in terms of a generating special basis. These special bases parameterize the classes of equivalence of biregular diffeomorphism of quaternionic tori, and are used to define a fundamental set (see (7.2)) for this equivalence relation, as the subset of
[TABLE]
We will not define a special basis here in the Introduction (see Definition 6.10), but we want to say that special bases have properties that urge a comparison with the complex case of elliptic curves, like the following: if is a special basis of a rank- lattice, i.e., if , then, in particular
- (1)
; 2. (2)
, for all ; 3. (3)
, for all such that .
The fundamental set of the equivalence relation of biregular diffeomorphism among quaternionic tori has some boundary points that are equivalent. In fact there are different elements belonging to the boundary of the fundamental set, that correspond to the same torus; as an example we can take the distinct points and : the two special bases and generate the same lattice (the ring of Lipschitz quaternions) and hence the same torus. However, in (7.4) we define the proper subset of the fundamental set , which coincides with the interior of , and which is a moduli space for the subset of equivalence classes of the so called tame tori. The complete quotient of the boundary , with respect to the equivalence relation of biregular diffeomorphism of the corresponding tori, is still unknown. However, as it happens in the complex case of elliptic curves, the classification of all the boundary tori of having non trivial groups of biregular automorphisms is an important step towards the understanding of the subtle features of the geometry of the moduli space.
In this perspective, by exploiting the classification of the finite subgroups of unitary quaternions, we identify all the groups that can play the role of groups of biregular automorphisms of tori, i.e.,
[TABLE]
called respectively tetrahedral, -dihedral, -dihedral, trivial-cyclic, cyclic-dihedral and cyclic group. We then find those points of the boundary of the fundamental set which correspond to tori whose group of biregular automorphisms either contains, or is isomorphic to, one of the groups listed above. We then prove
Theorem 1.1**.**
The only quaternionic tori with a non trivial group of biregular automorphisms correspond to the following elements of :
- •
* with and [ ];*
- •
* with and [ ].*
The most structured groups of biregular automorphisms appear in the tori with the richest symmetries: examples are the torus generated by the lattice of Lipschitz quaternions and the one generated by the lattice of Hurwitz quaternions, whose groups of automorphisms are the -dihedral group and the tetrahedral group , respectively. Notice that the complex counterpart of the tori listed in Theorem 1.1 consists of the * harmonic* and equianharmonic tori, having moduli and , respectively.
Appendices present a computational approach to the study of the modulus of a torus: for example, in the last appendix, an algorithm is produced that checks if a given basis of a lattice is tame.
To conclude the Introduction, we point out that the theory of slice regular functions, presented in detail in the monograph [18], has been applied to the study of a non-commutative functional calculus, (see for example the monograph [9] and references therein) and to address the problem of the construction and classification of orthogonal complex structures in open subsets of the space of quaternions (see [17]). Recent results of geometric theory of regular functions appear in [3], [4], [5], [6], [7], [12], [13], [21], [22], [23]. Paper [14] is strictly related to the topic of the present article.
2. Preliminary results
In this section we will briefly present those results on slice regular functions that are essential for what follows.
The -dimensional real algebra of quaternions is denoted by . An element in can be expressed in terms of the standard basis, denoted by , as , where are imaginary units, , related by the multiplication rule . To every non-real quaternion we can associate an imaginary unit, with the map
[TABLE]
If instead , we can set to be any arbitrary imaginary unit. In this way, for any there exist, and are unique, , with ( if ), such that
[TABLE]
The set of all imaginary units is denoted by ,
[TABLE]
and, from a topological point of view, it is a -dimensional sphere sitting in the -dimensional space of purely imaginary quaternions. The symbol will denote the open unit ball of the space of quaternions, and the -sphere of all the points of its boundary will be denoted by .
Remark 2.1**.**
It is worthwhile recalling that the two possible multiplications of a quaternion by any fixed element , that is and , both represent Euclidean rigid motions (rotations) of .Moreover, if is any fixed non zero quaternion, then both multiplications, and , can be decomposed as the composition of a Euclidean rigid motion of and a multiplication by a strictly positive real number: the reason is immediate, since with .
To each element of there corresponds a copy of the complex plane, namely . All these complex planes, also called slices, intersect along the real axis, and their union gives back the space of quaternions,
[TABLE]
Since , with the symbol we will mean The following definition appears in [19, 20].
Definition 2.2**.**
Let be a domain in and let be a function. For all let us consider and . The function is called (slice) regular if, for all , the restriction has continuous derivatives and the function defined by
[TABLE]
vanishes identically.
The same articles introduce the Cullen (or slice) derivative of a slice regular function as
[TABLE]
for
Remark 2.3**.**
The Cullen derivative of a slice regular function turns out to be still a slice regular function (see, e.g, [18, Definition 1.7, page 2], [20]).
Using the Cullen derivative, it is possible to characterize the slice regular functions defined in the entire space , or on a ball centered at , as follows (see, e.g., [18]).
Theorem 2.4**.**
A function is regular in if and only if has a power series expansion
[TABLE]
converging in . Moreover its Cullen derivative can be expressed as
[TABLE]
in .
The existence of the power series expansion yields a Liouville Theorem, that we will use in the sequel:
Theorem 2.5** (Liouville).**
Let be slice regular. If is bounded then is constant.
3. Lattices in the space of quaternions
Let (with ) be -linearly independent vectors in .
Definition 3.1**.**
The additive subgroup of generated by is called a rank- lattice, generated by .
We will focus our attention on (topologically) discrete subgroups of , for which the following result holds:
Lemma 3.2**.**
Let be a discrete (infinite) subgroup of . Then has no accumulation points.
Proof Since is discrete, there are no accumulation points of belonging to . By contradiction, assume that there exists an accumulation point of , belonging to . Then there exists a sequence converging to . Since is a Cauchy sequence, for all there exist such that and that
[TABLE]
If we define inequality (3.1) would imply that is not isolated, being the limit of when
As a straightforward consequence we obtain:
Corollary 3.3**.**
Let be a subgroup of , let and let be the closure of . Then is discrete if and only if, for all , the intersection is a finite set.
The following classical characterization of discrete subgroups of will be used as a basic fact in the sequel (for a proof see, e.g., [29, Theorem 6.1, page 136]).
Theorem 3.4**.**
A subgroup of is a lattice if and only if it is discrete.
This last theorem implies that the study of all possible quotient spaces of with respect to a discrete additive subgroup is reduced to the case in which is a lattice. With the aim of classifying these quotients, let denote the direct product of copies of the unit circle of , and call it the -dimensional torus.
It is well known that, given the rank of a lattice in , there exists only “one” quotient, up to real diffeomorphisms (see, e.g., [29, Theorem 6.4, page 140]):
Theorem 3.5**.**
Let be a rank- lattice in (with ). Then the group is isomorphic to .
The case of a rank- lattice is the one in which the quotient originates “the” real -dimensional torus:
Corollary 3.6**.**
Let be a rank- lattice in . Then the group is isomorphic to .
As we can see, up to real diffeomorphisms the classification is quite simple. Following the guidelines of the classical theory of complex elliptic functions we will work at the classification of -(real)-dimensional, quaternionic tori, up to biregular diffeomorphisms. In the next section we will define slice quaternionic structures on tori, that will be the object of our classification.
4. A regular quaternionic structure on a -(real)-dimensional torus
Since quaternionic regular affine transformations form a group with respect to composition, and analogously to what happens for complex tori, the field induces on a quaternionic torus a structure of quaternionic manifold. This structure will be called a regular quaternionic structure or simply a quaternionic structure on . From now on, a torus endowed with a quaternionic structure will be called a quaternionic torus. Moreover, the -(real)-dimensional torus will always be denoted simply by .
To construct such a structure, we will first of all consider the classical atlas of the real torus , and then adopt the procedure used in the complex case (see, e.g., [8, 16, 31]).Let be a rank- lattice of , generated by . Consider the canonical projection and, for any , an open neighborhood of small enough to make an homeomorphism of onto its image . The atlas will consist of the local coordinate systems . If we suppose that, for , the intersection is (open and) connected, then the change of coordinates is such that for fixed . Hence the change of coordinates is a regular function. Therefore we obtain a quaternionic structure on . Using the classical approach, regular maps between quaternionic tori can be defined in the natural manner, as well as biregular diffeomorphisms between quaternionic tori, and biregular automorphisms of a quaternionic torus. We can then proceed to study the quaternionic tori up to biregular diffeomorphisms, and give the following:
Definition 4.1**.**
If there is a biregular diffeomorphism of a -(real)-dimensional torus onto a (-(real)-dimensional) torus we will say that the two tori are equivalent.
To proceed, we recall that the group of biregular transformations (or automorphisms) of consists of all slice regular affine transformations, that is
[TABLE]
(see [30]).
The result stated in the next proposition has a complete analog in the complex setting, [16, Theorem 4.1, page 10]. Nevertheless we will produce a proof, to acquire familiarity with the quaternionic environment and to establish notations to be used.
Proposition 4.2**.**
Let and be two rank- lattices in , let and be the projections on the quotient tori. For any such that there exists a biregular diffeomorphism of onto which allows the equality . Conversely, for any biregular diffeomorphism of onto , there exists such that and .
Proof.
Let . Since we have and hence we can suppose . By definition of regular map between tori, to show that induces a biregular diffeomorphism of onto ,
[TABLE]
it is enough to show that implies . Indeed, if then and hence .
To prove the converse statement, we start by recalling that the map lifts to a continuous map , in such a way that the diagram (4.1) commutes. Moreover the map is regular since is a regular map of onto .
For any , consider . Since lifts a map between the quotients, maps -equivalent points into -equivalent points. Hence the image of is contained in the (discrete, see Theorem 3.4) lattice and, being continuous, is therefore constant. At this point it is clear that, taking the Cullen derivative, we obtain , for all . Thus the map is regular (see Remark 2.3) and -periodic, which makes it bounded. By the Liouville Theorem for regular functions (see Theorem 2.5) the Cullen derivative of is constant. Since expands as a power series (see Theorem 2.4)
[TABLE]
converging in the entire , we obtain (again by Theorem 2.4)
[TABLE]
and hence
[TABLE]
is a first degree regular polynomial. Again, since lifts a map between quotients, necessarily . If the inclusion is proper, then is not injective: indeed if some satisfies then there exists and such that and , with .
Now we know that , that is . The map defined by induces the map . Indeed
[TABLE]
This concludes the proof. ∎
5. Equivalence of quaternionic tori
To classify the -(real)-dimensional, quaternionic tori, up to biregular diffeomorphisms, we start with the following:
Theorem 5.1**.**
Two rank- lattices of the space generated respectively by the bases and , determine equivalent tori if and only if there exist and a linear transformation such that
[TABLE]
Proof.
By Proposition 4.2, if is a biregular diffeomorphism of onto , then there exists a biregular transformation of such that the diagram (4.1) commutes. Since is biregular on then with and As we pointed out in the proof of Proposition 4.2, without loss of generality, we can suppose both that and that the function maps the set of generators of to a set of generators of Taking into account that
[TABLE]
there exists a matrix
[TABLE]
with integer entries, such that
[TABLE]
or, more concisely,
[TABLE]
The same argument applied in the opposite direction, implies the existence of a matrix with integer entries such that
[TABLE]
and hence, substituting equation (5.3) into equation (5.2), we get
[TABLE]
which implies and hence that (and ) is such that , i.e. (and ) belongs to
On the other side, suppose there exists a matrix of this form:
[TABLE]
such that
[TABLE]
then we can compute in four different ways:
[TABLE]
for all Simple computations show that:
[TABLE]
[TABLE]
[TABLE]
[TABLE]
Hence defines a biregular diffeomorphism between and ∎
It is natural at this point to give the following
Definition 5.2**.**
Two rank- lattices of the space are called equivalent if the generated quaternionic tori and are equivalent. A basis of a rank- lattice and a basis of a rank- lattice are called equivalent if and are equivalent lattices, i.e. if (according to Theorem 5.1) there exist and a linear transformation such that
[TABLE]
Notice that two (different) equivalent bases and of rank- lattices may generate exactly the same lattice, and hence exactly the same quaternionic torus. This happens when there exists a linear transformation such that
[TABLE]
i.e., when in (5.4).
6. Minkowski-Siegel Reduction Algorithm: reduced and special bases
In this section we will specialize to the quaternionic setting the general Minkowski-Siegel Reduction Algorithm presented in [24, Section 4], [26, Section 9], and use it to construct reduced Gram matrices and reduced bases associated to lattices. In turn, reduced bases will be used to find special bases for lattices, useful in the sequel to identify and parameterize equivalence classes of quaternionic tori.
We explicitly present here some basic facts of this algorithmic construction, both to make the paper as much self-contained as possible, and to have a starting point for the proofs of the results that will follow.
Let denote the usual scalar product of . Let and be two quaternions. We set, and use in what follows,
[TABLE]
Let be a basis of the lattice . For any the squared norm of the element can be expressed by where the matrix
[TABLE]
is symmetric and positive definite, and is usually called the Gram matrix associated to the basis . In this setting and with the notations established, we will use the following procedure (see, e.g., [26], page 122):
Algorithm 6.1** **(Minkowski-Siegel Reduction Algorithm).
This algorithm acts on a Gram matrix and produces a matrix belonging to and a Gram matrix . The produced matrix has, and is in fact characterized by, the properties which follow.
Here are the steps of the algorithm:
- •
The Gram matrix (of a certain basis ) is given.
- •
Consider the function defined as
[TABLE]
By our assumption, is (the restriction to of) a positive definite quadratic form, and hence it attains its strictly positive minimum value at a point .
- •
To proceed, we need to recall that there exist infinitely many matrices of having the first row equal to (see e.g. [24, Section 4, pages 191-192], [26, Section 9, pages 122-123] for a proof of this assertion, and of the analogous ones, used in this algorithm). With this in mind, we consider the function obtained by restricting to the elements such that there exists a matrix of having the first two rows equal to and , respectively. Let be a point in which attains its strictly positive minimum value. Up to a change of sign, we can assume that .
- •
In the next step, we consider the restriction of to the elements such that there exists a matrix of having the first three rows equal to , and , respectively. Let be a point in which attains its strictly positive minimum value. Again, up to a change of sign, we can assume that .
- •
Finally, we take the restriction of to the elements such that there exists a matrix of having the four rows equal to , , and , respectively, and set to be a point in which has a strictly positive minimum value. As before, we can assume .
- •
The output of the algorithm consists of the matrix
[TABLE]
(belonging to by construction) and of the Gram matrix .
As we mentioned, the importance of the Minkowski-Siegel Reduction Algorithm stays in the features of the matrices ( and) that it produces; in fact, following [24, Section 4] and [26, Section 9], we can give the next definition.
Definition 6.2**.**
If is a Gram matrix obtained by applying to a given Gram matrix the Minkowski-Siegel Reduction Algorithm 6.1, then is called a reduced Gram matrix (relative to ). The symbol will denote the set of all reduced Gram matrices.
It turns out that there are two necessary and sufficient conditions that characterize the elements of the set of reduced Gram matrices; we recall these conditions here (see [26, equations (1), page 123]):
Proposition 6.3**.**
A Gram matrix is a reduced Gram matrix if and only if the two following sets of conditions hold:
- B1)
* for all *
- B2)
for all fixed , we have for any integer vector such that are without common divisors.
We point out, and we will use it in the sequel, the fact that conditions B2) are equivalent to B2)′:
- B2)′
for all fixed , if a vector has the property that then necessarily have common divisors.
Remark 6.4**.**
Let be a rank- lattice. Consider any basis of whose Gram matrix is . The Minkowski-Siegel Reduction Algorithm, applied to the Gram matrix , produces a matrix of which can be used to define the four elements
[TABLE]
The elements form a basis of since the matrix belongs to , with its inverse. Notice that
[TABLE]
and therefore that the two bases and are equivalent (in particular they generate the same lattice). We conclude the remark by pointing out that the Gram matrix associated to the basis is obtained as (recall formula (6.1)):
[TABLE]
and is therefore independent of the choice of a particular basis among those that have the same Gram matrix . In fact, in this sense, the matrix depends only on the Gram matrix .
To classify rank- lattices and generated tori, we will define the set of bases naturally emphasized by Algorithm 6.1.
Definition 6.5**.**
Let be a rank- lattice in . A basis of will be called a reduced basis if its Gram matrix is reduced.
A direct application of the Minkowski-Siegel Reduction Algorithm and Remark 6.4 prove a first result in the study of equivalence of lattices.
Theorem 6.6**.**
Let be a rank- lattice and let be a basis of . Then there exists a matrix of such that
[TABLE]
is a reduced basis of the lattice . As a consequence and we can suppose that the lattice is generated by a reduced basis.
We will now present the basic features of reduced Gram matrices (and bases).
Proposition 6.7**.**
If is a reduced Gram matrix, then the two following conditions hold:
- (1)
, for all such that ; 2. (2)
, for all such that .
Proof.
To verify , we use condition B2) at step , applied to the vector () where is the -th vector of the standard basis of . Inequalities are obtained by applying the same condition B2) at step , to the vector (), where are, respectively, the -th and -th vector of the standard basis of (see also [26, page 123]). ∎
Remark 6.8**.**
Concerning conditions B2) on the Gram matrix , we observe that: for each , if is the -th element of the standard basis of , then we obtain the obvious equality that gives no conditions.
We restate here a deep result that appears in [26, theorem stated at page 139], adapting it to our setting and notations.
Theorem 6.9**.**
The set of all reduced Gram matrices is a convex cone in with . If denotes the set of all Gram matrices, then
[TABLE]
If , and , then . Only for a finite set of matrices it is possible that .
The fact that is not obvious, and a non constructive proof is given in [26, page 137].
Now, our second step in the classification of rank- lattices and quaternionic tori makes use of a proper subset of the set of reduced bases.
Definition 6.10**.**
A reduced basis of a rank- lattice with the property that will be called a special basis.
Theorem 6.11**.**
Let be a rank- lattice and let be a basis of . Then is equivalent to a special basis of a rank- lattice . As a consequence and are equivalent, and hence they generate equivalent tori.
Proof.
By Theorem 6.6, there there exist a matrix of and a reduced basis of the lattice such that
[TABLE]
Therefore
[TABLE]
and, by Definition 5.2, the bases and are equivalent. This latter basis is special, since it is obtained multiplying the reduced basis on the right by , which corresponds to applying a rigid motion composed with a “positive” homothety of , see Remark 2.1. Indeed such a transformation maintains the fact that the Gram matrix is reduced (see Proposition 6.3). ∎
At this point it is possible to associate to each class of equivalence of quaternionic tori at least a special basis of a rank- lattice, according to
Corollary 6.12**.**
Let be a quaternionic torus. Then, up to biregular diffeomorphisms, we can suppose that where the lattice is generated by a special basis .
We will now pass to identify a natural and useful subset of the possible bases for rank- lattices. Let and let be a matrix with real coefficients. We set the notation to denote the quaternion whose real components are
[TABLE]
The following is a useful elementary result of linear algebra:
Proposition 6.13**.**
If two bases and have the same Gram matrix, then there exists an orthogonal matrix such that for
Proof.
Since the two bases have the same Gram matrix we have for . Let be the matrix which transforms the first basis into the second one, then , for all . Hence is an isometry with respect to the standard scalar product and the assertion follows. ∎
We will end this section with some remarks. A lattice is called normalized if and if every element of has norm greater or equal than . It can be proved that conditions (1)-(2) of Proposition 6.7 together with B1) are sufficient for a Gram matrix to be a reduced Gram matrix if and only if the associated lattice is normalized. To clarify what we mean, we provide an example of a non-normalized lattice having a basis whose Gram matrix satisfies conditions (1)-(2) of Proposition 6.7 together with B1), but is not reduced. In fact is such that an integer combination of three vectors of is inside To see this, notice that if and , then is a basis for whose (approximated) Gram matrix
[TABLE]
satisfies conditions (1)-(2) of Proposition 6.7 and B1). Nevertheless it is easy to see that and hence is not reduced.
7. A moduli space for quaternionic tori. Tame tori
The aim of this section is to find a fundamental set, and possibly a moduli space, to “parameterize” the equivalence classes of quaternionic tori, with respect to the action of biregular diffeomorphisms. We will then study the families of tame lattices and tame tori, whose definition is inspired by Theorem 6.9, and whose moduli correspond to the interior of the fundamental set .
We will start by identifying a useful subset of the set of reduced Gram matrices.
Remark 7.1**.**
In the sequel we will always consider reduced Gram matrices associated to special bases, i.e. matrices of the form
[TABLE]
where . This means, in particular, that we restrict to reduced Gram matrices belonging to an affine hyperplane of . We point out that we will consider, in the boundary of the set of reduced Gram matrices, only those elements that represent rank- lattices, and hence only definite positive matrices. Instead, as it appears in [26, page 136], when considering the entire set of reduced Gram matrices as a subset of the space of symmetric matrices, then its boundary contains also semi-positive definite, reduced matrices.
The promised space of “parameters” for the equivalence classes of biregular diffeomorphism of quaternionic tori is defined as follows.
Definition 7.2**.**
The set defined as
[TABLE]
is called the fundamental set of quaternionic tori. If we identify elements of when they correspond to special bases originating regularly diffeomorphic tori, then as customary the quotient set is called moduli space of quaternionic tori. Let be a point of , and let be the lattice generated by the special basis . We will say that is (a representative of) the modulus of any quaternionic torus equivalent to .
Corollary 6.12 guarantees an important property of the fundamental set:
Proposition 7.3**.**
Every quaternionic torus has (at least) a modulus in . In other words: for every quaternionic torus , there exists such that is equivalent to , where is the lattice generated by the special basis .
With obvious notations, set now
[TABLE]
and define
[TABLE]
where means the set of all , for all The fundamental set has a natural symmetry, namely we have that
Proposition 7.4**.**
The fundamental set and the set coincide. Equivalently, acts on . Moreover, if , then all elements of its orbit with respect to the action of ,
[TABLE]
correspond to special bases having the same reduced Gram matrix as .
Proof.
The proof is a direct computation. ∎
The geometric symmetry of the fundamental set stated in Proposition 7.4 is interesting, and suggests a remark and a few considerations, that help to identify similarities in the moduli of tori.
Remark 7.5**.**
Denote as usual by the standard basis for the space of quaternions. Recall that, for every two unitary quaternions with , the set is also a (positively oriented) basis for the space , having the same multiplication rules of the basis . Let us consider two lattices and generated, respectively, by the special bases and . Let the coefficients of the elements of with respect to the basis coincide with the coefficients of the elements of with respect to the basis . Then, in view of the last statement of Proposition 7.4, the two generated tori and - notwithstanding not equivalent according to our definition - have the same reduced Gram matrix and completely similar structures.
Let be a lattice having a special basis . Then there exists a basis of such that . What is stated in Proposition 7.4 and in Remark 7.5 allows us to study only the case of lattices having a special basis with , where is the second element of a (positively oriented) basis of . It is easy to see (and in any case we will see it later on, in this paper) that there are different elements belonging to that correspond to the same equivalence class of quaternionic tori, or equivalently that there are quaternionic tori having more than one representative in . However, this last phenomenon is not present in the case of the family of quaternionic tori that we are going to define.
Definition 7.6**.**
Let be a rank- lattice in .
- (1)
The lattice is called a tame lattice if there exists a reduced basis of whose Gram matrix is an interior point of . Such a basis will be called a tame basis. 2. (2)
A quaternionic torus is called a tame torus if there exists a tame lattice such that is equivalent to .
Here is an easy criterion to decide if a given torus is tame or not.
Proposition 7.7**.**
Let be a lattice and be a reduced basis for . Consider the torus . Then is a tame torus, if, and only if, are the unique reduced bases for .
Proof.
If the torus is tame, then suppose that there are two different reduced bases, and for the tame lattice , with . As a consequence, there exists such that
[TABLE]
If and denote the reduced Gram matrices associated, respectively, to the bases and , then
[TABLE]
Therefore, and belong to the boundary of by Theorem 6.9, and hence the torus is not tame. To prove the converse, suppose that is not tame, i.e. that the Gram matrix associated to belongs to . Therefore, equality holds either in B1) or in B2). In the first case, since for some , the two consecutive vectors and are orthogonal. We can then consider a second reduced basis obtained by substituting with , … , with . If instead equality holds in B2), the Minkowski-Siegel Reduction Algorithm directly implies the existence of a second reduced basis.
∎
We will now find, inside the fundamental set , a moduli space for the classes of equivalence of tame quaternionic tori. In fact if we set
[TABLE]
then, with the aid of a preliminary lemma, we can prove a uniqueness result for the modulus of a tame torus.
Lemma 7.8**.**
Let two lattices and of be generated respectively, by the special bases and If with is an automorphism of such that then
Proof.
Since , we have that and hence Moreover, since are linearly independent vectors which generate the lattice then there exist such that . This equality implies that But because is an element of Hence ∎
Theorem 7.9**.**
The set is a moduli space for the equivalence classes of tame tori. In other words, every tame torus has exactly one representative in (its modulus).
Proof.
Suppose that the two elements and correspond to equivalent tame tori. If this is the case, then (see Definition 5.2) there exist and a quaternion such that
[TABLE]
Lemma 7.8 implies that . Since multiplication by the unitary quaternion is an Euclidean rotation (see Remark 2.1), then has the same (reduced) Gram matrix as , and hence it is a reduced basis. Now, since and are both reduced bases, then by Proposition 7.7 we reach the conclusion that and hence that . ∎
8. On the groups of automorphisms of “boundary” tori
According to Theorem 6.9, a reduced Gram matrix belongs to if, and only if, there exists a reduced Gram matrix such that
[TABLE]
for some , . Notice that to each of these reduced Gram matrices there correspond infinitely many -nonequivalent bases (see (7.3)). Therefore the study of the equivalence classes of non tame tori consists in the identification and classification of reduced Gram matrices belonging to the boundary of , and corresponding to non equivalent special bases. We plan to address this fascinating problem in a forthcoming paper.
However, as it happens in the complex case, the first interesting and fundamental step in this direction is the search and classification of boundary tori with non trivial groups of (biregular) automorphisms. These tori, which are the quaternionic counterpart of tori with complex multiplication (classically indicated as harmonic and equianharmonic), will be found and classified in the rest of this section.
Remark 8.1**.**
Every vector of lies in an invariant real plane of the rotation (see Remark 2.1), where This fact can be verified directly as follows: for any quaternion we define and notice that, since and are perpendicular vectors in so are and Moreover thus
[TABLE]
[TABLE]
Therefore, the plane containing the vectors and is invariant, and the rotation in this plane is of an angle [15, page 37].
Lemma 8.2**.**
Let be a lattice of generated by the special basis and let with be an automorphism of such that Then has finite order, i.e. there exists such that and the order of divides either or
Proof.
Since is an element of the lattice and since maps onto it follows that similarly for all it holds that By compactness, the sequence of unit vectors in has a convergent subsequence. Unless is a finite set, this is in contradiction with Lemma 3.2 and Theorem 3.4 which assert respectively that is a (closed) discrete subgroup of . In order to prove the second assertion, we use what is stated in Remark 8.1: since and are elements of it follows that the complex plane , which contains and is invariant by right multiplication by i.e. the integer combinations of and form a rank- sublattice of contained in the complex plane , with as an automorphism restricted to the sublattice ; therefore is a root of unity of order where divides either or because in the complex setting these are the only possibilities (see, e.g., [15, page 82]).∎
Proposition 4.2 directly suggests how to define the automorphisms of a quaternionic torus.
Definition 8.3**.**
Let be the quaternionic torus associated to the rank- lattice . The group of biregular automorphisms of the torus is defined as
[TABLE]
We point out that the group of biregular automorphisms of the torus can also be interpreted as the group of biregular automorphisms of a rank- lattice fixing the point .
Proposition 8.4**.**
Let be a rank- lattice of containing . Let be the associated quaternionic torus, and let
[TABLE]
Then:
- (1)
the set is a subgroup (with respect to quaternionic multiplication) of the group of unitary quaternions; 2. (2)
the group is isomorphic to ; 3. (3)
for any fixed , each acts as a permutation on the finite set of all vectors of .
Proof.
The group structure of the set with respect to the quaternionic multiplication is inherited by the one of with respect to composition. The fact that each acts as a permutation on vectors of fixed norm is straightforward by Lemma 7.8 and 8.2 and by the fact that an automorphism of maps onto itself. ∎
We now pass to recall and list all the finite subgroups of the unitary quaternions, and refer the reader to the classical books [10], [11], [15] and [25] for the underlying theory.We point out that in the literature there is a diffused confusion in the use of notations that concern the groups we are dealing with; here we will mainly refer to, and use the notations of, the book [10] by Conway and Smith.
We begin by listing all (up to conjugation) finite subgroups of the group of rotations :
- (1)
the icosahedral group, , elements; 2. (2)
the octahedral group, , elements; 3. (3)
the tetrahedral group, , elements; 4. (4)
the dihedral group, , elements; 5. (5)
the cyclic group, , elements.
Every unitary quaternion is associated to a precise rotation of by means of the to correspondence that maps to the rotation (see, e.g., [10, Theorem 4, page 24]). As a consequence, every finite group of unitary quaternions is mapped to a group (isomorphic to) . The number of elements of is or times the number of elements of , according to whether is or is not in .
If denotes one of the finite subgroups of the group of rotations, then we set
[TABLE]
The only possible cases in which are those where . On the other hand, let us suppose that . In this case, can contain no order rotation : if then must be in . The only group without order elements is with odd; this gives rise to a group in isomorphic to . In fact, the following result holds (see, e.g., [10, Theorem 12, page 33]).
Theorem 8.5**.**
The finite subgroups of unitary quaternions are
[TABLE]
With the usual notations for quaternions, let , with , and let be a basis for having the usual multiplication rules. We then set
[TABLE]
Theorem 8.6**.**
The finite subgroups of unitary quaternions are generated as follows:
[TABLE]
Theorem 8.7**.**
There is no quaternionic torus whose group of automorphisms is isomorphic to 2\mathbb{I},\ 2\mathbb{O},\ 2D_{2n}\textnormal{(n\geq 4),}\ 2C_{n}\textnormal{(n\geq 4)},\ 1C_{n}\textnormal{(n odd)}.
Proof.
Since the subgroup contains an element of order , then has an element of order . Hence, by Lemma 8.2, it cannot be the group of automorphisms of a quaternionic torus. The same argument holds to exclude that has an element of order . Analogously the groups 2D_{2n}\textnormal{(n\geq 4),}\ 2C_{n}\textnormal{(n\geq 4)} are excluded since they both contain the element whose order is . Finally, the group 1C_{n}\textnormal{(n odd)} cannot be isomorphic to the group of automorphisms of a quaternonic torus since it does not contain . ∎
This last result reduces the possible groups of automorphisms of a quaternionic torus to the list
[TABLE]
Since, as it is well known and easy to check, the groups and are isomorphic, the final list of these groups becomes
[TABLE]
Remark 8.8**.**
We observe that the following group inclusions hold:
[TABLE]
[TABLE]
and
[TABLE]
For each group in the list (8.1), we will exhibit all tori whose group of automorphisms contains, or coincides with, . We will begin with the group , which appears for each quaternionic torus. As the reader may imagine, for reasons of neat presentation we will from now on suppose, without loss of generality, that the lattices which generate the tori involved have as a vector of minimum modulus.
Proposition 8.9**.**
The group is (isomorphic to) a subgroup of the group of biregular automorphisms of any quaternionic torus .
Proof.
Let be a rank- lattice of generated by the special basis and such that . Then consists of automorphisms of since generates the lattice . ∎
Proposition 8.10**.**
Let be a quaternionic torus. The group of biregular automorphisms of the torus contains a subgroup (isomorphic to) , if and only if there exists and a quaternion with such that is (a representative of) the modulus of .
Proof.
If there exists and with such that is (a representative of) the modulus of , then is a special basis of a lattice such that is equivalent to . Then the four bases
[TABLE]
all generate the lattice , and hence the subgroup is a subgroup of the group of automorphisms . On the other hand, if , then thanks to Theorem 8.6, Lemma 8.2 and Proposition 8.4, the vectors must belong to, and generate, the rank- sublattice of . Therefore (by the classification of rank- lattices of ), while using the Minkowski-Siegel Reduction Algorithm, we can choose in the special basis that generates . We can also suppose that the third vector is chosen (again according to Algorithm 6.1) among those vectors that can complete the special basis . As a consequence, must belong to together with . Since these three elements of have the same norm of , and since contains a subgroup isomorphic to , then the Minkowski-Siegel Reduction Algorithm 6.1 can produce a special basis that generates by using suitable and taking, automatically, ; this concludes the proof. ∎
Let be a standard basis for the skew field of quaternions. We recall that the ring of Lipschitz quaternions (or Lipschitz integers) consists of the set . The ring is, in turn, a subring of the ring of Hurwitz quaternions (or Hurwitz integers) . The surprising properties of these rings are described, for instance, in [10, Chapter II, Section 5].
Remark 8.11**.**
Concerning the proof of Proposition 8.10, notice that only in the case in which the lattice consists of the ring of Hurwitz integers , it can happen that there exists such that the special basis simply generates a proper sublattice of and not the whole for example if is an imaginary unit quaternion orthogonal to then the set generates the sublattice of the Lipschitz quaternions instead of the whole lattice of Hurwitz quaternions. According to Remark 8.8 and to the classification (8.1) of the finite subgroups of unitary quaternions which can be contained in , this is the only case in which a set of linearly independent vectors of type (with ) can generate a proper sublattice instead of the whole lattice. In this particular case, it is enough to change with another vector of which can be reached by means of the Minkowski-Siegel Reduction Algorithm and such that are not in the same multiplicative subgroup of we know that at least an of this kind exists (this last fact depends on the well known structure of the subgroups of ).
If the group of automorphisms of the torus contains a subgroup isomorphic to , and if the torus has a special basis of type with , then : this is a consequence of the classification of the rank- lattices of , and of the fact that, in these hypotheses, there are only four points in , all belonging to (see Proposition 8.4).
Definition 8.12**.**
A quaternionic torus whose group of biregular automorphisms is isomorphic to is called a cyclic-dihedral torus.
Proposition 8.13**.**
Let be a quaternionic torus. The group of biregular automorphisms of the torus contains a subgroup (isomorphic to) if and only if there exists and a quaternion with such that is (a representative of) the modulus of .
Proof.
If is (a representative of) the modulus of then is a special basis of a lattice such that is equivalent to . Then the six bases
[TABLE]
all generate the lattice , and hence the subgroup is a subgroup of the group of automorphisms . On the other hand, if , then thanks to Theorem 8.6, Lemma 8.2 and Proposition 8.4, there exists such that the vectors must belong to, and generate, the rank- sublattice of . Therefore (using the classification of the rank- lattices of ), while using the Minkowski-Siegel Reduction Algorithm to construct the special basis that generates , we can choose . We can also suppose that the third vector is chosen (according to Algorithm 6.1) among those vectors that can complete the special basis . As a consequence, the points must belong to together with . Since these five elements have the same norm of , and since contains a subgroup isomorphic to , then the Minkowski-Siegel Reduction Algorithm 6.1 can produce a special basis that generates by using suitable and, automatically, ; this completes the proof. ∎
If the group of automorphisms of the torus contains a subgroup isomorphic to , and if the torus has a special basis of type with , then : this is a consequence of the classification of the rank- lattices of , and of the fact that, in these hypotheses, there are only six points in , all belonging to (see Proposition 8.4).
Definition 8.14**.**
A quaternionic torus whose group of biregular automorphisms is isomorphic to is called a cyclic torus.
Proposition 8.15**.**
Let be a quaternionic torus. The group of biregular automorphisms of the torus is isomorphic to the group , if and only if, there exist with , such that the point is (a representative of) the modulus of .
Proof.
Let be the special basis associated to , and let be the generated lattice such that is equivalent to . Thanks to Proposition 8.10, we know that the multiplication by on the right generates a subgroup of . Using Theorem 8.6, we are left to prove that the multiplication by on the right generates a second subgroup of type of . To this aim notice that the four bases
[TABLE]
all generate the lattice . It is then easy to conclude that the subgroup is a subgroup of the group of automorphisms . On the other hand, if , then is a subgroup of , and hence thanks to Theorem 8.10, in the special basis that generates , we can choose and . Theorem 8.6, Lemma 8.2 and Proposition 8.4 imply that the vectors must belong to, and generate, the rank- sublattice of . Since and have the same norm of , then by the Minkowski-Siegel Reduction Algorithm used to construct a special basis for , we can suppose that and find the desired basis. To prove that , begin by noticing that is an orthonormal basis of ; it is then easy to see that and that this set has exactly the same cardinality of the group . Proposition 8.4 leads now to the conclusion. ∎
Definition 8.16**.**
A quaternionic torus whose group of biregular automorphisms is isomorphic to is called a -dihedral torus (or a dihedral torus of order ).
Remark 8.17**.**
As we already mentioned, the notations concerning finite subgroups of unit quaternions vary very much. We observe that the group that we (following [10, Subsection 3.5])called dihedral group of order 8, coincides with the so called multiplicative group of unit quaternions (and not with , sometimes called dihedral group of order 8, with the relations see [2, Theorem 3.4, page 164]).
Notice that the lattice of a dihedral torus of order is generated by the group and coincides with the ring of Lipschitz quaternions, defined after Proposition 8.10.
Proposition 8.18**.**
Let be a quaternionic torus. The group of biregular automorphisms of the torus is isomorphic to the group , if and only if, there exist , with , such that the point is (a representative of) the modulus of .
Proof.
Let be the special basis associated to and let be the generated lattice such that is equivalent to . Thanks to Proposition 8.13, we know that the multiplication by on the right generates a subgroup of . We are then left to prove that the multiplication by on the right generates a subgroup of type of . To this aim notice that the four bases
[TABLE]
all generate the lattice . We then conclude that the dihedral subgroup of order , , is a subgroup of the group of automorphisms . On the other hand, if , then is a subgroup of , and hence, thanks to Proposition 8.13, in the special basis that generates , we can choose and . Theorem 8.6, Lemma 8.2 and Proposition 8.4 imply that the vectors must belong to, and generate, the rank- sublattice of . Since and have the same norm of , then by the Minkowski-Siegel Reduction Algorithm used to construct a special basis for , we can suppose that and find the desired basis. To prove that , begin by noticing that is orthogonal to ; it is then easy to see that and that this set has exactly the same cardinality of the group . Proposition 8.4 leads now to the conclusion. ∎
Definition 8.19**.**
A quaternionic torus whose group of biregular automorphisms is isomorphic to is called a -dihedral torus (or a dihedral torus of order ).
Proposition 8.20**.**
Let be a quaternionic torus. The group of biregular automorphisms of the torus is isomorphic to the group , if and only if, for some with , setting and , the point is (a representative of) the modulus of .
Proof.
Notice, first of all, that if , then . Consider the special basis associated to and let be the generated lattice such that is equivalent to . Thanks to Theorem 8.6, we know that and generate a subgroup of . Set , and notice that belongs to . It is now necessary (and it is only a direct computation) to verify that the iterated multiplication by powers of and by powers of (on the right) maps the basis onto bases that generate the lattice . For example, as for the multiplication by on the right, we get that
[TABLE]
are all generating bases for . We then conclude that the subgroup is a subgroup of the group of automorphisms . On the other hand, if , then is a subgroup of , and hence, thanks to Proposition 8.13, in the special basis that generates , we can choose and . Theorem 8.6, Lemma 8.2 and Proposition 8.4 imply that the vectors must belong to, and generate, the rank- sublattice of . Since and have the same norm of , then by the Minkowski-Siegel Reduction Algorithm used to construct a special basis for , we can suppose that and find the desired basis. To prove that , begin by noticing that the elements belong to as well as the elements . Therefore the set has to have the same cardinality of the group . As in the previous cases, at this point Proposition 8.4 leads to the conclusion. ∎
Definition 8.21**.**
A quaternionic torus whose group of biregular automorphisms is isomorphic to is called a tetrahedral torus.
Notice that the lattice of the tetrahedral torus is the ring of the Hurwitz quaternions, defined after Proposition 8.10. This lattice is generated by the group .
The next result can be obtained directly as a consequence of the investigation performed up to now, on the possible groups of biregular automorphisms of “boundary” tori. Recall that, for a quaternionic torus , in Proposition 8.4 we have introduced the group .
Proposition 8.22**.**
Let be a rank- lattice of containing Let be the associated quaternionic torus. When the group is not reduced to , then it coincides with the biggest subgroup of .
Proof.
Suppose . Thanks to the classification (8.1), we get that has to coincide with or . If and is, by contradiction, strictly contained in a larger subgroup of , then using Remark 8.8 we obtain or . In both cases and generate a lattice associated to a different torus (see Propositions 8.15 and 8.20). If and is, by contradiction, strictly contained in a larger subgroup of , then using Remark 8.8 we obtain . In this last case generates a lattice associated to a different torus (see Proposition 8.20). In the remaining case in which the proof is totally analogous.
∎
Example 8.23**.**
We give an example which shows, in connection to Proposition 8.22, that when , then it can be strictly contained in a larger subgroup of . Let be the standard basis for the skew field of quaternions. Consider the lattice generated by the special basis If then the unitary vectors of are and they form a group isomorphic to .
We conclude this section by stating a summarizing result.
Theorem 8.24**.**
The cyclic, cyclic dihedral, -dihedral, -dihedral, tetrahedral tori defined in this section are the unique (up to biregular diffeomorphisms) tori with .
Proof.
Follows directly from Propositions 8.10, 8.13, 8.15, 8.18, 8.20. ∎
To close the Section, we recall that the lattices generating tori with or are classically called regular tessellations of ; the lattices generating tori with , or are called regular tessellations of
9. Appendix A: an algorithm to check if a basis is reduced
Let be the Gram matrix associated to a given basis of the rank- lattice Reordering the four vectors , without loss of generality, we can always suppose that To check that is a reduced basis, we will check the fact that is a reduced Gram matrix.
The first step of our algorithm is the easiest:
- Step 0:
We check if (for ) are all non negative quantities. If this is true, we proceed in the algorithm; otherwise we conclude that the Gram matrix , and the basis , are not reduced, and stop. Since is a symmetric, real and positive definite matrix, there exists a positive definite diagonal matrix
[TABLE]
and an orthogonal matrix such that Moreover, we can suppose that In order to verify if is a reduced Gram matrix we proceed as follows:
- Step 1:
Since the quadratic form can also be written as
[TABLE]
where are the ordered positive eigenvalues of and The geometric locus of vectors for which the diagonalized quadratic form is equal to is an ellipsoid having the length of the maximal axis of symmetry equal to Therefore, the quadruplets such that
[TABLE]
belong necessarily to the finite set where
[TABLE]
At this point we recall the first step of the construction of the Minkowski-Siegel Reduction Algorithm: we check if there exists a point in the finite set such that inequality (9.1) is fulfilled. If the answer is yes, then we conclude that the Gram matrix , and hence the basis , are not reduced, and stop. Otherwise we proceed in the algorithm.
- Step 2:
In this step we consider the quadratic equation
[TABLE]
Let
[TABLE]
and set to be the finite set . By Definition 6.2 and by condition B2)′, a reduced Gram matrix is such that: if there exists with
[TABLE]
then have common divisors. Therefore we check if B2)′ holds true. If the answer is no, then we conclude that the Gram matrix , and hence the basis , are not reduced, and stop. Otherwise we proceed in the algorithm.
- Step 3:
Similar procedure applies to the quadratic equation
[TABLE]
Let
[TABLE]
and set to be the finite set . By Definition 6.2 and by condition B2)′, a reduced Gram matrix is such that: if there exists with
[TABLE]
then have common divisors. Therefore we check if B2)′ holds true. If the answer is no, then we conclude that the Gram matrix , and hence the basis , are not reduced, and stop. Otherwise we proceed in the algorithm.
- Step 4:
In the last step our procedure is applied to the quadratic equation
[TABLE]
Let
[TABLE]
and set to be the finite set . By Definition 6.2 and by condition B2)′, a reduced Gram matrix is such that: if there exists with
[TABLE]
then must be different from . Therefore we check if B2)′ holds true. If the answer is no, then we conclude that the Gram matrix , and hence the basis , are not reduced. Otherwise we finally conclude that the Gram matrix , and hence the basis , are reduced.
A few significant examples, useful to illustrate the meaning of the results obtained, are the following.
Let be the standard basis for the skew field of quaternions.
Example 9.1**.**
An example of a cyclic-dihedral torus is the one associated to . Indeed, running the Algorithm presented in this section, we find out that Step 0 is satisfied and that the only vector of the lattice generated by the special basis inside the unit ball is the null vector; besides, on we only find the set of vectors .
Example 9.2**.**
A second example of a cyclic-dihedral torus is the one associated to . Indeed, running the Algorithm presented in this section, we find out that Step 0 is satisfied and all conditions in B2) are verified. The only vector of the lattice generated by the special basis inside the unit ball is the null vector; besides, on we only find the set of vectors On the sphere of radius there are 16 elements of and the product by permutes them, as explained in Proposition 8.4.
Example 9.3**.**
To present an example of a cyclic torus we use the one associated to , and hence to the lattice generated by the special basis . Indeed, running the Algorithm presented in this section, we find out that Step 0 is satisfied and the only vector of the lattice inside the unit ball is the null vector; besides, on we have vectors: of them are in and 4 other vectors correspond to .
10. Appendix B: an algorithm to check if a basis is tame
We want now to provide an algorithm to establish when a reduced basis is a tame basis.
Let be the reduced Gram matrix associated to a reduced basis of a rank- lattice We will use precisely the same notations as in the algorithm of the previous section. The first step of our new algorithm is the following:
- Step 0:
We check if (for ) are all strictly positive quantities. If this is true, we proceed in the algorithm; otherwise we conclude that the basis is not tame, and stop.
- Step 1:
Since the quadratic form can also be written as
[TABLE]
where are the ordered positive eigenvalues of and The geometric locus of vectors for which the diagonalized quadratic form is equal to is an ellipsoid having the length of the maximal axis of symmetry equal to Therefore, the quadruplets such that
[TABLE]
belong necessarily to the finite set , where
[TABLE]
At this point we recall the first step of the construction of the Minkowski-Siegel Reduction Algorithm: we check if there exists a point in the finite set such that equality (10.1) is fulfilled. If the answer is yes, then the integers have common divisors: indeed, if no non-trivial common divisor exists, we can find a vector in the lattice whose squared norm is equal to and which can substitute in If this is the case the basis is not unique and it is not tame. Otherwise we proceed in the algorithm.
- Step 2:
In this step we consider the quadratic equation
[TABLE]
Let
[TABLE]
and set to be the finite set . Suppose there exists no quadruplet with
[TABLE]
Then we go to the next step of the algorithm. Suppose that instead we find a (finite) set of quadruplets with
[TABLE]
We use now Definition 7.6, Proposition 7.7 and condition B2)′: if, for all elements , have common divisors, we go to the next step of the algorithm. Otherwise we conclude that the basis is not tame, and stop.
- Step 3:
Similar procedure applies to the quadratic equation
[TABLE]
Let
[TABLE]
and set to be the finite set . Suppose there exists no quadruplet with
[TABLE]
Then we go to the next step of the algorithm. Suppose that instead we find a (finite) set of quadruplets with
[TABLE]
We use again Definition 7.6, Proposition 7.7 and condition B2)′: if, for all elements , have common divisors, we go to the next step of the algorithm. Otherwise we conclude that the basis is not tame, and stop.
- Step 4:
In the last step our procedure is applied to the quadratic equation
[TABLE]
Let
[TABLE]
and set to be the finite set . Suppose there exists no quadruplet with
[TABLE]
Then we conclude that is tame. Suppose that instead we find a (finite) set of quadruplets with
[TABLE]
We use again Definition 7.6, Proposition 7.7 and condition B2)′: if, for all elements , , then the basis is not tame. Otherwise we conclude that the basis is tame.
To conclude, we present an explicit example of special tame lattice (i.e. a lattice whose reduced Gram matrix belongs to ).
Let be the standard basis for the skew field of quaternions.
Example 10.1**.**
The lattice generated by the special basis
[TABLE]
is a tame lattice. The proof follows by a direct application of the algorithm.
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