Corestricted group actions and eight-dimensional absolute valued algebras

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Abstract

We define and study the class of left reflection algebras, which is a subclass of eight-dimensional absolute valued algebras. We reduce its classification problem to the problem of finding a transversal for the action of a subgroup of O7 on O7 by conjugation. As a basis for this study, we give a general criterion for finding full subcategories of group action categories, which themselves arise from group actions.

Introduction

This paper is concerned with the classification of finite-dimensional absolute valued algebras. An algebra over a field k is a vector space A over k equipped with a k-bilinear multiplication A×AA,(x,y)xy. Neither associativity nor commutativity is in general assumed. A is called absolute valued if the vector space is real, non-zero and equipped with a norm such that xy=xy for all x,yA.

Finite-dimensional absolute valued algebras exist only in dimensions 1, 2, 4 and 8; except in dimension 8, they have been classified up to isomorphism and their automorphism groups have been determined. While this is easily done in dimensions 1 and 2, in the case of dimension 4, an exhaustive list and an isomorphism condition were provided in [13], and the list was refined in [5]. In [11], a description (in the sense of Dieterich, [10]) of the category of four-dimensional absolute valued algebras was given, expressing the classification problem in terms of the normal form problem for the action of SO3 on SO3×SO3 by simultaneous conjugation. This normal form problem was moreover solved, and the automorphism groups of the algebras were also determined. A more succinct proof of the description is given in [10]. The morphisms between absolute valued algebras of different dimensions at most 4 were determined in [2]. In dimension 8, conditions for when two algebras are isomorphic were obtained in [4], using triality. We formulate these conditions as a description of the category of eight-dimensional absolute valued algebras in Section 2. Since the classification problem has proved hard, we set out to systematically find suitable full subcategories for which the classification problem is feasible, in particular, where the generally difficult computations in connection with triality are avoided.

As a model, we consider, in Section 3, the full subcategories of eight-dimensional absolute valued algebras with a left unity, a right unity, and a non-zero central idempotent, respectively. These were treated in [14] and [15], where the classification problem was reduced to the normal form problem for the action of the automorphism group of the octonions on O7 by conjugation, thus avoiding triality computations. The subcategories were further described (in the sense of Dieterich) and classified up to isomorphism in [8], and we embed this description in that of Section 2.

A description of a groupoid1 C is an equivalence of categories between C and a groupoid arising from a group action. To construct full subcategories of C, we seek conditions under which the group action used in the description induces an action on a subset, which in turn gives a description of a full subcategory of C. (It is important that the subcategory be full, in order for a classification of it to be useful in classifying the larger category.) As it turns out, this occurs precisely when the subset imposes a certain dichotomy on the group. This is explored in Section 4, where a general framework is obtained, based on stabilizers of subsets with respect to a group action. One of the results there describes how the object class of a groupoid can be partitioned, giving rise to pairwise isomorphic subgroupoids. This simplifies the classification problem and gives additional structural insight.

In Section 5, we examine the subcategories of Section 3 in this new framework. A common feature of these subcategories is the triviality of the triality phenomenon. Using this knowledge and the methods of Section 4, we construct and describe a new subcategory of absolute valued algebras in Section 6; the subcategory of left reflection algebras. We reduce the classification problem for these algebras to a manageable, though somewhat computational, one, and in the final section, we classify some subclasses of such algebras to demonstrate the computations involved.

By [1], the norm in a finite-dimensional absolute valued algebra is uniquely determined by the algebra multiplication, and multiplicativity of the norm implies that an absolute valued algebra has no zero divisors and hence, if it is finite-dimensional, that it is a division algebra. In fact, a finite-dimensional real algebra is absolute valued if and only if it is a division algebra which is also a composition algebra. (We recall that a division algebra is a non-zero algebra D such that for each aD{0}, the left and right multiplication maps La:DD,xax and Ra:DD,xxa are bijective. A composition algebra over a field k with chark2 is a k-algebra with a non-degenerate multiplicative quadratic form.)

The class of all finite-dimensional absolute valued algebras forms a category A, in which the morphisms are all non-zero algebra homomorphisms. Thus A is a full subcategory of the category D(R) of finite-dimensional real division algebras. It is known that morphisms in D(R) are injective, and morphisms in A are isometries.

In 1947, Albert [1] characterized all finite-dimensional absolute valued algebras as follows.

Proposition 1.1

Every finite-dimensional absolute valued algebra is isomorphic to an orthogonal isotope A=Af,g of a unique A{R,C,H,O}, i.e. A=A as a vector space, and the multiplicationin A is given byxy=f(x)g(y) for all x,yA, where f and g are linear orthogonal operators on A, and juxtaposition denotes multiplication in A.

Moreover, Albert showed that the norm in A coincides with the norm in A.

Thus the objects of A are partitioned into four classes according to their dimension, and the class of d-dimensional algebras, d{1,2,4,8}, forms a full subgroupoid Ad of A. For d>1 we moreover have the following decomposition due to Darpö and Dieterich [9].2

Proposition 1.2

Let AAd with d{2,4,8}. For any a,bA{0},sgn(det(La))=sgn(det(Lb)),sgn(det(Ra))=sgn(det(Rb)). The double sign of A is the pair (i,j)C22 where i=sgn(det(La)) and j=sgn(det(Ra)) for any aA{0}. Moreover, for all d{2,4,8},Ad=(i,j)C22Adij where Adij is the full subcategory of Ad formed by all algebras having double sign (i,j).

In the classification of finite-dimensional real division algebras, a certain type of categories, more specifically, of groupoids, has proven useful. We recall their definition.

Definition 1.3

Let G be a group, X a set, and α:G×XX a left group action. The groupoid arising from α is the category XG with object set X and where, for each x,yX,XG(x,y)={(g,x,y)|gG,gx=y}.

It is clear that XG is a groupoid. The group action is implicit in the notation XG, and if the domain and codomain of a morphism (g,x,y) are clear from the context, the morphism is simply referred to by g.

Groupoids arising from group actions can, and will in this paper, be used to gain an understanding of finite-dimensional absolute valued algebras in the following way, due to [10].

Definition 1.4

Let d{2,4,8}. A description (in the sense of Dieterich) of a full subcategory CAd is a quadruple (G,X,α,F), where G is a group, X a set, α:G×XX a left group action, and F:GXC an equivalence of categories.

Once a description is obtained, the problem of classifying C is transformed to the normal form problem for α, i.e. the problem of finding a transversal for the orbits of α. It is therefore crucial that the quadruple (G,X,α,F) be given explicitly. In [10], descriptions are defined in the more general context of finite-dimensional real division algebras, which we will not need here.

We follow the convention that 0N, and use the notation Z+ for the set N{0}. For each nZ+ we denote by n̲ the set {1,2,,n}. As O denotes the algebra of the octonions, O denotes the hyperplane of its purely imaginary elements.

For a vector space V, we denote by P(V) the projective space of V, whose elements are the lines through the origin in V. An element in P(V) containing a non-zero vector v will be denoted by [v]. More generally, square brackets denote the linear span of a collection of vectors in V. If a basis is given, upper indices will always denote the coordinates of a vector in this basis; hence vi is the ith coordinate of v.

If V is normed and UV a subset, we denote by S(U) the set of all elements of U having norm 1. Unless otherwise stated, for each nN, Rn+1 is equipped with the Euclidean norm, and Sn=S(Rn+1) denotes the unit n-sphere.

The general linear group in dimension n over R will be denoted GLn=GL(Rn), which we identify with GLn(R) upon endowing Rn with a standard basis. Analogous notation will be used for its classical subgroups, notably On and SOn. We denote by O81 the group of all gO8=O(O) fixing 1O, and by O8+ and O8 the set of all fO8 with positive and negative determinant, respectively. (The elements of the cyclic group C2 are written as + and − rather than as 1 and −1.) The notation In will be used for the n×n identity matrix.

The symbol ≤ will denote the subgroup relation. For a group action α:G×XX, gG and x,yX, we write gx for α(g,x), and xαy or xy with respect to α to denote that x and y are in the same orbit.

Finally, given a functor F:AB between categories A and B, we denote by F|C the restriction of F to a subcategory C of A.

The study we are about to undertake makes frequent use of two concepts: the principle of triality, and the automorphism group AutO of the octonions. Both have been subject to profound research, which goes far beyond the scope of this paper. The aim of this section is to recall such facts about these concepts that will be needed here, in the form applicable to the problems at hand. For a more general approach, the reader is directed to the literature: both concepts are treated in the overview article of Baez [3], as well as in [7]; triality is further treated by Chevalley [6], while AutO and Cayley triples are dealt with in [16, Chapter 1]. Applications of these concepts to absolute valued algebras can be found in [4] and [8], to which we will refer in several places.

We will be concerned with the principle of triality as applied to SO8, which we quote here.

Proposition 1.5

For each ϕSO8 there exist ϕ1,ϕ2SO8 such that for each x,yO,ϕ(xy)=ϕ1(x)ϕ2(y). The pair (ϕ1,ϕ2) is unique up to (overall) sign.

Thus there exist two triality pairs ±(ϕ1,ϕ2) for each ϕSO8. Moreover, triality respects composition, i.e. if ϕ,ψSO8, and (ϕ1,ϕ2) and (ψ1,ψ2) are triality pairs for ϕ and ψ, respectively, then (ϕ1ψ1,ϕ2ψ2) is a triality pair for the product ϕψ, since for any x,yO,ϕ1ψ1(x)ϕ2ψ2(y)=ϕ(ψ1(x)ψ2(y))=ϕψ(xy). As (Id,Id) is a triality pair for IdSO8, we deduce that (ϕ11,ϕ21) is a triality pair for ϕ1. We moreover have the identitiesϕ1=Rϕ2(1)1ϕ,ϕ2=Lϕ1(1)1ϕ, where ϕi(1)1 is not to be confused with ϕi1(1) for i2̲.

Every automorphism of O has determinant 1. Thus AutOSO8, and an element ϕSO8 is an automorphism of O precisely when (ϕ,ϕ) is a triality pair for ϕ. We then say that the triality components of ϕ are trivial. The group AutO is an exceptional Lie group of type G2, which we will simply denote by G2. It has dimension 14, and may equivalently be characterized as the set of all ϕSO8 such thatϕ(1)=ϕ1(1)=ϕ2(1)=1 for some triality pair (ϕ1,ϕ2) of ϕ. The identity ϕ(1)=ϕ1(1)ϕ2(1) shows that if any two of ϕ(1), ϕ1(1) and ϕ2(1) equal 1, then so does the third. Since the group of all ϕSO8 such that ϕ(1)=1 is isomorphic to SO7, one may view G2 as a subgroup of SO7, which we will sometimes do for notational convenience.

Another way to characterize G2 is via Cayley triples.

Definition 1.6

A Cayley triple is an orthonormal triple (u,v,z)(O)3 such that zuv.

Some fundamental facts about Cayley triples are given in the following well-known result.

Proposition 1.7

Let (u,v,z)(O)3 be a Cayley triple.

  • (i)

    The algebra O is generated by (u,v,z).

  • (ii)

    (1,u,v,uv,z,uz,vz,(uv)z) is an orthonormal basis of O, called the basis induced by (u,v,z).

  • (iii)

    The group G2 corresponds bijectively to the set of all Cayley triples, the bijection being given by ϕ(ϕ(u),ϕ(v),ϕ(z)) for all ϕG2.

Cayley triples and induced bases will be used in the computations of Section 7.

Section snippets

Description of A8

Conditions for when two finite-dimensional absolute valued algebras are isomorphic are given in [4]. In this section we deduce from this a description of A8. In other words, we establish an equivalence of categories from a groupoid arising from a group action to A8. To begin with, we introduce the action, for which we define the quotient groupO8=(O8×O8)/{±(Id,Id)} and write [f,g] for the coset of (f,g)O8×O8.

Proposition 2.1

The map τ:SO8×O8O8 defined by(ϕ,[f,g])ϕ[f,g]=[ϕ1fϕ1,ϕ2gϕ1], where (ϕ1,ϕ2) is any

Algebras having a non-zero central idempotent or a one-sided unity

Consider the three full subcategories A8l, A8r and A8c of A8, the object classes of which areA8l={AA8|uA,xA,ux=x},A8r={AA8|uA,xA,xu=x},A8c={AA8|uZ(A){0},u2=u}, and consist of all algebras with a left unity, a right unity, and a non-zero central idempotent, respectively. Here, the centre Z(B) of an algebra B is defined byZ(B)={zB|bB,zb=bz}, and an element is called central if it belongs to the centre. The three categories defined above are studied in [8], where a classification

Corestricted group actions

In view of Proposition 3.2, one may ask under which conditions a description of a full subcategory CA8 can be obtained by restricting a description F:GXA8 of A8 to a subset YX. In precise terms, given the groupoid XG arising from a left action α of a group G on a set X, and given a subset YX, we seek conditions on Y under which there exists a subgroup HG such that

  • the restriction of α to H×Y admits a corestriction to Y, i.e. α induces a group action H×YY, and

  • the groupoid YH arising from

Applications to A8

We now apply the above to the setting of Section 3. In the light of Section 4, Proposition 3.2 may then be restated, in terms of groups and group actions, as follows.

Proposition 5.1

For each x{l,r,c}, let Yx={Gx(f)|fO81}. With respect to the triality action,St(Yx)=St(Yx)=G2 and SO8=G2Dest(Yx).

The definition of the functors Gx:G2O81SO8O8, x{l,r,c}, was given in Proposition 3.2.

Proof

Let x{l,r,c}. Then G2St(Yx) by definition of Gx. If ϕSO8 stabilizes Yx, then for all fO81 there exists fO81 such that ϕ(Gx(f)

Preliminaries

We now introduce a new class of algebras in A8, and apply the above framework to it. First is a notational definition.

Definition 6.1

Let V be a Euclidean space and UV a subspace. The linear operator σU:VV is defined as reflection in the subspace U, i.e. byσU(v)={vif vU,vif vU.

Note that σU=σU1, being a symmetric orthogonal operator. In this section we only consider cases where U=Ru for some uV{0}, in which case we write σu instead of σRu to denote the reflection in the hyperplane u. We note the

Classifying left reflection algebras

Let uS(O) be fixed. We are now ready to enter into the computation of a transversal for the action γu, i.e. the classification of left reflection algebras up to isomorphism. Hence, let C be the transversal obtained in [8] for the group action γ, and let B be the transversal obtained in Corollary 6.16 for the action β. (The group actions used in this section were defined in Definition 6.12.) In view of Section 6.2, what remains to be done is to compute, for each cC, the group Stγ(c), and

Conclusion and future perspectives

The procedure employed in Section 7 gives, if completed, an explicit classification of left reflection algebras. By the Cartan–Dieudonné Theorem, each gO81 is the product of n reflections for some 0n7. The cases n=0 and n=1 having been treated in [8] and above, respectively, one may attempt to use the above techniques to investigate the set of all algebras Of,g where f,gO81 and g is the product of n reflections, 2n7. These generalizations are, however, beyond the scope of this paper, and

Acknowledgement

The author wishes to express his gratitude to Professor Ernst Dieterich for his advice and helpful remarks, as well as for the contribution of Proposition 6.5 and Theorem 6.13.

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