Corestricted group actions and eight-dimensional absolute valued algebras
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 . 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 for all .
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 on 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 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 is an equivalence of categories between and a groupoid arising from a group action. To construct full subcategories of , 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 . (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 , the left and right multiplication maps and are bijective. A composition algebra over a field k with is a k-algebra with a non-degenerate multiplicative quadratic form.)
The class of all finite-dimensional absolute valued algebras forms a category , in which the morphisms are all non-zero algebra homomorphisms. Thus is a full subcategory of the category of finite-dimensional real division algebras. It is known that morphisms in are injective, and morphisms in 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 of a unique , i.e. as a vector space, and the multiplication ⋅ in A is given by for all , where f and g are linear orthogonal operators on A, and juxtaposition denotes multiplication in .
Moreover, Albert showed that the norm in A coincides with the norm in .
Thus the objects of are partitioned into four classes according to their dimension, and the class of d-dimensional algebras, , forms a full subgroupoid of . For we moreover have the following decomposition due to Darpö and Dieterich [9].2
Proposition 1.2 Let with . For any , The double sign of A is the pair where and for any . Moreover, for all , where is the full subcategory of formed by all algebras having double sign .
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 a left group action. The groupoid arising from α is the category with object set X and where, for each ,
It is clear that is a groupoid. The group action is implicit in the notation , and if the domain and codomain of a morphism 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 . A description (in the sense of Dieterich) of a full subcategory is a quadruple , where G is a group, X a set, a left group action, and an equivalence of categories.
Once a description is obtained, the problem of classifying 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 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 , and use the notation for the set . For each we denote by the set . As denotes the algebra of the octonions, denotes the hyperplane of its purely imaginary elements.
For a vector space V, we denote by the projective space of V, whose elements are the lines through the origin in V. An element in containing a non-zero vector v will be denoted by . 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 is the ith coordinate of v.
If V is normed and a subset, we denote by the set of all elements of U having norm 1. Unless otherwise stated, for each , is equipped with the Euclidean norm, and denotes the unit n-sphere.
The general linear group in dimension n over will be denoted , which we identify with upon endowing with a standard basis. Analogous notation will be used for its classical subgroups, notably and . We denote by the group of all fixing , and by and the set of all with positive and negative determinant, respectively. (The elements of the cyclic group are written as + and − rather than as 1 and −1.) The notation will be used for the identity matrix.
The symbol ≤ will denote the subgroup relation. For a group action , and , we write for , and or with respect to α to denote that x and y are in the same orbit.
Finally, given a functor between categories and , we denote by the restriction of to a subcategory of .
The study we are about to undertake makes frequent use of two concepts: the principle of triality, and the automorphism group 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 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 , which we quote here.
Proposition 1.5 For each there exist such that for each , The pair is unique up to (overall) sign.
Thus there exist two triality pairs for each . Moreover, triality respects composition, i.e. if , and and are triality pairs for ϕ and ψ, respectively, then is a triality pair for the product ϕψ, since for any , As is a triality pair for , we deduce that is a triality pair for . We moreover have the identities where is not to be confused with for .
Every automorphism of has determinant 1. Thus , and an element is an automorphism of precisely when is a triality pair for ϕ. We then say that the triality components of ϕ are trivial. The group is an exceptional Lie group of type , which we will simply denote by . It has dimension 14, and may equivalently be characterized as the set of all such that for some triality pair of ϕ. The identity shows that if any two of , and equal 1, then so does the third. Since the group of all such that is isomorphic to , one may view as a subgroup of , which we will sometimes do for notational convenience.
Another way to characterize is via Cayley triples.
Definition 1.6 A Cayley triple is an orthonormal triple such that .
Some fundamental facts about Cayley triples are given in the following well-known result.
Proposition 1.7 Let be a Cayley triple. The algebra is generated by . is an orthonormal basis of , called the basis induced by . The group corresponds bijectively to the set of all Cayley triples, the bijection being given by for all .
Cayley triples and induced bases will be used in the computations of Section 7.
Section snippets
Description of
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 . In other words, we establish an equivalence of categories from a groupoid arising from a group action to . To begin with, we introduce the action, for which we define the quotient group and write for the coset of .
Proposition 2.1 The map defined by where is any
Algebras having a non-zero central idempotent or a one-sided unity
Consider the three full subcategories , and of , the object classes of which are and consist of all algebras with a left unity, a right unity, and a non-zero central idempotent, respectively. Here, the centre of an algebra B is defined by 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 can be obtained by restricting a description of to a subset . In precise terms, given the groupoid arising from a left action α of a group G on a set X, and given a subset , we seek conditions on Y under which there exists a subgroup such that
- •
the restriction of α to admits a corestriction to Y, i.e. α induces a group action , and
- •
the groupoid arising from
Applications to
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 , let . With respect to the triality action, and .
The definition of the functors , , was given in Proposition 3.2.
Proof Let . Then by definition of . If stabilizes , then for all there exists such that
Preliminaries
We now introduce a new class of algebras in , and apply the above framework to it. First is a notational definition.
Definition 6.1 Let V be a Euclidean space and a subspace. The linear operator is defined as reflection in the subspace , i.e. by
Note that , being a symmetric orthogonal operator. In this section we only consider cases where for some , in which case we write instead of to denote the reflection in the hyperplane . We note the
Classifying left reflection algebras
Let be fixed. We are now ready to enter into the computation of a transversal for the action , 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 , the group , 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 is the product of n reflections for some . The cases and having been treated in [8] and above, respectively, one may attempt to use the above techniques to investigate the set of all algebras where and g is the product of n reflections, . 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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2015, Bulletin des Sciences MathematiquesCitation Excerpt :This has the following immediate consequence. The proof relies on Lemma 7.9 and Proposition 7.10 from [3], which are a matrix version of the following. We will not comment on the topological properties of the spaces and maps which we will consider, as they are of little interest in this context.
Composition algebras and outer automorphisms of algebraic groups via triality
2017, Communications in Algebra