We extend to arbitrary rings a definition of the octonion special linear group due to Baez. At the infinitesimal level, we get a Lie ring, which we describe over some large classes of rings, including all associative rings and all algebras over a field. As a corollary, we compute all the groups Baez defined.
The special linear groups and are, respectively, the double covers of and – the isometry groups of the hyperbolic plane and hyperbolic 3-space. The pattern continues with the quaternions , as shown by Kugo and Townsend in , and Sudbery deals with the final normed real division algebra, the octonions , in . Unfortunately, the way Sudbery defines the special linear group over only makes sense in dimensions two and three.
In his celebrated survey , Baez suggests a unified definition of for all m, and shows that it agrees with Sudbery’s definition when . He does not discuss the case , and it seems that until now no further investigation has been made.
Motivated by this, in Section 2, we reformulate Baez’s definition of the special linear group and algebra in a natural way that lends itself to computation, and note that it naturally extends to arbitrary nonassociative rings (in the present paper, we do not in general assume rings to be associative). We then determine the corresponding special linear ring (we do not necessarily get an algebra structure) for all associative rings. In Section 3, we cover the two-dimensional case for unital real composition algebras. In Section 4, we characterise , with , for a large class of algebras that includes . This allows us to compute Baez’s groups. In doing so, we find that in three dimensions his definition disagrees with Sudbery’s, which gives a real form of the exceptional Lie group .
An alternative definition for has been proposed by Hitchin . This definition is motivated by a dimension argument, and does not give a Lie group.
A composition algebra is a not necessarily unital or associative algebra C over a field , together with a nondegenerate quadratic form that is multiplicative in the sense that . Such algebras come with an anti-involution, which we call conjugation and denote by a bar, e.g. . They are also necessarily alternative. That is, the associator, given by
If the characteristic of is not two, then all unital composition algebras can be obtained from by the Cayley–Dickson construction (a description of which can be found in ), and have famously been classified by Jacobson . We are mainly interested in real composition algebras, and we state his classification in this case.
Theorem 2.1 (Jacobson).
The unital real composition algebras are exactly
|(ii)||(iii)||, with quadratic form|
|(iv)||(v)||, the matrices over , with|
|(vi)||(vii)||The split octonions|
We write for an orthonormal basis of a unital composition algebra of dimension d. We then have for all i, and whenever . Letting denote the left multiplication map , alternativity of C gives us
Moreover, whenever , we have
Let denote with the standard quadratic form of signature . The algebras in the left-hand column of Theorem 2.1 have signature , and those in the right-hand column have signature .
For a ring R, we write to mean the space of matrices with entries in R, and for an element of the standard basis. The trace of a matrix x is written , and left multiplication maps are again denoted . The definition of the octonion special linear group and algebra given by Baez is as follows [1, p. 177].
Definition 2.2 (Baez).
The octonion special linear algebra is the Lie algebra generated under commutators by the set . The octonion special linear group is the Lie group generated by exponentiating .
This definition is not well suited to computation, and we prefer to use the following, which is easily seen to agree with Definition 2.2 in the case .
For R a not necessarily associative or unital ring, is the ring generated by under commutators. Similarly, is the group generated by under composition.
Straight from the definition, we can obtain a nice description of the special linear algebra of an associative ring.
Let R be an associative (not necessarily unital) ring. Then there is an isomorphism
Since R is associative, we have for all , so we can identify with the matrix and consider . Now contains all matrices with all diagonal entries zero as these form the linear span of the generators. Furthermore, the commutator of two generators is
where δ denotes the Kronecker delta. If and are not both 1, then we get either zero or a generator. If both are 1, then we get . Clearly, this has trace lying in , and by varying a and b, we can get the whole of . Then varying i and j gives the right-hand side of the result. Note that the commutator of such a diagonal matrix with a generator is traceless, so all further commutators have trace in , and we are done. ∎
Theorem 2.4 shows that Definition 2.3 gives a true generalisation of the usual special linear algebra, for if R is a field, then , and, moreover, if , then , which gives the standard definition of [5, p. 52].
3 The two-dimensional case
Let C be a unital real composition algebra. For , the hermitian conjugate of x is . If , then x is said to be hermitian, and the set of such matrices is denoted . Note that all diagonal entries of a hermitian matrix lie in . We restrict our attention to the case , where alternativity of C ensures that the determinant map is a well defined quadratic form on (see [1, p. 176]).
If C has dimension d and signature , then, writing
for an element of C, we have that is isometric to via the map
We now define a representation of on . Let be a generator of , and for , set . This product is well defined because C is alternative, and we extend to in the obvious way.
The action of on is by isometries; i.e., if and , then .
It suffices to show that this holds for generators of . The two cases are similar, so we just do . Let , recalling that . Since for all , we obtain
It follows that there is a homomorphism of connected Lie groups
In order to analyse ψ, we describe a basis of , but first a remark.
If , then the same argument as the one above gives a homomorphism .
is based by the set
In particular, .
It follows from identities (2.1) and (2.2) that the set in question bases the subspace spanned by products of length at most two, so it suffices to show that this is the whole of . Products of length three are spanned by generators and elements δ and , where
There are three cases for δ, depending on the choice of , and .
is equal to either or . Then δ is a generator by identity (2.1).
are distinct. Then by identity (2.2).
Thus products of length three are spanned by generators, which completes the proof. ∎
The action of on induces an action of : if and y is a generator of , then . By definition, any element of acts trivially, so we calculate the action of the basis of Lemma 3.3 on an arbitrary . Here , , and below, the and are real. In several places, we find it convenient to write .
Together, these describe the action of every element of on . Now assume that y acts trivially. Writing y in the basis of Lemma 3.3 and using the four equations above (with respective sets of coefficients , , , and ), consider .
The upper left entry is , which must be zero for all and . Taking , gives , and then cycling z through the gives . Similarly, considering the lower right entry gives . We already have , and cycling z through the gives . Finally, considering the top right entry, we are left with
Taking gives , so all are zero. Then, successively considering , we find that all are zero. Thus we have . ∎
Knowing that has full rank is enough to prove the main theorem of this section.
If C is a unital real composition algebra of dimension d and signature , then .
Because it suffices to show that ψ is onto and has two-point kernel. By Lemma 3.3, we have
so, by Lemma 3.4, is onto. Hence ψ is onto. Indeed,
It remains to show that ψ has two-point kernel. Consider the real matrices
and the linear map , which acts as on . Because , the expression for associates, so
Hence consists of at least two elements.
If C is associative, then .
As in the proof of Theorem 2.4, we can consider elements of to be matrices. Then . Also, if acts trivially on , we have
Taking , in this gives and . Similarly, taking , gives and . Now we have
Taking gives . But , so . From this, we get , so for all . Hence , and thus . ∎
For the case , note that, since (see [4, pp. 335, 343]), any proper cover is a double cover.
This just leaves the case . The complexification is isomorphic to the bioctonions , a unital complex composition algebra. Thus is a real form of , which covers by Remark 3.2. Since (see [4, p. 343]), the covering is 2 : 1, with kernel . We thus have a commutative diagram
which completes the proof in the final case, . ∎
4 The general case
Let R be a commutative associative unital ring and A a finite-dimensional R-algebra that is free as an R-module, with basis . For example, A could be any finite-dimensional algebra over a field.
Write for the left multiplication algebra of A. That is, is the R-algebra generated by . Since A is free over R, we have
Under the above assumptions, if , then
In particular, .
The nonzero products of two generators are
Since , it follows by taking successive products that is the span of the set
Clearly all such matrices have trace in , and varying α and β gives the whole of . Varying i and j then gives the result. ∎
This reduces the problem of determining to that of finding . In the case of the -algebra , the group generated by left multiplications by units is [3, p. 92]. This -spans the full matrix algebra , so . We can thus calculate Baez’s groups.
If , then .
Theorem 4.1 gives . Exponentiating gives the result. ∎
Together with Theorem 3.5, this describes all the groups Baez defined. In fact, the same argument works for , and we similarly obtain for .
Corollary 4.2 shows that Baez’s definition of disagrees with Sudbery’s, which gives a real form of . For , Sudbery defines to be the left multiplications by traceless elements of , together with its derivations. The isomorphism with is due to Chevalley and Schafer . It is natural to ask what happens to this construction when n is greater than 3. In this case, it is shown in [9, Theorem 3.3] that the derivation algebra of is , but it is not clear how this interacts with the multiplication operators. In particular, when , the commutator of two such operators may fail to be a derivation.
I would like to thank Dmitriy Rumynin for introducing me to the problem and for his helpful comments and suggestions.
 F. R. Harvey, Spinors and Calibrations, Perspect. Math. 9, Academic Press, Boston, 1990. Search in Google Scholar
 R. D. Schafer, An Introduction to Nonassociative Algebras, Dover Publications, New York, 1995. Search in Google Scholar
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