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Publicly Available Published by De Gruyter November 8, 2019

The special linear group for nonassociative rings

  • Harry Petyt EMAIL logo
From the journal Journal of Group Theory


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.

1 Introduction

The special linear groups SL2() and SL2() are, respectively, the double covers of SO0(2,1) and SO0(3,1) – the isometry groups of the hyperbolic plane and hyperbolic 3-space. The pattern continues with the quaternions , as shown by Kugo and Townsend in [8], and Sudbery deals with the final normed real division algebra, the octonions 𝕆, in [11]. Unfortunately, the way Sudbery defines the special linear group over 𝕆 only makes sense in dimensions two and three.

In his celebrated survey [1], Baez suggests a unified definition of SLm(𝕆) for all m, and shows that it agrees with Sudbery’s definition when m=2. He does not discuss the case m>2, 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 SLm, with m>2, for a large class of algebras that includes 𝕆. 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 E6.

An alternative definition for SL2(𝕆) has been proposed by Hitchin [6]. This definition is motivated by a dimension argument, and does not give a Lie group.

2 Preliminaries

A composition algebra is a not necessarily unital or associative algebra C over a field 𝔽, together with a nondegenerate quadratic form ||2 that is multiplicative in the sense that |zw|2=|z|2|w|2. Such algebras come with an anti-involution, which we call conjugation and denote by a bar, e.g. z¯. They are also necessarily alternative. That is, the associator[,,]:C3C, given by


is alternating.

If the characteristic of 𝔽 is not two, then all unital composition algebras can be obtained from 𝔽 by the Cayley–Dickson construction (a description of which can be found in [10]), and have famously been classified by Jacobson [7]. 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)2, with quadratic form |(a,b)|2=ab
(iv)(v)M2(), the 2×2 matrices over , with ||2=det
(vi)𝕆(vii)The split octonions 𝕆

We write 1,e1,,ed-1 for an orthonormal basis of a unital composition algebra of dimension d. We then have ei2=±1 for all i, and eiej=-ejei whenever ij. Letting Lz:CC denote the left multiplication map wzw, alternativity of C gives us


Moreover, whenever eiej, we have


and hence


Let p,q denote p+q with the standard quadratic form of signature (p,q). The algebras in the left-hand column of Theorem 2.1 have signature (d,0), and those in the right-hand column have signature (d2,d2).

For a ring R, we write Mm(R) to mean the space of m×m matrices with entries in R, and Eij for an element of the standard basis. The trace of a matrix x is written trx, and left multiplication maps are again denoted Lx. 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 𝔰𝔩m(𝕆) is the Lie algebra generated under commutators by the set {Lx:xMm(𝕆),trx=0}. The octonion special linear group SLm(𝕆) is the Lie group generated by exponentiating 𝔰𝔩m(𝕆).

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 R=𝕆.

Definition 2.3.

For R a not necessarily associative or unital ring, 𝔰𝔩m(R) is the ring generated by {LaEij:aR,ij} under commutators. Similarly, SLm(R) is the group generated by {LI+aEij:aR,ij} under composition.

Straight from the definition, we can obtain a nice description of the special linear algebra of an associative ring.

Theorem 2.4.

Let R be an associative (not necessarily unital) ring. Then there is an isomorphism slm(R){xMm(R):trx[R,R]}


Since R is associative, we have LaLb=Lab for all a,bR, so we can identify LaEij with the matrix aEij and consider 𝔰𝔩m(R)Mm(R). Now 𝔰𝔩m(R) 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 δjk and δil are not both 1, then we get either zero or a generator. If both are 1, then we get abEii-baEjj. Clearly, this has trace lying in [R,R], and by varying a and b, we can get the whole of [R,R]. 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 [R,R], 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 [R,R]=0, and, moreover, if R=, then [,]={z:Re(z)=0}, which gives the standard definition of 𝔰𝔩m() [5, p. 52].

3 The two-dimensional case

Let C be a unital real composition algebra. For x=(xij)Mm(C), the hermitian conjugate of x is x*=(xji¯). If x*=x, then x is said to be hermitian, and the set of such matrices is denoted 𝔥m(C). Note that all diagonal entries of a hermitian matrix lie in . We restrict our attention to the case m=2, where alternativity of C ensures that the determinant map xx11x22-x12x12¯ is a well defined quadratic form on 𝔥2(C) (see [1, p. 176]).

If C has dimension d and signature (p,q), then, writing


for an element of C, we have that 𝔥2(C) is isometric to q+1,p+1 via the map


We now define a representation of SL2(C) on 𝔥2(C). Let y=LI+aEij be a generator of SL2(C), and for x𝔥2(C), set yx=(I+aEij)x(I+a¯Eji). This product is well defined because C is alternative, and we extend to SL2(C) in the obvious way.

Lemma 3.1.

The action of SL2(C) on h2(C) is by isometries; i.e., if xh2(C) and ySL2(C), then det(yx)=detx.


It suffices to show that this holds for generators of SL2(C). The two cases are similar, so we just do y=LI+aE21. Let x=(rzz¯s), recalling that r,s. Since ww¯=w¯w for all wC, we obtain


It follows that there is a homomorphism of connected Lie groups


In order to analyse ψ, we describe a basis of 𝔰𝔩2(C), but first a remark.

Remark 3.2.

If 𝔽=, then the same argument as the one above gives a homomorphism SL2()SO(d+2,).

Lemma 3.3.

𝔰𝔩2(C) is based by the set


In particular, dimSL2(C)=3+3(d-1)+(d-1)(d-2)2=(d+1)(d+2)2.


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 𝔰𝔩2(C). Products of length three are spanned by generators and elements δ and δT, where


There are three cases for δ, depending on the choice of ei, ej and ek.

  1. ej is equal to either ei or ek. Then δ is a generator by identity (2.1).

  2. ei=ekej. Using both identities (2.1) and (2.2), we see that δ is a generator.

  3. ei,ej,ek are distinct. Then δ=0 by identity (2.2).

Thus products of length three are spanned by generators, which completes the proof. ∎

Lemma 3.4.



The action of SL2(C) on 𝔥2(C) induces an action of 𝔰𝔩2(C): if x𝔥2(C) and y is a generator of 𝔰𝔩2(C), then yx=yx+xy*. By definition, any element of kerdψ acts trivially, so we calculate the action of the basis of Lemma 3.3 on an arbitrary x=(rzz¯s)𝔥2(C). Here r,s, z=z0+i=1d-1zieiC, and below, the λi and κij are real. In several places, we find it convenient to write w=λ0+i=1d-1λiei.

(c)  =2(rλ0wz-zw¯z¯w¯-wz¯-sλ0),
(d)  =i<j2κij(0ej2zjei-ei2ziej-ej2zjei+ei2ziej0).

Together, these describe the action of every element of 𝔰𝔩2(C) on 𝔥2(C). 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 λi, μi, νi, and κij), consider yx.

The upper left entry is 2Re((λ0+i=1d-1λiei)z¯)+2rν0, which must be zero for all r and zC. Taking z=0, r=1 gives ν0=0, and then cycling z through the ei gives λi=0. Similarly, considering the lower right entry gives 2Re((μ0+μiei)z)-2sν0=0. We already have ν0=0, and cycling z through the ei gives μi=0. Finally, considering the top right entry, we are left with


Taking z=1 gives 1d-14νiei=0, so all νi are zero. Then, successively considering z=e1,z=e2,, we find that all κ1j,κ2j, are zero. Thus we have kerdψ=0. ∎

Knowing that dψ has full rank is enough to prove the main theorem of this section.

Theorem 3.5.

If C is a unital real composition algebra of dimension d and signature (p,q), then SL2(C)Spin(p+1,q+1).


Because Spin(p+1,q+1)Spin(q+1,p+1) it suffices to show that ψ is onto and has two-point kernel. By Lemma 3.3, we have


so, by Lemma 3.4, dψ 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 ι=LaLbLcLbLaSL2(C), which acts as -I on C2. Because a,b,cM2(), the expression for ιx associates, so


Hence kerψ consists of at least two elements.

Claim 1.

If C is associative, then kerψ={1,ι}.


As in the proof of Theorem 2.4, we can consider elements of SL2(C) to be matrices. Then ι=-I. Also, if y=(yij)SL2(C) acts trivially on 𝔥2(C), we have


Taking r=1, s=0 in this gives y11y11¯=1 and y21=0. Similarly, taking r=0, s=1 gives y22y22¯=1 and y12=0. Now we have


Taking z=1 gives y11y22¯=1. But y11y11¯=1, so y22=y11. From this, we get z=y11zy11¯, so zy11=y11z for all zC. Hence y11=y22=±1, and thus y{1,ι}. ∎

For the case C=𝕆, note that, since π1(SO(9,1))=2 (see [4, pp. 335, 343]), any proper cover is a double cover.

This just leaves the case C=𝕆. The complexification 𝕆 is isomorphic to the bioctonions 𝕆, a unital complex composition algebra. Thus SL2(𝕆) is a real form of SL2(𝕆), which covers SO(10,) by Remark 3.2. Since π1(SO(10,))=2 (see [4, p. 343]), the covering is 2 : 1, with kernel {1,ι}. We thus have a commutative diagram

which completes the proof in the final case, C=𝕆. ∎

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 {e1,,en}. For example, A could be any finite-dimensional algebra over a field.

Write 𝔐A for the left multiplication algebra of A. That is, 𝔐A is the R-algebra generated by {Lrei:rR}. Since A is free over R, we have


Theorem 4.1.

Under the above assumptions, if m3, then


In particular, slm(A)slmn(R).


The nonzero products of two generators are


Since m3, it follows by taking successive products that 𝔰𝔩m(A) is the span of the set


Clearly all such matrices have trace in [𝔐A,𝔐A], and varying α and β gives the whole of [𝔐A,𝔐A]. Varying i and j then gives the result. ∎

This reduces the problem of determining 𝔰𝔩m(A) to that of finding 𝔐A. In the case of the -algebra 𝕆, the group generated by left multiplications by units is SO(8) [3, p. 92]. This -spans the full matrix algebra M8(), so 𝔐𝕆=M8(). We can thus calculate Baez’s groups.

Corollary 4.2.

If m3, then SLm(O)SL8m(R).


Theorem 4.1 gives 𝔰𝔩m(𝕆)𝔰𝔩8m(). Exponentiating gives the result. ∎

Together with Theorem 3.5, this describes all the groups Baez defined. In fact, the same argument works for 𝕆, and we similarly obtain SLm(𝕆)SL8m() for m3.

Remark 4.3.

Corollary 4.2 shows that Baez’s definition of SL3(𝕆) disagrees with Sudbery’s, which gives a real form of E6. For m=2,3, Sudbery defines 𝔰𝔩m(𝕆) to be the left multiplications by traceless elements of 𝔥m(𝕆), together with its derivations. The isomorphism with 𝔢6 is due to Chevalley and Schafer [2]. 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 𝔥m(𝕆) is 𝔤2𝔰𝔬m, but it is not clear how this interacts with the multiplication operators. In particular, when m>3, the commutator of two such operators may fail to be a derivation.

Communicated by Evgenii I. Khukhro


I would like to thank Dmitriy Rumynin for introducing me to the problem and for his helpful comments and suggestions.


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Received: 2019-05-24
Revised: 2019-09-30
Published Online: 2019-11-08
Published in Print: 2020-03-01

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