The special linear group for nonassociative rings

Theorem 2.1 (Jacobson).

The unital real composition algebras are exactly

Definition 2.2 (Baez).

The octonion special linear algebra 𝔰⁢𝔩m⁢(𝕆) is the Lie algebra generated under commutators by the set {Lx:x∈Mm⁢(𝕆),tr⁡x=0}. The octonion special linear group SLm⁡(𝕆) is the Lie group generated by exponentiating 𝔰⁢𝔩m⁢(𝕆).

Definition 2.3.

For R a not necessarily associative or unital ring, 𝔰⁢𝔩m⁢(R) is the ring generated by {La⁢Ei⁢j:a∈R,i≠j} under commutators. Similarly, SLm⁡(R) is the group generated by {LI+a⁢Ei⁢j:a∈R,i≠j} under composition.

Theorem 2.4.

Let R be an associative (not necessarily unital) ring. Then there is an isomorphism s⁢lm⁢(R)≅{x∈Mm⁢(R):tr⁡x∈[R,R]}

Proof.

Since R is associative, we have La⁢Lb=La⁢b for all a,b∈R, so we can identify La⁢Ei⁢j with the matrix a⁢Ei⁢j and consider 𝔰⁢𝔩m⁢(R)⊂Mm⁢(R). Now 𝔰⁢𝔩m⁢(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 δj⁢k and δi⁢l are not both 1, then we get either zero or a generator. If both are 1, then we get a⁢b⁢Ei⁢i-b⁢a⁢Ej⁢j. 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. ∎

Lemma 3.1.

The action of SL2⁡(C) on h2⁢(C) is by isometries; i.e., if x∈h2⁢(C) and y∈SL2⁡(C), then det⁡(y⋅x)=det⁡x.

Proof.

It suffices to show that this holds for generators of SL2⁡(C). The two cases are similar, so we just do y=LI+a⁢E21. Let x=(rzz¯s), recalling that r,s∈ℝ. Since w⁢w¯=w¯⁢w for all w∈C, we obtain

Remark 3.2.

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

Lemma 3.3.

𝔰⁢𝔩2⁢(C) is based by the set

In particular, dim⁡SL2⁡(C)=3+3⁢(d-1)+(d-1)⁢(d-2)2=(d+1)⁢(d+2)2.

Proof.

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 𝔰⁢𝔩2⁢(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=ek≠ej. 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.

ker⁡d⁢ψ=0.

Proof.

The action of SL2⁡(C) on 𝔥2⁢(C) induces an action of 𝔰⁢𝔩2⁢(C): if x∈𝔥2⁢(C) and y is a generator of 𝔰⁢𝔩2⁢(C), then y⋅x=y⁢x+x⁢y*. By definition, any element of ker⁡d⁢ψ 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∈ℝ, z=z0+∑i=1d-1zi⁢ei∈C, and below, the λi and κi⁢j are real. In several places, we find it convenient to write w=λ0+∑i=1d-1λi⁢ei.

Together, these describe the action of every element of 𝔰⁢𝔩2⁢(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 κi⁢j), consider y⋅x.

The upper left entry is 2⁢Re⁡((λ0+∑i=1d-1λi⁢ei)⁢z¯)+2⁢r⁢ν0, which must be zero for all r∈ℝ and z∈C. 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 2⁢Re⁡((μ0+∑μi⁢ei)⁢z)-2⁢s⁢ν0=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-14⁢νi⁢ei=0, so all νi are zero. Then, successively considering z=e1,z=e2,…, we find that all κ1⁢j,κ2⁢j,… are zero. Thus we have ker⁡d⁢ψ=0. ∎

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).

Proof.

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, d⁢ψ 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=La⁢Lb⁢Lc⁢Lb⁢La∈SL2⁡(C), which acts as -I on C2. Because a,b,c∈M2⁢(ℝ), the expression for ι⋅x associates, so

Hence ker⁡ψ consists of at least two elements.

For the case C=𝕆, note that, since π1⁢(SO⁢(9,1))=ℤ2 (see [4, pp. 335, 343]), any proper cover is a double cover.

This just leaves the case C=𝕆′. The complexification ℂ⊗𝕆′ is isomorphic to the bioctonions ℂ⊗𝕆, a unital complex composition algebra. Thus SL2⁡(𝕆′) is a real form of SL2⁡(ℂ⊗𝕆), which covers SO⁢(10,ℂ) by Remark 3.2. Since π1⁢(SO⁢(10,ℂ))=ℤ2 (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′. ∎

Theorem 4.1.

Under the above assumptions, if m≥3, then

In particular, s⁢lm⁢(A)⊂s⁢lm⁢n⁢(R).

Proof.

The nonzero products of two generators are

Since m≥3, it follows by taking successive products that 𝔰⁢𝔩m⁢(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. ∎

Corollary 4.2.

If m≥3, then SLm⁡(O)≅SL8⁢m⁡(R).

Proof.

Theorem 4.1 gives 𝔰⁢𝔩m⁢(𝕆)≅𝔰⁢𝔩8⁢m⁢(ℝ). Exponentiating gives the result. ∎

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 𝔰⁢𝔩m⁢(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 𝔤2⊕𝔰⁢𝔬m, 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.