The Special Linear Group for Nonassociative Rings

J. Group Theory 23 (2020), 327–335 DOI 10.1515/jgth-2019-0076 © de Gruyter 2020

The special linear group for nonassociative rings

Harry Petyt Communicated by Evgenii I. Khukhro

Abstract. 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.R/ and SL2.C/ 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 H, 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 D 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 corre- sponding 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. 328 H. Petyt

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 j j in the sense that zw 2 z 2 w 2. Such algebras come with an anti-involution, j j D j j j j which we call conjugation and denote by a bar, e.g. z. They are also necessarily N alternative. That is, the associator Œ ; ; C 3 C , given by W ! Œz; w; u .zw/u z.wu/; D 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 (i) R (ii) C (iii) R2, with quadratic form .a; b/ 2 ab j j D 2 (iv) H (v) M2.R/, the 2 2 matrices over R, with det j j D (vi) O (vii) The split octonions O0

We write 1; e1; : : : ; ed 1 for an orthonormal basis of a unital composition alge- 2 bra of dimension d. We then have e 1 for all i, and ei ej ej ei whenever i D ˙ D i j . Letting Lz C C denote the left multiplication map w zw, alterna- ¤ W ! 7! tivity of C gives us

Lei Lei Le2 1: (2.1) D i D ˙ Moreover, whenever ei ej , we have ¤ 0 Œei ; ej ; z Œej ; ei ; z Lei ej ej ei .z/ .Lei Lej Lej Lei /.z/ D C D C C and hence Le Le Le Le : (2.2) i j D j i p;q p q Let R denote R C 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 d d those in the right-hand column have signature . 2 ; 2 /. 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 tr x, and left multiplication maps are again denoted Lx. The deﬁni- tion of the octonion special linear group and algebra given by Baez is as follows [1, p. 177]. The special linear group for nonassociative rings 329

Deﬁnition 2.2 (Baez). The octonion special linear algebra slm.O/ is the Lie algebra generated under commutators by the set Lx x Mm.O/; tr x 0 . The ¹ W 2 D º 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 fol- lowing, which is easily seen to agree with Deﬁnition 2.2 in the case R O. D

Deﬁnition 2.3. For R a not necessarily associative or unital ring, slm.R/ is the ring generated by LaE a R; i j under commutators. Similarly, SLm.R/ ¹ ij W 2 ¤ º is the group generated by LI aEij a R; i j under composition. ¹ C W 2 ¤ º 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/ x Mm.R/ tr x ŒR; R Š ¹ 2 W 2 º Proof. Since R is associative, we have LaL L for all a; b R, so we can b D ab 2 identify LaE with the matrix aEij and consider slm.R/ Mm.R/. Now slm.R/ ij contains all matrices with all diagonal entries zero as these form the linear span of the generators. Furthermore, the commutator of two generators is

ŒaEij ; bE ı abE ı baE ; kl D jk il il kj where ı denotes the Kronecker delta. If ıjk and ıil 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 ŒR; R, and by varying a and b, we can get the whole of ŒR; 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 ŒR; 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 ŒR; R 0, and, moreover, if R H, D D then ŒH; H z H Re.z/ 0 , which gives the standard definition of slm.H/ D ¹ 2 W D º [5, p. 52].

3 The two-dimensional case

Let C be a unital real composition algebra. For x .xij / Mm.C /, the hermit- D 2 ian conjugate of x is x .xj i /. If x x, then x is said to be hermitian, and the D D set of such matrices is denoted hm.C /. Note that all diagonal entries of a hermit- 330 H. Petyt ian matrix lie in R. We restrict our attention to the case m 2, where alternativity D of C ensures that the determinant map x x11x22 x12x12 is a well deﬁned 7! quadratic form on h2.C / (see [1, p. 176]). If C has dimension d and signature .p; q/, then, writing

z z0 z1e1 zd 1ed 1 D C C C q 1;p 1 for an element of C , we have that h2.C / is isometric to R C C via the map ! r z Âr s r s Ã C ; ; z0; z1; : : : ; zd 1 : z s 7! 2 2 N

We now deﬁne a representation of SL2.C / on h2.C /. Let y LI aEij be D C a generator of SL2.C /, and for x h2.C /, set y x .I aEij /x.I aEj i /. 2 D C CN 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 x h2.C / 2 and y SL2.C /, then det.y x/ det x. 2 D Proof. It sufﬁces to show that this holds for generators of SL2.C /. The two cases r z are similar, so we just do y LI aE21 . Let x z s , recalling that r; s R. D C D 2 Since ww ww for all w C , we obtain N N DN 2 ! r ra z det.y x/ det N C rs zz det x: D ra z raa za az s D N D CN N CN N C C It follows that there is a homomorphism of connected Lie groups

SL2.C / SO0.q 1; p 1/: W ! C C

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

Remark 3.2. If F C, then the same argument as the one above gives a homo- D morphism SL2.C/ SO.d 2; C/. ! C

Lemma 3.3. sl2.C / is based by the set

LE ;LE ; ŒLE ;LE ; ¹ 12 21 12 21 ˛i Le E ; ˇi Le E ; i ŒLE ; ˇi ; "ij Œ˛i ; ˇj i < j D i 12 D i 21 D 12 D W º .d 1/.d 2/ .d 1/.d 2/ In particular, dim SL2.C / 3 3.d 1/ C C . D C C 2 D 2 The special linear group for nonassociative rings 331

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 sl2.C /. Products of length three are spanned by generators and elements ı and ıT , where ! 0 Le Le Le Le Le Le ı i j k C k j i : D 0 0

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

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

Case 2: ei e ej . Using both identities (2.1) and (2.2), we see that ı is a gen- D k ¤ erator.

Case 3: ei ; ej ; e are distinct. Then ı 0 by identity (2.2). k D Thus products of length three are spanned by generators, which completes the proof.

Lemma 3.4. ker d 0. D Proof. The action of SL2.C / on h2.C / induces an action of sl2.C /: if x h2.C / 2 and y is a generator of sl2.C /, then y x yx xy . By deﬁnition, any ele- D C ment of ker d acts trivially, so we calculate the action of the basis of Lemma 3.3 r z Pd 1 on an arbitrary x z s h2.C /. Here r; s R, z z0 i 1 zi ei C , and D N 2 2 D C D 2 below, the i and ij are real. In several places, we ﬁnd it convenient to write Pd 1 w 0 i 1 i ei . D C D d 1 ! Â X Ã 2 Re.wz/ sw 0LE i ˛i x LwE x N ; (a) 12 C D 12 D sw 0 i 1 D N d 1 ! Â X Ã 0 rw 0LE i ˇi x LwE x N ; (b) 21 C D 21 D rw 2 Re.wz/ i 1 D d 1 Â X Ã 0ŒLE ;LE i i x 12 21 C i 1 D ŒLE ;LwE x D 12 21 ! ! 0 rw 2 Re.z/ s LE N LwE N D 12 rw 2 Re.wz/ 21 s 0 ! r0 wz zw 2 N ; (c) D zw wz s0 N N N 332 H. Petyt

Â X Ã ij "ij x 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/. Š C C Proof. Because Spin.p 1; q 1/ Spin.q 1; p 1/ it sufﬁces to show that C C Š C C is onto and has two-point kernel. By Lemma 3.3, we have .d 1/.d 2/ dim SL2.C / C C dim SO0.q 1; p 1/; D 2 D C C so, by Lemma 3.4, d is onto. Hence is onto. Indeed,

.SL2.C // exp.d .sl2.C /// exp.so.q 1; p 1// SO0.q 1; p 1/: D D C C D C C The special linear group for nonassociative rings 333

It remains to show that has two-point kernel. Consider the real matrices ! ! ! 1 1 1 0 1 2 a ; b ; c D 0 1 D 1 1 D 0 1 2 and the linear map LaL LcL La SL2.C /, which acts as I on C . Be- D b b 2 cause a; b; c M2.R/, the expression for x associates, so 2 x . I /x. I/ x and ker : D D 2 Hence ker consists of at least two elements.

Claim 1. If C is associative, then ker 1; . D ¹ º Proof. As in the proof of Theorem 2.4, we can consider elements of SL2.C / to be matrices. Then I . Also, if y .yij / SL2.C / acts trivially on h2.C /, we D D 2 have ! ! ! r 0 r 0 ry11y11 sy12y12 ry11y21 sy12y22 y C C : 0 s D 0 s D ry21y11 sy22y12 ry21y21 sy22y22 C C Taking r 1, s 0 in this gives y11y11 1 and y21 0. Similarly, taking r 0, D D D D D s 1 gives y22y22 1 and y12 0. Now we have D D D ! ! ! ! 0 z y 0 0 z 0 y zy 11 11 22 : z 0 D 0 y22 z 0 D y22zy11 0 N N Taking z 1 gives y11y22 1. But y11y11 1, so y22 y11. From this, we D D D D get z y11zy11, so zy11 y11z for all z C . Hence y11 y22 1, and D D 2 D D ˙ thus y 1; . 2 ¹ º } For the case C O, note that, since 1.SO.9; 1// Z2 (see [4, pp. 335, 343]), D D any proper cover is a double cover. This just leaves the case C O . The complexiﬁcation C O is isomorphic D 0 ˝ 0 to the bioctonions C O, a unital complex composition algebra. Thus SL2.O / ˝ 0 is a real form of SL2.C O/, which covers SO.10; C/ by Remark 3.2. Since ˝ 1.SO.10; C// Z2 (see [4, p. 343]), the covering is 2 : 1, with kernel 1; . We D ¹ º thus have a commutative diagram

complexify SL2.O / SL2.C O/ 0 ˝ 2 1 W complexify SO0.5; 5/ SO.10; C/; which completes the proof in the ﬁnal case, C O . D 0 334 H. Petyt

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 ¹ º 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 Lre r R . Since A is free over R, we have ¹ i W 2 º n MA EndR.R / Mn.R/: D Theorem 4.1. Under the above assumptions, if m 3, then slm.A/ x Mm.MA/ tr x ŒMA; MA : Š ¹ 2 W 2 º In particular, slm.A/ slmn.R/. Proof. The nonzero products of two generators are

ŒLaE ;L LL L E LL L E ; ij bEj i D a b ii b a jj ŒLaE ;L LL L E : ij bEjk D a b ik Since m 3, it follows by taking successive products that slm.A/ is the span of the set

L˛E ˛ MA; i j L L ˛; ˇ MA; i j : ¹ ij W 2 ¤ º [ ¹ ˛ˇEii ˇ ˛Ejj W 2 ¤ º Clearly all such matrices have trace in ŒMA; MA, and varying ˛ and ˇ gives the whole of ŒMA; 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 R-algebra O, the group generated by left multiplications by units is SO.8/ [3, p. 92]. This R-spans the full matrix algebra M8.R/, so MO M8.R/. D We can thus calculate Baez’s groups.

Corollary 4.2. If m 3, then SLm.O/ SL8m.R/. Š Proof. Theorem 4.1 gives slm.O/ sl8m.R/. Exponentiating gives the result. Š 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.R/ 0 0 Š for m 3.

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/ D 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 nat- The special linear group for nonassociative rings 335 ural 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 som, ˚ 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.

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

Bibliography

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Received May 24, 2019; revised September 30, 2019.

Author information Harry Petyt, Mathematics Department, University of Bristol, Fry Building, Bristol, United Kingdom. E-mail: [email protected]