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The Special Linear Group for Nonassociative Rings

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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 hy- perbolic 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 alge- bra, 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 en- tries 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 al- gebra 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 exponentiat- ing 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 ei- ther 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 à 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.
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