On SO(3) as the Projective Group of the Space SO(3)
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ON SO(3, 3) AS THE PROJECTIVE GROUP OF THE SPACE SO(3) A. A. Akopyan∗ and A. V. Levichev†
UDC 514.14
The fractional linear action of SO(3, 3) on the projective space SO(3) is proven to be a (globally defined) projective action. Bibliography: 6 titles.
1. Introduction Originally, the choice of topics for the present article was motivated by Segal’s Chronometry and Levichev’s DLF theory: see [3, Sec. 7] for details on a (both general as well as DLF specific) notion of parallelization of a vector bundle. However, it is our opinion that the main statements (Theorems 1 through 4 below) are of general mathematical interest. We believe that the most interesting is our finding that SO(3, 3) can be viewed as the projective group of the (projective) space SO(3). We notice that there is a 2-cover P of S 1 × SO(3) (we denote this group by D 1/2 ) by the group U(2): P sends a matrix z into the pair (det z, p(u)), where p is the standard covering map from SU(2) onto SO(3), see (3.4) below, and u is a matrix in SU(2) such that z = du with d2 = det z (u is determined up to a sign). Both P and p are group homomorphisms. From the theoretical physics viewpoint, the fundamental role of p is well known. In regard to P , its mere existence allows one to present Segal’s chronometric theory (see [3] and references therein) starting with the “lowest level” possible (since the center of the group SO(3) is trivial). It is well known that the Lie group U(2) (with a bi-invariant Lorentzian metric on it) can be viewed as a conformal compactification of the Minkowski spacetime. This provides an approach to how to study physics of Segal’s compact cosmos D = U(2). The Lie group D 1/2 inherits a bi-invariant metric through the above-introduced homomorphism P . With this metric, we call D 1/2 the projective world. Thus, D 1/2 underlies Segal’s compact cosmos D = U(2). This gives another option of how to build physics in D: to start with that of D 1/2 . This last topic is to be explored elsewhere while the current presentation takes a turn towards geometry. We introduce the fractional linear action of SO(3, 3) on the projective space SO(3) and prove that this action is a (globally defined) projective action. (Throughout the article, SO(3, 3) stands for the component of the identity matrix.) One would probably anticipate such an observation: it is known (since long ago) that a projective transformation between two projective lines is a fractional linear one (see [4, p. 22]). The key ingredient of the proof is the commutative diagram that intertwines the linear action of SL(4) in R4 with the SO(3, 3) fractional linear action on SO(3). In that diagram, the element g˜ of SO(3, 3) corresponds to an element g of the real Lie group SL(4). 2. Fractional linear action of SO(3, 3) on SO(3) In this section, our main goal is to introduce a fractional linear action of SO(3, 3) on SO(3) and show that this action is globally defined. We denote by M6×6 the set of all 6 × 6 real ∗
Boston University, Boston, MA 02215, USA, e-mail: [email protected].
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Sobolev Inst
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