The Magic Square of Lie Groups: The 2 × 2 Case

Abstract

A unified treatment of the 2 × 2 analog of the Freudenthal-Tits magic square of Lie groups is given, providing an explicit representation in terms of matrix groups over composition algebras.Mathematics Subject Classification. 22E46, 17A35, 15A66.

The symplectic origin of conformal and Minkowski superspaces

“…The temptation is again to extend this statement to all division algebras. In [27,28,29], in fact, the authors realized SO(n + 2, 2) transformation by giving an explicit Clifford algebra description of SU(2, H ′ ⊗ K) that turns out to be equivalent to the symplectic description as Sp 4 (K) (see in particular [28] for a characterization of this group in the octonionic case and its connection with exceptional Lie groups); one can in conclusion establish the isomorphism Spin(n + 2, 2) ≅ Sp 4 (K) .…”

“…In [26], the authors constructed the compactified Minkowsky 3d superspace can be reinterpreted by noting the following isomorphism sp 4 (K) = so(n + 2, 2) withsp 4 (K) being the Sudbery symplectic algebra, where, with respect to the traditional definition, the transpose is replaced by hermitian conjugation. Recently, in [27], it was also proposed a Lie group version of the half split 2 × 2 magic square (see also [28,29,30] for further details and the relation with exceptional Lie algebras and groups). Inspired by these observations we then study in details a symplectic characterization of the 4 dimensional (compactified and real) Minkowski space and superspace respectively.…”

“…In [26], the authors constructed the compactified Minkowsky 3d superspace with sp 4 (K) being the Sudbery symplectic algebra, where, with respect to the traditional definition, the transpose is replaced by hermitian conjugation. Recently, in [27], it was also proposed a Lie group version of the half split 2 × 2 magic square (see also [28,29,30] for further details and the relation with exceptional Lie algebras and groups).…”

“…In Section 2 we review how finite Lorentz transformations of vectors in n + 2 dimensional Minkowski space can be characterized by means of "unit determinant" matrices over division algebras; this relation was originally presented in [37]. In Section 3 we discuss the Lie group version of the third row of 2 × 2 magic square [27]; in particular we show that certain type of symplectic transformations induce an O(n + 2, 2) rotation. In Section 4 we prove in details how the real form of the four dimensional Minkowski and conformal space can be obtained as a Lagrangian manifold containing the twistor space C 4 , while in Section 5 we extend this construction to the super case.…”

Klein and conformal superspaces, split algebras and spinor orbits

“…In [51], the compactified 3-dimensional Minkowski space M 2,1 was constructed, along with its N = 1 supersymmetric extension M 2,1|1 , in terms of a Lagrangian manifold over the twistor space R 4 , by exploiting the Lie group isomorphism Spin(3, 2) ∼ = Sp(4, R). Taking inspiration from the isomorphisms (1.2)-(1.5) and also relying on [41], in [19] a symplectic characterization of the 4-dimensional (compactified and real) Minkowski space M 3,1 and N = 1 Poincaré superspace M 3,1|1 was given, exploiting the Lie group isomorphism Spin(4, 2) ∼ = Sp(4, C). Therein, it was also argued the possibility to extend the approach also to the other critical dimensions D = 6 and 10, thus providing a uniform and elegant description of N = 1 Poincaré superspaces M q+1,1|1 in critical dimensions D = q + 2 in terms of the four normed division algebras A's.…”