Euler angles for G2

Abstract

We provide a simple coordinatization for the group G2, which is analogous to the Euler coordinatization for SU (2). We show how to obtain the general element of the group in a form emphasizing the structure of the fibration of G2 with fiber SO(4) and base H, the variety of quaternionic subalgebras of octonions. In particular this allows us to obtain a simple expression for the Haar measure on G2. Moreover, as a by-product it yields a concrete realization and an Einstein metric for H. * cacciatori@mi.infn.it † BLCerchiai@lbl.gov ‡ alberto.dellavedova@unimib.it § giovanni.ortenzi@unimib.it ¶ Graduate visitor (visitatore laureato), ascotti@mindspring.com

Mapping the geometry of the E6 group

“…This is particularly interesting, because recently it has been argued that this group could be the most promising for unification in GUT theories [3,6]. To parameterize the group we have used the generalized Euler angles method, a technique we introduced in [2,1] to give the most simple expression for the invariant measure on the group, while at the same time still being able to provide an explicit expression for the range of the parameters. Both these requirements are necessary in order to minimize the computation power needed for computer simulations, for example, of lattice models.…”

“…To determine the range of the parameters we will use the topological method we have developed in [2]. Let us first determine the volume of E 6 by means of the Macdonald formula.…”

“…QQ [7,1] = e [7]; QQ [7,2] = e [6]; QQ [7,3] = −e [8]; QQ [7,4] = e [5]; QQ [7,5] = −e [4]; QQ [7,6] = −e [2]; QQ [7,7] = −e [1]; QQ [7,8] %%%% construction of the matrices MT = { {{mt [1], 0, 0, 0, 0, 0, 0, 0}, {mt [2], mt [3], mt [4], mt [5], mt [6], mt [7], mt [8], mt [9]}, {mt [10], mt [11] C The matrices.…”

“…However for large dimensions it becomes quite hard to find a realization, which is able at the same time to provide both a reasonably simple form for the measure and an explicit determination for the range of the angles. In this paper we solve this problem for the exceptional Lie group E 6 , by constructing a generalization of the Euler parametrization, with a technique we have introduced in [2] and fully developed in [1]. In section 2 we explain how the e 6 algebra can be represented using a theorem due to Chevalley and Schafer.…”

Compact Lie groups: Euler constructions and generalized Dyson conjecture

“…A simple procedure to parameterize compact simple Lie groups is provided by the generalized Euler parameterization [4,5,6,7,8,9,10,26,27]. The role of Lie groups both in mathematics and in physics is very important and well known.…”