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Lie Groups and Algebras

Table of Contents

1. Overview

Lie groups are groups which are manifolds, so in particular we could consider a one-parameter smooth family of symmetries. But moreover, we could consider any symmetry "arbitrarily close" to the identity symmetry — these look like \(I + \varepsilon X\), and the collection of all such \(X\) guys form a Lie algebra. Intuitively, a "Lie algebra" is just "infinitesimal symmetries" which are "near" the identity symmetry.

There are some famous families of Lie groups:

  • \(GL(n, \mathbb{F})\) the set of \(n\times n\) invertible matrices with entries in the field \(\mathbb{F}\)
  • \(SL(n, \mathbb{F})\) the set of \(n\times n\) invertible matrices with entries in the field \(\mathbb{F}\) and determinant 1
  • \(O(n, \mathbb{F})\) the set of \(n\times n\) orthogonal matrices with entries in the field \(\mathbb{F}\)
  • \(SO(n, \mathbb{F})\) the set of \(n\times n\) orthogonal matrices with entries in the field \(\mathbb{F}\) and determinant 1
  • \(U(n)\) the set of \(n\times n\) unitary matrices with complex entries
  • \(SU(n)\) the set of \(n\times n\) unitary matrices with determinant 1, having complex entries.

We could also classify the "simple" Lie groups using Dynkin diagrams and root systems:

I am lying here, the notion of a "simple Lie group" is ambiguous: no one can agree on its definition. But there is a notion of a simple Lie algebra, namely \(\mathfrak{g}\) is a simple Lie algebra if its adjoint representation \(\mathrm{ad}\colon\mathfrak{g}\to\mathfrak{gl}(\mathfrak{g})\) is irreducible (i.e., does not contain proper ideals).

If we just take the connected Lie group corresponding to a simple Lie algebra, then we may have discrete normal Subgroups…which then does not correspond to the notion of a "simple group".

  • \(A_{n} = SU(n+1)\)
  • \(B_{n}=SO(2n+1)\)
  • \(C_{n}=Sp(n)\)
  • \(D_{n}=SO(2n)\)
  • Exceptional simple Lie groups — usually there are several Lie groups corresponding to the same exceptional Lie algebra, but the exceptions are:
    • \(E_{6}\)
    • \(E_{7}\)
    • \(E_{8}\) which can be obtained in a two-step shuffle (following Adams's Lectures on Exceptional Lie Groups):
      1. Construct the Lie algebra \(L=\mathfrak{spin}(16)+\Delta\) where \(\Delta\) is the irrep for \(\mathfrak{spin}(16)\)
      2. We have \(E_{8}=\mathrm{Aut}(L)\), possibly modulo some finite group.
    • \(F_{4}\) which is the isometry group of the Octonionic projective plane \(\mathbb{OP}^{2}\)
    • \(G_{2}\) which is the automorphism group of the Octonion algebra (it's also the subgroup of \(SO(7)\) which preserves any chosen particular vector in its 8-dimensional [real] spinor representation); there are technically three groups identifiable as \(G_{2}\).

The "two-step shuffle" used to construct the \(E_{8}\) group works for other exceptional groups. The underlying mechanism may be justified using Killing Spinors (see arXiv:0706.2829).

The Dynkin diagrams then give us a way to encode the Cartan matrix \(a_{ij}\), which is then used to construct the Lie algebra using the Serre generators [= Chevalley basis = Cartan-Weyl basis] ("ladder operators" and other observables). Consequently, Lie algebras play a critical role in quantum field theory.

2. References

  • C H Barton, A Sudbery,
    "Magic squares and matrix models of Lie algebras".
    arXiv:math/0203010, 44 pages.
  • Fabio Bernardoni, Sergio L. Cacciatori, Bianca L. Cerchiai, Antonio Scotti,
    "Mapping the geometry of the F4 group".
    arXiv:0705.3978, 50 pages.
  • Fabio Bernardoni, Sergio L. Cacciatori, Bianca L. Cerchiai, Antonio Scotti,
    "Mapping the geometry of the E6 group".
    arXiv:0710.0356, 30 pages.
  • Sergio L. Cacciatori,
    "A simple parametrization for G2".
    arXiv:math-ph/0503054, 9 pages — discusses \(G_{2}\) as an \(SU(3)\) fibration
  • Sergio L. Cacciatori, Bianca L. Cerchiai, Alberto Della Vedova, Giovanni Ortenzi, Antonio Scotti,
    "Euler angles for G2".
    arXiv:hep-th/0503106, 21 pages.
  • Sergio L. Cacciatori, Francesco Dalla Piazza, Antonio Scotti,
    "E7 groups from octonionic magic square".
    arXiv:1007.4758, 23 pages.
  • S. L. Cacciatori, F. Dalla Piazza, A. Scotti,
    "Compact Lie groups: Euler constructions and generalized Dyson conjecture".
    arXiv:1207.1262, 11 pages.
  • S. L. Cacciatori, F. Dalla Piazza, A. Scotti,
    "A Simple E8 Construction".
    arXiv:1207.3623, 3 pages.
  • Sergio L. Cacciatori, Bianca L. Cerchiai, Alessio Marrani,
    "Squaring the Magic".
    arXiv:1208.6153, 21 pages.
  • José Figueroa-O'Farrill,
    "A geometric construction of the exceptional Lie algebras F4 and E8".
    arXiv:0706.2829, 12 pages.
  • Merab Gogberashvili, Alexandre Gurchumelia,
    "Split Octonions and Triality in (4+4)-Space".
    arXiv:2012.02255, 14 pages — interesting for Spin(4, 4)
  • Kirill Krasnov,
    "Spin(11,3), particles and octonions".
    arXiv:2104.01786, 27 pages.
  • David A. Richter,
    "Triacontagonal coordinates for the E(8) root system".
    arXiv:0704.3091, 4 pages.
  • Ichiro Yokota,
    "Exceptional Lie groups".
    arXiv:0902.0431, 204 pages.

2.1. Triality

  • Jonathan M. Evans,
    "Trialities and Exceptional Lie Algebras: DECONSTRUCTING the Magic Square".
    arXiv:0910.1828, 34 pages.
  • Merab Gogberashvili, Alexandre Gurchumelia,
    "Split Octonions and Triality in (4+4)-Space".
    arXiv:2012.02255, 14 pages — interesting for Spin(4, 4)

2.2. Particle Physics Related

Last Updated 2023-07-12 Wed 09:27.