G₂ (mathematics)

The Lie algebra ${\displaystyle {\mathfrak {g}}_{2}}$, being the smallest exceptional simple Lie algebra, was the first of these to be discovered in the attempt to classify simple Lie algebras. On May 23, 1887, Wilhelm Killing wrote a letter to Friedrich Engel saying that he had found a 14-dimensional simple Lie algebra, which we now call ${\displaystyle {\mathfrak {g}}_{2}}$.[1]

In 1893, Élie Cartan published a note describing an open set in ${\displaystyle \mathbb {C} ^{5}}$ equipped with a 2-dimensional distribution—that is, a smoothly varying field of 2-dimensional subspaces of the tangent space—for which the Lie algebra ${\displaystyle {\mathfrak {g}}_{2}}$ appears as the infinitesimal symmetries.[2] In the same year, in the same journal, Engel noticed the same thing. Later it was discovered that the 2-dimensional distribution is closely related to a ball rolling on another ball. The space of configurations of the rolling ball is 5-dimensional, with a 2-dimensional distribution that describes motions of the ball where it rolls without slipping or twisting.[3][4]

In 1900, Engel discovered that a generic antisymmetric trilinear form (or 3-form) on a 7-dimensional complex vector space is preserved by a group isomorphic to the complex form of G₂.[5]

In 1908 Cartan mentioned that the automorphism group of the octonions is a 14-dimensional simple Lie group.[6] In 1914 he stated that this is the compact real form of G₂.[7]

In older books and papers, G₂ is sometimes denoted by E₂.

There are 3 simple real Lie algebras associated with this root system:

• The underlying real Lie algebra of the complex Lie algebra G₂ has dimension 28. It has complex conjugation as an outer automorphism and is simply connected. The maximal compact subgroup of its associated group is the compact form of G₂.
• The Lie algebra of the compact form is 14-dimensional. The associated Lie group has no outer automorphisms, no center, and is simply connected and compact.
• The Lie algebra of the non-compact (split) form has dimension 14. The associated simple Lie group has fundamental group of order 2 and its outer automorphism group is the trivial group. Its maximal compact subgroup is SU(2) × SU(2)/(−1,−1). It has a non-algebraic double cover that is simply connected.

G₂ is the automorphism group of the following two polynomials in 7 non-commutative variables.

${\displaystyle C_{1}=t^{2}+u^{2}+v^{2}+w^{2}+x^{2}+y^{2}+z^{2}}$

${\displaystyle C_{2}=tuv+wtx+ywu+zyt+vzw+xvy+uxz}$ (± permutations)

which comes from the octonion algebra. The variables must be non-commutative otherwise the second polynomial would be identically zero.

Adding a representation of the 14 generators with coefficients A, ..., N gives the matrix:

${\displaystyle A\lambda _{1}+\cdots +N\lambda _{14}={\begin{bmatrix}0&C&-B&E&-D&-G&F-M\\-C&0&A&F&-G+N&D-K&-E-L\\B&-A&0&-N&M&L&-K\\-E&-F&N&0&-A+H&-B+I&C-J\\D&G-N&-M&A-H&0&J&I\\G&K-D&-L&B-I&-J&0&-H\\-F+M&E+L&K&-C+J&-I&H&0\end{bmatrix}}}$

It is exactly the Lie algebra of the group

${\displaystyle G_{2}=\{g\in SO(7):g^{*}\varphi =\varphi ,\varphi =\omega ^{123}+\omega ^{145}+\omega ^{167}+\omega ^{246}-\omega ^{257}-\omega ^{347}-\omega ^{356}\}}$

The characters of finite-dimensional representations of the real and complex Lie algebras and Lie groups are all given by the Weyl character formula. The dimensions of the smallest irreducible representations are (sequence A104599 in the OEIS):

1, 7, 14, 27, 64, 77 (twice), 182, 189, 273, 286, 378, 448, 714, 729, 748, 896, 924, 1254, 1547, 1728, 1729, 2079 (twice), 2261, 2926, 3003, 3289, 3542, 4096, 4914, 4928 (twice), 5005, 5103, 6630, 7293, 7371, 7722, 8372, 9177, 9660, 10206, 10556, 11571, 11648, 12096, 13090….

The 14-dimensional representation is the adjoint representation, and the 7-dimensional one is action of G₂ on the imaginary octonions.

There are two non-isomorphic irreducible representations of dimensions 77, 2079, 4928, 28652, etc. The fundamental representations are those with dimensions 14 and 7 (corresponding to the two nodes in the Dynkin diagram in the order such that the triple arrow points from the first to the second).

Vogan (1994) described the (infinite-dimensional) unitary irreducible representations of the split real form of G₂.

The group G₂(q) is the points of the algebraic group G₂ over the finite field F_q. These finite groups were first introduced by Leonard Eugene Dickson in Dickson (1901) for odd q and Dickson (1905) for even q. The order of G₂(q) is q⁶(q⁶ − 1)(q² − 1). When q ≠ 2, the group is simple, and when q = 2, it has a simple subgroup of index 2 isomorphic to ²A₂(3²), and is the automorphism group of a maximal order of the octonions. The Janko group J₁ was first constructed as a subgroup of G₂(11). Ree (1960) introduced twisted Ree groups ²G₂(q) of order q³(q³ + 1)(q − 1) for q = 3²ⁿ⁺¹, an odd power of 3.

• Cartan matrix
• Dynkin diagram
• Exceptional Jordan algebra
• Fundamental representation
• G₂-structure
• Lie group
• Seven-dimensional cross product
• Simple Lie group

• Adams, J. Frank (1996), Lectures on exceptional Lie groups, Chicago Lectures in Mathematics, University of Chicago Press, ISBN 978-0-226-00526-3, MR 1428422
• Baez, John (2002), "The Octonions", Bull. Amer. Math. Soc., 39 (2): 145–205, arXiv:math/0105155, doi:10.1090/S0273-0979-01-00934-X.

See section 4.1: G₂; an online HTML version of which is available at http://math.ucr.edu/home/baez/octonions/node14.html.

• Bryant, Robert (1987), "Metrics with Exceptional Holonomy", Annals of Mathematics, 2, 126 (3): 525–576, doi:10.2307/1971360, JSTOR 1971360
• Dickson, Leonard Eugene (1901), "Theory of Linear Groups in An Arbitrary Field", Transactions of the American Mathematical Society, Providence, R.I.: American Mathematical Society, 2 (4): 363–394, doi:10.1090/S0002-9947-1901-1500573-3, ISSN 0002-9947, JSTOR 1986251, Reprinted in volume II of his collected papers Leonard E. Dickson reported groups of type G₂ in fields of odd characteristic.
• Dickson, L. E. (1905), "A new system of simple groups", Math. Ann., 60: 137–150, doi:10.1007/BF01447497 Leonard E. Dickson reported groups of type G₂ in fields of even characteristic.
• Ree, Rimhak (1960), "A family of simple groups associated with the simple Lie algebra of type (G₂)", Bulletin of the American Mathematical Society, 66 (6): 508–510, doi:10.1090/S0002-9904-1960-10523-X, ISSN 0002-9904, MR 0125155
• Vogan, David A. Jr. (1994), "The unitary dual of G₂", Inventiones Mathematicae, 116 (1): 677–791, Bibcode:1994InMat.116..677V, doi:10.1007/BF01231578, ISSN 0020-9910, MR 1253210