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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, G2 is the name of three simple
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
s (a complex form, a compact real form and a split real form), their
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
s \mathfrak_2, as well as some
algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Man ...
s. They are the smallest of the five exceptional
simple Lie group In mathematics, a simple Lie group is a connected non-abelian Lie group ''G'' which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of simple Lie algebras and Riemannian symme ...
s. G2 has rank 2 and dimension 14. It has two
fundamental representation In representation theory of Lie groups and Lie algebras, a fundamental representation is an irreducible finite-dimensional representation of a semisimple Lie group or Lie algebra whose highest weight is a fundamental weight. For example, the defini ...
s, with dimension 7 and 14. The compact form of G2 can be described as the
automorphism group In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...
of the
octonion algebra In mathematics, an octonion algebra or Cayley algebra over a field ''F'' is a composition algebra over ''F'' that has dimension 8 over ''F''. In other words, it is a unital non-associative algebra ''A'' over ''F'' with a non-degenerate quadratic ...
or, equivalently, as the subgroup of SO(7) that preserves any chosen particular vector in its 8-dimensional
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
spinor In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a slight ...
representation (a
spin representation In mathematics, the spin representations are particular projective representations of the orthogonal group, orthogonal or special orthogonal groups in arbitrary dimension and metric signature, signature (i.e., including indefinite orthogonal groups ...
).


History

The Lie algebra \mathfrak_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 Wilhelm Karl Joseph Killing (10 May 1847 – 11 February 1923) was a German mathematician who made important contributions to the theories of Lie algebras, Lie groups, and non-Euclidean geometry. Life Killing studied at the University of Mü ...
wrote a letter to Friedrich Engel saying that he had found a 14-dimensional simple Lie algebra, which we now call \mathfrak_2. In 1893,
Élie Cartan Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometry. ...
published a note describing an open set in \mathbb^5 equipped with a 2-dimensional
distribution Distribution may refer to: Mathematics *Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations * Probability distribution, the probability of a particular value or value range of a vari ...
—that is, a smoothly varying field of 2-dimensional subspaces of the tangent space—for which the Lie algebra \mathfrak_2 appears as the infinitesimal symmetries. 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. 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. In 1908 Cartan mentioned that the automorphism group of the octonions is a 14-dimensional simple Lie group. In 1914 he stated that this is the compact real form of G2. In older books and papers, G2 is sometimes denoted by E2.


Real forms

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 In mathematics, the outer automorphism group of a group, , is the quotient, , where is the automorphism group of and ) is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted . If is trivial and has a t ...
is the trivial group. Its maximal compact subgroup is . It has a non-algebraic double cover that is simply connected.


Algebra


Dynkin diagram and Cartan matrix

The
Dynkin diagram In the mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of graph with some edges doubled or tripled (drawn as a double or triple line). Dynkin diagrams arise in the classification of semisimple Lie algebras ...
for ''G''2 is given by . Its
Cartan matrix In mathematics, the term Cartan matrix has three meanings. All of these are named after the French mathematician Élie Cartan. Amusingly, the Cartan matrices in the context of Lie algebras were first investigated by Wilhelm Killing, whereas the ...
is: : \left begin 2 & -3 \\ -1 & 2 \end\right


Roots of G2

A set of simple roots for can be read directly from the Cartan matrix above. These are (2,−3) and (−1, 2), however the integer lattice spanned by those is not the one pictured above (from obvious reason: the hexagonal lattice on the plane cannot be generated by integer vectors). The diagram above is obtained from a different pair roots: \alpha = \left( \sqrt, 0 \right) and \beta = \left(\sqrt\cos,\sin\right) = \frac\left(\sqrt,1 \right). The remaining (positive) roots are A = α + β, B = 3α + β, α + A = 2α + β, and A + B = 3α + 2β. Although they do
span Span may refer to: Science, technology and engineering * Span (unit), the width of a human hand * Span (engineering), a section between two intermediate supports * Wingspan, the distance between the wingtips of a bird or aircraft * Sorbitan ester ...
a 2-dimensional space, as drawn, it is much more symmetric to consider them as vectors in a 2-dimensional subspace of a three-dimensional space. In this identification α corresponds to e₁−e₂, β to −e₁ + 2e₂−e₃, A to e₂−e₃ and so on. In euclidean coordinates these vectors look as follows: The corresponding set of simple roots is: :e₁−e₂ = (1,−1,0), and −e₁+2e₂−e₃ = (−1,2,−1) Note: α and A together form root system ''identical'' to A₂, while the system formed by β and B is ''isomorphic'' to A₂.


Weyl/Coxeter group

Its
Weyl Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is ass ...
/
Coxeter Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington to ...
group G = W(G_2) is the
dihedral group In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, ge ...
D_6 of order 12. It has minimal faithful degree \mu(G) = 5.


Special holonomy

G2 is one of the possible special groups that can appear as the
holonomy In differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus ...
group of a
Riemannian metric In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ''T ...
. The
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
s of G2 holonomy are also called G2-manifolds.


Polynomial invariant

G2 is the automorphism group of the following two polynomials in 7 non-commutative variables. :C_1 = t^2+u^2+v^2+w^2+x^2+y^2+z^2 :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.


Generators

Adding a representation of the 14 generators with coefficients ''A'', ..., ''N'' gives the matrix: :A\lambda_1+\cdots+N\lambda_= \begin 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 It is exactly the Lie algebra of the group : G_2=\


Representations

The characters of finite-dimensional representations of the real and complex Lie algebras and Lie groups are all given by the
Weyl character formula In mathematics, the Weyl character formula in representation theory describes the character theory, characters of irreducible representations of compact Lie groups in terms of their highest weights. It was proved by . There is a closely related fo ...
. The dimensions of the smallest irreducible representations are : :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 In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is GL(n ...
, 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, 30107, etc. The
fundamental representation In representation theory of Lie groups and Lie algebras, a fundamental representation is an irreducible finite-dimensional representation of a semisimple Lie group or Lie algebra whose highest weight is a fundamental weight. For example, the defini ...
s are those with dimensions 14 and 7 (corresponding to the two nodes in the
Dynkin diagram In the mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of graph with some edges doubled or tripled (drawn as a double or triple line). Dynkin diagrams arise in the classification of semisimple Lie algebras ...
in the order such that the triple arrow points from the first to the second). described the (infinite-dimensional) unitary irreducible representations of the split real form of G2.


Finite groups

The group G2(''q'') is the points of the algebraic group G2 over the
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
F''q''. These finite groups were first introduced by
Leonard Eugene Dickson Leonard Eugene Dickson (January 22, 1874 – January 17, 1954) was an American mathematician. He was one of the first American researchers in abstract algebra, in particular the theory of finite fields and classical groups, and is also remem ...
in for odd ''q'' and for even ''q''. The order of G2(''q'') is . When , the group is
simple Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by Johnn ...
, and when , it has a simple subgroup of
index Index (or its plural form indices) may refer to: Arts, entertainment, and media Fictional entities * Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index'' * The Index, an item on a Halo megastru ...
2 isomorphic to 2''A''2(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). introduced twisted
Ree group In mathematics, a Ree group is a group of Lie type over a finite field constructed by from an exceptional automorphism of a Dynkin diagram that reverses the direction of the multiple bonds, generalizing the Suzuki groups found by Suzuki using a dif ...
s 2G2(''q'') of order for , an odd power of 3.


See also

*
Cartan matrix In mathematics, the term Cartan matrix has three meanings. All of these are named after the French mathematician Élie Cartan. Amusingly, the Cartan matrices in the context of Lie algebras were first investigated by Wilhelm Killing, whereas the ...
*
Dynkin diagram In the mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of graph with some edges doubled or tripled (drawn as a double or triple line). Dynkin diagrams arise in the classification of semisimple Lie algebras ...
*
Exceptional Jordan algebra In mathematics, an Albert algebra is a 27-dimensional exceptional Jordan algebra. They are named after Abraham Adrian Albert, who pioneered the study of non-associative algebras, usually working over the real numbers. Over the real numbers, there a ...
*
Fundamental representation In representation theory of Lie groups and Lie algebras, a fundamental representation is an irreducible finite-dimensional representation of a semisimple Lie group or Lie algebra whose highest weight is a fundamental weight. For example, the defini ...
* G2-structure *
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
*
Seven-dimensional cross product In mathematics, the seven-dimensional cross product is a bilinear operation on vectors in seven-dimensional Euclidean space. It assigns to any two vectors a, b in a vector also in . Like the cross product in three dimensions, the seven-dime ...
*
Simple Lie group In mathematics, a simple Lie group is a connected non-abelian Lie group ''G'' which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of simple Lie algebras and Riemannian symme ...


References

* * . ::See section 4.1: G2; an online HTML version of which is available at http://math.ucr.edu/home/baez/octonions/node14.html. * * Leonard E. Dickson reported groups of type G2 in fields of odd characteristic. * Leonard E. Dickson reported groups of type G2 in fields of even characteristic. * * {{String theory topics , state=collapsed Algebraic groups Lie groups Octonions Exceptional Lie algebras