G2 (mathematics)

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In mathematics, G2 is the name of three simple Lie groups (a complex form, a compact real form and a split real form), their Lie algebras as well as some algebraic groups. They are the smallest of the five exceptional simple Lie groups. G2 has rank 2 and dimension 14. It has two fundamental representations, with dimension 7 and 14.

Contents

The compact form of G2 can be described as the automorphism group of the octonion algebra or, equivalently, as the subgroup of SO(7) that preserves any chosen particular vector in its 8-dimensional real spinor representation (a spin representation).

History

The Lie algebra , 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 . [1]

In 1893, Élie Cartan published a note describing an open set in 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 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.

Real forms

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

Algebra

Dynkin diagram and Cartan matrix

The Dynkin diagram for G2 is given by Dynkin diagram G2.png .

Its Cartan matrix is:

Roots of G2

Root system G2.svg
The 12 vector root system of G2 in 2 dimensions.
3-cube t1.svg
The A2 Coxeter plane projection of the 12 vertices of the cuboctahedron contain the same 2D vector arrangement.
G2Coxeter.svg
Graph of G2 as a subgroup of F4 and E8 projected into the Coxeter plane

Although they span 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.

(1,1,0), (1,1,0)
(1,0,1), (1,0,1)
(0,1,1), (0,1,1)
(2,1,1), (2,1,1)
(1,2,1), (1,2,1)
(1,1,2), (1,1,2)

One set of simple roots, for Dyn2-node n1.pngDyn2-6a.pngDyn2-node n2.png is:

(0,1,1), (1,2,1)

Weyl/Coxeter group

Its Weyl/Coxeter group is the dihedral group of order 12. It has minimal faithful degree .

Special holonomy

G2 is one of the possible special groups that can appear as the holonomy group of a Riemannian metric. The manifolds 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.

(± 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:

It is exactly the Lie algebra of the group

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. 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, 30107, 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.

Finite groups

The group G2(q) is the points of the algebraic group G2 over the finite field Fq. 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.

See also

Related Research Articles

In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface O or blackboard bold . Octonions have eight dimensions; twice the number of dimensions of the quaternions, of which they are an extension. They are noncommutative and nonassociative, but satisfy a weaker form of associativity; namely, they are alternative. They are also power associative.

Symplectic group Mathematical group

In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted Sp(2n, F) and Sp(n) for positive integer n and field F. The latter is called the compact symplectic group and is also denoted by . Many authors prefer slightly different notations, usually differing by factors of 2. The notation used here is consistent with the size of the most common matrices which represent the groups. In Cartan's classification of the simple Lie algebras, the Lie algebra of the complex group Sp(2n, C) is denoted Cn, and Sp(n) is the compact real form of Sp(2n, C). Note that when we refer to the (compact) symplectic group it is implied that we are talking about the collection of (compact) symplectic groups, indexed by their dimension n.

Simple Lie group Connected non-abelian Lie group lacking nontrivial connected normal subgroups

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 symmetric spaces.

F<sub>4</sub> (mathematics)

In mathematics, F4 is the name of a Lie group and also its Lie algebra f4. It is one of the five exceptional simple Lie groups. F4 has rank 4 and dimension 52. The compact form is simply connected and its outer automorphism group is the trivial group. Its fundamental representation is 26-dimensional.

E<sub>6</sub> (mathematics) 78-dimensional exceptional simple Lie group

In mathematics, E6 is the name of some closely related Lie groups, linear algebraic groups or their Lie algebras , all of which have dimension 78; the same notation E6 is used for the corresponding root lattice, which has rank 6. The designation E6 comes from the Cartan–Killing classification of the complex simple Lie algebras (see Élie Cartan § Work). This classifies Lie algebras into four infinite series labeled An, Bn, Cn, Dn, and five exceptional cases labeled E6, E7, E8, F4, and G2. The E6 algebra is thus one of the five exceptional cases.

In the mathematical field of differential geometry, a Cartan connection is a flexible generalization of the notion of an affine connection. It may also be regarded as a specialization of the general concept of a principal connection, in which the geometry of the principal bundle is tied to the geometry of the base manifold using a solder form. Cartan connections describe the geometry of manifolds modelled on homogeneous spaces.

E<sub>8</sub> (mathematics) 248-dimensional exceptional simple Lie group

In mathematics, E8 is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding root lattice, which has rank 8. The designation E8 comes from the Cartan–Killing classification of the complex simple Lie algebras, which fall into four infinite series labeled An, Bn, Cn, Dn, and five exceptional cases labeled G2, F4, E6, E7, and E8. The E8 algebra is the largest and most complicated of these exceptional cases.

E<sub>7</sub> (mathematics)

In mathematics, E7 is the name of several closely related Lie groups, linear algebraic groups or their Lie algebras e7, all of which have dimension 133; the same notation E7 is used for the corresponding root lattice, which has rank 7. The designation E7 comes from the Cartan–Killing classification of the complex simple Lie algebras, which fall into four infinite series labeled An, Bn, Cn, Dn, and five exceptional cases labeled E6, E7, E8, F4, and G2. The E7 algebra is thus one of the five exceptional cases.

Reductive group

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In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra of a Lie algebra that is self-normalising. They were introduced by Élie Cartan in his doctoral thesis. It controls the representation theory of a semi-simple Lie algebra over a field of characteristic .

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In mathematics, a zonal spherical function or often just spherical function is a function on a locally compact group G with compact subgroup K that arises as the matrix coefficient of a K-invariant vector in an irreducible representation of G. The key examples are the matrix coefficients of the spherical principal series, the irreducible representations appearing in the decomposition of the unitary representation of G on L2(G/K). In this case the commutant of G is generated by the algebra of biinvariant functions on G with respect to K acting by right convolution. It is commutative if in addition G/K is a symmetric space, for example when G is a connected semisimple Lie group with finite centre and K is a maximal compact subgroup. The matrix coefficients of the spherical principal series describe precisely the spectrum of the corresponding C* algebra generated by the biinvariant functions of compact support, often called a Hecke algebra. The spectrum of the commutative Banach *-algebra of biinvariant L1 functions is larger; when G is a semisimple Lie group with maximal compact subgroup K, additional characters come from matrix coefficients of the complementary series, obtained by analytic continuation of the spherical principal series.

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Complexification (Lie group) Universal construction of a complex Lie group from a real Lie group

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In mathematics, symmetric cones, sometimes called domains of positivity, are open convex self-dual cones in Euclidean space which have a transitive group of symmetries, i.e. invertible operators that take the cone onto itself. By the Koecher–Vinberg theorem these correspond to the cone of squares in finite-dimensional real Euclidean Jordan algebras, originally studied and classified by Jordan, von Neumann & Wigner (1934). The tube domain associated with a symmetric cone is a noncompact Hermitian symmetric space of tube type. All the algebraic and geometric structures associated with the symmetric space can be expressed naturally in terms of the Jordan algebra. The other irreducible Hermitian symmetric spaces of noncompact type correspond to Siegel domains of the second kind. These can be described in terms of more complicated structures called Jordan triple systems, which generalize Jordan algebras without identity.

References

  1. Agricola, Ilka (2008). "Old and new on the exceptional group G2" (PDF). Notices of the American Mathematical Society. 55 (8): 922–929. MR   2441524.
  2. Élie Cartan (1893). "Sur la structure des groupes simples finis et continus". C. R. Acad. Sci. 116: 784–786.
  3. Gil Bor and Richard Montgomery (2009). "G2 and the "rolling distribution"". L'Enseignement Mathématique. 55: 157–196. arXiv: math/0612469 . doi:10.4171/lem/55-1-8.
  4. John Baez and John Huerta (2014). "G2 and the rolling ball". Trans. Amer. Math. Soc. 366 (10): 5257–5293. arXiv: 1205.2447 . doi:10.1090/s0002-9947-2014-05977-1.
  5. Friedrich Engel (1900). "Ein neues, dem linearen Komplexe analoges Gebilde". Leipz. Ber. 52: 63–76, 220–239.
  6. Élie Cartan (1908). "Nombres complexes". Encyclopedie des Sciences Mathematiques. Paris: Gauthier-Villars. pp. 329–468.
  7. Élie Cartan (1914), "Les groupes reels simples finis et continus", Ann. Sci. École Norm. Sup., 31: 255–262
See section 4.1: G2; an online HTML version of which is available at http://math.ucr.edu/home/baez/octonions/node14.html.