| Algebraic structure → Group theory
|Lie groups and Lie algebras
In mathematics, G2 is 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.
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).
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 .
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. 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.
There are 3 simple real Lie algebras associated with this root system:
The Dynkin diagram for G2 is given by .
Its Cartan matrix is:
The 12 vector root system of G2 in 2 dimensions.
The A2 Coxeter plane projection of the 12 vertices of the cuboctahedron contain the same 2D vector arrangement.
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.
One set of simple roots, for is:
Its Weyl/Coxeter group is the dihedral group of order 12. It has minimal faithful degree .
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.
G2 is the automorphism group of the following two polynomials in 7 non-commutative variables.
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:
It is exactly the Lie algebra of the group
There are 480 different representations of corresponding to the 480 representations of octonions. The calibrated form, has 30 different forms and each has 16 different signed variations. Each of the signed variations generate signed differences of and each is an automorphism of all 16 corresponding octonions. Hence there are really only 30 different representations of . These can all be constructed with Clifford algebra using an invertible form for octonions. For other signed variations of , this form has remainders that classify 6 other non-associative algebras that show partial symmetry. An analogous calibration in leads to sedenions and at least 11 other related algebras.
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 ):
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.
The embeddings of the maximal subgroups of G2 up to dimension 77 are shown to the right.
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.
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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.
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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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This is a glossary for the terminology applied in the mathematical theories of Lie groups and Lie algebras. For the topics in the representation theory of Lie groups and Lie algebras, see Glossary of representation theory. Because of the lack of other options, the glossary also includes some generalizations such as quantum group.