Algebraic structure → Group theory Group theory 

Lie groups and Lie algebras 

In mathematics, G_{2} 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. G_{2} has rank 2 and dimension 14. It has two fundamental representations, with dimension 7 and 14.
The compact form of G_{2} 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 8dimensional 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 14dimensional simple Lie algebra, which we now call .^{ [1] }
In 1893, Élie Cartan published a note describing an open set in equipped with a 2dimensional distribution—that is, a smoothly varying field of 2dimensional 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 2dimensional distribution is closely related to a ball rolling on another ball. The space of configurations of the rolling ball is 5dimensional, with a 2dimensional 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 3form) on a 7dimensional complex vector space is preserved by a group isomorphic to the complex form of G_{2}.^{ [5] }
In 1908 Cartan mentioned that the automorphism group of the octonions is a 14dimensional simple Lie group.^{ [6] } In 1914 he stated that this is the compact real form of G_{2}.^{ [7] }
In older books and papers, G_{2} is sometimes denoted by E_{2}.
There are 3 simple real Lie algebras associated with this root system:
The Dynkin diagram for G_{2} is given by .
Its Cartan matrix is:
The 12 vector root system of G_{2} in 2 dimensions.  The A_{2} 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 2dimensional space, as drawn, it is much more symmetric to consider them as vectors in a 2dimensional subspace of a threedimensional space.


One set of simple roots, for is:
Its Weyl/Coxeter group is the dihedral group of order 12. It has minimal faithful degree .
G_{2} is one of the possible special groups that can appear as the holonomy group of a Riemannian metric. The manifolds of G_{2} holonomy are also called G_{2}manifolds.
G_{2} is the automorphism group of the following two polynomials in 7 noncommutative variables.
which comes from the octonion algebra. The variables must be noncommutative 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^{ [8] } using an invertible form for octonions. For other signed variations of , this form has remainders that classify 6 other nonassociative algebras that show partial symmetry. An analogous calibration in leads to sedenions and at least 11 other related algebras.
The characters of finitedimensional 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 14dimensional representation is the adjoint representation, and the 7dimensional one is action of G_{2} on the imaginary octonions.
There are two nonisomorphic 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 (infinitedimensional) unitary irreducible representations of the split real form of G_{2}.
The embeddings of the maximal subgroups of G_{2} up to dimension 77 are shown to the right.
The group G_{2}(q) is the points of the algebraic group G_{2} 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 G_{2}(q) is q^{6}(q^{6} − 1)(q^{2} − 1). When q ≠ 2, the group is simple, and when q = 2, it has a simple subgroup of index 2 isomorphic to ^{2}A_{2}(3^{2}), and is the automorphism group of a maximal order of the octonions. The Janko group J_{1} was first constructed as a subgroup of G_{2}(11). Ree (1960) introduced twisted Ree groups ^{2}G_{2}(q) of order q^{3}(q^{3} + 1)(q − 1) for q = 3^{2n+1}, an odd power of 3.
In mathematics, a Lie group is a group that is also a differentiable manifold.
In mathematics, the orthogonal group in dimension , denoted , is the group of distancepreserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. The orthogonal group is sometimes called the general orthogonal group, by analogy with the general linear group. Equivalently, it is the group of orthogonal matrices, where the group operation is given by matrix multiplication. The orthogonal group is an algebraic group and a Lie group. It is compact.
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 (usually C or R). 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 C_{n}, 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.
In mathematics, the special unitary group of degree n, denoted SU(n), is the Lie group of n × n unitary matrices with determinant 1.
In mathematics, a simple Lie group is a connected nonabelian 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.
In mathematics, E_{6} 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 E_{6} is used for the corresponding root lattice, which has rank 6. The designation E_{6} 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 A_{n}, B_{n}, C_{n}, D_{n}, and five exceptional cases labeled E_{6}, E_{7}, E_{8}, F_{4}, and G_{2}. The E_{6} 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.
In differential geometry, the holonomy of a connection on a smooth manifold is a general geometrical consequence of the curvature of the connection measuring the extent to which parallel transport around closed loops fails to preserve the geometrical data being transported. For flat connections, the associated holonomy is a type of monodromy and is an inherently global notion. For curved connections, holonomy has nontrivial local and global features.
In mathematics, E_{8} 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 E_{8} comes from the Cartan–Killing classification of the complex simple Lie algebras, which fall into four infinite series labeled A_{n}, B_{n}, C_{n}, D_{n}, and five exceptional cases labeled G_{2}, F_{4}, E_{6}, E_{7}, and E_{8}. The E_{8} algebra is the largest and most complicated of these exceptional cases.
In mathematics, E_{7} is the name of several closely related Lie groups, linear algebraic groups or their Lie algebras e_{7}, all of which have dimension 133; the same notation E_{7} is used for the corresponding root lattice, which has rank 7. The designation E_{7} comes from the Cartan–Killing classification of the complex simple Lie algebras, which fall into four infinite series labeled A_{n}, B_{n}, C_{n}, D_{n}, and five exceptional cases labeled E_{6}, E_{7}, E_{8}, F_{4}, and G_{2}. The E_{7} algebra is thus one of the five exceptional cases.
In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group G over a perfect field is reductive if it has a representation that has a finite kernel and is a direct sum of irreducible representations. Reductive groups include some of the most important groups in mathematics, such as the general linear group GL(n) of invertible matrices, the special orthogonal group SO(n), and the symplectic group Sp(2n). Simple algebraic groups and (more generally) semisimple algebraic groups are reductive.
In mathematics, Hurwitz's automorphisms theorem bounds the order of the group of automorphisms, via orientationpreserving conformal mappings, of a compact Riemann surface of genus g > 1, stating that the number of such automorphisms cannot exceed 84(g − 1). A group for which the maximum is achieved is called a Hurwitz group, and the corresponding Riemann surface a Hurwitz surface. Because compact Riemann surfaces are synonymous with nonsingular complex projective algebraic curves, a Hurwitz surface can also be called a Hurwitz curve. The theorem is named after Adolf Hurwitz, who proved it in (Hurwitz 1893).
In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra of a Lie algebra that is selfnormalising. They were introduced by Élie Cartan in his doctoral thesis. It controls the representation theory of a semisimple Lie algebra over a field of characteristic .
In mathematics, an affine Lie algebra is an infinitedimensional Lie algebra that is constructed in a canonical fashion out of a finitedimensional simple Lie algebra. Given an affine Lie algebra, one can also form the associated affine KacMoody algebra, as described below. From a purely mathematical point of view, affine Lie algebras are interesting because their representation theory, like representation theory of finitedimensional semisimple Lie algebras, is much better understood than that of general Kac–Moody algebras. As observed by Victor Kac, the character formula for representations of affine Lie algebras implies certain combinatorial identities, the Macdonald identities.
In mathematics, the spin representations are particular projective representations of the orthogonal or special orthogonal groups in arbitrary dimension and signature. More precisely, they are two equivalent representations of the spin groups, which are double covers of the special orthogonal groups. They are usually studied over the real or complex numbers, but they can be defined over other fields.
In mathematics, the classical groups are defined as the special linear groups over the reals R, the complex numbers C and the quaternions H together with special automorphism groups of symmetric or skewsymmetric bilinear forms and Hermitian or skewHermitian sesquilinear forms defined on real, complex and quaternionic finitedimensional vector spaces. Of these, the complex classical Lie groups are four infinite families of Lie groups that together with the exceptional groups exhaust the classification of simple Lie groups. The compact classical groups are compact real forms of the complex classical groups. The finite analogues of the classical groups are the classical groups of Lie type. The term "classical group" was coined by Hermann Weyl, it being the title of his 1939 monograph The Classical Groups.
In mathematics, a zonal spherical function or often just spherical function is a function on a locally compact group G with compact subgroup K (often a maximal compact subgroup) that arises as the matrix coefficient of a Kinvariant 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 L^{2}(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 L^{1} 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.
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations. The theory of matrices and linear operators is wellunderstood, so representations of more abstract objects in terms of familiar linear algebra objects helps glean properties and sometimes simplify calculations on more abstract theories.
In mathematics, the complexification or universal complexification of a real Lie group is given by a continuous homomorphism of the group into a complex Lie group with the universal property that every continuous homomorphism of the original group into another complex Lie group extends compatibly to a complex analytic homomorphism between the complex Lie groups. The complexification, which always exists, is unique up to unique isomorphism. Its Lie algebra is a quotient of the complexification of the Lie algebra of the original group. They are isomorphic if the original group has a quotient by a discrete normal subgroup which is linear.
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.