Algebraic structure → Group theory Group theory 

Lie groups 

In mathematics, G_{2} 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. 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
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 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, 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.
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 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, 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, F_{4} is the name of a Lie group and also its Lie algebra f_{4}. It is one of the five exceptional simple Lie groups. F_{4} 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 26dimensional.
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 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 with finite kernel which 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 semisimple algebraic groups are reductive.
In mathematics, the Cartan decomposition is a decomposition of a semisimple Lie group or Lie algebra, which plays an important role in their structure theory and representation theory. It generalizes the polar decomposition or singular value decomposition of matrices. Its history can be traced to the 1880s work of Élie Cartan and Wilhelm Killing.
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, a symmetric space is a Riemannian manifold whose group of symmetries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geometry, leading to consequences in the theory of holonomy; or algebraically through Lie theory, which allowed Cartan to give a complete classification. Symmetric spaces commonly occur in differential geometry, representation theory and harmonic analysis.
In the theory of von Neumann algebras, a part of the mathematical field of functional analysis, Tomita–Takesaki theory is a method for constructing modular automorphisms of von Neumann algebras from the polar decomposition of a certain involution. It is essential for the theory of type III factors, and has led to a good structure theory for these previously intractable objects.
In mathematics, a Hermitian symmetric space is a Hermitian manifold which at every point has an inversion symmetry preserving the Hermitian structure. First studied by Élie Cartan, they form a natural generalization of the notion of Riemannian symmetric space from real manifolds to complex manifolds.
In mathematics, the Freudenthal magic square is a construction relating several Lie algebras. It is named after Hans Freudenthal and Jacques Tits, who developed the idea independently. It associates a Lie algebra to a pair of division algebras A, B. The resulting Lie algebras have Dynkin diagrams according to the table at right. The "magic" of the Freudenthal magic square is that the constructed Lie algebra is symmetric in A and B, despite the original construction not being symmetric, though Vinberg's symmetric method gives a symmetric construction.
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 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.
In mathematics, Kostant's convexity theorem, introduced by Bertram Kostant (1973), states that the projection of every coadjoint orbit of a connected compact Lie group into the dual of a Cartan subalgebra is a convex set. It is a special case of a more general result for symmetric spaces. Kostant's theorem is a generalization of a result of Schur (1923), Horn (1954) and Thompson (1972) for hermitian matrices. They proved that the projection onto the diagonal matrices of the space of all n by n complex selfadjoint matrices with given eigenvalues Λ = is the convex polytope with vertices all permutations of the coordinates of Λ.
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.
In mathematics, symmetric cones, sometimes called domains of positivity, are open convex selfdual 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 finitedimensional 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.