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Exceptional Lie Algebra
In mathematics, an exceptional Lie algebra is a complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ... simple Lie algebra whose Dynkin diagram is of exceptional (nonclassical) type. There are exactly five of them: \mathfrak_2, \mathfrak_4, \mathfrak_6, \mathfrak_7, \mathfrak_8; their respective dimensions are 14, 52, 78, 133, 248. The corresponding diagrams are: * G2 : * F4 : * E6 : * E7 : * E8 : In contrast, simple Lie algebras that are not exceptional are called classical Lie algebras (there are infinitely many of them). Construction There is no simple universally accepted way to construct exceptional Lie algebras; in fact, they were discovered only in the process of the classification program. Here are some constructions: *§ 22.1-2 of give a detailed construction ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Lie Algebra
In mathematics, a complex Lie algebra is a Lie algebra over the complex numbers. Given a complex Lie algebra \mathfrak, its conjugate \overline is a complex Lie algebra with the same underlying real vector space but with i = \sqrt acting as -i instead. As a real Lie algebra, a complex Lie algebra \mathfrak is trivially isomorphic to its conjugate. A complex Lie algebra is isomorphic to its conjugate if and only if it admits a real form (and is said to be defined over the real numbers). Real form Given a complex Lie algebra \mathfrak, a real Lie algebra \mathfrak_0 is said to be a real form of \mathfrak if the complexification \mathfrak_0 \otimes_\mathbb is isomorphic to \mathfrak. A real form \mathfrak_0 is abelian (resp. nilpotent, solvable, semisimple) if and only if \mathfrak is abelian (resp. nilpotent, solvable, semisimple). On the other hand, a real form \mathfrak_0 is simple if and only if either \mathfrak is simple or \mathfrak is of the form \mathfrak \times \overli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simple Lie Algebra
In algebra, a simple Lie algebra is a Lie algebra that is non-abelian and contains no nonzero proper ideals. The classification of real simple Lie algebras is one of the major achievements of Wilhelm Killing and Élie Cartan. A direct sum of simple Lie algebras is called a semisimple Lie algebra. A simple Lie group is a connected Lie group whose Lie algebra is simple. Complex simple Lie algebras A finite-dimensional simple complex Lie algebra is isomorphic to either of the following: \mathfrak_n \mathbb, \mathfrak_n \mathbb, \mathfrak_ \mathbb ( classical Lie algebras) or one of the five exceptional Lie algebras. To each finite-dimensional complex semisimple Lie algebra In mathematics, a Lie algebra is semisimple if it is a direct sum of modules, direct sum of Simple Lie algebra, simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper Lie algebra#Subalgebras.2C ideals ... \mathfrak, there exists a corresponding diagram ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynkin Diagram
In the Mathematics, mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of Graph (discrete mathematics), graph with some edges doubled or tripled (drawn as a double or triple line). Dynkin diagrams arise in the classification of semisimple Lie algebras over algebraically closed fields, in the classification of Weyl groups and other finite reflection groups, and in other contexts. Various properties of the Dynkin diagram (such as whether it contains multiple edges, or its symmetries) correspond to important features of the associated Lie algebra. The term "Dynkin diagram" can be ambiguous. In some cases, Dynkin diagrams are assumed to be directed graph, directed, in which case they correspond to root systems and semi-simple Lie algebras, while in other cases they are assumed to be undirected graph, undirected, in which case they correspond to Weyl groups. In this article, "Dynkin diagram" means ''directed'' Dynkin diagram, and ''undirected'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
G2 (mathematics)
In mathematics, G2 is three simple Lie groups (a complex form, a compact real form and a split real form), their Lie algebras \mathfrak_2, 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, octonion algebra or, equivalently, as the subgroup of SO(7) that preserves any chosen particular vector in its 8-dimensional Real representation, real spinor Group representation, representation (a spin representation). 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 wrote a letter to Friedrich Engel (mathematician), Friedrich Engel saying that he had found a 14-dimensional simple Lie algebra, which we now ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
F4 (mathematics)
In mathematics, F4 is 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. The compact real form of F4 is the isometry group of a 16-dimensional Riemannian manifold known as the octonionic projective plane OP2. This can be seen systematically using a construction known as the Freudenthal magic square, ''magic square'', due to Hans Freudenthal and Jacques Tits. There are list of simple Lie groups, 3 real forms: a compact one, a split one, and a third one. They are the isometry groups of the three real Albert algebras. The F4 Lie algebra may be constructed by adding 16 generators transforming as a spinor to the 36-dimensional Lie algebra so(9), in analogy with the construction of E8 (mathematics), E8. In older books and papers, F4 is sometimes denoted by E4. Alg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
E6 (mathematics)
In mathematics, E6 is the name of some closely related Lie groups, linear algebraic groups or their Lie algebras \mathfrak_6, 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 ). This classifies Lie algebras into four infinite series labeled A''n'', B''n'', C''n'', D''n'', and five exceptional cases labeled E6, E7, E8, F4, and G2. The E6 algebra is thus one of the five exceptional cases. The fundamental group of the adjoint form of E6 (as a complex or compact Lie group) is the cyclic group Z/3Z, and its outer automorphism group is the cyclic group Z/2Z. For the simply-connected form, its fundamental representation is 27-dimensional, and a basis is given by the 27 lines on a cubic surface. The dual representation, which is inequivalent, is also 27-dimensional. In particle physics, E6 plays a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
E7 (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 A''n'', B''n'', C''n'', D''n'', and five exceptional cases labeled E6, E7, E8, F4, and G2. The E7 algebra is thus one of the five exceptional cases. The fundamental group of the (adjoint) complex form, compact real form, or any algebraic version of E7 is the cyclic group Z/2Z, and its outer automorphism group is the trivial group. The dimension of its fundamental representation is 56. Real and complex forms There is a unique complex Lie algebra of type E7, corresponding to a complex group of complex dimension 133. The complex adjoint Lie group E7 of complex dimension 133 can be considered ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
E8 (mathematics)
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 A''n'', B''n'', C''n'', D''n'', and five exceptional cases labeled G2, F4, E6, E7, and E8. The E8 algebra is the largest and most complicated of these exceptional cases. Basic description The Lie group E8 has dimension 248. Its rank, which is the dimension of its maximal torus, is eight. Therefore, the vectors of the root system are in eight-dimensional Euclidean space: they are described explicitly later in this article. The Weyl group of E8, which is the group of symmetries of the maximal torus that are induced by conjugations in the whole group, has order 2357 = . The compact group E8 is u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classical Lie Algebra
The classical Lie algebras are finite-dimensional Lie algebras over a field which can be classified into four types A_n , B_n , C_n and D_n , where for \mathfrak(n) the general linear Lie algebra and I_n the n \times n identity matrix: * A_n := \mathfrak(n+1) = \ , the ''special linear Lie algebra''; * B_n := \mathfrak(2n+1) = \ , the ''odd orthogonal Lie algebra''; * C_n := \mathfrak(2n) = \ , the ''symplectic Lie algebra''; and * D_n := \mathfrak(2n) = \ , the ''even orthogonal Lie algebra''. Except for the low-dimensional cases D_1 = \mathfrak(2) and D_2 = \mathfrak(4) , the classical Lie algebras are simple. The Moyal algebra is an infinite-dimensional Lie algebra that contains all classical Lie algebras as subalgebras. See also * Simple Lie algebra * Classical group In mathematics, the classical groups are defined as the special linear groups over the reals \mathbb, the complex numbers \mathbb and the quaternions \mathbb together with special automo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacques Tits
Jacques Tits () (12 August 1930 – 5 December 2021) was a Belgian-born French mathematician who worked on group theory and incidence geometry. He introduced Tits buildings, the Tits alternative, the Tits group, and the Tits metric. Early life and education Tits was born in Uccle, Belgium to Léon Tits, a professor, and Lousia André. Jacques attended the Athénée of Uccle and the Free University of Brussels (1834–1969), Free University of Brussels. His thesis advisor was , and Tits graduated with his doctorate in 1950 with the dissertation ''Généralisation des groupes projectifs basés sur la notion de transitivité''. Career Tits held professorships at the Free University of Brussels (now split into the Université libre de Bruxelles and the Vrije Universiteit Brussel) (1962–1964), the University of Bonn (1964–1974) and the Collège de France in Paris, until becoming emeritus in 2000. He changed his citizenship to French in 1974 in order to teach at the Collège de F ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Algebras
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identity. In other words, a Lie algebra is an algebra over a field for which the multiplication operation (called the Lie bracket) is alternating and satisfies the Jacobi identity. The Lie bracket of two vectors x and y is denoted ,y/math>. A Lie algebra is typically a non-associative algebra. However, every associative algebra gives rise to a Lie algebra, consisting of the same vector space with the commutator Lie bracket, ,y= xy - yx . Lie algebras are closely related to Lie groups, which are groups that are also smooth manifolds: every Lie group gives rise to a Lie algebra, which is the tangent space at the identity. (In this case, the Lie bracket measures the failure of commutativity for the Lie group.) Conversely, to any finite-dime ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
