Faster access than browser!
46 relations: Adjoint representation, Ado's theorem, Élie Cartan, Cartan subalgebra, Characteristic (algebra), Classification of finite simple groups, Compact group, Degenerate bilinear form, Derivation (differential algebra), Direct sum, Direct sum of modules, Dynkin diagram, E6 (mathematics), E7 (mathematics), E8 (mathematics), Eugene Dynkin, Exceptional isomorphism, F4 (mathematics), G2 (mathematics), Jordan–Chevalley decomposition, Killing form, Levi decomposition, Lie algebra, Lie algebra representation, Linear Lie algebra, Mathematics, Maximal torus, Orthogonal group, Radical of a Lie algebra, Real form (Lie theory), Reductive Lie algebra, Root system, Satake diagram, Semi-simplicity, Simple Lie group, Solvable Lie algebra, Special linear Lie algebra, Special unitary group, Split Lie algebra, Springer Science+Business Media, Symplectic group, Weight (representation theory), Weyl character formula, Weyl's theorem on complete reducibility, Whitehead's lemma (Lie algebras), Wilhelm Killing.
Adjoint representation
In mathematics, the adjoint representation (or adjoint action) of a Lie group G is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space.
New!!: Semisimple Lie algebra and Adjoint representation · See more »
Ado's theorem
In abstract algebra, Ado's theorem is a theorem characterizing finite-dimensional Lie algebras.
New!!: Semisimple Lie algebra and Ado's theorem · See more »
Élie Cartan
Élie Joseph Cartan, ForMemRS (9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups and their geometric applications.
New!!: Semisimple Lie algebra and Élie Cartan · See more »
Cartan subalgebra
In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra \mathfrak of a Lie algebra \mathfrak that is self-normalising (if \in \mathfrak for all X \in \mathfrak, then Y \in \mathfrak).
New!!: Semisimple Lie algebra and Cartan subalgebra · See more »
Characteristic (algebra)
In mathematics, the characteristic of a ring R, often denoted char(R), is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0) if the sum does indeed eventually attain 0.
New!!: Semisimple Lie algebra and Characteristic (algebra) · See more »
Classification of finite simple groups
In mathematics, the classification of the finite simple groups is a theorem stating that every finite simple group belongs to one of four broad classes described below.
New!!: Semisimple Lie algebra and Classification of finite simple groups · See more »
Compact group
In mathematics, a compact (topological) group is a topological group whose topology is compact.
New!!: Semisimple Lie algebra and Compact group · See more »
Degenerate bilinear form
In mathematics, specifically linear algebra, a degenerate bilinear form on a vector space V is a bilinear form such that the map from V to V∗ (the dual space of V) given by is not an isomorphism.
New!!: Semisimple Lie algebra and Degenerate bilinear form · See more »
Derivation (differential algebra)
In mathematics, a derivation is a function on an algebra which generalizes certain features of the derivative operator.
New!!: Semisimple Lie algebra and Derivation (differential algebra) · See more »
Direct sum
The direct sum is an operation from abstract algebra, a branch of mathematics.
New!!: Semisimple Lie algebra and Direct sum · See more »
Direct sum of modules
In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module.
New!!: Semisimple Lie algebra and Direct sum of modules · See more »
Dynkin diagram
In the mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of graph with some edges doubled or tripled (drawn as a double or triple line).
New!!: Semisimple Lie algebra and Dynkin diagram · See more »
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.
New!!: Semisimple Lie algebra and E6 (mathematics) · See more »
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.
New!!: Semisimple Lie algebra and E7 (mathematics) · See more »
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.
New!!: Semisimple Lie algebra and E8 (mathematics) · See more »
Eugene Dynkin
Eugene Borisovich Dynkin (Евге́ний Бори́сович Ды́нкин; 11 May 1924 – 14 November 2014) was a Soviet and American mathematician.
New!!: Semisimple Lie algebra and Eugene Dynkin · See more »
Exceptional isomorphism
In mathematics, an exceptional isomorphism, also called an accidental isomorphism, is an isomorphism between members ai and bj of two families, usually infinite, of mathematical objects, that is not an example of a pattern of such isomorphisms.
New!!: Semisimple Lie algebra and Exceptional isomorphism · See more »
F4 (mathematics)
In mathematics, F4 is the name of a Lie group and also its Lie algebra f4.
New!!: Semisimple Lie algebra and F4 (mathematics) · See more »
G2 (mathematics)
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 \mathfrak_2, as well as some algebraic groups.
New!!: Semisimple Lie algebra and G2 (mathematics) · See more »
Jordan–Chevalley decomposition
In mathematics, the Jordan–Chevalley decomposition, named after Camille Jordan and Claude Chevalley, expresses a linear operator as the sum of its commuting semisimple part and its nilpotent parts.
New!!: Semisimple Lie algebra and Jordan–Chevalley decomposition · See more »
Killing form
In mathematics, the Killing form, named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras.
New!!: Semisimple Lie algebra and Killing form · See more »
Levi decomposition
In Lie theory and representation theory, the Levi decomposition, conjectured by Killing and Cartan and proved by, states that any finite-dimensional real Lie algebra g is the semidirect product of a solvable ideal and a semisimple subalgebra.
New!!: Semisimple Lie algebra and Levi decomposition · See more »
Lie algebra
In mathematics, a Lie algebra (pronounced "Lee") is a vector space \mathfrak g together with a non-associative, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g; (x, y) \mapsto, called the Lie bracket, satisfying the Jacobi identity.
New!!: Semisimple Lie algebra and Lie algebra · See more »
Lie algebra representation
In the mathematical field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices (or endomorphisms of a vector space) in such a way that the Lie bracket is given by the commutator.
New!!: Semisimple Lie algebra and Lie algebra representation · See more »
Linear Lie algebra
In algebra, a linear Lie algebra is a subalgebra \mathfrak of the Lie algebra \mathfrak(V) consisting of endomorphisms of a vector space V. In other words, a linear Lie algebra is the image of a Lie algebra representation.
New!!: Semisimple Lie algebra and Linear Lie algebra · See more »
Mathematics
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
New!!: Semisimple Lie algebra and Mathematics · See more »
Maximal torus
In the mathematical theory of compact Lie groups a special role is played by torus subgroups, in particular by the maximal torus subgroups.
New!!: Semisimple Lie algebra and Maximal torus · See more »
Orthogonal group
In mathematics, the orthogonal group in dimension, denoted, is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations.
New!!: Semisimple Lie algebra and Orthogonal group · See more »
Radical of a Lie algebra
In the mathematical field of Lie theory, the radical of a Lie algebra \mathfrak is the largest solvable ideal of \mathfrak.
New!!: Semisimple Lie algebra and Radical of a Lie algebra · See more »
Real form (Lie theory)
In mathematics, the notion of a real form relates objects defined over the field of real and complex numbers.
New!!: Semisimple Lie algebra and Real form (Lie theory) · See more »
Reductive Lie algebra
In mathematics, a Lie algebra is reductive if its adjoint representation is completely reducible, whence the name.
New!!: Semisimple Lie algebra and Reductive Lie algebra · See more »
Root system
In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties.
New!!: Semisimple Lie algebra and Root system · See more »
Satake diagram
In the mathematical study of Lie algebras and Lie groups, a Satake diagram is a generalization of a Dynkin diagram introduced by whose configurations classify simple Lie algebras over the field of real numbers.
New!!: Semisimple Lie algebra and Satake diagram · See more »
Semi-simplicity
In mathematics, semi-simplicity is a widespread concept in disciplines such as linear algebra, abstract algebra, representation theory, category theory, and algebraic geometry.
New!!: Semisimple Lie algebra and Semi-simplicity · See more »
Simple Lie group
In group theory, a simple Lie group is a connected non-abelian Lie group G which does not have nontrivial connected normal subgroups.
New!!: Semisimple Lie algebra and Simple Lie group · See more »
Solvable Lie algebra
In mathematics, a Lie algebra \mathfrak is solvable if its derived series terminates in the zero subalgebra.
New!!: Semisimple Lie algebra and Solvable Lie algebra · See more »
Special linear Lie algebra
In mathematics, the special linear Lie algebra of order n (denoted \mathfrak_n(F) or \mathfrak(n, F)) is the Lie algebra of n \times n matrices with trace zero and with the Lie bracket.
New!!: Semisimple Lie algebra and Special linear Lie algebra · See more »
Special unitary group
In mathematics, the special unitary group of degree, denoted, is the Lie group of unitary matrices with determinant 1.
New!!: Semisimple Lie algebra and Special unitary group · See more »
Split Lie algebra
In the mathematical field of Lie theory, a split Lie algebra is a pair (\mathfrak, \mathfrak) where \mathfrak is a Lie algebra and \mathfrak is a splitting Cartan subalgebra, where "splitting" means that for all x \in \mathfrak, \operatorname_ x is triangularizable.
New!!: Semisimple Lie algebra and Split Lie algebra · See more »
Springer Science+Business Media
Springer Science+Business Media or Springer, part of Springer Nature since 2015, is a global publishing company that publishes books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
New!!: Semisimple Lie algebra and Springer Science+Business Media · See more »
Symplectic group
In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted and, the latter is called the compact symplectic group.
New!!: Semisimple Lie algebra and Symplectic group · See more »
Weight (representation theory)
In the mathematical field of representation theory, a weight of an algebra A over a field F is an algebra homomorphism from A to F, or equivalently, a one-dimensional representation of A over F. It is the algebra analogue of a multiplicative character of a group.
New!!: Semisimple Lie algebra and Weight (representation theory) · See more »
Weyl character formula
In mathematics, the Weyl character formula in representation theory describes the characters of irreducible representations of compact Lie groups in terms of their highest weights.
New!!: Semisimple Lie algebra and Weyl character formula · See more »
Weyl's theorem on complete reducibility
In algebra, Weyl's theorem on complete reducibility is a fundamental result in the theory of Lie algebra representations.
New!!: Semisimple Lie algebra and Weyl's theorem on complete reducibility · See more »
Whitehead's lemma (Lie algebras)
In algebra, Whitehead's lemma on a Lie algebra representation (named after J. H. C. Whitehead) is an important step toward the proof of Weyl's theorem on complete reducibility.
New!!: Semisimple Lie algebra and Whitehead's lemma (Lie algebras) · See more »
Wilhelm Killing
Wilhelm Karl Joseph Killing (10 May 1847 – 11 February 1923) was a German mathematician who made important contributions to the theories of Lie algebras, Lie groups, and non-Euclidean geometry.
New!!: Semisimple Lie algebra and Wilhelm Killing · See more »
Redirects here:
Semi-simple Lie algebra, Semi-simple Lie group, Semisimple Lie group, Semisimple lie algebra.
References
[1] https://en.wikipedia.org/wiki/Semisimple_Lie_algebra
