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Radical Of Lie Algebra
In the mathematical field of Lie theory, the radical of a Lie algebra \mathfrak is the largest solvable ideal of \mathfrak.. The radical, denoted by (\mathfrak), fits into the exact sequence :0 \to (\mathfrak) \to \mathfrak g \to \mathfrak/(\mathfrak) \to 0. where \mathfrak/(\mathfrak) is semisimple. When the ground field has characteristic zero and \mathfrak g has finite dimension, Levi's theorem states that this exact sequence splits; i.e., there exists a (necessarily semisimple) subalgebra of \mathfrak g that is isomorphic to the semisimple quotient \mathfrak/(\mathfrak) via the restriction of the quotient map \mathfrak g \to \mathfrak/(\mathfrak). A similar notion is a Borel subalgebra, which is a (not necessarily unique) maximal solvable subalgebra. Definition Let k be a field and let \mathfrak be a finite-dimensional Lie algebra over k. There exists a unique maximal solvable ideal, called the ''radical,'' for the following reason. Firstly let \mathfrak and \mathfrak be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Theory
In mathematics, the mathematician Sophus Lie ( ) initiated lines of study involving integration of differential equations, transformation groups, and contact of spheres that have come to be called Lie theory. For instance, the latter subject is Lie sphere geometry. This article addresses his approach to transformation groups, which is one of the areas of mathematics, and was worked out by Wilhelm Killing and Élie Cartan. The foundation of Lie theory is the exponential map relating Lie algebras to Lie groups which is called the Lie group–Lie algebra correspondence. The subject is part of differential geometry since Lie groups are differentiable manifolds. Lie groups evolve out of the identity (1) and the tangent vectors to one-parameter subgroups generate the Lie algebra. The structure of a Lie group is implicit in its algebra, and the structure of the Lie algebra is expressed by root systems and root data. Lie theory has been particularly useful in mathematical physics s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identity. The Lie bracket of two vectors x and y is denoted [x,y]. The vector space \mathfrak g together with this operation is a non-associative algebra, meaning that the Lie bracket is not necessarily associative property, associative. Lie algebras are closely related to Lie groups, which are group (mathematics), groups that are also smooth manifolds: any Lie group gives rise to a Lie algebra, which is its tangent space at the identity. Conversely, to any finite-dimensional Lie algebra over real or complex numbers, there is a corresponding connected space, connected Lie group unique up to finite coverings (Lie's third theorem). This Lie group–Lie algebra correspondence, correspondence allows one ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Solvable Lie Algebra
In mathematics, a Lie algebra \mathfrak is solvable if its derived series terminates in the zero subalgebra. The ''derived Lie algebra'' of the Lie algebra \mathfrak is the subalgebra of \mathfrak, denoted : mathfrak,\mathfrak/math> that consists of all linear combinations of Lie brackets of pairs of elements of \mathfrak. The ''derived series'' is the sequence of subalgebras : \mathfrak \geq mathfrak,\mathfrak\geq \mathfrak,\mathfrak mathfrak,\mathfrak \geq [ \mathfrak,\mathfrak mathfrak,\mathfrak, \mathfrak,\mathfrak mathfrak,\mathfrak] \geq ... If the derived series eventually arrives at the zero subalgebra, then the Lie algebra is called solvable. The derived series for Lie algebras is analogous to the derived series for commutator subgroups in group theory, and solvable Lie algebras are analogs of solvable groups. Any nilpotent Lie algebra_is_a_fortiori.html" ;"title="mathfrak,\mathfrak ... is a fortiori">mathfrak,\mathfrak ... is a fortiori solvable but the converse is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ideal (Lie Algebra)
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. The Lie bracket of two vectors x and y is denoted ,y/math>. The vector space \mathfrak g together with this operation is a non-associative algebra, meaning that the Lie bracket is not necessarily associative. Lie algebras are closely related to Lie groups, which are groups that are also smooth manifolds: any Lie group gives rise to a Lie algebra, which is its tangent space at the identity. Conversely, to any finite-dimensional Lie algebra over real or complex numbers, there is a corresponding connected Lie group unique up to finite coverings ( Lie's third theorem). This correspondence allows one to study the structure and classification of Lie groups in terms of Lie algebras. In physics, Lie groups appear as symmetry groups of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semisimple Lie Algebra
In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals). Throughout the article, unless otherwise stated, a Lie algebra is a finite-dimensional Lie algebra over a field of characteristic 0. For such a Lie algebra \mathfrak g, if nonzero, the following conditions are equivalent: *\mathfrak g is semisimple; *the Killing form, κ(x,y) = tr(ad(''x'')ad(''y'')), is non-degenerate; *\mathfrak g has no non-zero abelian ideals; *\mathfrak g has no non-zero solvable ideals; * the radical (maximal solvable ideal) of \mathfrak g is zero. Significance The significance of semisimplicity comes firstly from the Levi decomposition, which states that every finite dimensional Lie algebra is the semidirect product of a solvable ideal (its radical) and a semisimple algebra. In particular, there is no nonzero Lie algebra that is both solvable and semisimple. Semisimple L ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Levi's Theorem
In Lie theory and representation theory, the Levi decomposition, conjectured by Wilhelm Killing and Élie 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. One is its radical, a maximal solvable ideal, and the other is a semisimple subalgebra, called a Levi subalgebra. The Levi decomposition implies that any finite-dimensional Lie algebra is a semidirect product of a solvable Lie algebra and a semisimple Lie algebra. When viewed as a factor-algebra of ''g'', this semisimple Lie algebra is also called the Levi factor of ''g''. To a certain extent, the decomposition can be used to reduce problems about finite-dimensional Lie algebras and Lie groups to separate problems about Lie algebras in these two special classes, solvable and semisimple. Moreover, Malcev (1942) showed that any two Levi subalgebras are conjugate by an (inner) automorphism of the form :\exp(\mathrm(z ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Borel Subalgebra
In mathematics, specifically in representation theory, a Borel subalgebra of a Lie algebra \mathfrak is a maximal solvable subalgebra. The notion is named after Armand Borel. If the Lie algebra \mathfrak is the Lie algebra of a complex Lie group, then a Borel subalgebra is the Lie algebra of a Borel subgroup. Borel subalgebra associated to a flag Let \mathfrak g = \mathfrak(V) be the Lie algebra of the endomorphisms of a finite-dimensional vector space ''V'' over the complex numbers. Then to specify a Borel subalgebra of \mathfrak g amounts to specify a flag of ''V''; given a flag V = V_0 \supset V_1 \supset \cdots \supset V_n = 0, the subspace \mathfrak b = \ is a Borel subalgebra, and conversely, each Borel subalgebra is of that form by Lie's theorem. Hence, the Borel subalgebras are classified by the flag variety of ''V''. Borel subalgebra relative to a base of a root system Let \mathfrak g be a complex semisimple Lie algebra, \mathfrak h a Cartan subalgebra and ''R'' the r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Extension
In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If Q and N are two groups, then G is an extension of Q by N if there is a short exact sequence :1\to N\;\overset\;G\;\overset\;Q \to 1. If G is an extension of Q by N, then G is a group, \iota(N) is a normal subgroup of G and the quotient group G/\iota(N) is isomorphic to the group Q. Group extensions arise in the context of the extension problem, where the groups Q and N are known and the properties of G are to be determined. Note that the phrasing "G is an extension of N by Q" is also used by some. Since any finite group G possesses a maximal normal subgroup N with simple factor group G/N, all finite groups may be constructed as a series of extensions with finite simple groups. This fact was a motivation for completing the classification of finite simple groups. An extension is called a central extension if the subgroup N lies in the center o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semisimple Lie Algebra
In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals). Throughout the article, unless otherwise stated, a Lie algebra is a finite-dimensional Lie algebra over a field of characteristic 0. For such a Lie algebra \mathfrak g, if nonzero, the following conditions are equivalent: *\mathfrak g is semisimple; *the Killing form, κ(x,y) = tr(ad(''x'')ad(''y'')), is non-degenerate; *\mathfrak g has no non-zero abelian ideals; *\mathfrak g has no non-zero solvable ideals; * the radical (maximal solvable ideal) of \mathfrak g is zero. Significance The significance of semisimplicity comes firstly from the Levi decomposition, which states that every finite dimensional Lie algebra is the semidirect product of a solvable ideal (its radical) and a semisimple algebra. In particular, there is no nonzero Lie algebra that is both solvable and semisimple. Semisimple L ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reductive Lie Algebra
In mathematics, a Lie algebra is reductive if its adjoint representation is completely reducible, whence the name. More concretely, a Lie algebra is reductive if it is a direct sum of a semisimple Lie algebra and an abelian Lie algebra: \mathfrak = \mathfrak \oplus \mathfrak; there are alternative characterizations, given below. Examples The most basic example is the Lie algebra \mathfrak_n of n \times n matrices with the commutator as Lie bracket, or more abstractly as the endomorphism algebra of an ''n''-dimensional vector space, \mathfrak(V). This is the Lie algebra of the general linear group GL(''n''), and is reductive as it decomposes as \mathfrak_n = \mathfrak_n \oplus \mathfrak, corresponding to traceless matrices and scalar matrices. Any semisimple Lie algebra or abelian Lie algebra is ''a fortiori'' reductive. Over the real numbers, compact Lie algebras are reductive. Definitions A Lie algebra \mathfrak over a field of characteristic 0 is called reductive if any ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Levi Decomposition
In Lie theory and representation theory, the Levi decomposition, conjectured by Wilhelm Killing and Élie 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. One is its radical, a maximal solvable ideal, and the other is a semisimple subalgebra, called a Levi subalgebra. The Levi decomposition implies that any finite-dimensional Lie algebra is a semidirect product of a solvable Lie algebra and a semisimple Lie algebra. When viewed as a factor-algebra of ''g'', this semisimple Lie algebra is also called the Levi factor of ''g''. To a certain extent, the decomposition can be used to reduce problems about finite-dimensional Lie algebras and Lie groups to separate problems about Lie algebras in these two special classes, solvable and semisimple. Moreover, Malcev (1942) showed that any two Levi subalgebras are conjugate by an (inner) automorphism of the form :\exp(\mathrm(z ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
