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Glossary Of Lie Algebras
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. Notations: *Throughout the glossary, ( \cdot, \cdot ) denotes the inner product of a Euclidean space ''E'' and \langle \cdot, \cdot \rangle denotes the rescaled inner product :: \langle \beta, \alpha \rangle = \frac \, \forall \alpha, \beta \in E. A B C D E F G H I J K L N M P Q R S \mu \iff \lambda - \mu > 0 \iff \,\, \, \exists k_1, k_2, ..., k_n \in \mathbb^+, \, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Glossary
A glossary (from grc, γλῶσσα, ''glossa''; language, speech, wording) also known as a vocabulary or clavis, is an alphabetical list of Term (language), terms in a particular domain of knowledge with the definitions for those terms. Traditionally, a glossary appears at the end of a book and includes terms within that book that are either newly introduced, uncommon, or specialized. While glossaries are most commonly associated with non-fiction books, in some cases, fiction novels may come with a glossary for unfamiliar terms. A bilingual glossary is a list of terms in one language defined in a second language or Gloss (annotation), glossed by synonyms (or at least near-synonyms) in another language. In a general sense, a glossary contains explanations of concepts relevant to a certain field of study or action. In this sense, the term is related to the notion of ontology. Automatic methods have been also provided that transform a glossary into an ontology or a computational ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Armand Borel
Armand Borel (21 May 1923 – 11 August 2003) was a Swiss mathematician, born in La Chaux-de-Fonds, and was a permanent professor at the Institute for Advanced Study in Princeton, New Jersey, United States from 1957 to 1993. He worked in algebraic topology, in the theory of Lie groups, and was one of the creators of the contemporary theory of linear algebraic groups. Biography He studied at the ETH Zürich, where he came under the influence of the topologist Heinz Hopf and Lie-group theorist Eduard Stiefel. He was in Paris from 1949: he applied the Leray spectral sequence to the topology of Lie groups and their classifying spaces, under the influence of Jean Leray and Henri Cartan. With Hirzebruch, he significantly developed the theory of characteristic classes in the early 1950s. He collaborated with Jacques Tits in fundamental work on algebraic groups, and with Harish-Chandra on their arithmetic subgroups. In an algebraic group ''G'' a ''Borel subgroup'' ''H'' is one mini ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clebsch–Gordan Coefficients
In physics, the Clebsch–Gordan (CG) coefficients are numbers that arise in angular momentum coupling in quantum mechanics. They appear as the expansion coefficients of total angular momentum eigenstates in an uncoupled tensor product basis. In more mathematical terms, the CG coefficients are used in representation theory, particularly of compact Lie groups, to perform the explicit direct sum decomposition of the tensor product of two irreducible representations (i.e., a reducible representation into irreducible representations, in cases where the numbers and types of irreducible components are already known abstractly). The name derives from the German mathematicians Alfred Clebsch and Paul Gordan, who encountered an equivalent problem in invariant theory. From a vector calculus perspective, the CG coefficients associated with the SO(3) group can be defined simply in terms of integrals of products of spherical harmonics and their complex conjugates. The addition of spins in qu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Casimir Invariant
In mathematics, a Casimir element (also known as a Casimir invariant or Casimir operator) is a distinguished element of the center of the universal enveloping algebra of a Lie algebra. A prototypical example is the squared angular momentum operator, which is a Casimir element of the three-dimensional rotation group. The Casimir element is named after Hendrik Casimir, who identified them in his description of rigid body dynamics in 1931. Definition The most commonly-used Casimir invariant is the quadratic invariant. It is the simplest to define, and so is given first. However, one may also have Casimir invariants of higher order, which correspond to homogeneous symmetric polynomials of higher order. Quadratic Casimir element Suppose that \mathfrak is an n-dimensional Lie algebra. Let ''B'' be a nondegenerate bilinear form on \mathfrak that is invariant under the adjoint action of \mathfrak on itself, meaning that B(\operatorname_XY, Z) + B(Y, \operatorname_X Z) = 0 for all ''X ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartan Decomposition
In mathematics, the Cartan decomposition is a decomposition of a Semisimple Lie algebra, 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. Cartan involutions on Lie algebras Let \mathfrak be a real semisimple Lie algebra and let B(\cdot,\cdot) be its Killing form. An Involution (mathematics), involution on \mathfrak is a Lie algebra automorphism \theta of \mathfrak whose square is equal to the identity. Such an involution is called a ''Cartan involution'' on \mathfrak if B_\theta(X,Y) := -B(X,\theta Y) is a positive definite bilinear form. Two involutions \theta_1 and \theta_2 are considered equivalent if they differ only by an inner automorphism. Any real semisimple Lie algebra has a Cartan involution, and any two Cartan involutions are equi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartan Subgroup
In algebraic geometry, a Cartan subgroup of a connected linear algebraic group over an algebraically closed field is the centralizer of a maximal torus (which turns out to be connected). Cartan subgroups are nilpotent and are all conjugate. Examples * For a finite field ''F'', the group of diagonal matrices \begin a & 0 \\ 0 & b \end where ''a'' and ''b'' are elements of ''F*''. This is called the split Cartan subgroup of GL2(''F''). * For a finite field ''F'', every maximal commutative semisimple subgroup of GL2(''F'') is a Cartan subgroup (and conversely). See also *Borel subgroup In the theory of algebraic groups, a Borel subgroup of an algebraic group ''G'' is a maximal Zariski closed and connected solvable algebraic subgroup. For example, in the general linear group ''GLn'' (''n x n'' invertible matrices), the subgroup ... References * * * * {{algebra-stub Algebraic geometry Linear algebraic groups ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartan Matrix
In mathematics, the term Cartan matrix has three meanings. All of these are named after the French mathematician Élie Cartan. Amusingly, the Cartan matrices in the context of Lie algebras were first investigated by Wilhelm Killing, whereas the Killing form is due to Cartan. Lie algebras A (symmetrizable) generalized Cartan matrix is a square matrix A = (a_) with integral entries such that # For diagonal entries, a_ = 2 . # For non-diagonal entries, a_ \leq 0 . # a_ = 0 if and only if a_ = 0 # A can be written as DS, where D is a diagonal matrix, and S is a symmetric matrix. For example, the Cartan matrix for ''G''2 can be decomposed as such: : \begin 2 & -3 \\ -1 & 2 \end = \begin 3&0\\ 0&1 \end\begin \frac & -1 \\ -1 & 2 \end. The third condition is not independent but is really a consequence of the first and fourth conditions. We can always choose a ''D'' with positive diagonal entries. In that case, if ''S'' in the ab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartan's Criterion
In mathematics, Cartan's criterion gives conditions for a Lie algebra in characteristic 0 to be solvable, which implies a related criterion for the Lie algebra to be semisimple. It is based on the notion of the Killing form, a symmetric bilinear form on \mathfrak defined by the formula : B(u,v)=\operatorname(\operatorname(u)\operatorname(v)), where tr denotes the trace of a linear operator. The criterion was introduced by .Cartan, Chapitre IV, Théorème 1 Cartan's criterion for solvability Cartan's criterion for solvability states: :''A Lie subalgebra \mathfrak of endomorphisms of a finite-dimensional vector space over a field of characteristic zero is solvable if and only if \operatorname(ab)=0 whenever a\in\mathfrak,b\in mathfrak,\mathfrak'' The fact that \operatorname(ab)=0 in the solvable case follows from Lie's theorem that puts \mathfrak g in the upper triangular form over the algebraic closure of the ground field (the trace can be computed after extending the ground f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartan Criterion For Solvability
In mathematics, Cartan's criterion gives conditions for a Lie algebra in characteristic 0 to be solvable, which implies a related criterion for the Lie algebra to be semisimple. It is based on the notion of the Killing form, a symmetric bilinear form on \mathfrak defined by the formula : B(u,v)=\operatorname(\operatorname(u)\operatorname(v)), where tr denotes the trace of a linear operator. The criterion was introduced by .Cartan, Chapitre IV, Théorème 1 Cartan's criterion for solvability Cartan's criterion for solvability states: :''A Lie subalgebra \mathfrak of endomorphisms of a finite-dimensional vector space over a field of characteristic zero is solvable if and only if \operatorname(ab)=0 whenever a\in\mathfrak,b\in mathfrak,\mathfrak'' The fact that \operatorname(ab)=0 in the solvable case follows from Lie's theorem that puts \mathfrak g in the upper triangular form over the algebraic closure of the ground field (the trace can be computed after extending the ground f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 ,Y\in \mathfrak for all X \in \mathfrak, then Y \in \mathfrak). They were introduced by Élie Cartan in his doctoral thesis. It controls the representation theory of a semi-simple Lie algebra \mathfrak over a field of characteristic 0 . In a finite-dimensional semisimple Lie algebra over an algebraically closed field of characteristic zero (e.g., a Cartan subalgebra is the same thing as a maximal abelian subalgebra consisting of elements ''x'' such that the adjoint endomorphism \operatorname(x) : \mathfrak \to \mathfrak is semisimple (i.e., diagonalizable). Sometimes this characterization is simply taken as the definition of a Cartan subalgebra.pg 231 In general, a subalgebra is called toral if it consists of semisimple elements. Over an algebraically closed field, a toral subalgebra is automatically abelian. Thus, over ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Élie Cartan
Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometry. He also made significant contributions to general relativity and indirectly to quantum mechanics. He is widely regarded as one of the greatest mathematicians of the twentieth century. His son Henri Cartan was an influential mathematician working in algebraic topology. Life Élie Cartan was born 9 April 1869 in the village of Dolomieu, Isère to Joseph Cartan (1837–1917) and Anne Cottaz (1841–1927). Joseph Cartan was the village blacksmith; Élie Cartan recalled that his childhood had passed under "blows of the anvil, which started every morning from dawn", and that "his mother, during those rare minutes when she was free from taking care of the children and the house, was working with a spinning-wheel". Élie had an elder sister Je ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bruhat Decomposition
In mathematics, the Bruhat decomposition (introduced by François Bruhat for classical groups and by Claude Chevalley in general) ''G'' = ''BWB'' of certain algebraic groups ''G'' into cells can be regarded as a general expression of the principle of Gauss–Jordan elimination, which generically writes a matrix as a product of an upper triangular and lower triangular matrices—but with exceptional cases. It is related to the Schubert cell decomposition of flag varieties: see Weyl group for this. More generally, any group with a (''B'', ''N'') pair has a Bruhat decomposition. Definitions *''G'' is a connected, reductive algebraic group over an algebraically closed field. *''B'' is a Borel subgroup of ''G'' *''W'' is a Weyl group of ''G'' corresponding to a maximal torus of ''B''. The Bruhat decomposition of ''G'' is the decomposition :G=BWB =\bigsqcup_BwB of ''G'' as a disjoint union of double cosets of ''B'' parameterized by the elements of the Weyl group ''W''. (Note that alt ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
