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Sl2-triple
In the theory of Lie algebras, an ''sl''2-triple is a triple of elements of a Lie algebra that satisfy the commutation relations between the standard generators of the special linear Lie algebra ''sl''2. This notion plays an important role in the theory of semisimple Lie algebras, especially in regard to their nilpotent orbits. Definition Elements of a Lie algebra ''g'' form an ''sl''2-triple if : ,e= 2e, \quad ,f= -2f, \quad ,f= h. These commutation relations are satisfied by the generators : h = \begin 1 & 0\\ 0 & -1 \end, \quad e = \begin 0 & 1\\ 0 & 0 \end, \quad f = \begin 0 & 0\\ 1 & 0 \end of the Lie algebra ''sl''2 of 2 by 2 matrices with zero trace. It follows that ''sl''2-triples in ''g'' are in a bijective correspondence with the Lie algebra homomorphisms from ''sl''2 into ''g''. The alternative notation for the elements of an ''sl''2-triple is , with ''H'' corresponding to ''h'', ''X'' corresponding to ''e'', and ''Y'' corresponding to ''f''. H is cal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nilpotent Orbit
In mathematics, nilpotent orbits are generalizations of nilpotent matrices that play an important role in representation theory of real and complex semisimple Lie groups and semisimple Lie algebras. Definition An element ''X'' of a semisimple Lie algebra ''g'' is called nilpotent if its adjoint endomorphism : ''ad X'': ''g'' → ''g'', ''ad X''(''Y'') = 'X'',''Y'' is nilpotent, that is, (''ad X'')''n'' = 0 for large enough ''n''. Equivalently, ''X'' is nilpotent if its characteristic polynomial ''p''''ad X''(''t'') is equal to ''t''dim ''g''. A semisimple Lie group or algebraic group ''G'' acts on its Lie algebra via the adjoint representation, and the property of being nilpotent is invariant under this action. A nilpotent orbit is an orbit of the adjoint action such that any (equivalently, all) of its elements is (are) nilpotent. Examples Nilpotent n\times n matrices with complex entries form the main motivating case for th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nilpotent Orbit
In mathematics, nilpotent orbits are generalizations of nilpotent matrices that play an important role in representation theory of real and complex semisimple Lie groups and semisimple Lie algebras. Definition An element ''X'' of a semisimple Lie algebra ''g'' is called nilpotent if its adjoint endomorphism : ''ad X'': ''g'' → ''g'', ''ad X''(''Y'') = 'X'',''Y'' is nilpotent, that is, (''ad X'')''n'' = 0 for large enough ''n''. Equivalently, ''X'' is nilpotent if its characteristic polynomial ''p''''ad X''(''t'') is equal to ''t''dim ''g''. A semisimple Lie group or algebraic group ''G'' acts on its Lie algebra via the adjoint representation, and the property of being nilpotent is invariant under this action. A nilpotent orbit is an orbit of the adjoint action such that any (equivalently, all) of its elements is (are) nilpotent. Examples Nilpotent n\times n matrices with complex entries form the main motivating case for th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Principal Subalgebra
In mathematics, a principal subalgebra of a complex simple Lie algebra is a 3-dimensional simple subalgebra whose non-zero elements are regular. A finite-dimensional complex simple Lie algebra has a unique conjugacy class In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other wo ... of principal subalgebras, each of which is the span of an sl2-triple. References *https://aiolatest.com/application-of-abstract-algebra-in-real-life/ * Lie algebras Representation theory {{Algebra-stub ... [...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] |
Regular Element Of A Lie Algebra
In mathematics, a regular element of a Lie algebra or Lie group is an element whose centralizer has dimension as small as possible. For example, in a complex semisimple Lie algebra, an element X \in \mathfrak is regular if its centralizer in \mathfrak has dimension equal to the rank of \mathfrak, which in turn equals the dimension of some Cartan subalgebra \mathfrak (note that in earlier papers, an element of a complex semisimple Lie algebra was termed regular if it is semisimple and the kernel of its adjoint representation is a Cartan subalgebra). An element g \in G a Lie group is regular if its centralizer has dimension equal to the rank of G . Basic case In the specific case of \mathfrak_n(\mathbb), the Lie algebra of n \times n matrices over an algebraically closed field \mathbb(such as the complex numbers), a regular element M is an element whose Jordan normal form contains a single Jordan block for each eigenvalue (in other words, the geometric multiplicity of each eige ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ernest Vinberg
Ernest Borisovich Vinberg (russian: Эрне́ст Бори́сович Ви́нберг; 26 July 1937 – 12 May 2020) was a Soviet and Russian mathematician, who worked on Lie groups and algebraic groups, discrete subgroups of Lie groups, invariant theory, and representation theory. He introduced Vinberg's algorithm and the Koecher–Vinberg theorem. He was a recipient of the 1997 Humboldt Prize. He was on the executive committee of the Moscow Mathematical Society. In 1983, he was an Invited Speaker with a talk on ''Discrete reflection groups in Lobachevsky spaces'' at the International Congress of Mathematicians in Warsaw. In 2010, he was elected an International Honorary Member of the American Academy of Arts and Sciences. Ernest Vinberg died from pneumonia caused by COVID-19 on 12 May 2020. Selected publications * * * editor and co-author: (contains ''Construction of the exceptional simple Lie algebras'') * with A. L. Onishchik:2012 pbk edition* with V. V. Gorbatsevi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weyl Group
In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated by reflections through the hyperplanes orthogonal to the roots, and as such is a finite reflection group. In fact it turns out that ''most'' finite reflection groups are Weyl groups. Abstractly, Weyl groups are finite Coxeter groups, and are important examples of these. The Weyl group of a semisimple Lie group, a semisimple Lie algebra, a semisimple linear algebraic group, etc. is the Weyl group of the root system of that group or algebra. Definition and examples Let \Phi be a root system in a Euclidean space V. For each root \alpha\in\Phi, let s_\alpha denote the reflection about the hyperplane perpendicular to \alpha, which is given explicitly as :s_\alpha(v)=v-2\frac\alpha, where (\cdot,\cdot) is the inner product on V. The Weyl group W of \Phi is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 ,Y=XY-YX. This algebra is well studied and understood, and is often used as a model for the study of other Lie algebras. The Lie group that it generates is the special linear group. Applications The Lie algebra \mathfrak_2(\mathbb) is central to the study of special relativity, general relativity and supersymmetry: its fundamental representation is the so-called spinor representation, while its adjoint representation generates the Lorentz group SO(3,1) of special relativity. The algebra \mathfrak_2(\mathbb) plays an important role in the study of chaos and fractals, as it generates the Möbius group SL(2,R), which describes the automorphisms of the hyperbolic plane, the simplest Riemann surface of negative curvature; by contrast, SL(2,C) describes the automorphisms of the hyperbolic 3-dimensional ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Root System
In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representation theory of semisimple Lie algebras. Since Lie groups (and some analogues such as algebraic groups) and Lie algebras have become important in many parts of mathematics during the twentieth century, the apparently special nature of root systems belies the number of areas in which they are applied. Further, the classification scheme for root systems, by Dynkin diagrams, occurs in parts of mathematics with no overt connection to Lie theory (such as singularity theory). Finally, root systems are important for their own sake, as in spectral graph theory. Definitions and examples As a first example, consider the six vectors in 2-dimensional Euclidean space, R2, as shown in the image at the right; call them roots. These vectors Linear span, s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Algebraic Group
In mathematics, a linear algebraic group is a subgroup of the group of invertible n\times n matrices (under matrix multiplication) that is defined by polynomial equations. An example is the orthogonal group, defined by the relation M^TM = I_n where M^T is the transpose of M. Many Lie groups can be viewed as linear algebraic groups over the field of real or complex numbers. (For example, every compact Lie group can be regarded as a linear algebraic group over R (necessarily R-anisotropic and reductive), as can many noncompact groups such as the simple Lie group SL(''n'',R).) The simple Lie groups were classified by Wilhelm Killing and Élie Cartan in the 1880s and 1890s. At that time, no special use was made of the fact that the group structure can be defined by polynomials, that is, that these are algebraic groups. The founders of the theory of algebraic groups include Maurer, Chevalley, and . In the 1950s, Armand Borel constructed much of the theory of algebraic groups as it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hasse Diagram
In order theory, a Hasse diagram (; ) is a type of mathematical diagram used to represent a finite partially ordered set, in the form of a drawing of its transitive reduction. Concretely, for a partially ordered set ''(S, ≤)'' one represents each element of ''S'' as a vertex in the plane and draws a line segment or curve that goes ''upward'' from ''x'' to ''y'' whenever ''y'' ≠ ''x'' and ''y'' covers ''x'' (that is, whenever ''x'' ≤ ''y'' and there is no ''z'' such that ''x'' ≤ ''z'' ≤ ''y''). These curves may cross each other but must not touch any vertices other than their endpoints. Such a diagram, with labeled vertices, uniquely determines its partial order. The diagrams are named after Helmut Hasse (1898–1979); according to , they are so called because of the effective use Hasse made of them. However, Hasse was not the first to use these diagrams. One example that predates Hasse can be found in . Although Hasse diagrams were originally devised as a technique for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Coxeter Group
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example. However, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections. Coxeter groups were introduced in 1934 as abstractions of reflection groups , and finite Coxeter groups were classified in 1935 . Coxeter groups find applications in many areas of mathematics. Examples of finite Coxeter groups include the symmetry groups of regular polytopes, and the Weyl groups of simple Lie algebras. Examples of infinite Coxeter groups include the triangle groups corresponding to regular tessellations of the Euclidean plane and the hyperbolic plane, and the Weyl groups of infinite-dimensional Kac–Moody algebras. Standard re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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