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Trace Field
In mathematics, the trace field of a linear group is the field generated by the traces of its elements. It is mostly studied for Kleinian and Fuchsian groups, though related objects are used in the theory of lattices in Lie groups, often under the name ''field of definition''. Fuchsian and Kleinian groups Trace field and invariant trace fields for Fuchsian groups Fuchsian groups are discrete subgroups of \mathrm_2(\mathbb R). The trace of an element in \mathrm_2(\mathbb R) is well-defined up to sign (by taking the trace of an arbitrary preimage in \mathrm_2(\mathbb R)) and the ''trace field'' of \Gamma is the field generated over \mathbb Q by the traces of all elements of \Gamma (see for example in ). The ''invariant trace field'' is equal to the trace field of the subgroup \Gamma^ generated by all squares of elements of \Gamma (a finite-index subgroup of \Gamma). The invariant trace field of Fuchsian groups is stable under taking commensurable groups. This is not the c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Group
In mathematics, a matrix group is a group ''G'' consisting of invertible matrices over a specified field ''K'', with the operation of matrix multiplication. A linear group is a group that is isomorphic to a matrix group (that is, admitting a faithful, finite-dimensional representation over ''K''). Any finite group is linear, because it can be realized by permutation matrices using Cayley's theorem. Among infinite groups, linear groups form an interesting and tractable class. Examples of groups that are not linear include groups which are "too big" (for example, the group of permutations of an infinite set), or which exhibit some pathological behavior (for example, finitely generated infinite torsion groups). Definition and basic examples A group ''G'' is said to be ''linear'' if there exists a field ''K'', an integer ''d'' and an injective homomorphism from ''G'' to the general linear group GL''d''(''K'') (a faithful linear representation of dimension ''d'' over ''K''): if ne ... [...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] |
Arithmetic Group
In mathematics, an arithmetic group is a group obtained as the integer points of an algebraic group, for example \mathrm_2(\Z). They arise naturally in the study of arithmetic properties of quadratic forms and other classical topics in number theory. They also give rise to very interesting examples of Riemannian manifolds and hence are objects of interest in differential geometry and topology. Finally, these two topics join in the theory of automorphic forms which is fundamental in modern number theory. History One of the origins of the mathematical theory of arithmetic groups is algebraic number theory. The classical reduction theory of quadratic and Hermitian forms by Charles Hermite, Hermann Minkowski and others can be seen as computing fundamental domains for the action of certain arithmetic groups on the relevant symmetric spaces. The topic was related to Minkowski's geometry of numbers and the early development of the study of arithmetic invariant of number fields such as the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field Of Definition
In mathematics, the field of definition of an algebraic variety ''V'' is essentially the smallest field to which the coefficients of the polynomials defining ''V'' can belong. Given polynomials, with coefficients in a field ''K'', it may not be obvious whether there is a smaller field ''k'', and other polynomials defined over ''k'', which still define ''V''. The issue of field of definition is of concern in diophantine geometry. Notation Throughout this article, ''k'' denotes a field. The algebraic closure of a field is denoted by adding a superscript of "alg", e.g. the algebraic closure of ''k'' is ''k''alg. The symbols Q, R, C, and F''p'' represent, respectively, the field of rational numbers, the field of real numbers, the field of complex numbers, and the finite field containing ''p'' elements. Affine ''n''-space over a field ''F'' is denoted by A''n''(''F''). Definitions for affine and projective varieties Results and definitions stated below, for affine varieties, can ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Point
In number theory and algebraic geometry, a rational point of an algebraic variety is a point whose coordinates belong to a given field. If the field is not mentioned, the field of rational numbers is generally understood. If the field is the field of real numbers, a rational point is more commonly called a real point. Understanding rational points is a central goal of number theory and Diophantine geometry. For example, Fermat's Last Theorem may be restated as: for , the Fermat curve of equation x^n+y^n=1 has no other rational points than , , and, if is even, and . Definition Given a field ''k'', and an algebraically closed extension ''K'' of ''k'', an affine variety ''X'' over ''k'' is the set of common zeros in K^n of a collection of polynomials with coefficients in ''k'': :f_1(x_1,\ldots,x_n)=0,\ldots, f_r(x_1,\dots,x_n)=0. These common zeros are called the ''points'' of ''X''. A ''k''-rational point (or ''k''-point) of ''X'' is a point of ''X'' that belongs to ''k''''n'', ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Group
In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Many groups of geometric transformations are algebraic groups; for example, orthogonal groups, general linear groups, projective groups, Euclidean groups, etc. Many matrix groups are also algebraic. Other algebraic groups occur naturally in algebraic geometry, such as elliptic curves and Jacobian varieties. An important class of algebraic groups is given by the affine algebraic groups, those whose underlying algebraic variety is an affine variety; they are exactly the algebraic subgroups of the general linear group, and are therefore also called ''linear algebraic groups''. Another class is formed by the abelian varieties, which are the algebraic groups whose underlying variety is a projective variety. Chevalley's structure theorem states ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a field that contains \mathbb and has finite dimension when considered as a vector space over The study of algebraic number fields, and, more generally, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory. This study reveals hidden structures behind usual rational numbers, by using algebraic methods. Definition Prerequisites The notion of algebraic number field relies on the concept of a field. A field consists of a set of elements together with two operations, namely addition, and multiplication, and some distributivity assumptions. A prominent example of a field is the field of rational numbers, commonly denoted together with its usual operations of addition and multiplication. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local Rigidity
Local rigidity theorems in the theory of discrete subgroups of Lie groups are results which show that small deformations of certain such subgroups are always trivial. It is different from Mostow rigidity and weaker (but holds more frequently) than superrigidity. History The first such theorem was proven by Atle Selberg for co-compact discrete subgroups of the unimodular groups \mathrm_n(\mathbb R). Shortly afterwards a similar statement was proven by Eugenio Calabi in the setting of fundamental groups of compact hyperbolic manifolds. Finally, the theorem was extended to all co-compact subgroups of semisimple Lie groups by André Weil. The extension to non-cocompact lattices was made later by Howard Garland and Madabusi Santanam Raghunathan. The result is now sometimes referred to as Calabi—Weil (or just Weil) rigidity. Statement Deformations of subgroups Let \Gamma be a group generated by a finite number of elements g_1, \ldots, g_n and G a Lie group. Then the map \mathr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semisimple Lie Group
In mathematics, a Lie algebra is semisimple if it is a direct sum of modules, direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper Lie algebra#Subalgebras.2C ideals and homomorphisms, ideals). Throughout the article, unless otherwise stated, a Lie algebra is a finite-dimensional Lie algebra over a field of Characteristic (algebra), 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 Lie algebra, solvable ideals; * the Radical of a Lie algebra, 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 i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zariski Topology
In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is not Hausdorff. This topology was introduced primarily by Oscar Zariski and later generalized for making the set of prime ideals of a commutative ring (called the spectrum of the ring) a topological space. The Zariski topology allows tools from topology to be used to study algebraic varieties, even when the underlying field is not a topological field. This is one of the basic ideas of scheme theory, which allows one to build general algebraic varieties by gluing together affine varieties in a way similar to that in manifold theory, where manifolds are built by gluing together charts, which are open subsets of real affine spaces. The Zariski topology of an algebraic variety is the topology whose closed sets are the algebraic subsets of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Special Linear Group
In mathematics, the special linear group of degree ''n'' over a field ''F'' is the set of matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion. This is the normal subgroup of the general linear group given by the kernel of the determinant :\det\colon \operatorname(n, F) \to F^\times. where ''F''× is the multiplicative group of ''F'' (that is, ''F'' excluding 0). These elements are "special" in that they form an algebraic subvariety of the general linear group – they satisfy a polynomial equation (since the determinant is polynomial in the entries). When ''F'' is a finite field of order ''q'', the notation is sometimes used. Geometric interpretation The special linear group can be characterized as the group of ''volume and orientation preserving'' linear transformations of R''n''; this corresponds to the interpretation of the determinant as measuring change in volume and orientation. Lie subgroup When ''F'' is R ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trace (mathematics)
In linear algebra, the trace of a square matrix , denoted , is defined to be the sum of elements on the main diagonal (from the upper left to the lower right) of . The trace is only defined for a square matrix (). It can be proved that the trace of a matrix is the sum of its (complex) eigenvalues (counted with multiplicities). It can also be proved that for any two matrices and . This implies that similar matrices have the same trace. As a consequence one can define the trace of a linear operator mapping a finite-dimensional vector space into itself, since all matrices describing such an operator with respect to a basis are similar. The trace is related to the derivative of the determinant (see Jacobi's formula). Definition The trace of an square matrix is defined as \operatorname(\mathbf) = \sum_^n a_ = a_ + a_ + \dots + a_ where denotes the entry on the th row and th column of . The entries of can be real numbers or (more generally) complex numbers. The trace is not def ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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