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Algebraic Torus
In mathematics, an algebraic torus, where a one dimensional torus is typically denoted by \mathbf G_, \mathbb_m, or \mathbb, is a type of commutative affine algebraic group commonly found in projective algebraic geometry and toric geometry. Higher dimensional algebraic tori can be modelled as a product of algebraic groups \mathbf G_. These groups were named by analogy with the theory of ''tori'' in Lie group theory (see Cartan subgroup). For example, over the complex numbers \mathbb the algebraic torus \mathbf G_ is isomorphic to the group scheme \mathbb^* = \text(\mathbb ,t^, which is the scheme theoretic analogue of the Lie group U(1) \subset \mathbb. In fact, any \mathbf G_-action on a complex vector space can be pulled back to a U(1)-action from the inclusion U(1) \subset \mathbb^* as real manifolds. Tori are of fundamental importance in the theory of algebraic groups and Lie groups and in the study of the geometric objects associated to them such as symmetric spaces and buil ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
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 t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field Norm
In mathematics, the (field) norm is a particular mapping defined in field theory, which maps elements of a larger field into a subfield. Formal definition Let ''K'' be a field and ''L'' a finite extension (and hence an algebraic extension) of ''K''. The field ''L'' is then a finite dimensional vector space over ''K''. Multiplication by α, an element of ''L'', :m_\alpha\colon L\to L :m_\alpha (x) = \alpha x, is a ''K''-linear transformation of this vector space into itself. The norm, N''L''/''K''(''α''), is defined as the determinant of this linear transformation. If ''L''/''K'' is a Galois extension, one may compute the norm of α ∈ ''L'' as the product of all the Galois conjugates of α: :\operatorname_(\alpha)=\prod_ \sigma(\alpha), where Gal(''L''/''K'') denotes the Galois group of ''L''/''K''. (Note that there may be a repetition in the terms of the product.) For a general field extension ''L''/''K'', and nonzero α in ''L'', let ''σ''(''α''), ..., σ(''α'') ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inner Form
In mathematics, an inner form of an algebraic group G over a field K is another algebraic group H such that there exists an isomorphism \phi between G and H defined over \overline K (this means that H is a ''K-form'' of G) and in addition, for every Galois automorphism \sigma \in \mathrm(\overline K/K) the automorphism \phi^ \circ \phi^ is an inner automorphism of G(\overline K) (i.e. conjugation by an element of G(\overline K)). Through the correspondence between K-forms and the Galois cohomology H^1(\mathrm(\overline K/K), \mathrm(G)) this means that H is associated to an element of the subset H^1(\mathrm(\overline K/K), \mathrm(G)) where \mathrm(G) is the subgroup of inner automorphisms of G. Being inner forms of each other is an equivalence relation on the set of K-forms of a given algebraic group. A form which is not inner is called an outer form. In practice, to check whether a group is an inner or outer form one looks at the action of the Galois group \mathrm(\overline K/K) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasi-split Group
In mathematics, a quasi-split group over a field is a reductive group with a Borel subgroup defined over the field. Simply connected quasi-split groups over a field correspond to actions of the absolute Galois group on a Dynkin diagram. Examples All split groups (those with a split maximal torus) are quasi-split. These correspond to quasi-split groups where the action of the Galois group on the Dynkin diagram is trivial. showed that all simple algebraic groups over finite fields are quasi-split. Over the real numbers, the quasi-split groups include the split groups and the complex groups, together with the orthogonal groups ''O''''n'',''n''+2, the unitary groups ''SU''''n'',''n'' and ''SU''''n'',''n''+1, and the form of ''E''6 with signature 2. References *{{Citation , last1=Lang , first1=Serge , author1-link=Serge Lang , title=Algebraic groups over finite fields , jstor=2372673 , mr=0086367 , year=1956 , journal=American Journal of Mathematics The ''American Journal of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Galois Cohomology
In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. A Galois group ''G'' associated to a field extension ''L''/''K'' acts in a natural way on some abelian groups, for example those constructed directly from ''L'', but also through other Galois representations that may be derived by more abstract means. Galois cohomology accounts for the way in which taking Galois-invariant elements fails to be an exact functor. History The current theory of Galois cohomology came together around 1950, when it was realised that the Galois cohomology of ideal class groups in algebraic number theory was one way to formulate class field theory, at the time it was in the process of ridding itself of connections to L-functions. Galois cohomology makes no assumption that Galois groups are abelian groups, so this was a non-abelian theory. It was formulated abstractly as a theory of class ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tits Index
In the mathematics, mathematical study of Lie algebras and Lie groups, a Satake diagram is a generalization of a Dynkin diagram introduced by whose configurations classify semisimple Lie algebra, simple Lie algebras over the field (mathematics), field of real numbers. The Satake diagrams associated to a Dynkin diagram classify real form (Lie theory), real forms of the complex Lie algebra corresponding to the Dynkin diagram. More generally, the Tits index or Satake–Tits diagram of a reductive algebraic group over a field is a generalization of the Satake diagram to arbitrary fields, introduced by , that reduces the classification of reductive algebraic groups to that of Linear algebraic group#Examples, anisotropic Reductive group, reductive algebraic groups. Satake diagrams are not the same as Vogan diagrams of a Lie group, although they look similar. Definition A Satake diagram is obtained from a Dynkin diagram by blackening some vertices, and connecting other vertices in pai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynkin Diagram
In the Mathematics, mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of Graph (discrete mathematics), graph with some edges doubled or tripled (drawn as a double or triple line). Dynkin diagrams arise in the classification of semisimple Lie algebras over algebraically closed fields, in the classification of Weyl groups and other finite reflection groups, and in other contexts. Various properties of the Dynkin diagram (such as whether it contains multiple edges, or its symmetries) correspond to important features of the associated Lie algebra. The term "Dynkin diagram" can be ambiguous. In some cases, Dynkin diagrams are assumed to be directed graph, directed, in which case they correspond to root systems and semi-simple Lie algebras, while in other cases they are assumed to be undirected graph, undirected, in which case they correspond to Weyl groups. In this article, "Dynkin diagram" means ''directed'' Dynkin diagram, and ''undirected'' ... [...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 span the w ... [...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, ov ... [...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 Lie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Galois Module
In mathematics, a Galois module is a ''G''-module, with ''G'' being the Galois group of some extension of fields. The term Galois representation is frequently used when the ''G''-module is a vector space over a field or a free module over a ring in representation theory, but can also be used as a synonym for ''G''-module. The study of Galois modules for extensions of local or global fields and their group cohomology is an important tool in number theory. Examples *Given a field ''K'', the multiplicative group (''Ks'')× of a separable closure of ''K'' is a Galois module for the absolute Galois group. Its second cohomology group is isomorphic to the Brauer group of ''K'' (by Hilbert's theorem 90, its first cohomology group is zero). *If ''X'' is a smooth proper scheme over a field ''K'' then the ℓ-adic cohomology groups of its geometric fibre are Galois modules for the absolute Galois group of ''K''. Ramification theory Let ''K'' be a valued field (with valuation deno ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
