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Gelfand–Kirillov Dimension
In algebra, the Gelfand–Kirillov dimension (or GK dimension) of a right module ''M'' over a ''k''-algebra ''A'' is: :\operatorname = \sup_ \limsup_ \log_n \dim_k M_0 V^n where the supremum is taken over all finite-dimensional subspaces V \subset A and M_0 \subset M. An algebra is said to have ''polynomial growth'' if its Gelfand–Kirillov dimension is finite. Basic facts *The Gelfand–Kirillov dimension of a finitely generated commutative algebra ''A'' over a field is the Krull dimension of ''A'' (or equivalently the transcendence degree of the field of fractions of ''A'' over the base field.) *In particular, the GK dimension of the polynomial ring k _1, \dots, x_n/math> Is ''n''. *(Warfield) For any real number ''r'' ≥ 2, there exists a finitely generated algebra whose GK dimension is ''r''. In the theory of D-Modules Given a right module ''M'' over the Weyl algebra A_n, the Gelfand–Kirillov dimension of ''M'' over the Weyl algebra coincides with the dimension of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebra
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic operations other than the standard arithmetic operations, such as addition and multiplication. Elementary algebra is the main form of algebra taught in schools. It examines mathematical statements using variables for unspecified values and seeks to determine for which values the statements are true. To do so, it uses different methods of transforming equations to isolate variables. Linear algebra is a closely related field that investigates linear equations and combinations of them called '' systems of linear equations''. It provides methods to find the values that solve all equations in the system at the same time, and to study the set of these solutions. Abstract algebra studies algebraic structures, which consist of a set of mathemati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial Ring
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, often a field. Often, the term "polynomial ring" refers implicitly to the special case of a polynomial ring in one indeterminate over a field. The importance of such polynomial rings relies on the high number of properties that they have in common with the ring of the integers. Polynomial rings occur and are often fundamental in many parts of mathematics such as number theory, commutative algebra, and algebraic geometry. In ring theory, many classes of rings, such as unique factorization domains, regular rings, group rings, rings of formal power series, Ore polynomials, graded rings, have been introduced for generalizing some properties of polynomial rings. A closely related notion is that of the ring of polynomial functions on a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proceedings Of The American Mathematical Society
''Proceedings of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society. The journal is devoted to shorter research articles. As a requirement, all articles must be at most 15 printed pages. According to the ''Journal Citation Reports'', the journal has a 2018 impact factor of 0.813. Scope ''Proceedings of the American Mathematical Society'' publishes articles from all areas of pure and applied mathematics, including topology, geometry, analysis, algebra, number theory, combinatorics, logic, probability and statistics. Abstracting and indexing This journal is indexed in the following databases: 2011. American Mathematical Society. * [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Langlands Program
In mathematics, the geometric Langlands correspondence relates algebraic geometry and representation theory. It is a reformulation of the Langlands correspondence obtained by replacing the number fields appearing in the original number theoretic version by function fields and applying techniques from algebraic geometry. The correspondence is named for the Canadian mathematician Robert Langlands, who formulated the original form of it in the late 1960s. The geometric Langlands conjecture asserts the existence of the geometric Langlands correspondence. The existence of the geometric Langlands correspondence in the specific case of general linear groups over function fields was proven by Laurent Lafforgue in 2002, where it follows as a consequence of Lafforgue's theorem. Background In mathematics, the classical Langlands correspondence is a collection of results and conjectures relating number theory and representation theory. Formulated by Robert Langlands in the late 1960s, th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
D-module
In mathematics, a ''D''-module is a module (mathematics), module over a ring (mathematics), ring ''D'' of differential operators. The major interest of such ''D''-modules is as an approach to the theory of linear partial differential equations. Since around 1970, ''D''-module theory has been built up, mainly as a response to the ideas of Mikio Sato on algebraic analysis, and expanding on the work of Sato and Joseph Bernstein on the Bernstein–Sato polynomial. Early major results were the Kashiwara constructibility theorem and Kashiwara index theorem of Masaki Kashiwara. The methods of ''D''-module theory have always been drawn from sheaf theory and other techniques with inspiration from the work of Alexander Grothendieck in algebraic geometry. This approach is global in character, and differs from the functional analysis techniques traditionally used to study differential operators. The strongest results are obtained for over-determined systems (holonomic systems), and on the cha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Short Exact Sequence
In mathematics, an exact sequence is a sequence of morphisms between objects (for example, Group (mathematics), groups, Ring (mathematics), rings, Module (mathematics), modules, and, more generally, objects of an abelian category) such that the Image (mathematics), image of one morphism equals the kernel (algebra), kernel of the next. Definition In the context of group theory, a sequence :G_0\;\xrightarrow\; G_1 \;\xrightarrow\; G_2 \;\xrightarrow\; \cdots \;\xrightarrow\; G_n of groups and group homomorphisms is said to be exact at G_i if \operatorname(f_i)=\ker(f_). The sequence is called exact if it is exact at each G_i for all 1\leq i [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Polynomial
In commutative algebra, the Hilbert function, the Hilbert polynomial, and the Hilbert series of a graded commutative algebra finitely generated over a field are three strongly related notions which measure the growth of the dimension of the homogeneous components of the algebra. These notions have been extended to filtered algebras, and graded or filtered modules over these algebras, as well as to coherent sheaves over projective schemes. The typical situations where these notions are used are the following: * The quotient by a homogeneous ideal of a multivariate polynomial ring, graded by the total degree. * The quotient by an ideal of a multivariate polynomial ring, filtered by the total degree. * The filtration of a local ring by the powers of its maximal ideal. In this case the Hilbert polynomial is called the Hilbert–Samuel polynomial. The Hilbert series of an algebra or a module is a special case of the Hilbert–Poincaré series of a graded vector space. The Hil ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weyl Algebra
In abstract algebra, the Weyl algebras are abstracted from the ring of differential operators with polynomial coefficients. They are named after Hermann Weyl, who introduced them to study the Heisenberg uncertainty principle in quantum mechanics. In the simplest case, these are differential operators. Let F be a field, and let F /math> be the ring of polynomials in one variable with coefficients in F. Then the corresponding Weyl algebra consists of differential operators of form : f_m(x) \partial_x^m + f_(x) \partial_x^ + \cdots + f_1(x) \partial_x + f_0(x) This is the first Weyl algebra A_1. The ''n''-th Weyl algebra A_n are constructed similarly. Alternatively, A_1 can be constructed as the quotient of the free algebra on two generators, ''q'' and ''p'', by the ideal generated by ( ,q- 1). Similarly, A_n is obtained by quotienting the free algebra on ''2n'' generators by the ideal generated by ( _i,q_j- \delta_), \quad \forall i, j = 1, \dots, nwhere \delta_ is the K ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and in many other branches of mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives. The set of real numbers, sometimes called "the reals", is traditionally denoted by a bold , often using blackboard bold, . The adjective ''real'', used in the 17th century by René Descartes, distinguishes real numbers from imaginary numbers such as the square roots of . The real numbers include the rational numbers, such as the integer and the fraction . The rest of the real numbers are called irrational numbers. Some irrational numbers (as well as all the rationals) a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field Of Fractions
In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field of rational numbers. Intuitively, it consists of ratios between integral domain elements. The field of fractions of an integral domain R is sometimes denoted by \operatorname(R) or \operatorname(R), and the construction is sometimes also called the fraction field, field of quotients, or quotient field of R. All four are in common usage, but are not to be confused with the quotient of a ring by an ideal, which is a quite different concept. For a commutative ring that is not an integral domain, the analogous construction is called the localization or ring of quotients. Definition Given an integral domain R and letting R^* = R \setminus \, we define an equivalence relation on R \times R^* by letting (n,d) \sim (m,b) whenever nb = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Module (mathematics)
In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a (not necessarily commutative) ring. The concept of a ''module'' also generalizes the notion of an abelian group, since the abelian groups are exactly the modules over the ring of integers. Like a vector space, a module is an additive abelian group, and scalar multiplication is distributive over the operations of addition between elements of the ring or module and is compatible with the ring multiplication. Modules are very closely related to the representation theory of groups. They are also one of the central notions of commutative algebra and homological algebra, and are used widely in algebraic geometry and algebraic topology. Introduction and definition Motivation In a vector space, the set of scalars is a field and acts on the vectors by scalar multiplication, subject to certain axioms such as the distributive law. In a module, the scal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transcendence Degree
In mathematics, a transcendental extension L/K is a field extension such that there exists an element in the field L that is transcendental over the field K; that is, an element that is not a root of any univariate polynomial with coefficients in K. In other words, a transcendental extension is a field extension that is not algebraic. For example, \mathbb and \mathbb are both transcendental extensions of \mathbb. A transcendence basis of a field extension L/K (or a transcendence basis of L over K) is a maximal algebraically independent subset of L over K. Transcendence bases share many properties with bases of vector spaces. In particular, all transcendence bases of a field extension have the same cardinality, called the transcendence degree of the extension. Thus, a field extension is a transcendental extension if and only if its transcendence degree is nonzero. Transcendental extensions are widely used in algebraic geometry. For example, the dimension of an algebraic varie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
