|
Almost Commutative Ring
In algebra, a filtered ring ''A'' is said to be almost commutative if the associated graded ring \operatornameA = \oplus A_i/ is commutative. Basic examples of almost commutative rings involve differential operators. For example, the enveloping algebra of a complex Lie algebra is almost commutative by the PBW theorem. Similarly, a Weyl algebra is almost commutative. See also * Ore condition *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 \sub ... References * Victor GinzburgLectures on D-modules {{algebra-stub Ring theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary algebra deals with the manipulation of variables (commonly represented by Roman letters) as if they were numbers and is therefore essential in all applications of mathematics. Abstract algebra is the name given, mostly in education, to the study of algebraic structures such as groups, rings, and fields (the term is no more in common use outside educational context). Linear algebra, which deals with linear equations and linear mappings, is used for modern presentations of geometry, and has many practical applications (in weather forecasting, for example). There are many areas of mathematics that belong to algebra, some having "algebra" in their name, such as commutative algebra, and some not, such as Galois theory. The word ''algebra'' is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Filtered Ring
In mathematics, a filtered algebra is a generalization of the notion of a graded algebra. Examples appear in many branches of mathematics, especially in homological algebra and representation theory. A filtered algebra over the field k is an algebra (A,\cdot) over k that has an increasing sequence \ \subseteq F_0 \subseteq F_1 \subseteq \cdots \subseteq F_i \subseteq \cdots \subseteq A of subspaces of A such that :A=\bigcup_ F_ and that is compatible with the multiplication in the following sense: : \forall m,n \in \mathbb,\quad F_m\cdot F_n\subseteq F_. Associated graded algebra In general there is the following construction that produces a graded algebra out of a filtered algebra. If A is a filtered algebra then the ''associated graded algebra'' \mathcal(A) is defined as follows: The multiplication is well-defined and endows \mathcal(A) with the structure of a graded algebra, with gradation \_. Furthermore if A is associative then so is \mathcal(A). Also if A is uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graded Ring
In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that R_i R_j \subseteq R_. The index set is usually the set of nonnegative integers or the set of integers, but can be any monoid. The direct sum decomposition is usually referred to as gradation or grading. A graded module is defined similarly (see below for the precise definition). It generalizes graded vector spaces. A graded module that is also a graded ring is called a graded algebra. A graded ring could also be viewed as a graded \Z-algebra. The associativity is not important (in fact not used at all) in the definition of a graded ring; hence, the notion applies to non-associative algebras as well; e.g., one can consider a graded Lie algebra. First properties Generally, the index set of a graded ring is assumed to be the set of nonnegative integers, unless otherwise explicitly specified. This is the case in this article. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutative Ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not specific to commutative rings. This distinction results from the high number of fundamental properties of commutative rings that do not extend to noncommutative rings. Definition and first examples Definition A ''ring'' is a set R equipped with two binary operations, i.e. operations combining any two elements of the ring to a third. They are called ''addition'' and ''multiplication'' and commonly denoted by "+" and "\cdot"; e.g. a+b and a \cdot b. To form a ring these two operations have to satisfy a number of properties: the ring has to be an abelian group under addition as well as a monoid under multiplication, where multiplication distributes over addition; i.e., a \cdot \left(b + c\right) = \left(a \cdot b\right) + \left(a \cdot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and returns another function (in the style of a higher-order function in computer science). This article considers mainly linear differential operators, which are the most common type. However, non-linear differential operators also exist, such as the Schwarzian derivative. Definition An order-m linear differential operator is a map A from a function space \mathcal_1 to another function space \mathcal_2 that can be written as: A = \sum_a_\alpha(x) D^\alpha\ , where \alpha = (\alpha_1,\alpha_2,\cdots,\alpha_n) is a multi-index of non-negative integers, , \alpha, = \alpha_1 + \alpha_2 + \cdots + \alpha_n, and for each \alpha, a_\alpha(x) is a function on some open domain in ''n''-dimensional space. The operator D^\alpha is interpreted as D^\alp ... [...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] |
PBW Theorem
PBW may refer to: * Philadelphia-Baltimore-Washington Stock Exchange * Peanut Butter Wolf, American hip hop record producer * Proton beam writing, a lithography process * Play by Web, Play-by-post role-playing game * Prosopography of the Byzantine World The Prosopography of the Byzantine World (PBW) is a project to create a prosopographical database of individuals named in textual sources in the Byzantine Empire and surrounding areas in the period from 642 to 1265. The project is a collaboration b ..., a prosopographical database project * Poincaré-Birkhoff-Witt theorem, a result in mathematics {{disambig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weyl Algebra
In abstract algebra, the Weyl algebra is the ring of differential operators with polynomial coefficients (in one variable), namely expressions of the form : f_m(X) \partial_X^m + f_(X) \partial_X^ + \cdots + f_1(X) \partial_X + f_0(X). More precisely, let ''F'' be the underlying field, and let ''F'' 'X''be the ring of polynomials in one variable, ''X'', with coefficients in ''F''. Then each ''fi'' lies in ''F'' 'X'' ''∂X'' is the derivative with respect to ''X''. The algebra is generated by ''X'' and ''∂X''. The Weyl algebra is an example of a simple ring that is not a matrix ring over a division ring. It is also a noncommutative example of a domain, and an example of an Ore extension. The Weyl algebra is isomorphic to the quotient of the free algebra on two generators, ''X'' and ''Y'', by the ideal generated by the element :YX - XY = 1~. The Weyl algebra is the first in an infinite family of algebras, also known as Weyl algebras. The ''n''-th Weyl algebra, ''An'', is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ore Condition
In mathematics, especially in the area of algebra known as ring theory, the Ore condition is a condition introduced by Øystein Ore, in connection with the question of extending beyond commutative rings the construction of a field of fractions, or more generally localization of a ring. The ''right Ore condition'' for a multiplicative subset ''S'' of a ring ''R'' is that for and , the intersection . A (non-commutative) domain for which the set of non-zero elements satisfies the right Ore condition is called a right Ore domain. The left case is defined similarly. General idea The goal is to construct the right ring of fractions ''R'' 'S''−1with respect to a multiplicative subset ''S''. In other words, we want to work with elements of the form ''as''−1 and have a ring structure on the set ''R'' 'S''−1 The problem is that there is no obvious interpretation of the product (''as''−1)(''bt''−1); indeed, we need a method to "move" ''s''−1 past ''b''. This means that we need t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 ''M ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Victor Ginzburg
Victor Ginzburg (born 1957) is a Russian American mathematician who works in representation theory and in noncommutative geometry. He is known for his contributions to geometric representation theory, especially, for his works on representations of quantum groups and Hecke algebras, and on the geometric Langlands program (Satake equivalence of categories). He is currently a Professor of Mathematics at the University of Chicago. Career Ginzburg received his Ph.D. at Moscow State University in 1985, under the direction of Alexandre Kirillov and Israel Gelfand. Ginzburg wrote a textbook ''Representation theory and complex geometry'' with Neil Chriss on geometric representation theory. A paper by Alexander Beilinson, Ginzburg, and Wolfgang Soergel introduced the concept of Koszul duality (cf. Koszul algebra) and the technique of "mixed categories" to representation theory. Furthermore, Ginzburg and Mikhail Kapranov developed Koszul duality theory for operads. In noncommutative g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
