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Sklyanin Algebra
In mathematics, specifically the field of algebra, Sklyanin algebras are a class of noncommutative algebra named after Evgeny Sklyanin. This class of algebras was first studied in the classification of Artin-Schelter regular algebras of global dimension 3 in the 1980s. Sklyanin algebras can be grouped into two different types, the non-degenerate Sklyanin algebras and the degenerate Sklyanin algebras, which have very different properties. A need to understand the non-degenerate Sklyanin algebras better has led to the development of the study of point modules in noncommutative geometry. Formal definition Let k be a field with a primitive cube root of unity. Let \mathfrak be the following subset of the projective plane \textbf_k^2: \mathfrak = \ \sqcup \. Each point :b:c\in \textbf_k^2 gives rise to a (quadratic 3-dimensional) Sklyanin algebra, S_ = k \langle x,y,z \rangle / (f_1, f_2, f_3), where, f_1 = ayz + bzy + cx^2, \quad f_2 = azx + bxz + cy^2, \quad f_3 = axy + b yx + ... [...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 ... [...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] |
Elliptic Curve
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If the field's characteristic is different from 2 and 3, then the curve can be described as a plane algebraic curve which consists of solutions for: :y^2 = x^3 + ax + b for some coefficients and in . The curve is required to be non-singular, which means that the curve has no cusps or self-intersections. (This is equivalent to the condition , that is, being square-free in .) It is always understood that the curve is really sitting in the projective plane, with the point being the unique point at infinity. Many sources define an elliptic curve to be simply a curve given by an equation of this form. (When the coefficient field has characteristic 2 or 3, the above equation is not quite general enough to include all non-singular cubic cu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Noncommutative Ring
In mathematics, a noncommutative ring is a ring whose multiplication is not commutative; that is, there exist ''a'' and ''b'' in the ring such that ''ab'' and ''ba'' are different. Equivalently, a ''noncommutative ring'' is a ring that is not a commutative ring. Noncommutative algebra is the part of ring theory devoted to study of properties of the noncommutative rings, including the properties that apply also to commutative rings. Sometimes the term ''noncommutative ring'' is used instead of ''ring'' to refer to a unspecified ring which is not necessarily commutative, and hence may be commutative. Generally, this is for emphasizing that the studied properties are not restricted to commutative rings, as, in many contexts, ''ring'' is used as a shortcut for ''commutative ring''. Although some authors do not assume that rings have a multiplicative identity, in this article we make that assumption unless stated otherwise. Examples Some examples of noncommutative rings: * The ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Geometry
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts. The basic intuitions are that projective space has more points than Euclidean space, for a given dimension, and that geometric transformations are permitted that transform the extra points (called "points at infinity") to Euclidean points, and vice-versa. Properties meaningful for projective geometry are respected by this new idea of transformation, which is more radical in its effects than can be expressed by a transformation matrix and translations (the affine transformations). The first issue for geometers is what kind of geometry is adequate for a novel situation. It is not possible to refer to angles in projective geometry as it is in Euclidean geometry, because angle is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word isomorphism is derived from the Ancient Greek: ἴσος ''isos'' "equal", and μορφή ''morphe'' "form" or "shape". The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties (excluding further information such as additional structure or names of objects). Thus isomorphic structures cannot be distinguished from the point of view of structure only, and may be identified. In mathematical jargon, one says that two objects are . An automorphism is an isomorphism from a structure to itself. An isomorphism between two structures is a canonical isomorphism (a canonical map that is an isomorphism) if there is only one isomorphism between the two structures (as it is the case for solutions of a univer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Center (algebra)
The term center or centre is used in various contexts in abstract algebra to denote the set of all those elements that commutative operation, commute with all other elements. * The center of a group ''G'' consists of all those elements ''x'' in ''G'' such that ''xg'' = ''gx'' for all ''g'' in ''G''. This is a normal subgroup of ''G''. * The similarly named notion for a semigroup is defined likewise and it is a subsemigroup. * The center (ring theory), center of a ring (mathematics), ring (or an associative algebra) ''R'' is the subset of ''R'' consisting of all those elements ''x'' of ''R'' such that ''xr'' = ''rx'' for all ''r'' in ''R''., Exercise 22.22 The center is a commutative ring, commutative subring of ''R''. * The center of a Lie algebra ''L'' consists of all those elements ''x'' in ''L'' such that [''x'',''a''] = 0 for all ''a'' in ''L''. This is an ideal (ring theory), ideal of the Lie algebra ''L''. See also *Centralizer and normalizer *Center (category theory) Refere ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Element
In mathematics, an element ''x'' of a *-algebra is normal if it satisfies x x^ = x^ x. This definition stems from the definition of a normal linear operator in functional analysis, where a linear operator ''A'' from a Hilbert space into itself is called unitary if A A^ = A^ A, where the adjoint of ''A'' is ''A'' and the domain of ''A'' is the same as that of ''A''. See normal operator for a detailed discussion. If the Hilbert space is finite-dimensional and an orthonormal basis has been chosen, then the operator ''A'' is normal if and only if the matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ... describing ''A'' with respect to this basis is a normal matrix. See also * * * References * * * Abstract algebra Linear algebra {{Algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Advances In Mathematics
''Advances in Mathematics'' is a peer-reviewed scientific journal covering research on pure mathematics. It was established in 1961 by Gian-Carlo Rota. The journal publishes 18 issues each year, in three volumes. At the origin, the journal aimed at publishing articles addressed to a broader "mathematical community", and not only to mathematicians in the author's field. Herbert Busemann writes, in the preface of the first issue, "The need for expository articles addressing either all mathematicians or only those in somewhat related fields has long been felt, but little has been done outside of the USSR. The serial publication ''Advances in Mathematics'' was created in response to this demand." Abstracting and indexing The journal is abstracted and indexed in: * |
Koszul Algebra
In abstract algebra, a Koszul algebra R is a graded k-algebra over which the ground field k has a linear minimal graded free resolution, ''i.e.'', there exists an exact sequence: :\cdots \rightarrow R(-i)^ \rightarrow \cdots \rightarrow R(-2)^ \rightarrow R(-1)^ \rightarrow R \rightarrow k \rightarrow 0. Here, R(-j) is the graded algebra R with grading shifted up by j, ''i.e.'' R(-j)_i = R_. The exponents b_i refer to the b_i-fold direct sum. Choosing bases for the free modules in the resolution, the chain maps are given by matrices, and the definition requires the matrix entries to be zero or linear forms. An example of a Koszul algebra is a polynomial ring over a field, for which the Koszul complex is the minimal graded free resolution of the ground field. There are Koszul algebras whose ground fields have infinite minimal graded free resolutions, ''e.g'', R = k ,y(xy) . The concept is named after the French mathematician Jean-Louis Koszul. See also *Koszul duality *Comple ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Noetherian Ring
In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noetherian respectively. That is, every increasing sequence I_1\subseteq I_2 \subseteq I_3 \subseteq \cdots of left (or right) ideals has a largest element; that is, there exists an such that: I_=I_=\cdots. Equivalently, a ring is left-Noetherian (resp. right-Noetherian) if every left ideal (resp. right-ideal) is finitely generated. A ring is Noetherian if it is both left- and right-Noetherian. Noetherian rings are fundamental in both commutative and noncommutative ring theory since many rings that are encountered in mathematics are Noetherian (in particular the ring of integers, polynomial rings, and rings of algebraic integers in number fields), and many general theorems on rings rely heavily on Noetherian property (for example, the Lasker ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Series And 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 Hilbert ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
