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Noncommutative Algebraic Geometry
Noncommutative algebraic geometry is a branch of mathematics, and more specifically a direction in noncommutative geometry, that studies the geometric properties of formal duals of non-commutative algebraic objects such as rings as well as geometric objects derived from them (e.g. by gluing along localizations or taking noncommutative stack quotients). For example, noncommutative algebraic geometry is supposed to extend a notion of an algebraic scheme by suitable gluing of spectra of noncommutative rings; depending on how literally and how generally this aim (and a notion of spectrum) is understood in noncommutative setting, this has been achieved in various level of success. The noncommutative ring generalizes here a commutative ring of regular functions on a commutative scheme. Functions on usual spaces in the traditional (commutative) algebraic geometry have a product defined by pointwise multiplication; as the values of these functions commute, the functions also com ... [...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 poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homological Algebra
Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topology) and abstract algebra (theory of modules and syzygies) at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert. Homological algebra is the study of homological functors and the intricate algebraic structures that they entail; its development was closely intertwined with the emergence of category theory. A central concept is that of chain complexes, which can be studied through both their homology and cohomology. Homological algebra affords the means to extract information contained in these complexes and present it in the form of homological invariants of rings, modules, topological spaces, and other 'tangible' mathematical objects. A powerful tool for doing this is provided by spectral sequences. It has played a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primitive Ideal
In mathematics, specifically ring theory, a left primitive ideal is the annihilator of a (nonzero) simple left module. A right primitive ideal is defined similarly. Left and right primitive ideals are always two-sided ideals. Primitive ideals are prime. The quotient of a ring by a left primitive ideal is a left primitive ring. For commutative rings the primitive ideals are maximal, and so commutative primitive rings are all fields. Primitive spectrum The primitive spectrum of a ring is a non-commutative analogA primitive ideal tends to be more of interest than a prime ideal in non-commutative ring theory. of the prime spectrum of a commutative ring. Let ''A'' be a ring and \operatorname(A) the set of all primitive ideals of ''A''. Then there is a topology on \operatorname(A), called the Jacobson topology, defined so that the closure of a subset ''T'' is the set of primitive ideals of ''A'' containing the intersection of elements of ''T''. Now, suppose ''A'' is an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Irreducible Representations
In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho, _W,W), with W \subset V closed under the action of \. Every finite-dimensional unitary representation on a Hilbert space V is the direct sum of irreducible representations. Irreducible representations are always indecomposable (i.e. cannot be decomposed further into a direct sum of representations), but converse may not hold, e.g. the two-dimensional representation of the real numbers acting by upper triangular unipotent matrices is indecomposable but reducible. History Group representation theory was generalized by Richard Brauer from the 1940s to give modular representation theory, in which the matrix operators act on a vector space over a field K of arbitrary characteristic, rather than a vector space over the field of real numbers or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Representation Theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations (for example, matrix addition, matrix multiplication). The theory of matrices and linear operators is well-understood, so representations of more abstract objects in terms of familiar linear algebra objects helps glean properties and sometimes simplify calculations on more abstract theories. The algebraic objects amenable to such a description include groups, associative algebras and Lie algebras. The most prominent of these (and historically the first) is the representation theory of groups, in which elements of a group are represented by invertible matrices in such a way that the group operatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michael Artin
Michael Artin (; born 28 June 1934) is a German-American mathematician and a professor emeritus in the Massachusetts Institute of Technology mathematics department, known for his contributions to algebraic geometry.Faculty profile , MIT mathematics department, retrieved 2011-01-03 Life and career Michael Artin or Artinian was born in , Germany, and brought up in . His parents were Natalia Naumovna Jasny (Natascha) and |
Dixmier
Jacques Dixmier (born 24 May 1924) is a French mathematician. He worked on operator algebras, especially C*-algebras, and wrote several of the standard reference books on them, and introduced the Dixmier trace and the Dixmier mapping. Biography Dixmier received his Ph.D. in 1949 from the University of Paris, and his students include Alain Connes. In 1949 upon the initiative of Jean-Pierre Serre and Pierre Samuel, Dixmier became a member of Bourbaki, in which he made essential contributions to the Bourbaki volume on Lie algebras. After retiring as professor emeritus from the University of Paris VI, he spent five years at the Institut des Hautes Études Scientifiques. Often, there is made the erroneous claim that Dixmier originated the name ''von Neumann algebra'' for the operator algebras introduced by John von Neumann, but Dixmier said in an interview that the name originated from a proposal by Jean Dieudonné. Dixmier was an invited speaker at the International Congress o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Descent Theory
In mathematics, the idea of descent extends the intuitive idea of 'gluing' in topology. Since the topologists' glue is the use of equivalence relations on topological spaces, the theory starts with some ideas on identification. Descent of vector bundles The case of the construction of vector bundles from data on a disjoint union of topological spaces is a straightforward place to start. Suppose ''X'' is a topological space covered by open sets ''Xi''. Let ''Y'' be the disjoint union of the ''Xi'', so that there is a natural mapping :p: Y \rightarrow X. We think of ''Y'' as 'above' ''X'', with the ''Xi'' projection 'down' onto ''X''. With this language, ''descent'' implies a vector bundle on ''Y ''(so, a bundle given on each ''Xi''), and our concern is to 'glue' those bundles ''Vi'', to make a single bundle ''V'' on X. What we mean is that ''V'' should, when restricted to ''Xi'', give back ''Vi'', up to a bundle isomorphism. The data needed is then this: on each overlap :X_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-commutative Localization
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primitive Spectrum
In mathematics, specifically ring theory, a left primitive ideal is the Annihilator (ring theory), annihilator of a (nonzero) simple module, simple left module (mathematics), module. A right primitive ideal is defined similarly. Left and right primitive ideals are always two-sided ideals. Primitive ideals are prime ideal, prime. The quotient ring, quotient of a ring (mathematics), ring by a left primitive ideal is a left primitive ring. For commutative rings the primitive ideals are maximal ideal, maximal, and so commutative primitive rings are all field (mathematics), fields. Primitive spectrum The primitive spectrum of a ring is a non-commutative analogA primitive ideal tends to be more of interest than a prime ideal in non-commutative ring theory. of the prime spectrum of a commutative ring. Let ''A'' be a ring and \operatorname(A) the set (mathematics), set of all primitive ideals of ''A''. Then there is a topological space, topology on \operatorname(A), called the Jacobso ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simple Ring
In abstract algebra, a branch of mathematics, a simple ring is a non-zero ring that has no two-sided ideal besides the zero ideal and itself. In particular, a commutative ring is a simple ring if and only if it is a field. The center of a simple ring is necessarily a field. It follows that a simple ring is an associative algebra over this field. So, simple algebra and ''simple ring'' are synonyms. Several references (e.g., Lang (2002) or Bourbaki (2012)) require in addition that a simple ring be left or right Artinian (or equivalently semi-simple). Under such terminology a non-zero ring with no non-trivial two-sided ideals is called quasi-simple. Rings which are simple as rings but are not a simple module over themselves do exist: a full matrix ring over a field does not have any nontrivial ideals (since any ideal of M_n(R) is of the form M_n(I) with I an ideal of R), but has nontrivial left ideals (for example, the sets of matrices which have some fixed zero columns). Accordin ... [...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'', ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
