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Additive K-theory
In mathematics, additive K-theory means some version of algebraic K-theory in which, according to Spencer Bloch, the general linear group ''GL'' has everywhere been replaced by its Lie algebra ''gl''. It is not, therefore, one theory but a way of creating additive or infinitesimal analogues of multiplicative theories. Formulation Following Boris Feigin and Boris Tsygan,B. Feigin, B. Tsygan. ''Additive K-theory'', LNM 1289, Springer let A be an algebra over a field k of characteristic zero and let (A) be the algebra of infinite matrices over A with only finitely many nonzero entries. Then the Lie algebra homology : H_\cdot ((A),k) has a natural structure of a Hopf algebra. The space of its primitive elements of degree i is denoted by K^+_i(A) and called the i-th additive K-functor of ''A''. The additive K-functors are related to cyclic homology In noncommutative geometry and related branches of mathematics, cyclic homology and cyclic cohomology are certain (co)homolo ... [...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] |
Algebraic K-theory
Algebraic ''K''-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called ''K''-groups. These are groups in the sense of abstract algebra. They contain detailed information about the original object but are notoriously difficult to compute; for example, an important outstanding problem is to compute the ''K''-groups of the integers. ''K''-theory was discovered in the late 1950s by Alexander Grothendieck in his study of intersection theory on algebraic varieties. In the modern language, Grothendieck defined only ''K''0, the zeroth ''K''-group, but even this single group has plenty of applications, such as the Grothendieck–Riemann–Roch theorem. Intersection theory is still a motivating force in the development of (higher) algebraic ''K''-theory through its links with motivic cohomology and specifically Chow groups. The subject also includes classical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spencer Bloch
Spencer Janney Bloch (born May 22, 1944; New York City) is an American mathematician known for his contributions to algebraic geometry and algebraic ''K''-theory. Bloch is a R. M. Hutchins Distinguished Service Professor Emeritus in the Department of Mathematics of the University of Chicago. He is a member of the U.S. National Academy of Sciences and a Fellow of the American Academy of Arts and SciencesScholars, visiting faculty, leaders represent Chicago as AAAS fellows The University of Chicago Chronicle, April 30, 2009, Vol. 28 No. 15. Accessed January 12, 2010 and of the . At the |
General Linear Group
In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible, with identity matrix as the identity element of the group. The group is so named because the columns (and also the rows) of an invertible matrix are linearly independent, hence the vectors/points they define are in general linear position, and matrices in the general linear group take points in general linear position to points in general linear position. To be more precise, it is necessary to specify what kind of objects may appear in the entries of the matrix. For example, the general linear group over R (the set of real numbers) is the group of invertible matrices of real numbers, and is denoted by GL''n''(R) or . More generally, the general linear group of degree ''n'' over any ... [...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] |
Boris Feigin
Boris Lvovich Feigin (russian: Бори́с Льво́вич Фе́йгин) (born 20 November 1953) is a Russian mathematician. His research has spanned representation theory, mathematical physics, algebraic geometry, Lie groups and Lie algebras, conformal field theory, homological and homotopical algebra. In 1969 Feigin graduated from the Moscow Mathematical School No. 2 (Andrei Zelevinsky was among his classmates). From 1969 until 1974 he was a student in the Faculty of Mechanics and Mathematics at Moscow State University (MSU) under joint supervision of Dmitry Fuchs and Israel Gelfand. His diploma thesis was dedicated to characteristic classes of flags of foliations. Feigin was not accepted to the graduate school of MSU due to increasingly anti-semitic policies at that institution at that time. After working as a computer programmer in industry for some time, he was accepted in 1976 to the graduate school of Yaroslavl State University and defended his thesis "Cohomology ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boris Tsygan
Boris may refer to: People * Boris (given name), a male given name *:''See'': List of people with given name Boris * Boris (surname) * Boris I of Bulgaria (died 907), the first Christian ruler of the First Bulgarian Empire, canonized after his death * Boris II of Bulgaria (c. 931–977), ruler of the First Bulgarian Empire * Boris III of Bulgaria (1894–1943), ruler of the Kingdom of Bulgaria in the first half of the 20th century * Boris, Prince of Tarnovo (born 1997), Spanish-born Bulgarian royal * Boris and Gleb (died 1015), the first saints canonized in Kievan Rus * Boris (singer) (born 1965), pseudonym of French singer Philippe Dhondt Arts and media * Boris (band), a Japanese experimental rock trio * ''Boris'' (EP), by Yezda Urfa, 1975 * "Boris" (song), by the Melvins, 1991 * ''Boris'' (TV series), a 2007–2009 Italian comedy series * '' Boris: The Film'', a 2011 Italian film based on the TV series * '' Boris: The Rise of Boris Johnson'', a 2006 biography by Andrew G ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Characteristic Zero
In mathematics, the characteristic of a ring , often denoted , is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive identity the ring is said to have characteristic zero. That is, is the smallest positive number such that: :\underbrace_ = 0 if such a number exists, and otherwise. Motivation The special definition of the characteristic zero is motivated by the equivalent definitions characterized in the next section, where the characteristic zero is not required to be considered separately. The characteristic may also be taken to be the exponent of the ring's additive group, that is, the smallest positive integer such that: :\underbrace_ = 0 for every element of the ring (again, if exists; otherwise zero). Some authors do not include the multiplicative identity element in their requirements for a ring (see Multiplicative identity and the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Algebra Homology
In mathematics, Lie algebra cohomology is a cohomology theory for Lie algebras. It was first introduced in 1929 by Élie Cartan to study the topology of Lie groups and homogeneous spaces by relating cohomological methods of Georges de Rham to properties of the Lie algebra. It was later extended by to coefficients in an arbitrary Lie module. Motivation If G is a compact simply connected Lie group, then it is determined by its Lie algebra, so it should be possible to calculate its cohomology from the Lie algebra. This can be done as follows. Its cohomology is the de Rham cohomology of the complex of differential forms on G. Using an averaging process, this complex can be replaced by the complex of left-invariant differential forms. The left-invariant forms, meanwhile, are determined by their values at the identity, so that the space of left-invariant differential forms can be identified with the exterior algebra of the Lie algebra, with a suitable differential. The construction of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hopf Algebra
Hopf is a German surname. Notable people with the surname include: *Eberhard Hopf (1902–1983), Austrian mathematician *Hans Hopf (1916–1993), German tenor *Heinz Hopf (1894–1971), German mathematician *Heinz Hopf (actor) (1934–2001), Swedish actor *Ludwig Hopf (1884–1939), German physicist *Maria Hopf Maria Hopf (13 September 1913 – 24 August 2008) was a pioneering archaeobotanist, based at the RGZM, Mainz. Career Hopf studied botany from 1941–44, receiving her doctorate in 1947 on the subject of soil microbes. She then worked in phyto ... (1914-2008), German botanist and archaeologist {{surname, Hopf German-language surnames ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primitive Element (co-algebra)
In algebra, a primitive element of a co-algebra ''C'' (over an element ''g'') is an element ''x'' that satisfies :\mu(x) = x \otimes g + g \otimes x where \mu is the co-multiplication and ''g'' is an element of ''C'' that maps to the multiplicative identity 1 of the base field under the co-unit (''g'' is called ''group-like''). If ''C'' is a bi-algebra In mathematics, a bialgebra over a Field (mathematics), field ''K'' is a vector space over ''K'' which is both a unital algebra, unital associative algebra and a coalgebra, counital coassociative coalgebra. The algebraic and coalgebraic structures ..., i.e., a co-algebra that is also an algebra (with certain compatibility conditions satisfied), then one usually takes ''g'' to be 1, the multiplicative identity of ''C''. The bi-algebra ''C'' is said to be primitively generated if it is generated by primitive elements (as an algebra). If ''C'' is a bi-algebra, then the set of primitive elements form a Lie algebra with the usual comm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclic Homology
In noncommutative geometry and related branches of mathematics, cyclic homology and cyclic cohomology are certain (co)homology theories for associative algebras which generalize the de Rham (co)homology of manifolds. These notions were independently introduced by Boris Tsygan (homology) and Alain Connes (cohomology) in the 1980s. These invariants have many interesting relationships with several older branches of mathematics, including de Rham theory, Hochschild (co)homology, group cohomology, and the K-theory. Contributors to the development of the theory include Max Karoubi, Yuri L. Daletskii, Boris Feigin, Jean-Luc Brylinski, Mariusz Wodzicki, Jean-Louis Loday, Victor Nistor, Daniel Quillen, Joachim Cuntz, Ryszard Nest, Ralf Meyer, and Michael Puschnigg. Hints about definition The first definition of the cyclic homology of a ring ''A'' over a field of characteristic zero, denoted :''HC''''n''(''A'') or ''H''''n''λ(''A''), proceeded by the means of the following explicit c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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