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Ado's Theorem
In abstract algebra, Ado's theorem is a theorem characterizing finite-dimensional Lie algebras. Statement Ado's theorem states that every finite-dimensional Lie algebra ''L'' over a field ''K'' of characteristic zero can be viewed as a Lie algebra of square matrices under the commutator bracket. More precisely, the theorem states that ''L'' has a linear representation ρ over ''K'', on a finite-dimensional vector space ''V'', that is a faithful representation, making ''L'' isomorphic to a subalgebra of the endomorphisms of ''V''. History The theorem was proved in 1935 by Igor Dmitrievich Ado of Kazan State University, a student of Nikolai Chebotaryov. The restriction on the characteristic was later removed by Kenkichi Iwasawa (see also the below Gerhard Hochschild paper for a proof). Implications While for the Lie algebras associated to classical groups there is nothing new in this, the general case is a deeper result. Applied to the real Lie algebra of a Lie group ''G' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathematics), modules, vector spaces, lattice (order), lattices, and algebra over a field, algebras over a field. The term ''abstract algebra'' was coined in the early 20th century to distinguish it from older parts of algebra, and more specifically from elementary algebra, the use of variable (mathematics), variables to represent numbers in computation and reasoning. The abstract perspective on algebra has become so fundamental to advanced mathematics that it is simply called "algebra", while the term "abstract algebra" is seldom used except in mathematical education, pedagogy. Algebraic structures, with their associated homomorphisms, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kenkichi Iwasawa
Kenkichi Iwasawa ( ''Iwasawa Kenkichi'', September 11, 1917 – October 26, 1998) was a Japanese mathematician who is known for his influence on algebraic number theory. Biography Iwasawa was born in Shinshuku-mura, a town near Kiryū, in Gunma Prefecture. He attended elementary school there, but later moved to Tokyo to attend Musashi High School. From 1937 to 1940 Iwasawa studied as an undergraduate at the University of Tokyo, after which he entered graduate school at the same institution and became an assistant in the Department of Mathematics. In 1945 he was awarded a Doctor of Science degree. However, this same year Iwasawa became sick with pleurisy, and was unable to return to his position at the university until April 1947. From 1949 to 1955 he worked as assistant professor at the University of Tokyo. In 1950, Iwasawa was invited to Cambridge, Massachusetts to give a lecture at the International Congress of Mathematicians on his method to study Dedekind zeta functions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Terence Tao
Terence Chi-Shen Tao (; born 17 July 1975) is an Australian-American mathematician, Fields medalist, and professor of mathematics at the University of California, Los Angeles (UCLA), where he holds the James and Carol Collins Chair in the College of Letters and Sciences. His research includes topics in harmonic analysis, partial differential equations, algebraic combinatorics, arithmetic combinatorics, geometric combinatorics, probability theory, compressed sensing and analytic number theory. Tao was born to Chinese immigrant parents and raised in Adelaide. Tao won the Fields Medal in 2006 and won the Royal Medal and Breakthrough Prize in Mathematics in 2014, and is a 2006 MacArthur Fellow. Tao has been the author or co-author of over three hundred research papers, and is widely regarded as one of the greatest living mathematicians. Life and career Family Tao's parents are first generation immigrants from Hong Kong to Australia.'' Wen Wei Po'', Page A4, 24 August ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nathan Jacobson
Nathan Jacobson (October 5, 1910 – December 5, 1999) was an American mathematician. Biography Born Nachman Arbiser in Warsaw, Jacobson emigrated to America with his family in 1918. He graduated from the University of Alabama in 1930 and was awarded a doctorate in mathematics from Princeton University in 1934. While working on his thesis, ''Non-commutative polynomials and cyclic algebras'', he was advised by Joseph Wedderburn. Jacobson taught and researched at Bryn Mawr College (1935–1936), the University of Chicago (1936–1937), the University of North Carolina at Chapel Hill (1937–1943), and Johns Hopkins University (1943–1947) before joining Yale University in 1947. He remained at Yale until his retirement. He was a member of the National Academy of Sciences and the American Academy of Arts and Sciences. He served as president of the American Mathematical Society from 1971 to 1973, and was awarded their highest honour, the Leroy P. Steele prize for lifetime achievemen ... [...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] |
Annals Of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the founding editor-in-chief. It was "intended to afford a medium for the presentation and analysis of any and all questions of interest or importance in pure and applied Mathematics, embracing especially all new and interesting discoveries in theoretical and practical astronomy, mechanical philosophy, and engineering". It was published in Des Moines, Iowa, and was the earliest American mathematics journal to be published continuously for more than a year or two. This incarnation of the journal ceased publication after its tenth year, in 1883, giving as an explanation Hendricks' declining health, but Hendricks made arrangements to have it taken over by new management, and it was continued from March 1884 as the ''Annals of Mathematics''. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Group
In mathematics, a matrix group is a group ''G'' consisting of invertible matrices over a specified field ''K'', with the operation of matrix multiplication. A linear group is a group that is isomorphic to a matrix group (that is, admitting a faithful, finite-dimensional representation over ''K''). Any finite group is linear, because it can be realized by permutation matrices using Cayley's theorem. Among infinite groups, linear groups form an interesting and tractable class. Examples of groups that are not linear include groups which are "too big" (for example, the group of permutations of an infinite set), or which exhibit some pathological behavior (for example, finitely generated infinite torsion groups). Definition and basic examples A group ''G'' is said to be ''linear'' if there exists a field ''K'', an integer ''d'' and an injective homomorphism from ''G'' to the general linear group GL''d''(''K'') (a faithful linear representation of dimension ''d'' over ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local Isomorphism
Local may refer to: Geography and transportation * Local (train), a train serving local traffic demand * Local, Missouri, a community in the United States Arts, entertainment, and media * ''Local'' (comics), a limited series comic book by Brian Wood and Ryan Kelly * ''Local'' (novel), a 2001 novel by Jaideep Varma * ''The Local'' (film), a 2008 action-drama film * ''The Local'', English-language news websites in several European countries Computing * .local, a network address component Mathematics * Local property, a property which occurs on ''sufficiently small'' or ''arbitrarily small'' neighborhoods of points * Local ring, type of ring in commutative algebra Other uses * Pub, a drinking establishment, known as a "local" to its regulars See also * * * Local group (other) * Locale (other) * Localism (other) * Locality (other) * Localization (other) * Locus (other) * Lokal (other) Lokal may refer to: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Group
In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additional properties it must have to be thought of as a "transformation" in the abstract sense, for instance multiplication and the taking of inverses (to allow division), or equivalently, the concept of addition and subtraction. Combining these two ideas, one obtains a continuous group where multiplying points and their inverses is continuous. If the multiplication and taking of inverses are smoothness, smooth (differentiable) as well, one obtains a Lie group. Lie groups provide a natural model for the concept of continuous symmetry, a celebrated example of which is the circle group. Rotating a circle is an example of a continuous symmetry. For an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classical Group
In mathematics, the classical groups are defined as the special linear groups over the reals \mathbb, the complex numbers \mathbb and the quaternions \mathbb together with special automorphism groups of Bilinear form#Symmetric, skew-symmetric and alternating forms, symmetric or Bilinear form#Symmetric, skew-symmetric and alternating forms, skew-symmetric bilinear forms and Sesquilinear form#Hermitian form, Hermitian or Sesquilinear form#Skew-Hermitian form, skew-Hermitian sesquilinear forms defined on real, complex and quaternionic finite-dimensional vector spaces. Of these, the complex classical Lie groups are four infinite families of Lie groups that together with the Simple_Lie_group#Exceptional_cases, exceptional groups exhaust the classification of simple Lie groups. The compact classical groups are compact real forms of the complex classical groups. The finite analogues of the classical groups are the classical groups of Lie type. The term "classical group" was coined by Her ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gerhard Hochschild
Gerhard Paul Hochschild (April 29, 1915 in Berlin – July 8, 2010 in El Cerrito, California) was a German-born American mathematician who worked on Lie groups, algebraic groups, homological algebra and algebraic number theory. Early life On April 29, 1915, Hochschild was born to a middle-class Jewish family in Berlin, Germany, the son of Lilli and Heinrich Hochschild. Hochschild had an older brother. His father was a patent attorney who had an engineering degree. After the rise of the National Socialist German Workers' Party in 1933, his father sent him to South Africa where he was able to enroll in school with funding from the Hochschild Family Foundation established by Berthold Hochschild, a cousin of his grandfather. Education In 1936, Hochschild earned a BS degree in mathematics from University of Cape Town in Union of South Africa. In 1937, Hochschild earned a MS degree in mathematics from University of Cape Town. In 1941, Hochschild earned his PhD in mathematics fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nikolai Chebotaryov
Nikolai Grigorievich Chebotaryov (often spelled Chebotarov or Chebotarev; ; ; – 2 July 1947) was a Soviet mathematician. He is best known for the Chebotaryov density theorem. He was a student of Dmitry Grave. Chebotaryov worked on the algebra of polynomials, in particular examining the distribution of the zeros. He also studied Galois theory and wrote a textbook on the subject titled ''Basic Galois Theory''. His ideas were used by Emil Artin to prove the Artin reciprocity law. He worked with his student Anatoly Dorodnov on a generalization of the quadrature of the lune, and proved the conjecture now known as the Chebotarev theorem on roots of unity. Early life Nikolai Chebotaryov was born on 15 June 1894 in Kamianets-Podilskyi, Russian Empire (now in Ukraine). He entered the department of physics and mathematics at Kiev University in 1912. In 1928, he became a professor at Kazan University Kazan Federal University (; ) is a public research university located in Kazan, Russ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
