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Moss Sweedler
Moss Eisenberg Sweedler (born 29 April 1942, in Brooklyn) is an American mathematician, known for Sweedler's Hopf algebra, Sweedler's notation, measuring coalgebras, and his proof, with Harry Prince Allen, of a conjecture of Nathan Jacobson. Education and career Sweedler received his Ph.D. from the Massachusetts Institute of Technology in 1965. His thesis, ''Commutative Hopf Algebras with Antipode,'' was written under the direction of thesis advisor Bertram Kostant. Sweedler wrote ''Hopf Algebras'' (1969), which became the standard reference book on Hopf algebras. He, with Harry P. Allen, used Hopf algebras to prove in 1969 a famous 25-year-old conjecture of Jacobson about the forms of generalized Witt algebras over algebraically closed fields of finite characteristic. From 1965 to the mid 1980s Sweeder worked on commutative algebra and related disciplines. Since the mid 1980s Sweedler has worked primarily on computer algebra. His research resulted in his position as director of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brooklyn
Brooklyn () is a borough of New York City, coextensive with Kings County, in the U.S. state of New York. Kings County is the most populous county in the State of New York, and the second-most densely populated county in the United States, behind New York County (Manhattan). Brooklyn is also New York City's most populous borough,2010 Gazetteer for New York State . Retrieved September 18, 2016. with 2,736,074 residents in 2020. Named after the Dutch village of Breukelen, Brooklyn is located on the w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sweedler's Hopf Algebra
In mathematics, introduced an example of an infinite-dimensional Hopf algebra, and Sweedler's Hopf algebra ''H''4 is a certain 4-dimensional quotient of it that is neither commutative nor cocommutative. Definition The following infinite dimensional Hopf algebra was introduced by . The Hopf algebra is generated as an algebra by three elements ''x'', ''g'' and ''g''-1. The coproduct Δ is given by :Δ(g) = ''g'' ⊗''g'', Δ(''x'') = 1⊗''x'' + ''x'' ⊗''g'' The antipode ''S'' is given by :''S''(''x'') = –''x'' ''g''−1, ''S''(''g'') = ''g''−1 The counit ε is given by :ε(''x'')=0, ε(''g'') = 1 Sweedler's 4-dimensional Hopf algebra ''H''4 is the quotient of this by the relations :''x''2 = 0, ''g''2 = 1, ''gx'' = –''xg'' so it has a basis 1, ''x'', ''g'', ''xg'' . Note that Montgomery describes a slight variant of this Hopf algebra using the opposite coproduct, i.e. the coproduct described above composed with the tensor flip on ''H''4⊗''H''4. This Hopf algebr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coalgebra
In mathematics, coalgebras or cogebras are structures that are dual (in the category-theoretic sense of reversing arrows) to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagrams. Turning all arrows around, one obtains the axioms of coalgebras. Every coalgebra, by (vector space) duality, gives rise to an algebra, but not in general the other way. In finite dimensions, this duality goes in both directions ( see below). Coalgebras occur naturally in a number of contexts (for example, representation theory, universal enveloping algebras and group schemes). There are also F-coalgebras, with important applications in computer science. Informal discussion One frequently recurring example of coalgebras occurs in representation theory, and in particular, in the representation theory of the rotation group. A primary task, of practical use in physics, is to obtain combinations of systems with different states of angu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measuring Coalgebra
In algebra, a measuring coalgebra of two algebras ''A'' and ''B'' is a coalgebra In mathematics, coalgebras or cogebras are structures that are dual (in the category-theoretic sense of reversing arrows) to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagrams ... enrichment of the set of homomorphisms from ''A'' to ''B''. In other words, if coalgebras are thought of as a sort of linear analogue of sets, then the measuring coalgebra is a sort of linear analogue of the set of homomorphisms from ''A'' to ''B''. In particular its group-like elements are (essentially) the homomorphisms from ''A'' to ''B''. Measuring coalgebras were introduced by . Definition A coalgebra ''C'' with a linear map from ''C''×''A'' to ''B'' is said to measure ''A'' to ''B'' if it preserves the algebra product and identity (in the coalgebra sense). If we think of the elements of ''C'' as linear maps from ''A'' to ''B'', this means that ... [...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] |
Massachusetts Institute Of Technology
The Massachusetts Institute of Technology (MIT) is a private land-grant research university in Cambridge, Massachusetts. Established in 1861, MIT has played a key role in the development of modern technology and science, and is one of the most prestigious and highly ranked academic institutions in the world. Founded in response to the increasing industrialization of the United States, MIT adopted a European polytechnic university model and stressed laboratory instruction in applied science and engineering. MIT is one of three private land grant universities in the United States, the others being Cornell University and Tuskegee University. The institute has an urban campus that extends more than a mile (1.6 km) alongside the Charles River, and encompasses a number of major off-campus facilities such as the MIT Lincoln Laboratory, the Bates Center, and the Haystack Observatory, as well as affiliated laboratories such as the Broad and Whitehead Institutes. , 98 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bertram Kostant
Bertram Kostant (May 24, 1928 – February 2, 2017) was an American mathematician who worked in representation theory, differential geometry, and mathematical physics. Early life and education Kostant grew up in New York City, where he graduated from Stuyvesant High School in 1945. He went on to obtain an undergraduate degree in mathematics from Purdue University in 1950. He earned his Ph.D. from the University of Chicago in 1954, under the direction of Irving Segal, where he wrote a dissertation on representations of Lie groups. Career in mathematics After time at the Institute for Advanced Study, Princeton University, and the University of California, Berkeley, he joined the faculty at the Massachusetts Institute of Technology, where he remained until his retirement in 1993. Kostant's work has involved representation theory, Lie groups, Lie algebras, homogeneous spaces, differential geometry and mathematical physics, particularly symplectic geometry. He has given several l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computer Algebra
In mathematics and computer science, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the study and development of algorithms and software for manipulating mathematical expressions and other mathematical objects. Although computer algebra could be considered a subfield of scientific computing, they are generally considered as distinct fields because scientific computing is usually based on numerical computation with approximate floating point numbers, while symbolic computation emphasizes ''exact'' computation with expressions containing variables that have no given value and are manipulated as symbols. Software applications that perform symbolic calculations are called ''computer algebra systems'', with the term ''system'' alluding to the complexity of the main applications that include, at least, a method to represent mathematical data in a computer, a user programming language (usually different from the lang ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sustainment Center Of Excellence
The U.S. Army Sustainment Center of Excellence (SCoE) ronounced sko or sko-eis a subordinate organization under the Combined Arms Support Command (CASCOM) at Fort Lee, Virginia. SCoE definition SCoE is way of describing CASCOM Headquarters oversight of one of several multi-branch, multi-school Centers of Excellence (CoE) within the United States Army Training and Doctrine Command (TRADOC). TRADOC Centers of Excellence (CoEs) came about as a result of a 2005 Base Realignment and Closure (BRAC) Commission recommendation. According to TRADOC a CoE is, "a premier organization that creates the highest standards of achievement in an assigned sphere of expertise by generating synergy through effective and efficient combination and integration of functions while reinforcing the unique requirements and capabilities of the branches". CoEs with a similar structure to the SCoE are Fires (Fort Sill, OK), Maneuver (Fort Benning, GA) and Maneuver Support (Fort Leonard Wood, MO). SCoE h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
International Congress Of Mathematicians
The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU). The Fields Medals, the Nevanlinna Prize (to be renamed as the IMU Abacus Medal), the Gauss Prize, and the Chern Medal are awarded during the congress's opening ceremony. Each congress is memorialized by a printed set of Proceedings recording academic papers based on invited talks intended to be relevant to current topics of general interest. Being invited to talk at the ICM has been called "the equivalent ... of an induction to a hall of fame". History Felix Klein and Georg Cantor are credited with putting forward the idea of an international congress of mathematicians in the 1890s.A. John Coleman"Mathematics without borders": a book review ''CMS Notes'', vol 31, no. 3, April 1999, pp. 3-5 The University of Chicago, which had opened in 1892, organized an International Mathema ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
21st-century American Mathematicians
The 1st century was the century spanning AD 1 ( I) through AD 100 ( C) according to the Julian calendar. It is often written as the or to distinguish it from the 1st century BC (or BCE) which preceded it. The 1st century is considered part of the Classical era, epoch, or historical period. The 1st century also saw the appearance of Christianity. During this period, Europe, North Africa and the Near East fell under increasing domination by the Roman Empire, which continued expanding, most notably conquering Britain under the emperor Claudius (AD 43). The reforms introduced by Augustus during his long reign stabilized the empire after the turmoil of the previous century's civil wars. Later in the century the Julio-Claudian dynasty, which had been founded by Augustus, came to an end with the suicide of Nero in AD 68. There followed the famous Year of Four Emperors, a brief period of civil war and instability, which was finally brought to an end by Vespasian, ninth Roman emper ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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