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Noncommutative Symmetric Function
In mathematics, the noncommutative symmetric functions form a Hopf algebra NSymm analogous to the Hopf algebra of symmetric functions. The Hopf algebra NSymm was introduced by Israel M. Gelfand, Daniel Krob, Alain Lascoux, Bernard Leclerc, Vladimir Retakh, and Jean-Yves Thibon. It is noncommutative but cocommutative graded Hopf algebra. It has the Hopf algebra of symmetric functions as a quotient, and is a subalgebra of the Hopf algebra of permutations, and is the graded dual of the Hopf algebra of quasisymmetric function. Over the rational numbers it is isomorphic as a Hopf algebra to the universal enveloping algebra of the free Lie algebra on countably many variables. Definition The underlying algebra of the Hopf algebra of noncommutative symmetric functions is the free ring Z⟨''Z''1, ''Z''2,...⟩ generated by non-commuting variables ''Z''1, ''Z''2, ... The coproduct takes ''Z''''n'' to Σ ''Z''''i'' ⊗ ''Z''''n''–''i'', where ''Z' ... [...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 (1914-2008), German botanist and archaeologist {{surname, Hopf German-language surnames ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hopf Algebra Of Symmetric Functions
In algebra and in particular in algebraic combinatorics, the ring of symmetric functions is a specific limit of the rings of symmetric polynomials in ''n'' indeterminates, as ''n'' goes to infinity. This ring serves as universal structure in which relations between symmetric polynomials can be expressed in a way independent of the number ''n'' of indeterminates (but its elements are neither polynomials nor functions). Among other things, this ring plays an important role in the representation theory of the symmetric group. The ring of symmetric functions can be given a coproduct and a bilinear form making it into a positive selfadjoint graded Hopf algebra that is both commutative and cocommutative. Symmetric polynomials The study of symmetric functions is based on that of symmetric polynomials. In a polynomial ring in some finite set of indeterminates, a polynomial is called ''symmetric'' if it stays the same whenever the indeterminates are permuted in any way. More formally, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Israel M
Israel (; he, יִשְׂרָאֵל, ; ar, إِسْرَائِيل, ), officially the State of Israel ( he, מְדִינַת יִשְׂרָאֵל, label=none, translit=Medīnat Yīsrāʾēl; ), is a country in Western Asia. It is situated on the southeastern shore of the Mediterranean Sea and the northern shore of the Red Sea, and shares borders with Lebanon to the north, Syria to the northeast, Jordan to the east, and Egypt to the southwest. Israel also is bordered by the Palestinian territories of the West Bank and the Gaza Strip to the east and west, respectively. Tel Aviv is the economic and technological center of the country, while its seat of government is in its proclaimed capital of Jerusalem, although Israeli sovereignty over East Jerusalem is unrecognized internationally. The land held by present-day Israel witnessed some of the earliest human occupations outside Africa and was among the earliest known sites of agriculture. It was inhabited by the Canaanites ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alain Lascoux
Alain Lascoux (17 October 1944 – 20 October 2013) was a French mathematician at the University of Marne la Vallée and Nankai University. His research was primarily in algebraic combinatorics, particularly Hecke algebras and Young tableaux. Lascoux earned his doctorate in 1977 from the University of Paris. He worked for twenty years with Marcel-Paul Schützenberger on properties of the symmetric group. They wrote many articles together and had a major impact on the development of algebraic combinatorics. They succeeded in giving a combinatorial understanding of various algebraic and geometric questions in representation theory. Thus they introduced many new objects related to both fields like Schubert polynomials and Grothendieck polynomials. They were also the first to define the crystal graph structure on Young tableaux (though not under this name). Lascoux was an invited speaker at the 1998 International Congress of Mathematicians in Berlin, Germany. See also *LLT ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vladimir Retakh
Vladimir Solomonovich Retakh (russian: Ретах Владимир Соломонович; 20 May 1948) is a Russian-American mathematician who made important contributions to Noncommutative algebra and combinatorics among other areas. Biography Retakh graduated in 1970 from the Moscow State Pedagogical University. Beginning as an undergraduate Retakh regularly attended lectures and seminars at the Moscow State University most notably the Gelfand seminars. He obtained his PhD in 1973 under the mentorship of Dmitrii Abramovich Raikov. He joined the Gelfand group in 1986. His first position was at the central Research Institute for Engineering Buildings and later obtained his first academic position at the Council for Cybernetics of the Soviet Academy of Sciences in 1989. While at the Council for Cybernetics of the Soviet Academy of Sciences in 1990, Retakh had started working with Gelfand on their new program on Noncommutative determinants. Prior to immigrating to the US ... [...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: * [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |