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Leibniz Algebra
In mathematics, a (right) Leibniz algebra, named after Gottfried Wilhelm Leibniz, sometimes called a Loday algebra, after Jean-Louis Loday, is a module ''L'' over a commutative ring ''R'' with a bilinear product _ , _ The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...satisfying the Leibniz identity : a,bc] = ,c">,[b,c<_a>+__a,c.html" ;"title=",c.html" ;"title=",[b,c">,[b,c+ a,c">,c.html" ;"title=",[b,c">,[b,c+ a,cb \, In other words, right multiplication by any element ''c'' is a derivation (abstract algebra)">derivation. If in addition the bracket is alternating ([''a'', ''a''] = 0) then the Leibniz algebra is a Lie algebra. Indeed, in this case [''a'', ''b''] = −[''b'', ''a''] and the Leibniz's identity is equivalent t ... [...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] |
Gottfried Wilhelm Leibniz
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathematics. He wrote works on philosophy, theology, ethics, politics, law, history and philology. Leibniz also made major contributions to physics and technology, and anticipated notions that surfaced much later in probability theory, biology, medicine, geology, psychology, linguistics and computer science. In addition, he contributed to the field of library science: while serving as overseer of the Wolfenbüttel library in Germany, he devised a cataloging system that would have served as a guide for many of Europe's largest libraries. Leibniz's contributions to this vast array of subjects were scattered in various learned journals, in tens of thousands of letters and in unpublished manuscripts. He wrote in several languages, primaril ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jean-Louis Loday
Jean-Louis Loday (12 January 1946 – 6 June 2012) was a French mathematician who worked on cyclic homology and who introduced Leibniz algebras (sometimes called Loday algebras) and Zinbiel algebras. He occasionally used the pseudonym Guillaume William Zinbiel, formed by reversing the last name of Gottfried Wilhelm Leibniz. Education and career Loday studied at Lycée Louis-le-Grand and at École Normale Supérieure in Paris. He completed his Ph.D. at the University of Strasbourg in 1975 under the supervision of Max Karoubi, with a dissertation titled ''K-Théorie algébrique et représentations de groupes''. He went on to become a senior scientist at CNRS and a member of the Institute for Advanced Mathematical Research (IRMA) at the University of Strasbourg. Publications * * * * See also *Associahedron In mathematics, an associahedron is an -dimensional convex polytope in which each vertex corresponds to a way of correctly inserting opening and closing parent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Engel's Theorem
In representation theory, a branch of mathematics, Engel's theorem states that a finite-dimensional Lie algebra \mathfrak g is a nilpotent Lie algebra if and only if for each X \in \mathfrak g, the adjoint representation of a Lie algebra">adjoint map :\operatorname(X)\colon \mathfrak \to \mathfrak, given by \operatorname(X)(Y) = [X, Y], is a nilpotent endomorphism on \mathfrak; i.e., \operatorname(X)^k = 0 for some ''k''. It is a consequence of the theorem, also called Engel's theorem, which says that if a Lie algebra of matrices consists of nilpotent matrices, then the matrices can all be simultaneously brought to a strictly upper triangular form. Note that if we merely have a Lie algebra of matrices which is nilpotent ''as a Lie algebra'', then this conclusion does ''not'' follow (i.e. the naïve replacement in Lie's theorem of "solvable" with "nilpotent", and "upper triangular" with "strictly upper triangular", is false; this already fails for the one-dimensional Lie subalgebra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leibniz Homology
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathematics. He wrote works on philosophy, theology, ethics, politics, law, history and philology. Leibniz also made major contributions to physics and technology, and anticipated notions that surfaced much later in probability theory, biology, medicine, geology, psychology, linguistics and computer science. In addition, he contributed to the field of library science: while serving as overseer of the Wolfenbüttel library in Germany, he devised a cataloging system that would have served as a guide for many of Europe's largest libraries. Leibniz's contributions to this vast array of subjects were scattered in various learned journals, in tens of thousands of letters and in unpublished manuscripts. He wrote in several languages, primarily in Latin, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hochschild Homology
In mathematics, Hochschild homology (and cohomology) is a homology theory for associative algebras over rings. There is also a theory for Hochschild homology of certain functors. Hochschild cohomology was introduced by for algebras over a field, and extended to algebras over more general rings by . Definition of Hochschild homology of algebras Let ''k'' be a field, ''A'' an associative ''k''-algebra, and ''M'' an ''A''-bimodule. The enveloping algebra of ''A'' is the tensor product A^e=A\otimes A^o of ''A'' with its opposite algebra. Bimodules over ''A'' are essentially the same as modules over the enveloping algebra of ''A'', so in particular ''A'' and ''M'' can be considered as ''Ae''-modules. defined the Hochschild homology and cohomology group of ''A'' with coefficients in ''M'' in terms of the Tor functor and Ext functor by : HH_n(A,M) = \operatorname_n^(A, M) : HH^n(A,M) = \operatorname^n_(A, M) Hochschild complex Let ''k'' be a ring, ''A'' an associative ''k''-algebra t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zinbiel Algebra
In mathematics, a Zinbiel algebra or dual Leibniz algebra is a module over a commutative ring with a bilinear product satisfying the defining identity: :(a \circ b) \circ c = a \circ (b \circ c) + a \circ (c \circ b). Zinbiel algebras were introduced by . The name was proposed by Jean-Michel Lemaire as being "opposite" to Leibniz algebra. In any Zinbiel algebra, the symmetrised product :a \star b = a \circ b + b \circ a is associative. A Zinbiel algebra is the Koszul dual concept to a Leibniz algebra. The free Zinbiel algebra over ''V'' is the tensor algebra with product :(x_0 \otimes \cdots \otimes x_p) \circ (x_ \otimes \cdots \otimes x_) = x_0 \sum_ (x_1,\ldots,x_), where the sum is over all (p,q) shuffles Shuffling is a procedure used to randomize a deck of playing cards to provide an element of chance in card games. Shuffling is often followed by a cut, to help ensure that the shuffler has not manipulated the outcome. __TOC__ Techniques Ov .... References ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Koszul Duality
In mathematics, Koszul duality, named after the French mathematician Jean-Louis Koszul, is any of various kinds of dualities found in representation theory of Lie algebras, abstract algebras (semisimple algebra) and topology (e.g., equivariant cohomology). The prototype example, due to Joseph Bernstein, Israel Gelfand, and Sergei Gelfand, is the rough duality between the derived category of a symmetric algebra and that of an exterior algebra. The importance of the notion rests on the suspicion that Koszul duality seems quite ubiquitous in nature. Koszul duality for modules over Koszul algebras The simplest, and in a sense prototypical case of Koszul duality arises as follows: for a 1-dimensional vector space ''V'' over a field ''k'', with dual vector space V^*, the exterior algebra of ''V'' has two non-trivial components, namely :\bigwedge^1 V=V, \quad \bigwedge^0 V = k. This exterior algebra and the symmetric algebra of V^*, \operatorname(V^*), serve to build a two-step chain c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annales De L'Institut Fourier
The ''Annales de l'Institut Fourier'' is a French mathematical journal publishing papers in all fields of mathematics. It was established in 1949. The journal publishes one volume per year, consisting of six issues. The current editor-in-chief is Hervé Pajot. Articles are published either in English or in French. The journal is indexed in ''Mathematical Reviews'', ''Zentralblatt MATH'' and the Web of Science. According to the ''Journal Citation Reports'', the journal had a 2008 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a given journal, as ... of 0.804. 2008 Journal Citation Reports, Science Edition, Thomson Scientific, 2008. References External links * Mathematics journals Academic journals established in 1949 Multilingual journals Bimonthly journals Open access journa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
