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Chiral Algebra
In mathematics, a chiral algebra is an algebraic structure introduced by as a rigorous version of the rather vague concept of a chiral algebra in physics. In Chiral Algebras, Beilinson and Drinfeld introduced the notion of chiral algebra, which based on the pseudo-tensor category of D-modules. They give a 'coordinate independent' notion of vertex algebras, which are based on formal power series. Chiral algebras on curves are essentially conformal vertex algebras. Definition A chiral algebra on a smooth algebraic curve X is a right D-module \mathcal, equipped with a D-module homomorphism \mu : \mathcal \boxtimes \mathcal(\infty \Delta) \rightarrow \Delta_! \mathcal on X^2 and with an embedding \Omega \hookrightarrow \mathcal, satisfying the following conditions * \mu = -\sigma_ \circ \mu \circ \sigma_ ( Skew-symmetry) * \mu_ = \mu_ + \mu_ (Jacobi identity) * The unit map is compatible with the homomorphism \mu_\Omega: \Omega \boxtimes \Omega (\infty \Delta) \rightarrow \Delta_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexander Beilinson
Alexander A. Beilinson (born 1957) is the David and Mary Winton Green University professor at the University of Chicago and works on mathematics. His research has spanned representation theory, algebraic geometry and mathematical physics. In 1999, Beilinson was awarded the Ostrowski Prize with Helmut Hofer. In 2017, he was elected to the National Academy of Sciences. In 2018, he received the Wolf Prize in Mathematics and in 2020 the Shaw Prize in Mathematics. Early life and education Beilinson was born in Moscow of mostly Russian descent while his paternal grandfather was Jewish. Nevertheless he was discriminated because of his Jewish surname, and was not admitted to Moscow State University. He went to Pedagogical Institute instead and transferred to Moscow State University when he was a third year student. Work In 1978, Beilinson published a paper on coherent sheaves and several problems in linear algebra. His two-page note in the journal ''Functional Analysis and Its ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Canonical Bundle
In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n over a field is the line bundle \,\!\Omega^n = \omega, which is the nth exterior power of the cotangent bundle \Omega on V. Over the complex numbers, it is the determinant bundle of the holomorphic cotangent bundle T^*V. Equivalently, it is the line bundle of holomorphic n-forms on V. This is the dualising object for Serre duality on V. It may equally well be considered as an invertible sheaf. The canonical class is the divisor class of a Cartier divisor K on V giving rise to the canonical bundle — it is an equivalence class for linear equivalence on V, and any divisor in it may be called a canonical divisor. An anticanonical divisor is any divisor −K with K canonical. The anticanonical bundle is the corresponding inverse bundle \omega^. When the anticanonical bundle of V is ample, V is called a Fano variety. The adjunction formula Suppose that X is a smooth variety and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe became the first president while Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance over concerns about competing with the '' American Journal of Mathematics''. The result was the ''Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influentia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chiral Lie Algebra
In algebra, a chiral Lie algebra is a D-module on a curve with a certain structure of Lie algebra. It is related to an \mathcal_2-algebra via the Riemann–Hilbert correspondence In mathematics, the term Riemann–Hilbert correspondence refers to the correspondence between regular singular flat connections on algebraic vector bundles and representations of the fundamental group, and more generally to one of several generali .... See also * Chiral algebra * Chiral homology References * Lie algebras {{algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chiral Homology
In mathematics, chiral homology, introduced by Alexander Beilinson and Vladimir Drinfeld, is, in their words, "a “quantum” version of (the algebra of functions on) the space of global horizontal sections of an affine \mathcal_X-scheme (i.e., the space of global solutions of a system of non-linear differential equations)." Jacob Lurie's topological chiral homology gives an analog for manifolds. See also * Ran space *Chiral Lie algebra In algebra, a chiral Lie algebra is a D-module on a curve with a certain structure of Lie algebra. It is related to an \mathcal_2-algebra via the Riemann–Hilbert correspondence In mathematics, the term Riemann–Hilbert correspondence refers to ... * Factorization homology References *{{cite book, last1=Beilinson, first1=Alexander, authorlink1=Alexander Beilinson, last2=Drinfeld, first2=Vladimir, authorlink2=Vladimir Drinfeld , title=Chiral algebras, date=2004, publisher=American Mathematical Society, isbn=0-8218-3528-9, chapter=Chapte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Factorization Algebra
In mathematics and mathematical physics, a factorization algebra is an algebraic structure first introduced by Beilinson and Drinfel'd in an algebro-geometric setting as a reformulation of chiral algebras, and also studied in a more general setting by Costello and Gwilliam to study quantum field theory. Definition Prefactorization algebras A factorization algebra is a prefactorization algebra satisfying some properties, similar to sheafs being a presheaf with extra conditions. If M is a topological space, a prefactorization algebra \mathcal of vector spaces on M is an assignment of vector spaces \mathcal(U) to open sets U of M, along with the following conditions on the assignment: * For each inclusion U \subset V, there's a linear map m_V^U: \mathcal(U) \rightarrow \mathcal(V) * There is a linear map m_V^: \mathcal(U_1)\otimes \cdots \otimes \mathcal(U_n) \rightarrow \mathcal(V) for each finite collection of open sets with each U_i \subset V and the U_i pairwise disjo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Translation (geometry)
In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure, shape or space by the same Distance geometry, distance in a given direction (geometry), direction. A translation can also be interpreted as the addition of a constant vector space, vector to every point, or as shifting the Origin (mathematics), origin of the coordinate system. In a Euclidean space, any translation is an isometry. As a function If \mathbf is a fixed vector, known as the ''translation vector'', and \mathbf is the initial position of some object, then the translation function T_ will work as T_(\mathbf)=\mathbf+\mathbf. If T is a translation, then the image (mathematics), image of a subset A under the function (mathematics), function T is the translate of A by T . The translate of A by T_ is often written as A+\mathbf . Application in classical physics In classical physics, translational motion is movement that changes the Position (geometry), positio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group (mathematics)
In mathematics, a group is a Set (mathematics), set with an Binary operation, operation that combines any two elements of the set to produce a third element within the same set and the following conditions must hold: the operation is Associative property, associative, it has an identity element, and every element of the set has an inverse element. For example, the integers with the addition, addition operation form a group. The concept of a group was elaborated for handling, in a unified way, many mathematical structures such as numbers, geometric shapes and polynomial roots. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics. In geometry, groups arise naturally in the study of symmetries and geometric transformations: The symmetries of an object form a group, called the symmetry group of the object, and the transformations of a given type form a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Victor Kac
Victor Gershevich (Grigorievich) Kac (; born 19 December 1943) is a Soviet and American mathematician at MIT, known for his work in representation theory. He co-discovered Kac–Moody algebras, and used the Weyl–Kac character formula for them to reprove the Macdonald identities. He classified the finite-dimensional simple Lie superalgebras, and found the Kac determinant formula for the Virasoro algebra. He is also known for the Kac–Weisfeiler conjectures with Boris Weisfeiler. Biography Kac studied mathematics at Moscow State University, receiving his MS in 1965 and his PhD in 1968. From 1968 to 1976, he held a teaching position at the Moscow Institute of Electronic Machine Building (MIEM). He left the Soviet Union in 1977, becoming an associate professor of mathematics at MIT. In 1981, he was promoted to full professor. Kac received a Sloan Fellowship and the Medal of the Collège de France, both in 1981, and a Guggenheim Fellowship in 1986. He received the Wigner ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richard Borcherds
Richard Ewen Borcherds (; born 29 November 1959) is a British mathematician currently working in quantum field theory. He is known for his work in lattices, group theory, and infinite-dimensional algebras,James Lepowsky"The Work of Richard Borcherds" ''Notices of the American Mathematical Society'', Volume 46, Number 1 (January 1999). for which he was awarded the Fields Medal in 1998. He is well known for his proof of monstrous moonshine using ideas from string theory. Early life and education Borcherds was born in Cape Town, South Africa, but the family moved to Birmingham in the United Kingdom when he was six months old. Borcherds was educated at King Edward's School, Birmingham. As a student, Borcherds won a gold medal, silver medal, and special prize in the International Mathematical Olympiad. He attended university at Trinity College, Cambridge, where he studied under John Horton Conway. Career After receiving his doctorate in 1985, Borcherds has held various alternati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category (mathematics)
In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose objects are sets and whose arrows are functions. ''Category theory'' is a branch of mathematics that seeks to generalize all of mathematics in terms of categories, independent of what their objects and arrows represent. Virtually every branch of modern mathematics can be described in terms of categories, and doing so often reveals deep insights and similarities between seemingly different areas of mathematics. As such, category theory provides an alternative foundation for mathematics to set theory and other proposed axiomatic foundations. In general, the objects and arrows may be abstract entities of any kind, and the n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sheaf (mathematics)
In mathematics, a sheaf (: sheaves) is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could be the ring of continuous functions defined on that open set. Such data are well-behaved in that they can be restricted to smaller open sets, and also the data assigned to an open set are equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original open set (intuitively, every datum is the sum of its constituent data). The field of mathematics that studies sheaves is called sheaf theory. Sheaves are understood conceptually as general and abstract objects. Their precise definition is rather technical. They are specifically defined as sheaves of sets or as sheaves of rings, for example, depending on the type of data assigned to the open sets. There are also maps (or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
