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Quantum Enveloping Algebra
In mathematics, a quantum or quantized enveloping algebra is a ''q''-analog of a universal enveloping algebra. Given a Lie algebra \mathfrak, the quantum enveloping algebra is typically denoted as U_q(\mathfrak). The notation was introduced by Drinfeld and independently by Jimbo. Among the applications, studying the q \to 0 limit led to the discovery of crystal bases. The case of \mathfrak_2 Michio Jimbo considered the algebras with three generators related by the three commutators : ,e= 2e,\ ,f= -2f,\ ,f= \sinh(\eta h)/\sinh \eta. When \eta \to 0, these reduce to the commutators that define the special linear Lie algebra \mathfrak_2. In contrast, for nonzero \eta, the algebra defined by these relations is not a Lie algebra but instead an associative algebra that can be regarded as a deformation of the universal enveloping algebra of \mathfrak_2. See also *quantum group References * * External links Quantized enveloping algebraat the nLab Quantized enveloping alge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Q-analog
In mathematics, a ''q''-analog of a theorem, identity or expression is a generalization involving a new parameter ''q'' that returns the original theorem, identity or expression in the limit as . Typically, mathematicians are interested in ''q''-analogs that arise naturally, rather than in arbitrarily contriving ''q''-analogs of known results. The earliest ''q''-analog studied in detail is the basic hypergeometric series, which was introduced in the 19th century.Exton, H. (1983), ''q-Hypergeometric Functions and Applications'', New York: Halstead Press, Chichester: Ellis Horwood, 1983, , , ''q''-analogues are most frequently studied in the mathematical fields of combinatorics and special functions. In these settings, the limit is often formal, as is often discrete-valued (for example, it may represent a prime power). ''q''-analogs find applications in a number of areas, including the study of fractals and multi-fractal measures, and expressions for the entropy of chaotic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Universal Enveloping Algebra
In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra. Universal enveloping algebras are used in the representation theory of Lie groups and Lie algebras. For example, Verma modules can be constructed as quotients of the universal enveloping algebra. In addition, the enveloping algebra gives a precise definition for the Casimir operators. Because Casimir operators commute with all elements of a Lie algebra, they can be used to classify representations. The precise definition also allows the importation of Casimir operators into other areas of mathematics, specifically, those that have a differential algebra. They also play a central role in some recent developments in mathematics. In particular, their dual provides a commutative example of the objects studied in non-commutative geometry, the quantum groups. This dual can be shown, by the Gelfand–N ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graduate Texts In Mathematics
Graduate Texts in Mathematics (GTM) (ISSN 0072-5285) is a series of graduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are yellow books of a standard size (with variable numbers of pages). The GTM series is easily identified by a white band at the top of the book. The books in this series tend to be written at a more advanced level than the similar Undergraduate Texts in Mathematics series, although there is a fair amount of overlap between the two series in terms of material covered and difficulty level. List of books #''Introduction to Axiomatic Set Theory'', Gaisi Takeuti, Wilson M. Zaring (1982, 2nd ed., ) #''Measure and Category – A Survey of the Analogies between Topological and Measure Spaces'', John C. Oxtoby (1980, 2nd ed., ) #''Topological Vector Spaces'', H. H. Schaefer, M. P. Wolff (1999, 2nd ed., ) #''A Course in Homological Algebra'', Peter Hilton, Urs Stammbac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identity. The Lie bracket of two vectors x and y is denoted [x,y]. The vector space \mathfrak g together with this operation is a non-associative algebra, meaning that the Lie bracket is not necessarily associative property, associative. Lie algebras are closely related to Lie groups, which are group (mathematics), groups that are also smooth manifolds: any Lie group gives rise to a Lie algebra, which is its tangent space at the identity. Conversely, to any finite-dimensional Lie algebra over real or complex numbers, there is a corresponding connected space, connected Lie group unique up to finite coverings (Lie's third theorem). This Lie group–Lie algebra correspondence, correspondence allows one ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Crystal Base
A crystal base for a representation of a quantum group on a \Q(v)-vector space is not a base of that vector space but rather a \Q-base of L/vL where L is a \Q(v)-lattice in that vector spaces. Crystal bases appeared in the work of and also in the work of . They can be viewed as specializations as v \to 0 of the canonical basis defined by . Definition As a consequence of its defining relations, the quantum group U_q(G) can be regarded as a Hopf algebra over the field of all rational functions of an indeterminate ''q'' over \Q, denoted \Q(q). For simple root \alpha_i and non-negative integer n, define :\begin e_i^ = f_i^ &= 1 \\ e_i^ &= \frac \\ ptf_i^ &= \frac \end In an integrable module M, and for weight \lambda, a vector u \in M_ (i.e. a vector u in M with weight \lambda) can be uniquely decomposed into the sums :u = \sum_^\infty f_i^ u_n = \sum_^\infty e_i^ v_n, where u_n \in \ker(e_i) \cap M_, v_n \in \ker(f_i) \cap M_, u_n \ne 0 only if n + \frac \ge 0, and v_n \ne 0 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michio Jimbo
is a Japanese mathematician working in mathematical physics and is a professor of mathematics at Rikkyo University. He is a grandson of the linguist . Career After graduating from the University of Tokyo in 1974, he studied under Mikio Sato at the Research Institute for Mathematical Sciences in Kyoto University. He has made important contributions to mathematical physics, including (independently of Vladimir Drinfeld) the initial development of the study of quantum groups, the development of the theory of \tau-functions for the KP ( Kadomtsev–Petviashvili) integrable hierarchy, and other related integrable hierarchies , E. Date, M. Jimbo, M. Kashiwara and T. Miwa, "Operator approach to the Kadomtsev-Petviashvili equation III". ''J. Phys. Soc. Jap.'' 50 (11): 3806–3812 (1981). doi:10.1143/JPSJ.50.3806. M. Jimbo and T. Miwa, "Solitons and infinite-dimensional Lie algebras", ''Publ. Res. Inst. Math. Sci.'', 19(3):943–1001 (1983). and development of the theory of isomonodromi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Special Linear Lie Algebra
In mathematics, the special linear Lie algebra of order n (denoted \mathfrak_n(F) or \mathfrak(n, F)) is the Lie algebra of n \times n matrices with trace zero and with the Lie bracket ,Y=XY-YX. This algebra is well studied and understood, and is often used as a model for the study of other Lie algebras. The Lie group that it generates is the special linear group. Applications The Lie algebra \mathfrak_2(\mathbb) is central to the study of special relativity, general relativity and supersymmetry: its fundamental representation is the so-called spinor representation, while its adjoint representation generates the Lorentz group SO(3,1) of special relativity. The algebra \mathfrak_2(\mathbb) plays an important role in the study of chaos and fractals, as it generates the Möbius group SL(2,R), which describes the automorphisms of the hyperbolic plane, the simplest Riemann surface of negative curvature; by contrast, SL(2,C) describes the automorphisms of the hyperbolic 3-dimensional ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Associative Algebra
In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplication operations together give ''A'' the structure of a ring; the addition and scalar multiplication operations together give ''A'' the structure of a vector space over ''K''. In this article we will also use the term ''K''-algebra to mean an associative algebra over the field ''K''. A standard first example of a ''K''-algebra is a ring of square matrices over a field ''K'', with the usual matrix multiplication. A commutative algebra is an associative algebra that has a commutative multiplication, or, equivalently, an associative algebra that is also a commutative ring. In this article associative algebras are assumed to have a multiplicative identity, denoted 1; they are sometimes called unital associative algebras for clarification. I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Letters In Mathematical Physics
''Letters in Mathematical Physics'' is a peer-reviewed scientific journal in mathematical physics published by Springer Science+Business Media. It publishes letters and longer research articles, occasionally also articles containing topical reviews. It is essentially a platform for the rapid dissemination of short contributions in the field of mathematical physics. In addition, the journal publishes contributions to modern mathematics in fields which have a potential physical application, and developments in theoretical physics which have potential mathematical impact. The editors are Volker Bach, Edward Frenkel, Maxim Kontsevich, Dirk Kreimer, Nikita Nekrasov, Massimo Porrati, and Daniel Sternheimer. Abstracting and indexing The following services abstract or index ''Letters in Mathematical Physics'': Academic OneFile, Academic Search, Astrophysics Data System, Chemical Abstracts Service, Current Contents/Physical, Chemical and Earth Sciences, Current Index to Statistics, EBSC ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Group
In mathematics and theoretical physics, the term quantum group denotes one of a few different kinds of noncommutative algebras with additional structure. These include Drinfeld–Jimbo type quantum groups (which are quasitriangular Hopf algebras), compact matrix quantum groups (which are structures on unital separable C*-algebras), and bicrossproduct quantum groups. Despite their name, they do not themselves have a natural group structure, though they are in some sense 'close' to a group. The term "quantum group" first appeared in the theory of quantum integrable systems, which was then formalized by Vladimir Drinfeld and Michio Jimbo as a particular class of Hopf algebra. The same term is also used for other Hopf algebras that deform or are close to classical Lie groups or Lie algebras, such as a "bicrossproduct" class of quantum groups introduced by Shahn Majid a little after the work of Drinfeld and Jimbo. In Drinfeld's approach, quantum groups arise as Hopf algebras depe ... [...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 was the first president and Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance, due to 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 influential in in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
