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Integrable Module
In algebra, an integrable module (or integrable representation) of a Kac–Moody algebra \mathfrak g (a certain infinite-dimensional Lie algebra) is a representation of \mathfrak g such that (1) it is a sum of weight spaces and (2) the Chevalley generators e_i, f_i of \mathfrak g are locally nilpotent. For example, the adjoint representation In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is \m ... of a Kac–Moody algebra is integrable. Notes References * Abstract algebra {{algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kac–Moody Algebra
In mathematics, a Kac–Moody algebra (named for Victor Kac and Robert Moody, who independently and simultaneously discovered them in 1968) is a Lie algebra, usually infinite-dimensional, that can be defined by generators and relations through a generalized Cartan matrix. These algebras form a generalization of finite-dimensional semisimple Lie algebras, and many properties related to the structure of a Lie algebra such as its root system, irreducible representations, and connection to flag manifolds have natural analogues in the Kac–Moody setting. A class of Kac–Moody algebras called affine Lie algebras is of particular importance in mathematics and theoretical physics, especially two-dimensional conformal field theory and the theory of exactly solvable models. Kac discovered an elegant proof of certain combinatorial identities, the Macdonald identities, which is based on the representation theory of affine Kac–Moody algebras. Howard Garland and James Lepowsky demonstrated ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infinite-dimensional Lie Algebra
In mathematics, especially in Lie theory, E''n'' is the Kac–Moody algebra whose Dynkin diagram is a bifurcating graph with three branches of length 1, 2 and ''k'', with . In some older books and papers, ''E''2 and ''E''4 are used as names for ''G''2 and ''F''4. Finite-dimensional Lie algebras The E''n'' group is similar to the A''n'' group, except the nth node is connected to the 3rd node. So the Cartan matrix appears similar, −1 above and below the diagonal, except for the last row and column, have −1 in the third row and column. The determinant of the Cartan matrix for E''n'' is . *E3 is another name for the Lie algebra ''A''1''A''2 of dimension 11, with Cartan determinant 6. *:\left \begin 2 & -1 & 0 \\ -1 & 2 & 0 \\ 0 & 0 & 2 \end\right /math> *E4 is another name for the Lie algebra ''A''4 of dimension 24, with Cartan determinant 5. *:\left \begin 2 & -1 & 0 & 0 \\ -1 & 2 & -1& 0 \\ 0 & -1 & 2 & -1 \\ 0 & 0 & -1 & 2 \end\right /math> *E5 is ano ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Representation Of A Lie Algebra
Representation may refer to: Law and politics *Representation (politics), political activities undertaken by elected representatives, as well as other theories ** Representative democracy, type of democracy in which elected officials represent a group of people * Representation in contract law, a pre-contractual statement that may (if untrue) result in liability for misrepresentation * Labor representation, or worker representation, the work of a union representative who represents and defends the interests of fellow labor union members * Legal representation, provided by a barrister, lawyer, or other advocate * Lobbying or interest representation, attempts to influence the actions, policies, or decisions of officials * " No taxation without representation", a 1700s slogan that summarized one of the American colonists' 27 colonial grievances in the Thirteen Colonies, which was one of the major causes of the American Revolution * Permanent representation, a type of diplomatic missio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locally Nilpotent Endomorphism
In mathematics, a mathematical object is said to satisfy a property locally, if the property is satisfied on some limited, immediate portions of the object (e.g., on some ''sufficiently small'' or ''arbitrarily small'' neighborhoods of points). Properties of a point on a function Perhaps the best-known example of the idea of locality lies in the concept of local minimum (or local maximum), which is a point in a function whose functional value is the smallest (resp., largest) within an immediate neighborhood of points. This is to be contrasted with the idea of global minimum (or global maximum), which corresponds to the minimum (resp., maximum) of the function across its entire domain. Properties of a single space A topological space is sometimes said to exhibit a property locally, if the property is exhibited "near" each point in one of the following ways: # Each point has a neighborhood exhibiting the property; # Each point has a neighborhood base of sets exhibiting the propert ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adjoint Representation Of A Lie Algebra
In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is \mathrm(n, \mathbb), the Lie group of real ''n''-by-''n'' invertible matrices, then the adjoint representation is the group homomorphism that sends an invertible ''n''-by-''n'' matrix g to an endomorphism of the vector space of all linear transformations of \mathbb^n defined by: x \mapsto g x g^ . For any Lie group, this natural representation is obtained by linearizing (i.e. taking the differential of) the action of ''G'' on itself by conjugation. The adjoint representation can be defined for linear algebraic groups over arbitrary fields. Definition Let ''G'' be a Lie group, and let :\Psi: G \to \operatorname(G) be the mapping , with Aut(''G'') the automorphism group of ''G'' and given by the inner automorphism (conjugatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integrable
In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first integrals, that its motion is confined to a submanifold of much smaller dimensionality than that of its phase space. Three features are often referred to as characterizing integrable systems: * the existence of a ''maximal'' set of conserved quantities (the usual defining property of complete integrability) * the existence of algebraic invariants, having a basis in algebraic geometry (a property known sometimes as algebraic integrability) * the explicit determination of solutions in an explicit functional form (not an intrinsic property, but something often referred to as solvability) Integrable systems may be seen as very different in qualitative character from more ''generic'' dynamical systems, which are more typically chaotic sys ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessment to form Cambridge University Press and Assessment under Queen Elizabeth II's approval in August 2021. With a global sales presence, publishing hubs, and offices in more than 40 countries, it published over 50,000 titles by authors from over 100 countries. Its publications include more than 420 academic journals, monographs, reference works, school and university textbooks, and English language teaching and learning publications. It also published Bibles, runs a bookshop in Cambridge, sells through Amazon, and has a conference venues business in Cambridge at the Pitt Building and the Sir Geoffrey Cass Sports and Social Centre. It also served as the King's Printer. Cambridge University Press, as part of the University of Cambridge, was a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
