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Affine Lie algebra

Index Affine Lie algebra

In mathematics, an affine Lie algebra is an infinite-dimensional Lie algebra that is constructed in a canonical fashion out of a finite-dimensional simple Lie algebra. [1]

39 relations: Algebraic character, Anomaly (physics), Automorphism, Cartan matrix, Chern class, Conformal field theory, Current algebra, Dedekind eta function, Dynkin diagram, Fibration, Heterotic string theory, Kac–Moody algebra, Killing form, Langlands program, Laurent series, Lie algebra, Loop algebra, Macdonald identities, Mathematics, Modular group, Normal order, Null vector, Outer automorphism group, Q-analog, Representation theory, Semidirect product, Semisimple Lie algebra, Simple Lie group, String theory, Theoretical physics, Theta function, Two-dimensional conformal field theory, Verma module, Victor Kac, Weight (representation theory), Wess–Zumino–Witten model, Weyl character formula, Weyl group, Worldsheet.

Algebraic character

Algebraic character is a formal expression attached to a module in representation theory of semisimple Lie algebras that generalizes the character of a finite-dimensional representation and is analogous to the Harish-Chandra character of the representations of semisimple Lie groups.

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Anomaly (physics)

In quantum physics an anomaly or quantum anomaly is the failure of a symmetry of a theory's classical action to be a symmetry of any regularization of the full quantum theory.

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Automorphism

In mathematics, an automorphism is an isomorphism from a mathematical object to itself.

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Cartan matrix

In mathematics, the term Cartan matrix has three meanings.

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Chern class

In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles.

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Conformal field theory

A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations.

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Current algebra

Certain commutation relations among the current density operators in quantum field theories define an infinite-dimensional Lie algebra called a current algebra.

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Dedekind eta function

In mathematics, the Dedekind eta function, named after Richard Dedekind, is a modular form of weight 1/2 and is a function defined on the upper half-plane of complex numbers, where the imaginary part is positive.

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Dynkin diagram

In the mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of graph with some edges doubled or tripled (drawn as a double or triple line).

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Fibration

In topology, a branch of mathematics, a fibration is a generalization of the notion of a fiber bundle.

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Heterotic string theory

In string theory, a heterotic string is a closed string (or loop) which is a hybrid ('heterotic') of a superstring and a bosonic string.

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Kac–Moody algebra

In mathematics, a Kac–Moody algebra (named for Victor Kac and Robert Moody, who independently discovered them) is a Lie algebra, usually infinite-dimensional, that can be defined by generators and relations through a generalized Cartan matrix.

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Killing form

In mathematics, the Killing form, named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras.

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Langlands program

In mathematics, the Langlands program is a web of far-reaching and influential conjectures about connections between number theory and geometry.

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Laurent series

In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree.

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Lie algebra

In mathematics, a Lie algebra (pronounced "Lee") is a vector space \mathfrak g together with a non-associative, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g; (x, y) \mapsto, called the Lie bracket, satisfying the Jacobi identity.

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Loop algebra

In mathematics, loop algebras are certain types of Lie algebras, of particular interest in theoretical physics.

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Macdonald identities

In mathematics, the Macdonald identities are some infinite product identities associated to affine root systems, introduced by.

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Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

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Modular group

In mathematics, the modular group is the projective special linear group PSL(2,Z) of 2 x 2 matrices with integer coefficients and unit determinant.

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Normal order

In quantum field theory a product of quantum fields, or equivalently their creation and annihilation operators, is usually said to be normal ordered (also called Wick order) when all creation operators are to the left of all annihilation operators in the product.

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Null vector

In mathematics, given a vector space X with an associated quadratic form q, written, a null vector or isotropic vector is a non-zero element x of X for which.

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Outer automorphism group

In mathematics, the outer automorphism group of a group,, is the quotient,, where is the automorphism group of and) is the subgroup consisting of inner automorphisms.

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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.

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Representation theory

Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures.

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Semidirect product

In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product.

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Semisimple Lie algebra

In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras, i.e., non-abelian Lie algebras \mathfrak g whose only ideals are and \mathfrak g itself.

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Simple Lie group

In group theory, a simple Lie group is a connected non-abelian Lie group G which does not have nontrivial connected normal subgroups.

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String theory

In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings.

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Theoretical physics

Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena.

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Theta function

In mathematics, theta functions are special functions of several complex variables.

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Two-dimensional conformal field theory

A two-dimensional conformal field theory is a quantum field theory on a Euclidean two-dimensional space, that is invariant under local conformal transformations.

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Verma module

Verma modules, named after Daya-Nand Verma, are objects in the representation theory of Lie algebras, a branch of mathematics.

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Victor Kac

Victor Gershevich (Grigorievich) Kac (Виктор Гершевич (Григорьевич) Кац; born 19 December 1943 in Buguruslan, Russia, USSR) is a Soviet and American mathematician at MIT, known for his work in representation theory.

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Weight (representation theory)

In the mathematical field of representation theory, a weight of an algebra A over a field F is an algebra homomorphism from A to F, or equivalently, a one-dimensional representation of A over F. It is the algebra analogue of a multiplicative character of a group.

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Wess–Zumino–Witten model

In theoretical physics and mathematics, a Wess–Zumino–Witten (WZW) model, also called a Wess–Zumino–Novikov–Witten model, is a type of two-dimensional conformal field theory named after Julius Wess, Bruno Zumino, Sergei Novikov and Edward Witten.

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Weyl character formula

In mathematics, the Weyl character formula in representation theory describes the characters of irreducible representations of compact Lie groups in terms of their highest weights.

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Weyl group

In mathematics, in particular the theory of Lie algebras, the Weyl group of a root system Φ is a subgroup of the isometry group of the root system.

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Worldsheet

In string theory, a worldsheet is a two-dimensional manifold which describes the embedding of a string in spacetime.

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Affine Kac-Moody algebra, Affine Kac–Moody algebra, Affine lie algebra, Euclidean Lie algebra, Twisted affine Dynkin diagram, Twisted affine Lie algebra, Twisted affine algebra, Untwisted affine algebra.

References

[1] https://en.wikipedia.org/wiki/Affine_Lie_algebra

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