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36 relations: Affine action, Bruhat order, Cartan subalgebra, Coxeter group, Daya-Nand Verma, Dimension, Dimension (vector space), Direct sum of modules, Generalized flag variety, Generalized Verma module, Glossary of semisimple groups, Harish-Chandra isomorphism, Homomorphism, Infinitesimal character, Invariant differential operator, Irreducible representation, Isomorphism, Israel Gelfand, Joseph Bernstein, Kostant partition function, Lie algebra, Mathematics, Module (mathematics), Poincaré–Birkhoff–Witt theorem, Quotient group, Representation theory, Resolution (algebra), Root system, ScienceDirect, Semisimple Lie algebra, Surjective function, Tensor product of modules, Universal enveloping algebra, Weight (representation theory), Weyl group, Weyl module.
Affine action
Let W be the Weyl group of a semisimple Lie algebra \mathfrak (associate to fixed choice of a Cartan subalgebra \mathfrak).
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Bruhat order
In mathematics, the Bruhat order (also called strong order or strong Bruhat order or Chevalley order or Bruhat–Chevalley order or Chevalley–Bruhat order) is a partial order on the elements of a Coxeter group, that corresponds to the inclusion order on Schubert varieties.
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Cartan subalgebra
In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra \mathfrak of a Lie algebra \mathfrak that is self-normalising (if \in \mathfrak for all X \in \mathfrak, then Y \in \mathfrak).
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Coxeter group
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors).
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Daya-Nand Verma
Daya-Nand Verma (25 June 1933, Varanasi – 10 June 2012, Mumbai) was a mathematician at the Tata Institute of Fundamental Research during the period 1968-1993.
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Dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it.
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Dimension (vector space)
In mathematics, the dimension of a vector space V is the cardinality (i.e. the number of vectors) of a basis of V over its base field.
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Direct sum of modules
In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module.
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Generalized flag variety
In mathematics, a generalized flag variety (or simply flag variety) is a homogeneous space whose points are flags in a finite-dimensional vector space V over a field F. When F is the real or complex numbers, a generalized flag variety is a smooth or complex manifold, called a real or complex flag manifold.
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Generalized Verma module
In mathematics, generalized Verma modules are a generalization of a (true) Verma module, and are objects in the representation theory of Lie algebras.
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Glossary of semisimple groups
This is a glossary for the terminology applied in the mathematical theories of semisimple Lie groups.
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Harish-Chandra isomorphism
In mathematics, the Harish-Chandra isomorphism, introduced by, is an isomorphism of commutative rings constructed in the theory of Lie algebras.
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Homomorphism
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces).
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Infinitesimal character
In mathematics, the infinitesimal character of an irreducible representation ρ of a semisimple Lie group G on a vector space V is, roughly speaking, a mapping to scalars that encodes the process of first differentiating and then diagonalizing the representation.
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Invariant differential operator
In mathematics and theoretical physics, an invariant differential operator is a kind of mathematical map from some objects to an object of similar type.
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Irreducible representation
In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper subrepresentation (\rho|_W,W), W \subset V closed under the action of \. Every finite-dimensional unitary representation on a Hermitian vector space V is the direct sum of irreducible representations.
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Isomorphism
In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that can be reversed by an inverse morphism.
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Israel Gelfand
Israel Moiseevich Gelfand, also written Israïl Moyseyovich Gel'fand, or Izrail M. Gelfand (ישראל געלפֿאַנד, Изра́иль Моисе́евич Гельфа́нд; – 5 October 2009) was a prominent Soviet mathematician.
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Joseph Bernstein
Joseph Bernstein (sometimes spelled I. N. Bernshtein; יוס(י)ף נאומוביץ ברנשטיין; Иосиф Наумович Бернштейн, Iosif Naumovič Bernštejn; born 18 April 1945) is an Israeli mathematician working at Tel Aviv University.
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Kostant partition function
In representation theory, a branch of mathematics, the Kostant partition function, introduced by, of a root system \Delta is the number of ways one can represent a vector (weight) as a non-negative integer linear combination of the positive roots \Delta^+\subset\Delta.
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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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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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Module (mathematics)
In mathematics, a module is one of the fundamental algebraic structures used in abstract algebra.
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Poincaré–Birkhoff–Witt theorem
In mathematics, more specifically in abstract algebra, in the theory of Lie algebras, the Poincaré–Birkhoff–Witt theorem (or PBW theorem) is a result giving an explicit description of the universal enveloping algebra of a Lie algebra.
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Quotient group
A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves the group structure.
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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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Resolution (algebra)
In mathematics, and more specifically in homological algebra, a resolution (or left resolution; dually a coresolution or right resolution) is an exact sequence of modules (or, more generally, of objects of an abelian category), which is used to define invariants characterizing the structure of a specific module or object of this category.
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Root system
In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties.
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ScienceDirect
ScienceDirect is a website which provides subscription-based access to a large database of scientific and medical research.
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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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Surjective function
In mathematics, a function f from a set X to a set Y is surjective (or onto), or a surjection, if for every element y in the codomain Y of f there is at least one element x in the domain X of f such that f(x).
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Tensor product of modules
In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps (e.g. multiplication) to be carried out in terms of linear maps.
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Universal enveloping algebra
In mathematics, a universal enveloping algebra is the most general (unital, associative) algebra that contains all representations of a Lie algebra.
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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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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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Weyl module
In algebra, a Weyl module is a representation of a reductive algebraic group, introduced by and named after Hermann Weyl.
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Redirects here:
BGG resolution, Bgg resolution.
References
[1] https://en.wikipedia.org/wiki/Verma_module
