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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. Verma modules can be used in the classification of irreducible representations of a complex semisimple Lie algebra. Specifically, although Verma modules themselves are infinite dimensional, quotients of them can be used to construct finite-dimensional representations with highest weight \lambda, where \lambda is dominant and integral. Their homomorphisms correspond to invariant differential operators over flag manifolds. Informal construction We can explain the idea of a Verma module as follows. Let \mathfrak be a semisimple Lie algebra (over \mathbb, for simplicity). Let \mathfrak be a fixed Cartan subalgebra of \mathfrak and let R be the associated root system. Let R^+ be a fixed set of positive roots. For each \alpha\in R^+, choose a nonzero element X_\alpha for the corresponding root space \mathfrak_\alpha and a nonzero element Y_\alpha in the root ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. The construction of Verma modules appears in his Ph.D. thesis as a student of Nathan Jacobson at Yale University.Arun RaD-N. VERMA (1933-2012): A MEMORYms.unimelb.edu.au Select publications *. *. *. *. References External links Math Genealogy ProjectOberwolfach Photo Collection D_n_Verma Daya-Nand Verma, On-Line Encyclopedia of Integer Sequences The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to the ... {{DEFAULTSORT:Verma, Daya-Nand 1933 births 2012 deaths Yale University alumni Scientists from Varanasi 20th-century Indian mathematicians ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Highest Weight Module
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. The importance of the concept, however, stems from its application to representations of Lie algebras and hence also to representations of algebraic and Lie groups. In this context, a weight of a representation is a generalization of the notion of an eigenvalue, and the corresponding eigenspace is called a weight space. Motivation and general concept Given a set ''S'' of n\times n matrices over the same field, each of which is diagonalizable, and any two of which commute, it is always possible to simultaneously diagonalize all of the elements of ''S''.In fact, given a set of commuting matrices over an algebraically closed field, they are simultaneously triangularizable, without needing to assume that they are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Weight
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. The importance of the concept, however, stems from its application to representations of Lie algebras and hence also to representations of algebraic and Lie groups. In this context, a weight of a representation is a generalization of the notion of an eigenvalue, and the corresponding eigenspace is called a weight space. Motivation and general concept Given a set ''S'' of n\times n matrices over the same field, each of which is diagonalizable, and any two of which commute, it is always possible to simultaneously diagonalize all of the elements of ''S''.In fact, given a set of commuting matrices over an algebraically closed field, they are simultaneously triangularizable, without needing to assume that they are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 nontrivial subrepresentation (\rho, _W,W), with W \subset V closed under the action of \. Every finite-dimensional unitary representation on a Hilbert space V is the direct sum of irreducible representations. Irreducible representations are always indecomposable (i.e. cannot be decomposed further into a direct sum of representations), but converse may not hold, e.g. the two-dimensional representation of the real numbers acting by upper triangular unipotent matrices is indecomposable but reducible. History Group representation theory was generalized by Richard Brauer from the 1940s to give modular representation theory, in which the matrix operators act on a vector space over a field K of arbitrary characteristic, rather than a vector space over the field of real numbers or o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word isomorphism is derived from the Ancient Greek: ἴσος ''isos'' "equal", and μορφή ''morphe'' "form" or "shape". The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties (excluding further information such as additional structure or names of objects). Thus isomorphic structures cannot be distinguished from the point of view of structure only, and may be identified. In mathematical jargon, one says that two objects are . An automorphism is an isomorphism from a structure to itself. An isomorphism between two structures is a canonical isomorphism (a canonical map that is an isomorphism) if there is only one isomorphism between the two structures (as it is the case for solutions of a univer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Submodule
In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of ''module'' generalizes also the notion of abelian group, since the abelian groups are exactly the modules over the ring of integers. Like a vector space, a module is an additive abelian group, and scalar multiplication is distributive over the operation of addition between elements of the ring or module and is compatible with the ring multiplication. Modules are very closely related to the representation theory of groups. They are also one of the central notions of commutative algebra and homological algebra, and are used widely in algebraic geometry and algebraic topology. Introduction and definition Motivation In a vector space, the set of scalars is a field and acts on the vectors by scalar multiplication, subject to certain axioms such as the distributive law. In a module, the scalars need only be a ring, so the module conce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 some of the group structure (the rest of the structure is "factored" out). For example, the cyclic group of addition modulo ''n'' can be obtained from the group of integers under addition by identifying elements that differ by a multiple of n and defining a group structure that operates on each such class (known as a congruence class) as a single entity. It is part of the mathematical field known as group theory. For a congruence relation on a group, the equivalence class of the identity element is always a normal subgroup of the original group, and the other equivalence classes are precisely the cosets of that normal subgroup. The resulting quotient is written G\,/\,N, where G is the original group and N is the normal subgroup. (This is pronounced G\bmod N, where \mbox is short for modulo.) Much of the importance o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" and () meaning "form" or "shape". However, the word was apparently introduced to mathematics due to a (mis)translation of German meaning "similar" to meaning "same". The term "homomorphism" appeared as early as 1892, when it was attributed to the German mathematician Felix Klein (1849–1925). Homomorphisms of vector spaces are also called linear maps, and their study is the subject of linear algebra. The concept of homomorphism has been generalized, under the name of morphism, to many other structures that either do not have an underlying set, or are not algebraic. This generalization is the starting point of category theory. A homomorphism may also be an isomorphism, an endomorphism, an automorphism, etc. (see below). Each of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surjective
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of its domain. It is not required that be unique; the function may map one or more elements of to the same element of . The term ''surjective'' and the related terms ''injective'' and ''bijective'' were introduced by Nicolas Bourbaki, a group of mainly French 20th-century mathematicians who, under this pseudonym, wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. The French word '' sur'' means ''over'' or ''above'', and relates to the fact that the image of the domain of a surjective function completely covers the function's codomain. Any function induces a surjection by restricting its codomain to the image of its domain. Every surjective function has a right inverse assuming the axiom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. Kostant used it to rewrite the Weyl character formula as a formula (the Kostant multiplicity formula) for the multiplicity of a weight of an irreducible representation of a semisimple Lie algebra. An alternative formula, that is more computationally efficient in some cases, is Freudenthal's formula. The Kostant partition function can also be defined for Kac–Moody algebras and has similar properties. An example Consider the A2 root systems, with positive roots \alpha_1, \alpha_2, and \alpha_3:=\alpha_1+\alpha_2. If an element \mu can be expressed as a non-negative integer linear combination of \alpha_1, \alpha_2, and \alpha_3, then since \alpha_3=\alpha_1+\alpha_2, it can also be expressed as a non-negative intege ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Positive Root
In mathematics, a root system is a configuration of vector space, vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representation theory of semisimple Lie algebras. Since Lie groups (and some analogues such as algebraic groups) and Lie algebras have become important in many parts of mathematics during the twentieth century, the apparently special nature of root systems belies the number of areas in which they are applied. Further, the classification scheme for root systems, by Dynkin diagrams, occurs in parts of mathematics with no overt connection to Lie theory (such as singularity theory). Finally, root systems are important for their own sake, as in spectral graph theory. Definitions and examples As a first example, consider the six vectors in 2-dimensional Euclidean space, R2, as shown in the image at the right; call them roots. These vectors Li ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weight Space (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. The importance of the concept, however, stems from its application to representations of Lie algebras and hence also to representations of algebraic and Lie groups. In this context, a weight of a representation is a generalization of the notion of an eigenvalue, and the corresponding eigenspace is called a weight space. Motivation and general concept Given a set ''S'' of n\times n matrices over the same field, each of which is diagonalizable, and any two of which commute, it is always possible to simultaneously diagonalize all of the elements of ''S''.In fact, given a set of commuting matrices over an algebraically closed field, they are simultaneously triangularizable, without needing to assume that they are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
