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Jantzen Filtration
In representation theory, a Jantzen filtration is a filtration of a Verma module of a semisimple Lie algebra, or a Weyl module of a reductive algebraic group of positive characteristic. Jantzen filtrations were introduced by . Jantzen filtration for Verma modules If ''M''(λ) is a Verma module of a semisimple Lie algebra with highest weight λ, then the Janzen filtration is a decreasing filtration :M(\lambda)=M(\lambda)^0\supseteq M(\lambda)^1\supseteq M(\lambda)^2\supseteq\cdots. It has the following properties: *''M''(λ)1=''N''(λ), the unique maximal proper submodule of ''M''(λ) *The quotients ''M''(λ)''i''/''M''(λ)''i''+1 have non-degenerate contravariant bilinear forms. * The Jantzen sum formula holds: :\sum_\text(M(\lambda)^i) = \sum_\text(M(s_\alpha \cdot \lambda)) : where \text(\cdot) denotes the formal character. References * * *{{Citation , last1=Jantzen , first1=Jens Carsten , title=Moduln mit einem höchsten Gewicht , publisher=Springer-Verlag Spring ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations (for example, matrix addition, matrix multiplication). The theory of matrices and linear operators is well-understood, so representations of more abstract objects in terms of familiar linear algebra objects helps glean properties and sometimes simplify calculations on more abstract theories. The algebraic objects amenable to such a description include groups, associative algebras and Lie algebras. The most prominent of these (and historically the first) is the representation theory of groups, in which elements of a group are represented by invertible matrices in such a way that the group operation i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Filtration (mathematics)
In mathematics, a filtration \mathcal is an indexed family (S_i)_ of subobjects of a given algebraic structure S, with the index i running over some totally ordered index set I, subject to the condition that ::if i\leq j in I, then S_i\subseteq S_j. If the index i is the time parameter of some stochastic process, then the filtration can be interpreted as representing all historical but not future information available about the stochastic process, with the algebraic structure S_i gaining in complexity with time. Hence, a process that is adapted to a filtration \mathcal is also called non-anticipating, because it cannot "see into the future". Sometimes, as in a filtered algebra, there is instead the requirement that the S_i be subalgebras with respect to some operations (say, vector addition), but not with respect to other operations (say, multiplication) that satisfy only S_i \cdot S_j \subseteq S_, where the index set is the natural numbers; this is by analogy with a graded ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Semisimple Lie Algebra
In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals). Throughout the article, unless otherwise stated, a Lie algebra is a finite-dimensional Lie algebra over a field of characteristic 0. For such a Lie algebra \mathfrak g, if nonzero, the following conditions are equivalent: *\mathfrak g is semisimple; *the Killing form, κ(x,y) = tr(ad(''x'')ad(''y'')), is non-degenerate; *\mathfrak g has no non-zero abelian ideals; *\mathfrak g has no non-zero solvable ideals; * the radical (maximal solvable ideal) of \mathfrak g is zero. Significance The significance of semisimplicity comes firstly from the Levi decomposition, which states that every finite dimensional Lie algebra is the semidirect product of a solvable ideal (its radical) and a semisimple algebra. In particular, there is no nonzero Lie algebra that is both solvable and semisimple. Semisimple L ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weyl Module
In algebra, a Weyl module is a representation of a reductive algebraic group, introduced by and named after Hermann Weyl. In characteristic 0 these representations are irreducible, but in positive characteristic they can be reducible, and their decomposition into irreducible components can be hard to determine. See also *Borel–Weil–Bott theorem *Garnir relations In mathematics, the Garnir relations give a way of expressing a basis of the Specht modules ''V''λ in terms of standard polytabloids. Specht modules in terms of polytabloids Given a partition ''λ'' of ''n'', one has the Specht module ''V''λ. In ... Further reading * * *{{eom, id=Weyl_module, first=R., last= Dipper Representation theory Algebraic groups ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reductive Algebraic Group
In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direct sum of irreducible representations. Reductive groups include some of the most important groups in mathematics, such as the general linear group ''GL''(''n'') of invertible matrices, the special orthogonal group ''SO''(''n''), and the symplectic group ''Sp''(2''n''). Simple algebraic groups and (more generally) semisimple algebraic groups are reductive. Claude Chevalley showed that the classification of reductive groups is the same over any algebraically closed field. In particular, the simple algebraic groups are classified by Dynkin diagrams, as in the theory of compact Lie groups or complex semisimple Lie algebras. Reductive groups over an arbitrary field are harder to classify, but for many fields such as the real numbers R or a nu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Contravariant Form
Covariance and contravariance may refer to: * Covariance and contravariance of vectors, in mathematics and theoretical physics * Covariance and contravariance of functors, in category theory * Covariance and contravariance (computer science), whether a type system preserves the ordering ≤ of types See also * Covariance In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the ..., in probability theory and statistics, the measure of how much two random variables vary together * Covariance (other) {{Mathematical disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Character
An 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. Definition Let \mathfrak be a semisimple Lie algebra with a fixed Cartan subalgebra \mathfrak, and let the abelian group A=\mathbb \mathfrak^* consist of the (possibly infinite) formal integral linear combinations of e^, where \mu\in\mathfrak^*, the (complex) vector space of weights. Suppose that V is a locally-finite weight module. Then the algebraic character of V is an element of A defined by the formula: : ch(V)=\sum_\dim V_e^, where the sum is taken over all weight spaces of the module V. Example The algebraic character of the Verma module M_\lambda with the highest weight \lambda is given by the formula : ch(M_)=\frac, with the product taken over the set of positive roots. Properti ... [...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] |
Graduate Studies In Mathematics
Graduate Studies in Mathematics (GSM) is a series of graduate-level textbooks in mathematics published by the American Mathematical Society (AMS). The books in this series are published ihardcoverane-bookformats. List of books *1 ''The General Topology of Dynamical Systems'', Ethan Akin (1993, ) *2 ''Combinatorial Rigidity'', Jack Graver, Brigitte Servatius, Herman Servatius (1993, ) *3 ''An Introduction to Gröbner Bases'', William W. Adams, Philippe Loustaunau (1994, ) *4 ''The Integrals of Lebesgue, Denjoy, Perron, and Henstock'', Russell A. Gordon (1994, ) *5 ''Algebraic Curves and Riemann Surfaces'', Rick Miranda (1995, ) *6 ''Lectures on Quantum Groups'', Jens Carsten Jantzen (1996, ) *7 ''Algebraic Number Fields'', Gerald J. Janusz (1996, 2nd ed., ) *8 ''Discovering Modern Set Theory. I: The Basics'', Winfried Just, Martin Weese (1996, ) *9 ''An Invitation to Arithmetic Geometry'', Dino Lorenzini (1996, ) *10 ''Representations of Finite and Compact Groups'', Barry Simon (199 ... [...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] |
Lie Algebras
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an 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 ,y/math>. The vector space \mathfrak g together with this operation is a non-associative algebra, meaning that the Lie bracket is not necessarily associative. Lie algebras are closely related to Lie groups, which are 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 Lie group unique up to finite coverings (Lie's third theorem). This correspondence allows one to study the structure and classification of Lie groups in terms of Lie algebras. In physics, Lie groups appear as symmetry groups of ph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
