|
Zhu Algebra
In mathematics, the Zhu algebra and the closely related C2-algebra, introduced by Yongchang Zhu in his PhD thesis, are two associative algebras canonically constructed from a given vertex operator algebra. Many important representation theoretic properties of the vertex algebra are logically related to properties of its Zhu algebra or C2-algebra. Definitions Let V = \bigoplus_ V_ be a graded vertex operator algebra with V_ = \mathbb\mathbf and let Y(a, z) = \sum_ a_n z^ be the vertex operator associated to a \in V. Define C_2(V)\subset Vto be the subspace spanned by elements of the form a_ b for a,b \in V. An element a \in V is homogeneous with \operatorname a = n if a \in V_. There are two binary operations on Vdefined bya * b = \sum_ \binom a_b, ~~~~~ a \circ b = \sum_ \binom a_ bfor homogeneous elements and extended linearly to all of V. Define O(V)\subset Vto be the span of all elements a\circ b. The algebra A(V) := V/O(V) with the binary operation induced by * is an assoc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Associative Algebra
In mathematics, an associative algebra ''A'' over a commutative ring (often a field) ''K'' is a ring ''A'' together with a ring homomorphism from ''K'' into the center of ''A''. This is thus an algebraic structure with an addition, a multiplication, and a scalar multiplication (the multiplication by the image of the ring homomorphism of an element of ''K''). The addition and multiplication operations together give ''A'' the structure of a ring; the addition and scalar multiplication operations together give ''A'' the structure of a module or vector space over ''K''. In this article we will also use the term ''K''-algebra to mean an associative algebra over ''K''. A standard first example of a ''K''-algebra is a ring of square matrices over a commutative ring ''K'', with the usual matrix multiplication. A commutative algebra is an associative algebra for which the multiplication is commutative, or, equivalently, an associative algebra that is also a commutative ring. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutative Algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideal (ring theory), ideals, and module (mathematics), modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings include polynomial rings; rings of algebraic integers, including the ordinary integers \mathbb; and p-adic number, ''p''-adic integers. Commutative algebra is the main technical tool of algebraic geometry, and many results and concepts of commutative algebra are strongly related with geometrical concepts. The study of rings that are not necessarily commutative is known as noncommutative algebra; it includes ring theory, representation theory, and the theory of Banach algebras. Overview Commutative algebra is essentially the study of the rings occurring in algebraic number theory and algebraic geometry. Several concepts of commutative algebras have been developed in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nilpotent Orbit
In mathematics, nilpotent orbits are generalizations of nilpotent matrices that play an important role in representation theory of real and complex semisimple Lie groups and semisimple Lie algebras. Definition An element ''X'' of a semisimple Lie algebra ''g'' is called nilpotent if its adjoint endomorphism : ''ad X'': ''g'' → ''g'', ''ad X''(''Y'') = 'X'',''Y'' is nilpotent, that is, (''ad X'')''n'' = 0 for large enough ''n''. Equivalently, ''X'' is nilpotent if its characteristic polynomial ''p''''ad X''(''t'') is equal to ''t''dim ''g''. A semisimple Lie group or algebraic group ''G'' acts on its Lie algebra via the adjoint representation, and the property of being nilpotent is invariant under this action. A nilpotent orbit is an orbit of the adjoint action such that any (equivalently, all) of its elements is (are) nilpotent. Examples Nilpotent n\times n matrices with complex entries form the main motivating case for the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. Given an affine Lie algebra, one can also form the associated affine Kac-Moody algebra, as described below. From a purely mathematical point of view, affine Lie algebras are interesting because their representation theory, like representation theory of finite-dimensional semisimple Lie algebras, is much better understood than that of general Kac–Moody algebras. As observed by Victor Kac, the character formula for representations of affine Lie algebras implies certain combinatorial identities, the Macdonald identities. Affine Lie algebras play an important role in string theory and two-dimensional conformal field theory due to the way they are constructed: starting from a simple Lie algebra \mathfrak, one considers the loop algebra, L\mathfrak, formed by the \mathfrak-valued functions on a circle (interpreted as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
W-algebra
In conformal field theory and representation theory, a W-algebra is an associative algebra that generalizes the Virasoro algebra. W-algebras were introduced by Alexander Zamolodchikov, and the name "W-algebra" comes from the fact that Zamolodchikov used the letter W for one of the elements of one of his examples. Definition A W-algebra is an associative algebra that is generated by the modes of a finite number of meromorphic fields W^(z), including the energy-momentum tensor T(z)=W^(z). For h\neq 2, W^(z) is a primary field of conformal dimension h\in\frac12\mathbb^*. The generators (W^_n)_ of the algebra are related to the meromorphic fields by the mode expansions : W^(z) = \sum_ W^_n z^ The commutation relations of L_n=W^_n are given by the Virasoro algebra, which is parameterized by a central charge c\in \mathbb. This number is also called the central charge of the W-algebra. The commutation relations : _m, W^_n= ((h-1)m-n)W^_ are equivalent to the assumption that W^(z) i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cone (algebraic Geometry)
In algebraic geometry, a cone is a generalization of a vector bundle. Specifically, given a scheme ''X'', the relative Spec :C = \operatorname_X R of a quasi-coherent graded ''O''''X''-algebra ''R'' is called the cone or affine cone of ''R''. Similarly, the relative Proj :\mathbb(C) = \operatorname_X R is called the projective cone of ''C'' or ''R''. Note: The cone comes with the \mathbb_m-action due to the grading of ''R''; this action is a part of the data of a cone (whence the terminology). Examples *If ''X'' = Spec ''k'' is a point and ''R'' is a homogeneous coordinate ring, then the affine cone of ''R'' is the (usual) affine cone over the projective variety corresponding to ''R''. *If R = \bigoplus_0^\infty I^n/I^ for some ideal sheaf ''I'', then \operatorname_X R is the normal cone to the closed scheme determined by ''I''. *If R = \bigoplus_0^\infty L^ for some line bundle ''L'', then \operatorname_X R is the total space of the dual of ''L''. *More generally, given a ve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Action
In mathematics, a group action of a group G on a set S is a group homomorphism from G to some group (under function composition) of functions from S to itself. It is said that G acts on S. Many sets of transformations form a group under function composition; for example, the rotations around a point in the plane. It is often useful to consider the group as an abstract group, and to say that one has a group action of the abstract group that consists of performing the transformations of the group of transformations. The reason for distinguishing the group from the transformations is that, generally, a group of transformations of a structure acts also on various related structures; for example, the above rotation group also acts on triangles by transforming triangles into triangles. If a group acts on a structure, it will usually also act on objects built from that structure. For example, the group of Euclidean isometries acts on Euclidean space and also on the figures dra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Variety
In algebraic geometry, an affine variety or affine algebraic variety is a certain kind of algebraic variety that can be described as a subset of an affine space. More formally, an affine algebraic set is the set of the common zeros over an algebraically closed field of some family of polynomials in the polynomial ring k _1, \ldots,x_n An affine variety is an affine algebraic set which is not the union of two smaller algebraic sets; algebraically, this means that (the radical of) the ideal generated by the defining polynomials is prime. One-dimensional affine varieties are called affine algebraic curves, while two-dimensional ones are affine algebraic surfaces. Some texts use the term ''variety'' for any algebraic set, and ''irreducible variety'' an algebraic set whose defining ideal is prime (affine variety in the above sense). In some contexts (see, for example, Hilbert's Nullstellensatz), it is useful to distinguish the field in which the coefficients are considered, from ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Scheme
In commutative algebra, the prime spectrum (or simply the spectrum) of a commutative ring R is the set of all prime ideals of R, and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with a sheaf of rings. Zariski topology For any ideal I of R, define V_I to be the set of prime ideals containing I. We can put a topology on \operatorname(R) by defining the collection of closed sets to be :\big\. This topology is called the Zariski topology. A basis for the Zariski topology can be constructed as follows: For f\in R, define D_f to be the set of prime ideals of R not containing f. Then each D_f is an open subset of \operatorname(R), and \big\ is a basis for the Zariski topology. \operatorname(R) is a compact space, but almost never Hausdorff: In fact, the maximal ideals in R are precisely the closed points in this topology. By the same reasoning, \operatorname(R) is not, in general, a T1 space. However, \operat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; the modern approach generalizes this in a few different aspects. The fundamental objects of study in algebraic geometry are algebraic variety, algebraic varieties, which are geometric manifestations of solution set, solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are line (geometry), lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscate of Bernoulli, lemniscates and Cassini ovals. These are plane algebraic curves. A point of the plane lies on an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of points of special interest like singular point of a curve, singular p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category (mathematics)
In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose objects are sets and whose arrows are functions. ''Category theory'' is a branch of mathematics that seeks to generalize all of mathematics in terms of categories, independent of what their objects and arrows represent. Virtually every branch of modern mathematics can be described in terms of categories, and doing so often reveals deep insights and similarities between seemingly different areas of mathematics. As such, category theory provides an alternative foundation for mathematics to set theory and other proposed axiomatic foundations. In general, the objects and arrows may be abstract entities of any kind, and the n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vertex Operator Algebra
In mathematics, a vertex operator algebra (VOA) is an algebraic structure that plays an important role in two-dimensional conformal field theory and string theory. In addition to physical applications, vertex operator algebras have proven useful in purely mathematical contexts such as monstrous moonshine and the geometric Langlands correspondence. The related notion of vertex algebra was introduced by Richard Borcherds in 1986, motivated by a construction of an infinite-dimensional Lie algebra due to Igor Frenkel. In the course of this construction, one employs a Fock space that admits an action of vertex operators attached to elements of a unimodular lattice, lattice. Borcherds formulated the notion of vertex algebra by axiomatizing the relations between the lattice vertex operators, producing an algebraic structure that allows one to construct new Lie algebras by following Frenkel's method. The notion of vertex operator algebra was introduced as a modification of the notion ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
