|
Bifundamental Representation
In mathematics and theoretical physics, a bifundamental representation is a representation obtained as a tensor product of two fundamental or antifundamental representations. For example, the ''MN''-dimensional representation (''M'',''N'') of the group :SU(M) \times SU(N) is a bifundamental representation. These representations occur in quiver diagram In theoretical physics, a quiver diagram is a graph representing the matter content of a gauge theory that describes D-branes on orbifolds. Quiver diagrams may also be used to described \mathcal = 2 supersymmetric gauge theories in four dimens ...s. Abstract algebra {{algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. This is in contrast to experimental physics, which uses experimental tools to probe these phenomena. The advancement of science generally depends on the interplay between experimental studies and theory. In some cases, theoretical physics adheres to standards of mathematical rigour while giving little weight to experiments and observations.There is some debate as to whether or not theoretical physics uses mathematics to build intuition and illustrativeness to extract physical insight (especially when normal experience fails), rather than as a tool in formalizing theories. This links to the question of it using mathematics in a less formally rigorous, and more intuitive or heuristic way than, say, mathematical physics. For example, while developing special relativity, Albert Einstein was concerned wit ... [...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] |
Tensor Product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W denoted v \otimes w. An element of the form v \otimes w is called the tensor product of and . An element of V \otimes W is a tensor, and the tensor product of two vectors is sometimes called an ''elementary tensor'' or a ''decomposable tensor''. The elementary tensors span V \otimes W in the sense that every element of V \otimes W is a sum of elementary tensors. If bases are given for and , a basis of V \otimes W is formed by all tensor products of a basis element of and a basis element of . The tensor product of two vector spaces captures the properties of all bilinear maps in the sense that a bilinear map from V\times W into another vector space factors uniquely through a linear map V\otimes W\to Z (see Universal property). Tenso ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Representation
In representation theory of Lie groups and Lie algebras, a fundamental representation is an irreducible finite-dimensional representation of a semisimple Lie group or Lie algebra whose highest weight is a fundamental weight. For example, the defining module of a classical Lie group is a fundamental representation. Any finite-dimensional irreducible representation of a semisimple Lie group or Lie algebra can be constructed from the fundamental representations by a procedure due to Élie Cartan. Thus in a certain sense, the fundamental representations are the elementary building blocks for arbitrary finite-dimensional representations. Examples * In the case of the general linear group, all fundamental representations are exterior products of the defining module. * In the case of the special unitary group SU(''n''), the ''n'' − 1 fundamental representations are the wedge products \operatorname^k\ ^n consisting of the alternating tensors, for ''k'' = 1,&n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Antifundamental Representation
In mathematics differential geometry, an antifundamental representation of a Lie group is the complex conjugate of the fundamental representation In representation theory of Lie groups and Lie algebras, a fundamental representation is an irreducible representation, irreducible finite-dimensional representation of a semisimple Lie algebra, semisimple Lie group or Lie algebra whose highest weig ...,. although the distinction between the fundamental and the antifundamental representation is a matter of convention. However, these two are often non-equivalent, because each of them is a complex representation. References Representation theory of Lie groups {{differential-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quiver Diagram
In theoretical physics, a quiver diagram is a graph representing the matter content of a gauge theory that describes D-branes on orbifolds. Quiver diagrams may also be used to described \mathcal = 2 supersymmetric gauge theories in four dimensions. Each node of the graph corresponds to a factor ''U''(''N'') of the gauge group, and each link represents a field in the bifundamental representation :(M,\bar). The relevance of quiver diagrams for string theory was pointed out and studied by Michael Douglas and Greg Moore. While string theorists use the words ''quiver diagram'', many of their colleagues in particle physics call these diagrams ''mooses''. Definition For convenience, consider the supersymmetric \mathcal =1 gauge theory in four-dimensional spacetime. The quiver gauge theory is given by the following data: * Finite quiver Q * Each vertex v\in \operatorname (Q) corresponds to a compact Lie group G_. This can be the unitary group U(N), the special unitary group ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
