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Center Of A Ring
In algebra, the center of a ring ''R'' is the subring consisting of the elements ''x'' such that ''xy = yx'' for all elements ''y'' in ''R''. It is a commutative ring and is denoted as Z(R); "Z" stands for the German word ''Zentrum'', meaning "center". If ''R'' is a ring, then ''R'' is an associative algebra over its center. Conversely, if ''R'' is an associative algebra over a commutative subring ''S'', then ''S'' is a subring of the center of ''R'', and if ''S'' happens to be the center of ''R'', then the algebra ''R'' is called a central algebra. Examples *The center of a commutative ring ''R'' is ''R'' itself. *The center of a skew-field is a field. *The center of the (full) matrix ring with entries in a commutative ring ''R'' consists of ''R''-scalar multiples of the identity matrix. *Let ''F'' be a field extension of a field ''k'', and ''R'' an algebra over ''k''. Then Z\left(R \otimes_k F\right) = Z(R) \otimes_k F. *The center of the universal enveloping algebra of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathematics), modules, vector spaces, lattice (order), lattices, and algebra over a field, algebras over a field. The term ''abstract algebra'' was coined in the early 20th century to distinguish this area of study from older parts of algebra, and more specifically from elementary algebra, the use of variable (mathematics), variables to represent numbers in computation and reasoning. Algebraic structures, with their associated homomorphisms, form category (mathematics), mathematical categories. Category theory is a formalism that allows a unified way for expressing properties and constructions that are similar for various structures. Universal algebra is a related subject that studies types of algebraic structures as single objects. For exampl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Algebra
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Central Simple Algebra
In ring theory and related areas of mathematics a central simple algebra (CSA) over a field ''K'' is a finite-dimensional associative ''K''-algebra ''A'' which is simple, and for which the center is exactly ''K''. (Note that ''not'' every simple algebra is a central simple algebra over its center: for instance, if ''K'' is a field of characteristic 0, then the Weyl algebra K ,\partial_X/math> is a simple algebra with center ''K'', but is ''not'' a central simple algebra over ''K'' as it has infinite dimension as a ''K''-module.) For example, the complex numbers C form a CSA over themselves, but not over the real numbers R (the center of C is all of C, not just R). The quaternions H form a 4-dimensional CSA over R, and in fact represent the only non-trivial element of the Brauer group of the reals (see below). Given two central simple algebras ''A'' ~ ''M''(''n'',''S'') and ''B'' ~ ''M''(''m'',''T'') over the same field ''F'', ''A'' and ''B'' are called ''similar'' (or '' Braue ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Center Of A Group
In abstract algebra, the center of a group, , is the set of elements that commute with every element of . It is denoted , from German '' Zentrum,'' meaning ''center''. In set-builder notation, :. The center is a normal subgroup, . As a subgroup, it is always characteristic, but is not necessarily fully characteristic. The quotient group, , is isomorphic to the inner automorphism group, . A group is abelian if and only if . At the other extreme, a group is said to be centerless if is trivial; i.e., consists only of the identity element. The elements of the center are sometimes called central. As a subgroup The center of ''G'' is always a subgroup of . In particular: # contains the identity element of , because it commutes with every element of , by definition: , where is the identity; # If and are in , then so is , by associativity: for each ; i.e., is closed; # If is in , then so is as, for all in , commutes with : . Furthermore, the center of is al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simple Algebra
In abstract algebra, a branch of mathematics, a simple ring is a non-zero ring that has no two-sided ideal besides the zero ideal and itself. In particular, a commutative ring is a simple ring if and only if it is a field. The center of a simple ring is necessarily a field. It follows that a simple ring is an associative algebra over this field. So, simple algebra and ''simple ring'' are synonyms. Several references (e.g., Lang (2002) or Bourbaki (2012)) require in addition that a simple ring be left or right Artinian (or equivalently semi-simple). Under such terminology a non-zero ring with no non-trivial two-sided ideals is called quasi-simple. Rings which are simple as rings but are not a simple module over themselves do exist: a full matrix ring over a field does not have any nontrivial ideals (since any ideal of M_n(R) is of the form M_n(I) with I an ideal of R), but has nontrivial left ideals (for example, the sets of matrices which have some fixed zero columns). A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Algebra Representation
In the mathematical field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices (or endomorphisms of a vector space) in such a way that the Lie bracket is given by the commutator. In the language of physics, one looks for a vector space V together with a collection of operators on V satisfying some fixed set of commutation relations, such as the relations satisfied by the angular momentum operators. The notion is closely related to that of a representation of a Lie group. Roughly speaking, the representations of Lie algebras are the differentiated form of representations of Lie groups, while the representations of the universal cover of a Lie group are the integrated form of the representations of its Lie algebra. In the study of representations of a Lie algebra, a particular ring, called the universal enveloping algebra, associated with the Lie algebra plays an important role. The un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Casimir Element
In mathematics, a Casimir element (also known as a Casimir invariant or Casimir operator) is a distinguished element of the center of the universal enveloping algebra of a Lie algebra. A prototypical example is the squared angular momentum operator, which is a Casimir element of the three-dimensional rotation group. The Casimir element is named after Hendrik Casimir, who identified them in his description of rigid body dynamics in 1931. Definition The most commonly-used Casimir invariant is the quadratic invariant. It is the simplest to define, and so is given first. However, one may also have Casimir invariants of higher order, which correspond to homogeneous symmetric polynomials of higher order. Quadratic Casimir element Suppose that \mathfrak is an n-dimensional Lie algebra. Let ''B'' be a nondegenerate bilinear form on \mathfrak that is invariant under the adjoint action of \mathfrak on itself, meaning that B(\operatorname_XY, Z) + B(Y, \operatorname_X Z) = 0 for a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Representation Theory Of Lie Algebras
In the mathematical field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices (or endomorphisms of a vector space) in such a way that the Lie bracket is given by the commutator. In the language of physics, one looks for a vector space V together with a collection of operators on V satisfying some fixed set of commutation relations, such as the relations satisfied by the angular momentum operators. The notion is closely related to that of a representation of a Lie group. Roughly speaking, the representations of Lie algebras are the differentiated form of representations of Lie groups, while the representations of the universal cover of a Lie group are the integrated form of the representations of its Lie algebra. In the study of representations of a Lie algebra, a particular ring, called the universal enveloping algebra, associated with the Lie algebra plays an important role. The un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Universal Enveloping Algebra
In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra. Universal enveloping algebras are used in the representation theory of Lie groups and Lie algebras. For example, Verma modules can be constructed as quotients of the universal enveloping algebra. In addition, the enveloping algebra gives a precise definition for the Casimir operators. Because Casimir operators commute with all elements of a Lie algebra, they can be used to classify representations. The precise definition also allows the importation of Casimir operators into other areas of mathematics, specifically, those that have a differential algebra. They also play a central role in some recent developments in mathematics. In particular, their dual provides a commutative example of the objects studied in non-commutative geometry, the quantum groups. This dual can be shown, by the Gelfa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subring
In mathematics, a subring of ''R'' is a subset of a ring that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and which shares the same multiplicative identity as ''R''. For those who define rings without requiring the existence of a multiplicative identity, a subring of ''R'' is just a subset of ''R'' that is a ring for the operations of ''R'' (this does imply it contains the additive identity of ''R''). The latter gives a strictly weaker condition, even for rings that do have a multiplicative identity, so that for instance all ideals become subrings (and they may have a multiplicative identity that differs from the one of ''R''). With definition requiring a multiplicative identity (which is used in this article), the only ideal of ''R'' that is a subring of ''R'' is ''R'' itself. Definition A subring of a ring is a subset ''S'' of ''R'' that preserves the structure of the ring, i.e. a ring with . Equivalently, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field Extension
In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ''F''. For example, under the usual notions of addition and multiplication, the complex numbers are an extension field of the real numbers; the real numbers are a subfield of the complex numbers. Field extensions are fundamental in algebraic number theory, and in the study of polynomial roots through Galois theory, and are widely used in algebraic geometry. Subfield A subfield K of a field L is a subset K\subseteq L that is a field with respect to the field operations inherited from L. Equivalently, a subfield is a subset that contains 1, and is closed under the operations of addition, subtraction, multiplication, and taking the inverse of a nonzero element of K. As , the latter definition implies K and L have the same zero element ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Identity Matrix
In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. Terminology and notation The identity matrix is often denoted by I_n, or simply by I if the size is immaterial or can be trivially determined by the context. I_1 = \begin 1 \end ,\ I_2 = \begin 1 & 0 \\ 0 & 1 \end ,\ I_3 = \begin 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end ,\ \dots ,\ I_n = \begin 1 & 0 & 0 & \cdots & 0 \\ 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & 1 \end. The term unit matrix has also been widely used, but the term ''identity matrix'' is now standard. The term ''unit matrix'' is ambiguous, because it is also used for a matrix of ones and for any unit of the ring of all n\times n matrices. In some fields, such as group theory or quantum mechanics, the identity matrix is sometimes denoted by a boldface one, \mathbf, or called "id" (short for identity) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
