Kernel (algebra) - Kernel (algebra) In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. The definition of kernel takes various forms in various contexts. But in all of them, the kernel of a homomorphism is trivial (in a sense relevant to that context) if and only if the homomorphism is injective. The fundamental theorem on homomorphisms (or first isomorphism theorem) is a theorem, again taking various forms, that applies to the quotient algebra defined by the kernel. In this article, we first survey kernels for some important types of algebraic structures; then we give general definitions from universal algebra for generic algebraic structures. Table of contents showTocToggle("show","hide") 1 Survey.
Kleene algebra - Kleene algebra In mathematics, a Kleene algebra (named after Stephen Cole Kleene, pronounced "clay-knee") is either of two different things: A bounded distributive lattice with an involution satisfying certain laws weaker than those governing lattice-theoretic complementations. Thus every Boolean algebra is a Kleene algebra, but most Kleene algebras are not Boolean algebras. An algebraic structure that generalizes the operations known from regular expressions. The remainder of this article deals with this notion of Kleene algebra. Table of contents showTocToggle("show","hide") 1 Definition 2 Examples 3 Properties 4 History 5 References Definition Various inequivalent definitions of Kleene algebras and related structures have been given in the literature. See [1] for a survey. Here we will give the definition that seems to be the most common nowadays. A Kleene algebra.
VEGA computer algebra system - VEGA computer algebra system Vega is a system for manipulating discrete mathematical structures. The ongoing project is located at the Department of Theoretical Computer Science at IMFM. See http://vega.ijp.si/Htmldoc/vega03.html.
Incidence algebra - Incidence algebra A partially ordered set is locally finite precisely if every closed interval [a, b] within it is finite. For every locally finite poset and every field of scalars there is an incidence algebra, an associative algebra defined as follows. The members of the incidence algebra are the functions f assigning to each interval [a, b] a scalar f(a, b). On this underlying set one defines addition and scalar multiplication pointwise, and "multiplication" in the incidence algebra is a convolution defined by The multiplicative identity element of the incidence algebra is An incidence algebra is finite-dimensional if and only if the underlying poset is finite. The ζ function of an incidence algebra is the constant function ζ(a, b) = 1 for every interval [a, b]. One.
Virasoro algebra - Virasoro algebra The orientation-preserving diffeomorphism group of the circle, Diff(S1) admits a central extension called the Virasoro group. Its complexified Lie algebra is spanned by {Li}i in Z and c with Ln+L-n and c being real elements. c is called the central charge. The algebra satisfies [c,Ln]=0 [Lm,Ln]=(n-m)Lm+n+c/12 (m3-m)δm,-n. The factor of 1/12 is merely a matter of convention. Note that the Virasoro algebra generates both a centrally extended orientation-preserving diffeomorphism group and a centrally extended orientation-preserving homeomorphism group of the circle. The difference lies in the topology chosen. See also Kac-Moody algebra. This article is a stub. You can help Wikipedia by fixing it..
Von Neumann algebra - Von Neumann algebra A von Neumann algebra is a subspace of bounded Hilbert space operators closed under the weak/strong topology. It's also called a W* algebra. Von Neumann algebras are automatically C* algebras. The von Neumann bicommutant theorem gives another description of von Neumann algebras, using algebraical rather than topological properties. The relationship between commutative von Neumann algebras and locally compact measure spaces is analogous to that between commutative C* algebrass and compact Hausdorff spaces. Every commutative von Neumann algebra is isomorphic to L∞(X) for some locally compact measure space X, and for every locally compact measure space X, conversely, L∞(X) is a von Neumann algebra. Due to this analogy, the theory of von Neumann algebras has been called noncommutative measure theory, while the theory of C* algebrass.
Heyting algebra - Heyting algebra In mathematics, Heyting algebras are a generalization of Boolean algebras. Heyting algebras model intuitionistic logic, in which the law of excluded middle does not in general hold. Formally, a Heyting algebra is a bounded lattice L such that for all a and b in L there is a greatest element x of L such that a ∧ x ≤ b. This element is called the relative pseudo-complement of a with respect to b, and is denoted a⇒b (or a→b). We write ¬a for a⇒0. Heyting algebras are always distributive; this is sometimes stated as an axiom, but in fact it follows from the existence of relative pseudo-complements. Boolean algebras are those Heyting algebras in which x = ¬¬x for all x, or, equivalently, in which x ∨ ¬x = 1 for all x. In this.
Homological algebra - Homological algebra Homological algebra is that branch of mathematics which studies the methods of homology and cohomology in a general setting. These concepts originated in algebraic topology. Cohomology theories have been described for topological spaces, sheaves, and groupss; also for Lie algebras, C-star algebras. The study of modern algebraic geometry would be almost unthinkable without sheaf cohomology. There are also other homological functors that take their place in the theory, such as Ext and Tor. There have been attempts at 'non-commutative' theories, which extend first cohomology as torsors (which is important in Galois cohomology). Foundational aspects The methods of homological algebra start with use of the exact sequence to perform actual calculations. With a diverse set of applications in mind, it was natural to try to put.
Hopf algebra - Hopf algebra A Hopf algebra is a bialgebra with an antipode morphism satisfying . Hopf algebras are also called quantum groups because they can be interpreted as "quantizations" of groups. See also Algebra/set analogy. Examples Given a topological group G, we can form two different Hopf algebras over it. The first is the algebra of continuous functions from G to K whose product is the pointwise product. ε acting on a function gives its value at the identity and (Δ f)(x,y)=f(xy) for all x and y in G. (Sf)(x)=f(x-1). The coaction of this Hopf algebra upon noncommutative spaces is as a left (right) comodule. The other Hopf algebra we can construct is the convolution product algebra of distributions over G. This time, the action of this Hopf.
GAP computer algebra system - GAP computer algebra system GAP (Groups, Algorithms and Programming) is a computer algebra system for computational discrete algebra with particular emphasis on, but not restricted to computational group theory. GAP was developed at Lehrstuhl D für Mathematik (LDFM), RWTH Aachen, Germany from 1986 to 1997. After the retirement of J. Neubüser from the chair of LDFM, the development and maintenance of GAP is coordinated by the School of Mathematical and Computational Sciences at the University of St. Andrews, Scotland. Several users have contributed to the system via share packages which can be used in the same form as the main library. There are presently (Jul. '99) two versions in distribution: Version 4 .3 (GAP 4r3), incorporates several new basic features developed over the last years. In particular it.
Geometric algebra - Geometric algebra David Hestenes et al.'s geometric algebra is a radical reinterpretation of seemingly harmless Clifford algebras over the reals (or, stated ironically, return to the original name and interpretation intended by William Clifford). The key ingredient of this formulation is the (natural) correspondence between geometric entities and the elements of the associative algebra. Contrary to the claims its proponents make, the "mixing" of quantities of different grades is helpful only by the virtue of the computational advantages offered by associativity (and inverse of vectors), not particularly in visualization or conceptualization. The applications in physics are valuable, and Hestenes' unusual stand in favor of using real numbers only (expelling complex numbers) as the underlying field, has given the reformulation a sort of a distinction, given the fact.
Glossary of linear algebra - Glossary of linear algebra Vector space Subspace Linear combination Generating a vector space Linearly independent vectors Basis for a vector space Dimension of a vector space Normed vector space Inner product space Banach space Hilbert space Jordan Normal Form Some important topics: Least squares Gauss elimination method Gram-Schmidt Process.
Universal algebra - Universal algebra Universal algebra is the field of mathematics that studies the ideas common to all algebraic structures. Table of contents showTocToggle("show","hide") 1 Basic idea 2 Examples 2.1 Groups 2.2 Modules 3 Further issues Basic idea From the point of view of universal algebra, an algebra is a set A together with a collection of operations on A. An n-ary operation on A is a function that takes n elements of A and returns a single element of A. Thus, a 0-ary operation (or nullary operation) is simply an element of A, or a constant, often denoted by a letter like a. A 1-ary operation (or unary operation) is simply a function from A to A, often denoted by a symbol placed in front of its argument,.
Graded algebra - Graded algebra A graded algebra is an algebra generated when an outer product (wedge product) is defined in a vector space over the scalars . The outer product generates a set of new entities: the -vectors. As they are obtained by the outer product of linearly independent vectors, they are said to be of step or grade . -vectors are vectors in nature, so any -vector is a member of a vector subspace known as subspace of grade , denoted by ∧kVn. Each of this has a dimension of where is the binomial coefficient. Vectors are said to have step 1, so with dimension , and scalars are considered as the 0-step vector space ∧0Vn, and have dimension 1. The -vectors also generate a 1-dimensional vector space,.
Grassmann algebra - Grassmann algebra A Grassmann algebra (also known as an exterior algebra) is a unital associative algebra K generated by a set, S subject to the relation χξ+ξχ=0 for any χ,ξ in S. This definition amounts to saying that the generators are anti-commuting quantities (and otherwise 'as general as possible); it should be modified in case K has characteristic 2. The construction of such an algebra comes from the wedge product: take the vector space V that has S as basis, and the direct sum of all the exterior powers of V, using wedge product in each graded piece. If S is finite of cardinality n, the Grassmann algebra has as basis one wedge product for each subset of S, and each product made up by wedging elements.
Group algebra - Group algebra In mathematics, the group algebra of a group G means, firstly, its group ring over a given field. It may mean also, in case G is a topological group, some other ring of functions on G, the group ring being the ring of functions with finite support. Suppose G is locally compact, so that Haar measure exists on G. The product in the group algebra will then always be of convolution type; and the values of functions considered will be complex numbers. In different contexts the condition imposed might be to have compact support; or to be integrable, giving rings Co(G) and L1(G), either of which can serve as a group algebra in the sense that representations of G become modules over that ring..
Universal enveloping algebra - Universal enveloping algebra In mathematics, the universal enveloping algebra construction of abstract algebra is applied to a Lie algebra L in order to pass from a non-associative structure to a more familiar and associative algebra over a field U(L) while preserving the representation theory. That is, L is to be made into an algebra over the same field K, in such a way that L's representations become modules over U(L) in the usual sense. Table of contents showTocToggle("show","hide") 1 General construction 2 Examples in particular cases 3 Further description of structure General construction Noting that any associative K-algebra becomes a Lie algebra with the bracket [a,b] = a.b-b.a, a construction and precise characterisation suggests itself on general grounds (as adjoint functors). Starting with the tensor algebra T(L) on.
Fundamental theorem of algebra - Fundamental theorem of algebra The fundamental theorem of algebra (now considered something of a misnomer by many mathematicians) states that every complex polynomial of degree n has exactly n zeroes, counted with multiplicity. More formally, if (where the coefficients a0, ..., an−1 can be real or complex numbers), then there exist (not necessarily distinct) complex numbers z1, ..., zn such that This shows that the field of complex numbers, unlike the field of real numbers, is algebraically closed. An easy consequence is that the product of all the roots equals (−1)n a0 and the sum of all the roots equals -an−1. The theorem had been conjectured in the 17th century but could not be proved since the complex numbers had not yet been firmly grounded. The first rigorous.
Elementary algebra - Elementary algebra Elementary algebra is the most basic form of algebra taught to students who are presumed to have no knowledge of mathematics beyond the basic principles of arithmetic. While in arithmetic only numbers and their arithmetical operations occur, in algebra one also uses symbols (such as a, x, y) to denote numbers. This is useful because it allows the general formulation of arithmetical laws (such as a + b = b + a for all a and b), and thus is the first step to a systematic exploration of the properties of the real number system it allows to talk about "unknown" numbers, the formulation of equations and the study of how to solve these (for instance "find a number x such that 3x + 2.
Extension (algebra) - Extension (algebra) If G is a group/algebra over a field or any other algebraic structure, G' is an extension of G if there's an exact sequence . See also central extension, extension problem, field extension. This article is a stub. You can help Wikipedia by fixing it..
