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J-structure
In mathematics, a J-structure is an algebraic structure over a field (algebra), field related to a Jordan algebra. The concept was introduced by to develop a theory of Jordan algebras using linear algebraic groups and axioms taking the Jordan inversion as basic operation and Hua's identity as a basic relation. There is a classification of simple structures deriving from the classification of semisimple algebraic groups. Over fields of characteristic of a field, characteristic not equal to 2, the theory of J-structures is essentially the same as that of Jordan algebras. Definition Let ''V'' be a finite-dimensional vector space over a field ''K'' and ''j'' a rational map from ''V'' to itself, expressible in the form ''n''/''N'' with ''n'' a polynomial map from ''V'' to itself and ''N'' a polynomial in ''K''[''V'']. Let ''H'' be the subset of GL(''V'') × GL(''V'') containing the pairs (''g'',''h'') such that ''g''∘''j'' = ''j''∘''h'': it is a closed subgroup of the product an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jordan Algebra
In abstract algebra, a Jordan algebra is a nonassociative algebra over a field whose multiplication satisfies the following axioms: # xy = yx (commutative law) # (xy)(xx) = x(y(xx)) (). The product of two elements ''x'' and ''y'' in a Jordan algebra is also denoted ''x'' ∘ ''y'', particularly to avoid confusion with the product of a related associative algebra. The axioms imply that a Jordan algebra is power-associative, meaning that x^n = x \cdots x is independent of how we parenthesize this expression. They also imply that x^m (x^n y) = x^n(x^m y) for all positive integers ''m'' and ''n''. Thus, we may equivalently define a Jordan algebra to be a commutative, power-associative algebra such that for any element x, the operations of multiplying by powers x^n all commute. Jordan algebras were introduced by in an effort to formalize the notion of an algebra of observables in quantum electrodynamics. It was soon shown that the algebras were not useful in this context, howev ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Structure
In mathematics, an algebraic structure or algebraic system consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set of identities (known as ''axioms'') that these operations must satisfy. An algebraic structure may be based on other algebraic structures with operations and axioms involving several structures. For instance, a vector space involves a second structure called a field, and an operation called ''scalar multiplication'' between elements of the field (called '' scalars''), and elements of the vector space (called '' vectors''). Abstract algebra is the name that is commonly given to the study of algebraic structures. The general theory of algebraic structures has been formalized in universal algebra. Category theory is another formalization that includes also other mathematical structures and functions between structu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadratic Form
In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example, 4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to a fixed field , such as the real or complex numbers, and one speaks of a quadratic form ''over'' . Over the reals, a quadratic form is said to be '' definite'' if it takes the value zero only when all its variables are simultaneously zero; otherwise it is '' isotropic''. Quadratic forms occupy a central place in various branches of mathematics, including number theory, linear algebra, group theory ( orthogonal groups), differential geometry (the Riemannian metric, the second fundamental form), differential topology ( intersection forms of manifolds, especially four-manifolds), Lie theory (the Killing form), and statistics (where the exponent of a zero-mean multivariate normal distribution has the quadratic form -\mathbf^\math ... [...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, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isotope (Jordan Algebra)
In mathematics, a mutation, also called a homotope, of a unital Jordan algebra is a new Jordan algebra defined by a given element of the Jordan algebra. The mutation has a unit if and only if the given element is invertible, in which case the mutation is called a proper mutation or an isotope. Mutations were first introduced by Max Koecher in his Jordan algebraic approach to Hermitian symmetric spaces and bounded symmetric domains of tube type. Their functorial properties allow an explicit construction of the corresponding Hermitian symmetric space of compact type as a compactification of a finite-dimensional complex semisimple Jordan algebra. The automorphism group of the compactification becomes a complex subgroup, the complexification of its maximal compact subgroup. Both groups act transitively on the compactification. The theory has been extended to cover all Hermitian symmetric spaces using the theory of Jordan pairs or Jordan triple systems. Koecher obtained the results in t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Direct Sum
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently but analogously for different kinds of structures. As an example, the direct sum of two abelian groups A and B is another abelian group A\oplus B consisting of the ordered pairs (a,b) where a \in A and b \in B. To add ordered pairs, the sum is defined (a, b) + (c, d) to be (a + c, b + d); in other words, addition is defined coordinate-wise. For example, the direct sum \Reals \oplus \Reals , where \Reals is real coordinate space, is the Cartesian plane, \R ^2 . A similar process can be used to form the direct sum of two vector spaces or two modules. Direct sums can also be formed with any finite number of summands; for example, A \oplus B \oplus C, provided A, B, and C are the same kinds of algebraic structures (e.g., all abelian groups, or all vector spaces). That relies on the fact that the direct sum is associative up to isomorphism. That is, (A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Torus
In mathematics, an algebraic torus, where a one dimensional torus is typically denoted by \mathbf G_, \mathbb_m, or \mathbb, is a type of commutative affine algebraic group commonly found in Projective scheme, projective algebraic geometry and toric geometry. Higher dimensional algebraic tori can be modelled as a product of algebraic groups \mathbf G_. These Group (mathematics), groups were named by analogy with the theory of ''tori'' in Lie group theory (see Cartan subgroup). For example, over the complex numbers \mathbb the algebraic torus \mathbf G_ is isomorphic to the group scheme \mathbb^* = \text(\mathbb[t,t^]), which is the scheme theoretic analogue of the Lie group U(1) \subset \mathbb. In fact, any \mathbf G_-action on a complex vector space can be pulled back to a U(1)-action from the inclusion U(1) \subset \mathbb^* as real manifolds. Tori are of fundamental importance in the theory of algebraic groups and Lie groups and in the study of the geometric objects associated ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peirce Decomposition
Peirce may refer to: * Charles Sanders Peirce (1839–1914), American philosopher, founder of pragmatism Schools * Peirce College, Philadelphia, formerly known as Peirce College of Business, Peirce Junior College and Peirce School of Business Administration * Peirce School (also known as Old Peirce School), West Newton, Massachusetts * Helen C. Peirce School of International Studies, an elementary school in Chicago Others * Peirce (crater), a lunar crater * Peirce (given name), including a list of people with the given name * Peirce (surname), including a list of people with the surname See also * * * Pierce (other) * Peirse (other) {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mutation (algebra)
In the theory of algebras over a field, mutation is a construction of a new binary operation related to the multiplication of the algebra. In specific cases the resulting algebra may be referred to as a homotope or an isotope of the original. Definitions Let ''A'' be an algebra over a field ''F'' with multiplication (not assumed to be associative) denoted by juxtaposition. For an element ''a'' of ''A'', define the left ''a''-homotope A(a) to be the algebra with multiplication :x * y = (xa)y. \, Similarly define the left (''a'',''b'') mutation A(a,b) :x * y = (xa)y - (yb)x. \, Right homotope and mutation are defined analogously. Since the right (''p'',''q'') mutation of ''A'' is the left (−''q'', −''p'') mutation of the opposite algebra to ''A'', it suffices to study left mutations.Elduque & Myung (1994) p. 34 If ''A'' is a unital algebra and ''a'' is invertible, we refer to the isotope by ''a''. Properties * If ''A'' is associative then so is any homotope o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zariski Dense
In algebraic geometry and commutative algebra, the Zariski topology is a topology defined on geometric objects called varieties. It is very different from topologies that are commonly used in real or complex analysis; in particular, it is not Hausdorff. This topology was introduced primarily by Oscar Zariski and later generalized for making the set of prime ideals of a commutative ring (called the spectrum of the ring) a topological space. The Zariski topology allows tools from topology to be used to study algebraic varieties, even when the underlying field is not a topological field. This is one of the basic ideas of scheme theory, which allows one to build general algebraic varieties by gluing together affine varieties in a way similar to that in manifold theory, where manifolds are built by gluing together charts, which are open subsets of real affine spaces. The Zariski topology of an algebraic variety is the topology whose closed sets are the algebraic subsets of the vari ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dominant Morphism
In algebraic geometry, a morphism between algebraic varieties is a function between the varieties that is given locally by polynomials. It is also called a regular map. A morphism from an algebraic variety to the affine line is also called a regular function. A regular map whose inverse is also regular is called biregular, and the biregular maps are the isomorphisms of algebraic varieties. Because regular and biregular are very restrictive conditions – there are no non-constant regular functions on projective varieties – the concepts of rational and birational maps are widely used as well; they are partial functions that are defined locally by rational fractions instead of polynomials. An algebraic variety has naturally the structure of a locally ringed space; a morphism between algebraic varieties is precisely a morphism of the underlying locally ringed spaces. Definition If ''X'' and ''Y'' are closed subvarieties of \mathbb^n and \mathbb^m (so they are affine varietie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadratic Jordan Algebra
In mathematics, quadratic Jordan algebras are a generalization of Jordan algebras introduced by . The fundamental identities of the quadratic representation of a linear Jordan algebra are used as axioms to define a quadratic Jordan algebra over a field of arbitrary characteristic. There is a uniform description of finite-dimensional simple quadratic Jordan algebras, independent of characteristic. If 2 is invertible in the field of coefficients, the theory of quadratic Jordan algebras reduces to that of linear Jordan algebras. Definition A quadratic Jordan algebra consists of a vector space ''A'' over a field ''K'' with a distinguished element 1 and a quadratic map of ''A'' into the ''K''-endomorphisms of ''A'', ''a'' ↦ ''Q''(''a''), satisfying the conditions: * ; * ("fundamental identity"); * ("commutation identity"), where Further, these properties are required to hold under any extension of scalars. Elements An element ''a'' is invertible if is invertible and there exists ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
