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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 first introduced by to formalize the notion of an algebra of observables in quantum mechanics. They were originally called "r-number systems", but were renamed "Jordan algebras" by , ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Structure
In mathematics, an algebraic structure 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 structures of the same type (homomor ... [...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 . If K=\mathbb R, and the quadratic form takes zero only when all variables are simultaneously zero, then it is a definite quadratic form, otherwise it is an isotropic quadratic form. Quadratic forms occupy a central place in various branches of mathematics, including number theory, linear algebra, group theory (orthogonal group), differential geometry (Riemannian metric, second fundamental form), differential topology ( intersection forms of four-manifolds), and Lie theory (the Killing form). Quadratic forms are not to be confused with a quadratic equation, which has only one variable and includes terms of degree two or less. A quadratic form is ... [...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, o ... [...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 ... [...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. To see how the direct sum is used in abstract algebra, consider a more elementary kind of structure, the abelian group. 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, we define the sum (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. We can also form direct sums 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 spa ... [...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 algebraic geometry and toric geometry. Higher dimensional algebraic tori can be modelled as a product of algebraic groups \mathbf G_. These 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^, 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 to them such as symmetric spaces and buil ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peirce Decomposition
In ring theory, a Peirce decomposition is a decomposition of an algebra as a sum of eigenspaces of commuting idempotent elements. The Peirce decomposition for associative algebras was introduced by . A similar but more complicated Peirce decomposition for Jordan algebras was introduced by . Peirce decomposition for associative algebras If ''e'' is a commuting idempotent (''e''2 = ''e'' and ''e'' is in the center Center or centre may refer to: Mathematics *Center (geometry), the middle of an object * Center (algebra), used in various contexts ** Center (group theory) ** Center (ring theory) * Graph center, the set of all vertices of minimum eccentricity ... of ''A'') in an associative algebra ''A'', then the two-sided Peirce decomposition writes ''A'' as the direct sum of ''eAe'', ''eA''(1 − ''e''), (1 − ''e'')''Ae'', and (1 − ''e'')''A''(1 − ''e''). There are also left and right Peirce decompositions, where the left decomposition writes ''A'' as the ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zariski Dense
In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the 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 ... [...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 they are isomorphisms in the category of algebraic varieties. Because regular and biregular are very restrictive conditions – there are no non-constant regular functions on projective varieties – the weaker condition of a rational map and birational maps are frequently used as well. Definition If ''X'' and ''Y'' are closed subvarieties of \mathbb^n and \mathbb^m (so they are affine varieties), then a regular map f\colon X\to Y is the restriction of a polynomial map \mathbb^n\to \mathbb^m. Explicitly, it has the form: :f = (f_1, \dots, f_m) where the f_is are in the coordinate ring of ''X'': :k = k _1, \dots, x_nI, where ''I'' is the ideal defining ''X'' ... [...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] |
