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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] |
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] |
Ian G
Ian or Iain is a name of Scottish Gaelic origin, derived from the Hebrew given name (Yohanan, ') and corresponding to the English name John. The spelling Ian is an Anglicization of the Scottish Gaelic forename ''Iain''. It is a popular name in Scotland, where it originated, as well as other English-speaking countries. The name has fallen out of the top 100 male baby names in the United Kingdom, having peaked in popularity as one of the top 10 names throughout the 1960s. In 1900, Ian was the 180th most popular male baby name in England and Wales. , the name has been in the top 100 in the United States every year since 1982, peaking at 65 in 2003. Other Gaelic forms of "John" include "Seonaidh" ("Johnny" from Lowland Scots), "Seon" (from English), "Seathan", and "Seán" and " Eoin" (from Irish). Its Welsh counterpart is Ioan, its Cornish equivalent is Yowan and Breton equivalent is Yann. Notable people named Ian As a first name (alphabetical by family name) *Ian Agol (born 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
University Of Virginia
The University of Virginia (UVA) is a Public university#United States, public research university in Charlottesville, Virginia. Founded in 1819 by Thomas Jefferson, the university is ranked among the top academic institutions in the United States, with College admissions in the United States, highly selective admission. Set within the The Lawn, Academical Village, a World Heritage Site, UNESCO World Heritage Site, the university is referred to as a "Public Ivy" for offering an academic experience similar to that of an Ivy League university. It is known in part for certain rare characteristics among public universities such as #1800s, its historic foundations, #Honor system, student-run academic honor code, honor code, and Secret societies at the University of Virginia, secret societies. The original governing Board of Visitors included three List of presidents of the United States, U.S. presidents: Thomas Jefferson, Jefferson, James Madison, and James Monroe. The latter as si ... [...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] |
Mutation (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 Lie group, complex subgroup, the complexification (Lie group), 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 tri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mutation (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 Lie group, complex subgroup, the complexification (Lie group), 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 tri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polarization Of An Algebraic Form
In mathematics, in particular in algebra, polarization is a technique for expressing a homogeneous polynomial in a simpler fashion by adjoining more variables. Specifically, given a homogeneous polynomial, polarization produces a unique symmetric multilinear form from which the original polynomial can be recovered by evaluating along a certain diagonal. Although the technique is deceptively simple, it has applications in many areas of abstract mathematics: in particular to algebraic geometry, invariant theory, and representation theory. Polarization and related techniques form the foundations for Weyl's invariant theory. The technique The fundamental ideas are as follows. Let f(\mathbf) be a polynomial in n variables \mathbf = \left(u_1, u_2, \ldots, u_n\right). Suppose that f is homogeneous of degree d, which means that f(t \mathbf) = t^d f(\mathbf) \quad \text t. Let \mathbf^, \mathbf^, \ldots, \mathbf^ be a collection of indeterminates with \mathbf^ = \left(u^_1, u^_2, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Power Associativity
In mathematics, specifically in abstract algebra, power associativity is a property of a binary operation that is a weak form of associativity. Definition An algebra (or more generally a magma) is said to be power-associative if the subalgebra generated by any element is associative. Concretely, this means that if an element x is performed an operation * by itself several times, it doesn't matter in which order the operations are carried out, so for instance x*(x*(x*x)) = (x*(x*x))*x = (x*x)*(x*x). Examples and properties Every associative algebra is power-associative, but so are all other alternative algebras (like the octonions, which are non-associative) and even some non-alternative algebras like the sedenions and Okubo algebras. Any algebra whose elements are idempotent is also power-associative. Exponentiation to the power of any positive integer can be defined consistently whenever multiplication is power-associative. For example, there is no need to distinguish whether ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nathan Jacobson
Nathan Jacobson (October 5, 1910 – December 5, 1999) was an American mathematician. Biography Born Nachman Arbiser in Warsaw, Jacobson emigrated to America with his family in 1918. He graduated from the University of Alabama in 1930 and was awarded a doctorate in mathematics from Princeton University in 1934. While working on his thesis, ''Non-commutative polynomials and cyclic algebras'', he was advised by Joseph Wedderburn. Jacobson taught and researched at Bryn Mawr College (1935–1936), the University of Chicago (1936–1937), the University of North Carolina at Chapel Hill (1937–1943), and Johns Hopkins University (1943–1947) before joining Yale University in 1947. He remained at Yale until his retirement. He was a member of the National Academy of Sciences and the American Academy of Arts and Sciences. He served as president of the American Mathematical Society from 1971 to 1973, and was awarded their highest honour, the Leroy P. Steele prize for lifetime achievement, ... [...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] |
Max Koecher
Max Koecher (; 20 January 1924 in Weimar – 7 February 1990, Lengerich (Westfalen), Lengerich) was a German mathematician. Biography Koecher studied mathematics and physics at the Georg-August-Universität in Göttingen. In 1951, he received his doctorate under Max Deuring with his work on Dirichlet series with functional equation where he introduced Koecher–Maass series. He qualified in 1954 at the Westfälische Wilhelms University in Münster. From 1962 to 1970, Koecher was department chair at the University of Munich. He retired in 1989. His main research area was the theory of Jordan algebras, where he introduced the Kantor–Koecher–Tits construction and the Koecher–Vinberg theorem. He discovered the Koecher boundedness principle in the theory of Siegel modular forms. References * * * External links *Max Koecher on Wikimedia Commons {{DEFAULTSORT:Koecher, Max 20th-century German mathematicians 1990 deaths 1924 births ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Associative Enveloping Algebra
A non-associative algebra (or distributive algebra) is an algebra over a field where the binary multiplication operation is not assumed to be associative. That is, an algebraic structure ''A'' is a non-associative algebra over a field ''K'' if it is a vector space over ''K'' and is equipped with a ''K''- bilinear binary multiplication operation ''A'' × ''A'' → ''A'' which may or may not be associative. Examples include Lie algebras, Jordan algebras, the octonions, and three-dimensional Euclidean space equipped with the cross product operation. Since it is not assumed that the multiplication is associative, using parentheses to indicate the order of multiplications is necessary. For example, the expressions (''ab'')(''cd''), (''a''(''bc''))''d'' and ''a''(''b''(''cd'')) may all yield different answers. While this use of ''non-associative'' means that associativity is not assumed, it does not mean that associativity is disallowed. In other words, "non-associative" means "not nec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
