|
Biquaternion Algebra
In mathematics, a biquaternion algebra is a compound of quaternion algebras over a field. The biquaternions of William Rowan Hamilton (1844) and the related split-biquaternions and dual quaternions do not form biquaternion algebras in this sense. Definition Let ''F'' be a field of characteristic not equal to 2. A ''biquaternion algebra'' over ''F'' is a tensor product of two quaternion algebras.Lam (2005) p.60Szymiczek (1997) p.452 A biquaternion algebra is a central simple algebra of dimension 16 and degree 4 over the base field: it has exponent (order of its Brauer class in the Brauer group of ''F'') equal to 1 or 2. Albert's theorem Let ''A'' = (''a''1,''a''2) and ''B'' = (''b''1,''b''2) be quaternion algebras over ''F''. The Albert form for ''A'', ''B'' is :q = \left\langle\right\rangle \ . It can be regarded as the difference in the Witt ring of the ternary forms attached to the imaginary subspaces of ''A'' and ''B''.Knus et al (1991) p.192 The quaternion algebras are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quaternion Algebra
In mathematics, a quaternion algebra over a field ''F'' is a central simple algebra ''A'' over ''F''See Milies & Sehgal, An introduction to group rings, exercise 17, chapter 2. that has dimension 4 over ''F''. Every quaternion algebra becomes a matrix algebra by '' extending scalars'' (equivalently, tensoring with a field extension), i.e. for a suitable field extension ''K'' of ''F'', A \otimes_F K is isomorphic to the 2 × 2 matrix algebra over ''K''. The notion of a quaternion algebra can be seen as a generalization of Hamilton's quaternions to an arbitrary base field. The Hamilton quaternions are a quaternion algebra (in the above sense) over F = \mathbb, and indeed the only one over \mathbb apart from the 2 × 2 real matrix algebra, up to isomorphism. When F = \mathbb, then the biquaternions form the quaternion algebra over ''F''. Structure ''Quaternion algebra'' here means something more general than the algebra of Hamilton's quaternions. When th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Albert Form
In mathematics, a linked field is a field for which the quadratic forms attached to quaternion algebras have a common property. Linked quaternion algebras Let ''F'' be a field of characteristic not equal to 2. Let ''A'' = (''a''1,''a''2) and ''B'' = (''b''1,''b''2) be quaternion algebras over ''F''. The algebras ''A'' and ''B'' are linked quaternion algebras over ''F'' if there is ''x'' in ''F'' such that ''A'' is equivalent to (''x'',''y'') and ''B'' is equivalent to (''x'',''z''). The Albert form for ''A'', ''B'' is :q = \left\langle\right\rangle \ . It can be regarded as the difference in the Witt ring of the ternary forms attached to the imaginary subspaces of ''A'' and ''B''. The quaternion algebras are linked if and only if the Albert form is isotropic. Linked fields The field ''F'' is ''linked'' if any two quaternion algebras over ''F'' are linked. Every global and local field is linked since all quadratic forms of degree 6 over such fields are isotropic. The fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe was the first president and Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance, due to concerns about competing with the American Journal of Mathematics. The result was the ''Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influential in in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Witt Index
:''"Witt's theorem" or "the Witt theorem" may also refer to the Bourbaki–Witt fixed point theorem of order theory.'' In mathematics, Witt's theorem, named after Ernst Witt, is a basic result in the algebraic theory of quadratic forms: any isometry between two subspaces of a nonsingular quadratic space over a field ''k'' may be extended to an isometry of the whole space. An analogous statement holds also for skew-symmetric, Hermitian and skew-Hermitian bilinear forms over arbitrary fields. The theorem applies to classification of quadratic forms over ''k'' and in particular allows one to define the Witt group ''W''(''k'') which describes the "stable" theory of quadratic forms over the field ''k''. Statement Let be a finite-dimensional vector space over a field ''k'' of characteristic different from 2 together with a non-degenerate symmetric or skew-symmetric bilinear form. If is an isometry between two subspaces of ''V'' then ''f'' extends to an isometry of ''V''. Witt's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schur Index
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 ''Brauer e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Division Algebra
In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible. Definitions Formally, we start with a non-zero algebra ''D'' over a field. We call ''D'' a division algebra if for any element ''a'' in ''D'' and any non-zero element ''b'' in ''D'' there exists precisely one element ''x'' in ''D'' with ''a'' = ''bx'' and precisely one element ''y'' in ''D'' such that . For associative algebras, the definition can be simplified as follows: a non-zero associative algebra over a field is a division algebra if and only if it has a multiplicative identity element 1 and every non-zero element ''a'' has a multiplicative inverse (i.e. an element ''x'' with ). Associative division algebras The best-known examples of associative division algebras are the finite-dimensional real ones (that is, algebras over the field R of real numbers, which are finite- dimensional as a vector space ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abraham Adrian Albert
Abraham Adrian Albert (November 9, 1905 – June 6, 1972) was an American mathematician. In 1939, he received the American Mathematical Society's Cole Prize in Algebra for his work on Riemann matrices. He is best known for his work on the Albert–Brauer–Hasse–Noether theorem on finite-dimensional division algebras over number fields and as the developer of Albert algebras, which are also known as exceptional Jordan algebras. Professional overview A first generation American, he was born in Chicago and most associated with that city. He received his Bachelor of Science in 1926, Masters in 1927, and PhD in 1928, at the age of 22. All degrees were obtained from the University of Chicago. He married around the same time as his graduation. He spent his postdoctoral year at Princeton University and then from 1929 to 1931 he was an instructor at Columbia University. During this period he worked on Abelian varieties and their endomorphism algebras. He returned to Princeto ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isotropic Quadratic Form
In mathematics, a quadratic form over a field ''F'' is said to be isotropic if there is a non-zero vector on which the form evaluates to zero. Otherwise the quadratic form is anisotropic. More precisely, if ''q'' is a quadratic form on a vector space ''V'' over ''F'', then a non-zero vector ''v'' in ''V'' is said to be isotropic if . A quadratic form is isotropic if and only if there exists a non-zero isotropic vector (or null vector) for that quadratic form. Suppose that is quadratic space and ''W'' is a subspace of ''V''. Then ''W'' is called an isotropic subspace of ''V'' if ''some'' vector in it is isotropic, a totally isotropic subspace if ''all'' vectors in it are isotropic, and an anisotropic subspace if it does not contain ''any'' (non-zero) isotropic vectors. The of a quadratic space is the maximum of the dimensions of the totally isotropic subspaces. A quadratic form ''q'' on a finite-dimensional real vector space ''V'' is anisotropic if and only if ''q'' is a defin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linked Quaternion Algebras
In mathematics, a linked field is a field for which the quadratic forms attached to quaternion algebras have a common property. Linked quaternion algebras Let ''F'' be a field of characteristic not equal to 2. Let ''A'' = (''a''1,''a''2) and ''B'' = (''b''1,''b''2) be quaternion algebras over ''F''. The algebras ''A'' and ''B'' are linked quaternion algebras over ''F'' if there is ''x'' in ''F'' such that ''A'' is equivalent to (''x'',''y'') and ''B'' is equivalent to (''x'',''z''). The Albert form for ''A'', ''B'' is :q = \left\langle\right\rangle \ . It can be regarded as the difference in the Witt ring of the ternary forms attached to the imaginary subspaces of ''A'' and ''B''. The quaternion algebras are linked if and only if the Albert form is isotropic. Linked fields The field ''F'' is ''linked'' if any two quaternion algebras over ''F'' are linked. Every global and local field is linked since all quadratic forms of degree 6 over such fields are isotropic. The fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Witt Ring (forms)
In mathematics, a Witt group of a field, named after Ernst Witt, is an abelian group whose elements are represented by symmetric bilinear forms over the field. Definition Fix a field ''k'' of characteristic not equal to two. All vector spaces will be assumed to be finite-dimensional. We say that two spaces equipped with symmetric bilinear forms are equivalent if one can be obtained from the other by adding a metabolic quadratic space, that is, zero or more copies of a hyperbolic plane, the non-degenerate two-dimensional symmetric bilinear form with a norm 0 vector.Milnor & Husemoller (1973) p. 14 Each class is represented by the core form of a Witt decomposition.Lorenz (2008) p. 30 The Witt group of ''k'' is the abelian group ''W''(''k'') of equivalence classes of non-degenerate symmetric bilinear forms, with the group operation corresponding to the orthogonal direct sum of forms. It is additively generated by the classes of one-dimensional forms.Milnor & Husemoll ... [...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] |
Biquaternion
In abstract algebra, the biquaternions are the numbers , where , and are complex numbers, or variants thereof, and the elements of multiply as in the quaternion group and commute with their coefficients. There are three types of biquaternions corresponding to complex numbers and the variations thereof: * Biquaternions when the coefficients are complex numbers. * Split-biquaternions when the coefficients are split-complex numbers. * Dual quaternions when the coefficients are dual numbers. This article is about the ''ordinary biquaternions'' named by William Rowan Hamilton in 1844 (see ''Proceedings of the Royal Irish Academy'' 1844 & 1850 page 388). Some of the more prominent proponents of these biquaternions include Alexander Macfarlane, Arthur W. Conway, Ludwik Silberstein, and Cornelius Lanczos. As developed below, the unit quasi-sphere of the biquaternions provides a representation of the Lorentz group, which is the foundation of special relativity. The algebra of biquatern ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |