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Toric Hyperkahler Manifold
In mathematics, a hypertoric variety or toric hyperkähler variety is a quaternionic analog of a toric variety constructed by applying the hyper-Kähler quotient construction of to a torus acting on a quaternionic vector space In mathematics, a left (or right) quaternionic vector space is a left (or right) H-module (mathematics), module where H is the (non-commutative) division ring of quaternions. The space H''n'' of ''n''-tuples of quaternions is both a left and right .... gave a systematic description of hypertoric varieties. References * *{{citation, mr=0877637 , last1=Hitchin, first1= N. J., last2= Karlhede, first2= A., last3= Lindström, first3= U., last4= Roček, first4= M. , title=Hyper-Kähler metrics and supersymmetry , journal=Communications in Mathematical Physics, volume= 108 , year=1987, issue= 4, pages= 535–589, doi=10.1007/BF01214418, s2cid=120041594 , url=http://projecteuclid.org/euclid.cmp/1104116624 Algebraic geometry ... [...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 poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quaternion as the quotient of two '' directed lines'' in a three-dimensional space, or, equivalently, as the quotient of two vectors. Multiplication of quaternions is noncommutative. Quaternions are generally represented in the form :a + b\ \mathbf i + c\ \mathbf j +d\ \mathbf k where , and are real numbers; and , and are the ''basic quaternions''. Quaternions are used in pure mathematics, but also have practical uses in applied mathematics, particularly for calculations involving three-dimensional rotations, such as in three-dimensional computer graphics, computer vision, and crystallographic texture analysis. They can be used alongside other methods of rotation, such as Euler angles and rotation matrices, or as an alternative to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Toric Variety
In algebraic geometry, a toric variety or torus embedding is an algebraic variety containing an algebraic torus as an open dense subset, such that the action of the torus on itself extends to the whole variety. Some authors also require it to be normal. Toric varieties form an important and rich class of examples in algebraic geometry, which often provide a testing ground for theorems. The geometry of a toric variety is fully determined by the combinatorics of its associated fan, which often makes computations far more tractable. For a certain special, but still quite general class of toric varieties, this information is also encoded in a polytope, which creates a powerful connection of the subject with convex geometry. Familiar examples of toric varieties are affine space, projective spaces, products of projective spaces and bundles over projective space. Toric varieties from tori The original motivation to study toric varieties was to study torus embeddings. Given the algebrai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quaternionic Vector Space
In mathematics, a left (or right) quaternionic vector space is a left (or right) H-module where H is the (non-commutative) division ring of quaternions. The space H''n'' of ''n''-tuples of quaternions is both a left and right H-module using the componentwise left and right multiplication: : q (q_1,q_2,\ldots q_n) = (q q_1,q q_2,\ldots q q_n) : (q_1,q_2,\ldots q_n) q = (q_1 q, q_2 q,\ldots q_n q) for quaternions ''q'' and ''q''1, ''q''2, ... ''q''''n''. Since H is a division algebra, every finitely generated (left or right) H-module has a basis, and hence is isomorphic to H''n'' for some ''n''. See also * Vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ... * General linear group * Special linear group * SL(n,H) * Symplectic group References * Quaternions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
