mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra with the additional...
64 KB (9,177 words) - 21:08, 17 October 2024
Clifford algebra is algebra isomorphic to a full matrix ring over R, C, or H (the quaternions), or to a direct sum of two copies of such an algebra,...
20 KB (2,419 words) - 21:43, 2 September 2024
geometric algebra (also known as a Clifford algebra) is an algebra that can represent and manipulate geometrical objects such as vectors. Geometric algebra is...
93 KB (13,917 words) - 17:13, 18 October 2024
Spinor (section Clifford algebras)
spin group and its Lie algebra are embedded inside the Clifford algebra in a natural way, and in applications the Clifford algebra is often the easiest...
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Hypercomplex number (redirect from Hypercomplex algebra)
be a Clifford algebra over the reals—i.e. coefficients of elements of the algebra are to be real numbers. These algebras, called geometric algebras, form...
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the Clifford algebra, which is also referred to as the orthogonal Clifford algebra. The Weyl algebra is also referred to as the symplectic Clifford algebra...
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mathematics, a Clifford module is a representation of a Clifford algebra. In general a Clifford algebra C is a central simple algebra over some field...
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distillation and for quantum error correction. Magic state distillation Clifford algebra Clifford gates Nielsen, Michael A.; Chuang, Isaac L. (2010-12-09). Quantum...
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is now termed geometric algebra, a special case of the Clifford algebra named in his honour. The operations of geometric algebra have the effect of mirroring...
37 KB (4,225 words) - 22:37, 15 September 2024
In mathematics, a generalized Clifford algebra (GCA) is a unital associative algebra that generalizes the Clifford algebra, and goes back to the work of...
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Quaternion (category Composition algebras)
division algebra over the real numbers, and therefore a ring, also a division ring and a domain. It is a special case of a Clifford algebra, classified...
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by Lee (1948) and Chevalley (1954) using Clifford algebras. Hurwitz's theorem has been applied in algebraic topology to problems on vector fields on spheres...
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directed by Paul Flaherty Clifford (Muppet) Clifford algebra, a type of associative algebra, named after William Kingdon Clifford Clifford analysis, a mathematical...
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Clifford algebra used as forms in Geometric Algebra rather than matrices. Recognising the geometric operations, such as rotations, allows the algebra...
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[citation needed] The Weyl algebra An Azumaya algebra The Clifford algebras, which are useful in geometry and physics. Incidence algebras of locally finite partially...
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Gamma matrices (category Clifford algebras)
relations that ensure they generate a matrix representation of the Clifford algebra C l 1 , 3 ( R ) . {\displaystyle \ \mathrm {Cl} _{1,3}(\mathbb...
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{\displaystyle Z} -angle rotations are known. Magic state distillation Clifford algebra Gottesman, Daniel (1998-01-01). "Theory of fault-tolerant quantum computation"...
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Spin group (redirect from Spin algebra)
can be constructed as a subgroup of the invertible elements in the Clifford algebra Cl(n). A distinct article discusses the spin representations. The spin...
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like an algebra over a field. Algebra over an operad Alternative algebra Clifford algebra Composition algebra Differential algebra Free algebra Geometric...
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Pauli matrices (redirect from Pauli algebra)
special unitary group SU(2). The algebra generated by the three matrices σ1, σ2, σ3 is isomorphic to the Clifford algebra of R 3 , {\displaystyle \mathbb...
45 KB (7,437 words) - 13:49, 18 October 2024
Clifford analysis, using Clifford algebras named after William Kingdon Clifford, is the study of Dirac operators, and Dirac type operators in analysis...
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Octonion (redirect from Dixon algebra)
gauge from the complex Clifford algebra C ℓ 6 {\displaystyle \mathbb {C} \ell _{6}} ", Advances in Applied Clifford Algebras, 28: 52, arXiv:1702.04336...
40 KB (4,962 words) - 18:49, 12 October 2024
Plane-based geometric algebra is an application of Clifford algebra to modelling planes, lines, points, and rigid transformations. Generally this is with...
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Special unitary group (redirect from Special unitary Lie algebra)
Since the quaternions can be identified as the even subalgebra of the Clifford Algebra Cl(3), SU(2) is in fact identical to one of the symmetry groups of...
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category Symplectic Clifford algebra, see Weyl algebra Symplectic geometry Symplectic group, and corresponding symplectic Lie algebra Symplectic integrator...
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Pure spinor (section Clifford algebra and pure spinors)
spinors (or simple spinors) are spinors that are annihilated, under the Clifford algebra representation, by a maximal isotropic subspace of a vector space V...
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In mathematics, a Lie algebra (pronounced /liː/ LEE) is a vector space g {\displaystyle {\mathfrak {g}}} together with an operation called the Lie bracket...
61 KB (10,459 words) - 23:14, 17 September 2024
spacetime algebra (STA) is the application of Clifford algebra Cl1,3(R), or equivalently the geometric algebra G(M4) to physics. Spacetime algebra provides...
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interest in the theory of solitons and integrable systems. In the Clifford algebra C ℓ ( 2 ) {\displaystyle C\ell (2)} , the complex number z = x + i...
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Cross product (section Lie algebra)
as the cross product. In 1878, William Kingdon Clifford, known for a precursor to the Clifford algebra named in his honor, published Elements of Dynamic...
75 KB (11,475 words) - 18:39, 5 September 2024