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A Clifford algebra is a unital associative algebra that contains and is generated by a vector space V over a field K, where V is equipped with a quadratic form Q : V → K.The Clifford algebra Cl(V, Q) is the "freest" unital associative algebra generated by V subject to the condition [c] = , where the product on the left is that of the algebra, and the 1 on the right is the algebra's ...
In even dimension n, the Clifford algebra Cl n (C) is isomorphic to End(C N), which has its fundamental representation on Δ n := C N. A complex Dirac spinor is an element of Δ n. The term complex signifies that it is the element of a representation space of a complex Clifford algebra, rather than that is an element of a complex vector space.
In mathematics, a geometric algebra (also known as a Clifford algebra) is an algebra that can represent and manipulate geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the geometric product. Multiplication of vectors results in higher-dimensional objects called multivectors ...
In Clifford analysis one also considers differential operators on upper half space, the disc, or hyperbola with respect to the hyperbolic, or Poincaré metric. For upper half space one splits the Clifford algebra, Cl n into Cl n−1 + Cl n−1 e n. So for a in Cl n one may express a as b + ce n with a, b in Cl n−1.
William Kingdon Clifford (4 May 1845 – 3 March 1879) was a British mathematician and philosopher.Building on the work of Hermann Grassmann, he introduced what is now termed geometric algebra, a special case of the Clifford algebra named in his honour.
In 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 extension L of the field K over which the quadratic form Q defining C is defined. The abstract theory of Clifford modules was founded by a paper of M. F. Atiyah, R. Bott and Arnold S. Shapiro.
In mathematics, a generalized Clifford algebra (GCA) is a unital associative algebra that generalizes the Clifford algebra, and goes back to the work of Hermann Weyl, [1] who utilized and formalized these clock-and-shift operators introduced by J. J. Sylvester (1882), [2] and organized by Cartan (1898) [3] and Schwinger.
In multilinear algebra, a multivector, sometimes called Clifford number or multor, [1] is an element of the exterior algebra Λ(V) of a vector space V.This algebra is graded, associative and alternating, and consists of linear combinations of simple k-vectors [2] (also known as decomposable k-vectors [3] or k-blades) of the form