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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 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 ...
The Clifford product is the manifest ring product for the Clifford algebra, and all algebra homomorphisms in this article are with respect to this ring product. Other products defined within Clifford algebras, such as the exterior product, and other structure, such as the distinguished subspace of generators V, are not used here.
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 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.
In mathematics, a Clifford bundle is an algebra bundle whose fibers have the structure of a Clifford algebra and whose local trivializations respect the algebra structure. There is a natural Clifford bundle associated to any ( pseudo ) Riemannian manifold M which is called the Clifford bundle of M .
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. The operations of geometric algebra have the effect of mirroring, rotating, translating ...
The Clifford algebra is an associative algebra that can be constructed from Euclidean space and its inner product in a basis-independent way. Both the 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 to work with.