Summary
In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal transformations. Clifford algebras have important applications in a variety of fields including geometry, theoretical physics and . They are named after the English mathematician William Kingdon Clifford (1845–1879). The most familiar Clifford algebras, the orthogonal Clifford algebras, are also referred to as (pseudo-)Riemannian Clifford algebras, as distinct from symplectic Clifford algebras. 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 where the product on the left is that of the algebra, and the 1 is its multiplicative identity. The idea of being the "freest" or "most general" algebra subject to this identity can be formally expressed through the notion of a universal property, as done below. When V is a finite-dimensional real vector space and Q is nondegenerate, Cl(V, Q) may be identified by the label Clp,q(R), indicating that V has an orthogonal basis with p elements with ei2 = +1, q with ei2 = −1, and where R indicates that this is a Clifford algebra over the reals; i.e. coefficients of elements of the algebra are real numbers. This basis may be found by orthogonal diagonalization. The free algebra generated by V may be written as the tensor algebra ⨁n≥0 V ⊗ ⋯ ⊗ V, that is, the direct sum of the tensor product of n copies of V over all n. Therefore one obtains a Clifford algebra as the quotient of this tensor algebra by the two-sided ideal generated by elements of the form v ⊗ v − Q(v)1 for all elements v ∈ V.
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
Related courses (14)
MATH-432: Probability theory
The course is based on Durrett's text book Probability: Theory and Examples.
It takes the measure theory approach to probability theory, wherein expectations are simply abstract integrals.
MATH-334: Representation theory
Study the basics of representation theory of groups and associative algebras.
MATH-115(b): Advanced linear algebra II
L'objectif du cours est d'introduire les notions de base de l'algèbre linéaire et de démontrer rigoureusement les résultats principaux du sujet.
Show more
Related publications (33)