Clifford algebra is a mathematical structure that extends the concept of vector spaces and inner products into a more generalized algebraic framework. It is named after the mathematician William Kingdon Clifford, who developed the concept in the 19th century. ### Definition A Clifford algebra is generated from a vector space \( V \) equipped with a quadratic form.
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For a vector space over a field of characteristic different from two and a quadratic form , its Clifford algebra is the quotient of the tensor algebra by the relations . Writing gives .
For a real two-dimensional positive-definite form and an orthonormal basis , the generators satisfy and . The algebra has basis . Its Clifford modules provide spinor spaces, while products of suitable unit vectors represent rotations. The sign convention on must be stated, especially when comparing geometric and physical conventions.