Universal geometric algebra (original) (raw)
In mathematics, a universal geometric algebra is a type of geometric algebra generated by real vector spaces endowed with an indefinite quadratic form. Some authors restrict this to the infinite-dimensional case. When n = ∞, i.e. there are countably many dimensions, then is called simply the universal geometric algebra (UGA), which contains vector spaces such as Rp, q and their respective geometric algebras . UGA contains all finite-dimensional geometric algebras (GA). One may generate a finite-dimensional GA by choosing a unit pseudoscalar (I). The set of all vectors that satisfy
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| dbo:abstract | In mathematics, a universal geometric algebra is a type of geometric algebra generated by real vector spaces endowed with an indefinite quadratic form. Some authors restrict this to the infinite-dimensional case. The universal geometric algebra of order 22n is defined as the Clifford algebra of 2n-dimensional pseudo-Euclidean space Rn, n. This algebra is also called the "mother algebra". It has a nondegenerate signature. The vectors in this space generate the algebra through the geometric product. This product makes the manipulation of vectors more similar to the familiar algebraic rules, although non-commutative. When n = ∞, i.e. there are countably many dimensions, then is called simply the universal geometric algebra (UGA), which contains vector spaces such as Rp, q and their respective geometric algebras . UGA contains all finite-dimensional geometric algebras (GA). The elements of UGA are called multivectors. Every multivector can be written as the sum of several r-vectors. Some r-vectors are scalars (r = 0), vectors (r = 1) and bivectors (r = 2). One may generate a finite-dimensional GA by choosing a unit pseudoscalar (I). The set of all vectors that satisfy is a vector space. The geometric product of the vectors in this vector space then defines the GA, of which I is a member. Since every finite-dimensional GA has a unique I (up to a sign), one can define or characterize the GA by it. A pseudoscalar can be interpreted as an n-plane segment of unit area in an n-dimensional vector space. (en) |
| dbo:wikiPageExternalLink | https://books.google.com/books%3Fid=O7jO2pmUFOgC&pg=PA371 https://books.google.com/books%3Fid=VW4yt0WHdjoC%7Cdate=2003-05-29 https://books.google.com/books%3Fid=bUEcbyfW55YC%7Cyear=2008 https://books.google.com/books%3Fid=dScR5zwrheYC%7Cdate=1987-08-31 |
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| rdfs:comment | In mathematics, a universal geometric algebra is a type of geometric algebra generated by real vector spaces endowed with an indefinite quadratic form. Some authors restrict this to the infinite-dimensional case. When n = ∞, i.e. there are countably many dimensions, then is called simply the universal geometric algebra (UGA), which contains vector spaces such as Rp, q and their respective geometric algebras . UGA contains all finite-dimensional geometric algebras (GA). One may generate a finite-dimensional GA by choosing a unit pseudoscalar (I). The set of all vectors that satisfy (en) |
| rdfs:label | Universal geometric algebra (en) |
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