An elementary construction of the geometric algebra

Macdonald, Alan

doi:10.1007/bf03161249

Cited by 17 publications

(18 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Proof. The proof is based on Macdonald [5]. From the definition of the scalar product it is obvious that G is diagonal.…”

Section: The Main Resultsmentioning

confidence: 99%

“…The axioms define a canonical simplified representation of a multivector. The main lemma is demonstrated in [5] and is repeated here for convenience:…”

Section: Simplification Identitiesmentioning

confidence: 99%

“…2. Construction of the algebra Cℓ p,q,r This section gives a concise self-contained construction of the Clifford algebra based on the construction of Macdonald [5,10]. The construction is repeated here for consistency of the presentation.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A Symbolic Algorithm for Computation of Non-degenerate Clifford Algebra Matrix Representations

Prodanov¹

2019

Preprint

View full text Add to dashboard Cite

Clifford algebras are an active area of mathematical research. The main objective of the paper is to exhibit a construction of a matrix algebra isomorphic to a Clifford algebra of signature (p,q), which can be automatically implemented using general purpose linear algebra software. While this is not the most economical way of implementation for lower-dimensional algebras it offers a transparent mechanism of translation between a Clifford algebra and its isomorphic faithful real matrix representation. Examples of lower dimensional Clifford algebras are presented.

show abstract

“…Proof. The proof is based on Macdonald [5]. From the definition of the scalar product it is obvious that G is diagonal.…”

Section: The Main Resultsmentioning

confidence: 99%

“…The axioms define a canonical simplified representation of a multivector. The main lemma is demonstrated in [5] and is repeated here for convenience:…”

Section: Simplification Identitiesmentioning

confidence: 99%

See 1 more Smart Citation

A Symbolic Algorithm for Computation of Non-degenerate Clifford Algebra Matrix Representations

Prodanov¹

2019

Preprint

View full text Add to dashboard Cite

show abstract

“…The basics of geometric algebra can be found in [10][11][12]. Here we consider a vector space V over a field F. In all what follows, we set F = R. The dimension n of V is finite.…”

Section: Basic Conceptsmentioning

confidence: 99%

Chirality in Geometric Algebra

Petitjean

2021

Mathematics

View full text Add to dashboard Cite

show abstract

“…Geometric algebra is not an arbitrary abstraction. Rather, as its adherents (author included) like to insist, it is the natural language for discussing oriented lengths, areas, and volumes in physical space [18,19]. Its basic objects are multivectors, which can be visualized as (sums of, see below) k-dimensional parallelepipeds or oriented k-planes through the origin in N -dimensional space.…”

Section: Geometric Algebramentioning

confidence: 99%

Geometric calculus on pseudo-Riemannian manifolds

Schindler¹

2019

Preprint

View full text Add to dashboard Cite

This is the first entry in a planned series aiming to establish a modified, and simpler, formalism for studying the geometry of smooth manifolds with a metric, while remaining close to standard textbook treatments in terms of notation and concepts. The key step is extending the tangent space at each point from a vector space to a geometric algebra, which is a linear space incorporating vectors with dot and wedge multiplication, and extending the affine connection to a directional derivative acting naturally on fields of multivectors (elements of the geometric algebra). A short introduction to geometric algebra is included in the text. The theory that results from this extension is simpler and more powerful than either differential forms or tensor methods, in a number of ways. The multivector directional derivative obeys a powerful product rule. Simple conditions are obtained for metric-compatibility and torsion-freeness of the connection coefficients, and derivatives with torsion are treated generally. The covariant derivative of tensor fields is derived from a simple chain rule. Arbitrary metric signatures are treated in generality. The curved-manifold equivalents of the gradient, divergence, and curl operators are investigated, and the torsion-free curl is shown to be equivalent to the exterior derivative of differential forms. Unlike most traditional treatments, the entire formalism is developed in terms of completely arbitrary vector bases, which might be neither coordinate bases nor orthonormal. Methods of geometric algebra have previously been applied to vector calculus with great success, and extended to curved spaces in several ways (notably including "vector manifolds" and "gauge theory gravity"). Here we provide a new way of extending geometric calculus to curved manifolds, which more closely parallels standard Riemannian geometry.

show abstract

An elementary construction of the geometric algebra

Abstract: We give a simple, elementary, direct, and motivated construction of the geometric algebra over R n .Mathematics Subject Classification 2000. Primary 15A66.

Cited by 17 publications

References 4 publications

A Symbolic Algorithm for Computation of Non-degenerate Clifford Algebra Matrix Representations

A Symbolic Algorithm for Computation of Non-degenerate Clifford Algebra Matrix Representations

Chirality in Geometric Algebra

Geometric calculus on pseudo-Riemannian manifolds

Contact Info

Product

Resources

About