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arxiv 0907.5356 v1 pith:XQFMYEPW submitted 2009-07-30 math-ph math.MPmath.RA

Clifford algebra, geometric algebra, and applications

classification math-ph math.MPmath.RA
keywords algebraapplicationscliffordgeometricwellalgebrastheoryalgebraically
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These are lecture notes for a course on the theory of Clifford algebras, with special emphasis on their wide range of applications in mathematics and physics. Clifford algebra is introduced both through a conventional tensor algebra construction (then called geometric algebra) with geometric applications in mind, as well as in an algebraically more general form which is well suited for combinatorics, and for defining and understanding the numerous products and operations of the algebra. The various applications presented include vector space and projective geometry, orthogonal maps and spinors, normed division algebras, as well as simplicial complexes and graph theory.

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Cited by 3 Pith papers

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  1. Equivariant Quantum Cohomology of Grassmannians via the Clifford algebra

    math.CO 2026-06 unverdicted novelty 7.0

    Constructs equivariant quantum Satake map for Grassmannians via Clifford algebra to derive recurrence relations for equivariant Gromov-Witten invariants and prove Graham positivity.

  2. Generalize cross-ratios in n-dimensional Plane-Based Geometric Algebra

    cs.CG 2026-05 unverdicted novelty 7.0

    Generalizes projective cross-ratios to every grade of geometric object in n-dimensional PGA with explicit formulas, duality pairs, and recovery of classical invariants.

  3. On the geometric algebras of the Ising model

    cond-mat.stat-mech 2026-04 unverdicted novelty 3.0

    The Ising transfer matrix and its excitations are re-expressed as elements of a conformal Clifford algebra, with the eigenvalue problem becoming a dispersion relation for free Majorana quasiparticles.