Skip to main content
SpringerLink
Log in

Postfaces

  • Chapter
  • First Online:
Book cover Real Spinorial Groups

Part of the book series: SpringerBriefs in Mathematics ((BRIEFSMATH))

  • 428 Accesses

Abstract

This chapter is a collection of notes. Most of them are pointed out at suitable spots in the text of the preceding chapters, but an effort has been aimed to ensure that it may be read on its own as a closing chapter. The numbered sections, 6.1–6.5, correspond to Chaps. 1–5, respectively, while this prelude (6.0) corresponds to the Preface.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 44.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 59.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    https://mat-web.upc.edu/people/sebastia.xambo/agacse2015/.

  2. 2.

    Fortunately, part of Sancho’s teachings have recently been systematically written up by Juan Antonio Navarro [111].

  3. 3.

    https://www.mathunion.org/icmi/activities/klein-project.

  4. 4.

    https://mat-web.upc.edu/people/sebastia.xambo/santalo2016/.

  5. 5.

    https://mat-web.upc.edu/people/sebastia.xambo/GA/2016-Xambo-csasc.pdf.

  6. 6.

    It is the topology induced by any Euclidean norm ||v|| on E, an (auxiliary) notation that we will use henceforth. One way of choosing an auxiliary norm is by transferring the standard norm of \(\mathbb {R}^n\) by the isomorphism \(E\simeq \mathbb {R}^n\) induced by a basis of E. In practical terms, this means that to take the limit of a sequence of vectors in E is reduced to taking the limits in \(\mathbb {R}\) of the components of the vectors in the sequence. To avoid any confusion, recall that the magnitude |v| of v relative to a metric q (of arbitrary signature) is defined to be \(\sqrt {\varepsilon _{\boldsymbol {v}}q(\boldsymbol {v})}\). Only if q is positive definite can we use the magnitude also as an auxiliary norm.

  7. 7.

    If U and V  are vector subspaces of a vector space, then

    $$\displaystyle \begin{aligned} \dim(U+V)=\dim(U)+\dim(V)-\dim(U\cap V). \end{aligned}$$
  8. 8.

    http://www-history.mcs.st-andrews.ac.uk/Biographies/Grassmann.html.

  9. 9.

    In fact, in Perwass’ text Clifford only appears in the term “Clifford group,” which actually corresponds to (a variation of) the Lipschitz group.

References

  1. R. Abłamowicz (ed.), Clifford Algebras: Applications to Mathematics, Physics, and Engineering. Progress in Mathematical Physics, vol. 34 (Birkhäuser, Boston, 2004)

    Google Scholar 

  2. W. Achtner, From religion to dialectics and mathematics. Stud. Log. Grammar Rhetor. 44(1), 111–131 (2016)

    Google Scholar 

  3. J.F. Adams, Lectures on Lie Groups (Benjamin, Reading, 1969)

    MATH  Google Scholar 

  4. P. Angles, Conformal Groups in Geometry and Spin Structures. Progress in Mathematical Physics, vol. 50 (Birkhäuser, Boston, 2008), xvii+283 pp.

    Google Scholar 

  5. M. Arrayás, D. Bouwmeester, J.L. Trueba, Knots in electromagnetism. Phys. Rep. 667, 1–61 (2017)

    MathSciNet  MATH  Google Scholar 

  6. E. Artin, Geometric Algebra. Tracts in Pure and Applied Mathematics, vol. 3 (Interscience Publishers, New York, 1957)

    Google Scholar 

  7. M. Atiyah, R. Bott, A. Shapiro, Clifford modules. Topology 3(Suppl. 1), 3–38 (1964)

    MathSciNet  MATH  Google Scholar 

  8. W.E. Baylis, Electrodynamics. A Modern Geometric Approach. Progress in Mathematical Physics (Birkhäuser, Boston, 1999)

    Google Scholar 

  9. E. Bayro-Corrochano, Geometric Computing for Perception Action Systems: Concepts, Algorithms, and Scientific Applications (Springer, New York, 2001)

    MATH  Google Scholar 

  10. E. Bayro-Corrochano, Handbook of Geometric Computing: Applications in Pattern Recognition, Computer Vision, Neuralcomputing, and Robotics (Springer, Berlin, 2005), xv+779 pp.

    Google Scholar 

  11. E. Bayro-Corrochano, Geometric Computing: For Wavelet Transforms, Robot Vision, Learning, Control and Action (Springer, London, 2010)

    MATH  Google Scholar 

  12. E. Bayro-Corrochano, G. Scheuermann (eds.), Geometric Algebra Computing: In Engineering and Computer Science (Springer, London, 2010)

    MATH  Google Scholar 

  13. E. Bayro-Corrochano, G. Sobczyk, Geometric Algebra with Applications in Science and Engineering (Birkhäuser, Boston, 2001)

    MATH  Google Scholar 

  14. G. Bellamy, Lie Groups, Lie Algebras, and Their Representations, 2016. Lecture notes available at https://www.maths.gla.ac.uk/~gbellamy/lie.pdf

  15. I.M. Benn, R.W. Tucker, An Introduction to Spinors and Geometry with Applications in Physics (Adam Hilger, Bristol, 1987), x+358 pp.

    Google Scholar 

  16. J.P. Bourguignon, O. Hijazi, J.-L. Milhorat, A. Moroianu, A Spinorial Approach to Riemannian and Conformal Geometry. Monographs in Mathematics (European Mathematical Society, Zürich, 2015)

    Google Scholar 

  17. R. Brauer, H. Weyl, Spinors in n dimensions. Am. J. Math. 57, 230–254 (1935)

    MathSciNet  MATH  Google Scholar 

  18. T. Bröcker, T. Dieck, Representations of Compact Lie Groups. Graduate Texts in Mathematics, vol. 98 (Springer, New York, 1985)

    MATH  Google Scholar 

  19. P. Budinich, A. Trautman, The Spinorial Chessboard (Springer, Berlin, 1988)

    MATH  Google Scholar 

  20. D. Bump, Lie Groups. Graduate Texts in Mathematics, vol. 225 (Springer, Berlin, 2004)

    MATH  Google Scholar 

  21. J. Cameron, J. Lasenby, Oriented conformal geometric algebra. Adv. Appl. Clifford Algebr. 18(3–4), 523–538 (2008)

    MathSciNet  MATH  Google Scholar 

  22. J.F. Cariñena, A. Ibort, G. Marmo, G. Morandi, Geometry from Dynamics, Classical and Quantum (Springer, Berlin, 2015)

    MATH  Google Scholar 

  23. É. Cartan, Les groupes projectifs qui ne laissent invariante aucune multiplicité plane. Bull. Soc. Math. Fr. 41, 53–96 (1913)

    MATH  Google Scholar 

  24. É. Cartan, Leçons sur la théorie des spineurs (2 volumes). Actualités Scientifiques et Industrielles, vol. 643 (Hermann, Paris, 1937). Translated into English in 1966 (The theory of spinors, Hermann), and republished by Dover in 1981

    Google Scholar 

  25. R. Carter, G. Segal, I. Macdonald, Lectures on Lie Groups and Lie Algebras. LMS Student Texts, vol. 32 (Cambridge University Press, Cambridge, 1995). The book has three parts: Lie algebras and root systems (Carter, 1–44); Lie groups (Segal, 45–132); and Linear algebraic groups (Macdonald, 133–188)

    Google Scholar 

  26. D. Castelvecchi, The shape of things to come. Nature 547, 272–274 (2017)

    Google Scholar 

  27. C. Chevalley, The Algebraic Theory of Spinors (Columbia University Press, New York, 1954), viii+ 131 pp.

    Google Scholar 

  28. C. Chevalley, The Construction and Study of Certain Important Algebras (The Mathematical Society of Japan, Tokyo, 1955), vi+ 64 pp.

    Google Scholar 

  29. C. Chevalley, The Algebraic Theory of Spinors and Clifford Algebras. Collected Works, vol. 2 (Springer, Berlin, 1997). Includes [28] and [27], a Preface by C. Chevalley and P. Cartier, the review of [27] by J. Dieudonné, and a Postface by J.-P. Bourguignon with the title “Spinors in 1995”

    Google Scholar 

  30. W.K. Clifford, Applications of Grassmann’s extensive algebra. Am. J. Math. 1, 350–358 (1878)

    MathSciNet  MATH  Google Scholar 

  31. P. Colapinto, Spatial computing with conformal geometric algebra. Master’s thesis, University of California Santa Barbara, 2011. http://wolftype.com/versor/colapinto_masters_final_02.pdf

    Google Scholar 

  32. P. Colapinto, Articulating space: geometric algebra for parametric design – symmetry, kinematics, and curvature. PhD thesis, Media Arts and Technology Program, University of California Santa Barbara, 2016

    Google Scholar 

  33. A. Crumeyrolle, Orthogonal and Symplectic Clifford Algebras – Spinor Structures (Kluwer Academic Publishers, Dordrecht, 1990)

    MATH  Google Scholar 

  34. P.A.M. Dirac, The quantum theory of the electron, I, II. Proc. R. Soc. Lond. 118, A117:610–624 and A118:351–361 (1928)

    Google Scholar 

  35. D.Ž. Ðogović, K.H. Hofmann, The surjectivity question for the exponential function of real Lie groups: a status report. J. Lie Theory 7, 171–199 (1997)

    MathSciNet  MATH  Google Scholar 

  36. D.Z. Doković, The interior and the exterior of the image of the exponential map in classical Lie groups. J. Algebra 112, 90–109 (1988)

    Article  MathSciNet  Google Scholar 

  37. C. Doran, A. Lasenby, Geometric Algebra for Physicists (Cambridge University Press, Cambridge, 2003)

    Book  Google Scholar 

  38. L. Dorst, Structure preserving representation of Euclidean motions through Conformal Geometric Algebra, in Guide to Geometric Algebra in Practice, ed. by L. Dorst, J. Lasenby (Springer, Berlin, 2011)

    Chapter  Google Scholar 

  39. L. Dorst, J. Lasenby (eds.), Guide to Geometric Algebra in Practice (Springer, Berlin, 2011)

    MATH  Google Scholar 

  40. L. Dorst, C.J.L. Doran, J. Lasenby (eds.), Applications of Geometric Algebra in Computer Science and Engineering (Springer, Berlin, 2002)

    MATH  Google Scholar 

  41. L. Dorst, D. Fontijne, S. Mann, Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry (Morgan Kaufmann Publishers Inc., Amsterdam, 2007)

    Google Scholar 

  42. J.J. Duistermaat, J.A.C. Kolk, Lie Groups (Springer, Berlin, 2012)

    MATH  Google Scholar 

  43. J. Figueroa-O’Farrill, Spin geometry. http://empg.maths.ed.ac.uk/Activities/Spin/, versión 4/5/2010

  44. G.B. Folland, Quantum Field Theory. A Tourist Guide for Mathematicians. Mathematical Surveys and Monographs, vol. 149 (American Mathematical Society, Providence, 2008)

    Google Scholar 

  45. M. Forster, Friedrich Daniel Ernst Schleiermacher, in The Stanford Encyclopedia of Philosophy, ed. by E.N. Zalta, 2015 edition. https://plato.stanford.edu/archives/sum2015/entries/schleiermacher

  46. T. Frankel, The Geometry of Physics. An Introduction (Cambridge University Press, Cambridge, 2004)

    Google Scholar 

  47. W. Fulton, J. Harris, Representation Theory. A First Course. Graduate Texts in Mathematics (Springer, Berlin, 1991)

    Google Scholar 

  48. C. Furey, Charge quantization from a number operator. Phys. Lett. B 742, 195–199 (2015)

    Article  Google Scholar 

  49. C. Furey, A demonstration that electroweak theory can violate parity automatically (leptonic case). Int. J. Mod. Phys. A 33(4), 1830005 (10 pp.) (2018)

    Article  MathSciNet  Google Scholar 

  50. J.H. Gallier, Geometric Methods and Applications, For Computer Science and Engineering. Texts in Applied Mathematics, vol. 38, 2nd edn. (Springer, Berlin, 2011)

    Book  Google Scholar 

  51. J.H. Gallier, J. Quaintance, Notes on Differential Geometry of Lie Groups. Texts in Applied Mathematics, vol. 38 (Springer, Berlin, 2016)

    Google Scholar 

  52. D.J.H. Garling, Clifford Algebras: An Introduction. LMS Student Texts, vol. 78 (Cambridge University Press, Cambridge, 2011)

    Google Scholar 

  53. H. Georgi, Lie Algebras in Particle Physics (Westview/Perseus Books, Reading, 1999)

    Google Scholar 

  54. M. Gourdin, Basics of Lie Groups (Frontières, Gif-sur-Yvette, 1982)

    Google Scholar 

  55. H.G. Grassmann, Die lineale Ausdehnungslehre, ein neuer Zweig der Mathematik (Otto Wiegand, Leipzig, 1844)

    Google Scholar 

  56. H.G. Grassmann, Die Ausdehnungslehre. Vollständig und in strenger Form (Adolf Enslin, Berlin, 1862)

    Google Scholar 

  57. H.G. Grassmann, Extension Theory (American Mathematical Society, Providence, 2000). Translated from the German version Die Ausdehnungslehre von 1862 by Lloys C. Kannenberg

    Google Scholar 

  58. B. Hall, Lie Groups, Lie Algebras, and Representations: An Elementary Introduction. Graduate Texts in Mathematics, vol. 222 (Springer, Cham, 2003), xiv+351 pp. Second edition 2015, xiv+451 pp.

    Google Scholar 

  59. W.R. Hamilton, Lectures on Quaternions (Hodges and Smith, Dublin, 1853). http://ebooks.library.cornell.edu/m/math/index.php

  60. T.F. Havel, Distance geometry: theory, algorithms, and chemical applications. Encycl. Comput. Chem. 120, 723–742 (1998)

    Google Scholar 

  61. D. Hestenes, Space-Time Algebra (Gordon & Breach, New York, 1966). 2nd edition: Birkhäuser 2015, with a Foreword by A. Lasenby and new “Preface after fifty years” by the author

    Google Scholar 

  62. D. Hestenes, A unified language for mathematics and physics, in Clifford Algebras and Their Applications in Mathematical Physics, ed. by J.S.R. Chisholm, A.K. Commons (Reidel, Dordrecht, 1986), pp. 1–23

    Google Scholar 

  63. D. Hestenes, Grassmann’s vision, in Hermann Gunther Grasmann (1809–1877): Visionary Mathematician, Scientist and Neohumanist Scholar, ed. by G. Schubring (Kluwer, Boston, 1996), pp. 191–201

    Google Scholar 

  64. D. Hestenes, Real Dirac theory, 1996, in The Theory of the Electron, ed. by J. Keller, Z. Oziewicz (UNAM, México, 1996), pp. 1–50

    Google Scholar 

  65. D. Hestenes, New Foundations for Classical Mechanics. Fundamental Theories of Physics, vol. 99, 2nd edn. (Kluwer Academic Publishers, Dordrecht, 1999). 1st edition published 1990

    Google Scholar 

  66. D. Hestenes, Old wine in new bottles: a new algebraic framework for computational geometry, in Advances in Geometric Algebra with Applications in Science and Engineering, ed. by E. Bayro-Corrochano, G. Sobczyk (Birkhäuser, Boston, 2001), pp. 1–14

    Google Scholar 

  67. D. Hestenes, Point groups and space groups in geometric algebra, in Applications of Geometric Algebra in Computer Science and Engineering, ed. by L. Dorst, C. Doran, J. Lasenby (Birkhäuser, Boston, 2002), pp. 3–34

    Chapter  Google Scholar 

  68. D. Hestenes, Mysteries and insights of Dirac theory. Ann. Fond. Louis de Broglie 28(3), 390–408 (2003)

    MathSciNet  MATH  Google Scholar 

  69. D. Hestenes, Oersted Medal Lecture 2002: reforming the mathematical language of physics. Am. J. Phys. 71(2), 104–121 (2003)

    Article  Google Scholar 

  70. D. Hestenes, New tools for computational geometry and rejuvenation of screw theory, in Geometric Algebra Computing in Engineering and Computer Science, ed. by E. Bayro-Corrochano, G. Scheuermann (Springer, London, 2010), pp. 3–34

    Chapter  Google Scholar 

  71. D. Hestenes, Grassmann’s legacy, in From Past to Future: Grassmann’s Work in Context, ed. by H.-J. Petsche, A. Lewis, J. Liesen, S. Russ (Birkhäuser, Basel, 2011), pp. 243–260

    Chapter  Google Scholar 

  72. D. Hestenes, The shape of differential geometry in geometric calculus, in Guide to Geometric Algebra in Practice, ed. by L. Dorst, J. Lasenby (Springer, London, 2011), pp. 393–410

    Chapter  Google Scholar 

  73. D. Hestenes, The genesis of geometric algebra: a personal perspective. Adv. Appl. Clifford Algebr. 27(1), 351–379 (2017)

    Article  MathSciNet  Google Scholar 

  74. D. Hestenes, Deconstructing the electron clock, 2018. Preprint received on July 16, 2018. Can be accessed at http://www.ime.unicamp.br/~agacse2018/abstracts/InvitedSpeakers/Hestenes-Maxwell-Dirac.pdf

  75. D. Hestenes, Quantum Mechanics of the electron particle-clock, 2018. Preprint received on July 16, 2018. Can be accessed at http://www.ime.unicamp.br/~agacse2018/abstracts/InvitedSpeakers/Hestenes-ElectronClock.pdf

  76. D. Hestenes, J. Holt, The crystallographic space groups in geometric algebra. J. Math. Phys. 48, 023514 (2007)

    Article  MathSciNet  Google Scholar 

  77. D. Hestenes, G. Sobczyk, Clifford Algebra to Geometric Calculus (Reidel, Dordrecht, 1984)

    Book  Google Scholar 

  78. D. Hestenes, R. Ziegler, Projective geometry with Clifford algebra. Acta Appl. Math. 23, 25–63 (1991)

    Article  MathSciNet  Google Scholar 

  79. D. Hestenes, H. Li, A. Rockwood, Spherical conformal geometry with geometric algebra, in Geometric Computing with Clifford Algebras (Springer, Berlin, 2001), pp. 61–75

    MATH  Google Scholar 

  80. D. Hestenes, H. Li, A. Rockwood, New algebraic tools for classical geometry, in Geometric Computing with Clifford Algebras (Springer, Berlin, 2001), pp. 3–26

    Book  Google Scholar 

  81. D. Hildenbrand, Foundations of Geometric Algebra Computing (Springer, Berlin, 2012)

    MATH  Google Scholar 

  82. E. Hitzer, Three-dimensional quadrics in hybrid conformal geometric algebras of higher dimensions, in Early Proceedings of AGACSE 2018 (2018)

    Google Scholar 

  83. J. Hladic, Spinors in Physics. Graduate Texts in Contemporary Physics (Springer, New York, 1999). Translated by J. Michael Cole from the French edition “Les spineurs en physique” published in 1996 by Masson, Paris

    Google Scholar 

  84. J. Hrdina, A. Návrat, P. Vasik, Geometric algebra of conics (2018). Preprint

    Google Scholar 

  85. S. Huang, Y.Y. Qiao, G.C. Wen, Real and Complex Clifford Analysis. Advances in Complex Analysis and Its Applications (Springer, Berlin, 2006)

    Google Scholar 

  86. B. Jancewicz, Multivectors and Clifford Algebra in Electrodynamics (World Scientific, Singapore, 1988), xiv+316 pp.

    Google Scholar 

  87. N. Jeevanjee, An Introduction to Tensors and Group Theory for Physicists, 2nd edn. (Birkhäuser, Cham, 2014)

    Google Scholar 

  88. A. Kirillov Jr., An Introduction to Lie Groups and Lie Algebras. Cambridge Studies in Advanced Mathematics, vol. 113 (Cambridge University Press, Cambridge, 2008)

    Google Scholar 

  89. V.V. Kisil, Elliptic, parabolic and hyperbolic analytic function theory-1: geometry of invariants, 2006. https://arxiv.org/pdf/math/0512416v4.pdf

  90. V.V. Kisil, Starting with the Group \(\mathrm {SL}_2(\mathbb {R})\). Not. AMS 54(11), 1458–1465 (2007)

    Google Scholar 

  91. Y. Kuroe, T. Nitta, E. Hitzer, Applications of Clifford’s geometric algebra. SICE J. Control Meas. Syst. Integr. 4(1), 1–10 (2011)

    Article  Google Scholar 

  92. S. Lang, SL2(R). Graduate Texts in Mathematics (Springer, Berlin, 1985)

    Google Scholar 

  93. C. Lavor, S. Xambó-Descamps, I. Zaplana, A Geometric Algebra Invitation to Space-Time Physics, Robotics and Molecular Geometry. SBMA/Springerbrief (Springer, Cham, 2018)

    Google Scholar 

  94. H.B. Lawson, M.-L. Michelsohn, Spin Geometry (Princeton University Press, Princeton, 1989)

    MATH  Google Scholar 

  95. H. Li, Symbolic geometric reasoning with advanced invariant algebras, in International Conference on Mathematical Aspects of Computer and Information Sciences (Springer, Cham, 2015), pp. 35–49

    Google Scholar 

  96. H. Li, D. Hestenes, A. Rockwood, A universal model for conformal geometries of Euclidean, spherical and double-hyperbolic spaces, in Geometric Computing with Clifford Algebras (Springer, Berlin, 2001), pp. 77–104

    MATH  Google Scholar 

  97. H. Li, D. Hestenes, A. Rockwood, Generalized homogeneous coordinates for computational geometry, in Geometric Computing with Clifford Algebras (Springer, Berlin, 2001), pp. 27–59

    Book  Google Scholar 

  98. R. Lipschitz, Principes d’un calcul algébrique que contient comme espèces particulières le calcul des quantités imaginaires et des quaternions. C. R. Acad. Sci. Paris xli (1880). Reproduced in the Bull. Sci. Math. (2) 11, 115–120 (1887)

    Google Scholar 

  99. R. Lipschitz, Untersuchungen über die Summen von Quadraten (M. Cohen and Sohn, Pittsburgh, 1886). The first chapter, pp. 5–57. Translated into French by J. Molk: Recherches sur la transformation, par des substitutions réelles, d’un somme de deux ou trois carrés en elle-mêmme, J. Math. Pures Appl. 2, 163–183 (1886)

    Google Scholar 

  100. P. Lounesto, Clifford Algebras and Spinors. LMS Lecture Notes Series, vol. 286, 2nd edn. (Cambridge University Press, Cambridge, 2001)

    Google Scholar 

  101. S. MacLane, Mathematics Form and Function (Springer, New York, 1986)

    Book  Google Scholar 

  102. E. Meinrenken, A.M. Cohen, Clifford Algebras and Lie Theory (Springer, Berlin, 2013)

    Book  Google Scholar 

  103. H. Minkowski, Space and time, in The Principle of Relativity (Dover, New York, 1952). Translation of the communication<< Raum und Zeit>>  presented by the author to the 80th Convention of German Scientists and Doctors (Köln, 21 September 1908)

    Google Scholar 

  104. R. Mneimné, F. Testard, Introduction à la théorie des groupes classiques. Méthodes (Hermann, Paris, 1986)

    MATH  Google Scholar 

  105. J.A. Navarro, Notes for a Degree in Mathematics. Algebra and Geometry (2017). http://matematicas.unex.es/~navarro/degree.pdf. Based on Lectures of J. Sancho

  106. M. Nishikawa, On the exponential map of the group O(p, q)0. Mem. Fac. Sci. Kyushu Univ. 37(1), 63–69 (1983)

    MathSciNet  MATH  Google Scholar 

  107. W. Pauli, Zur Quantenmechanik des magnetischen Elektrons. Z. Phys. 42, 601–623 (1927)

    Article  Google Scholar 

  108. R. Penrose, The Road to Reality. A Complete Guide to the Laws of the Universe (Alfred A. Knopf, New York, 2005), xxviii+1099 pp.

    Google Scholar 

  109. C. Perwass, Geometric Algebra with Applications in Engineering. Geometry and Computing, vol. 4 (Springer, Berlin, 2009)

    Google Scholar 

  110. H.-J. Petsche, Grassmann. Vita Mathematica, vol. 13 (Birkhäuser, Basel, 2006), xxii+326 pp.

    Google Scholar 

  111. L.S. Pontryagin, Topological Groups. Russian Monographs and Texts on Advanced Mathematics and Physics, 2nd edn. (Gordon and Breach, New York, 1966). Translated from the Russian by Arlen Brown. xv+543 pp.

    Google Scholar 

  112. I.R. Porteous, Topological Geometry, 2nd edn. (Cambridge University Press, Cambridge, 1981) (1st edn., 1969)

    Book  Google Scholar 

  113. I.R. Porteous, Clifford Algebras and the Classical Groups (Cambridge University Press, Cambridge, 1995)

    Book  Google Scholar 

  114. M.M. Postnikov, Leçons de géométrie: Groupes et algebres de Lie (Éditions Mir, Moscou, 1985) (Translation of the 1982 Russian edition, by D. Embarek)

    Google Scholar 

  115. C. Procesi, Lie Groups. An Approach Through Invariants and Representations. Universitext (Springer, New York, 2007)

    Google Scholar 

  116. M. Riesz, Clifford Numbers and Spinors. Fundamental Theories of Physics, vol. 54 (Kluwer Academic Publishers, Dordrecht, 1997). An edition by E.F. Bolinder and P. Lounesto of M. Riesz Clifford numbers and spinors. Lecture Series No. 38k Institute for Fluid Dynamics and Applied Mathematics, University of Maryland (1958)

    Google Scholar 

  117. P. de M. Rios, E. Straume, Symbol Correspondence for Spin Systems (Birkhäuser, Basel, 2014)

    MATH  Google Scholar 

  118. W.A. Rodrigues Jr., E.C. de Oliveira, The Many Faces of Maxwell, Dirac and Einstein Equations. Lecture Notes in Physics, vol. 922, 2nd edn. (Springer, Berlin, 2016)

    Chapter  Google Scholar 

  119. D.H. Sattinger, O.L. Weaver, Lie Groups and Algebras with Applications to Physics, Geometry and Mechanics. Applied Mathematical Sciences, vol. 61 (Springer, New York, 1986)

    Book  Google Scholar 

  120. M. Schottenloher, A Mathematical Introduction to Conformal Field Theory. Lecture Notes in Physics, vol. 759 (Springer, New York, 2008). A much enlarged second edition appeared in 2008

    Google Scholar 

  121. I. Singer, J.A. Thorpe, Lecture Notes on Elementary Topology and Geometry. Lecture Notes in Mathematics, vol. 388 (Springer, New York, 1967)

    Chapter  Google Scholar 

  122. J. Snygg, Clifford Algebra—A Computational Tool for Physicists (Oxford University Press, New York, 1997)

    Google Scholar 

  123. J. Snygg, A New Approach to Differential Geometry Using Clifford’s Geometric Algebra (Birkhäuser, Boston, 2012)

    Book  Google Scholar 

  124. G. Sommer (ed.), Geometric Computing with Clifford Algebras: Theoretical Foundations and Applications in Computer Vision and Robotics (Springer, Berlin, 2001)

    MATH  Google Scholar 

  125. S. Sternberg, Group Theory and Physics (Cambridge University Press, Cambridge, 1994) (paperback 1995)

    MATH  Google Scholar 

  126. J. Stillwell, Mathematics and Its History. Undergraduate Texts in Mathematics (Springer, New York, 2010)

    Book  Google Scholar 

  127. O.C. Stoica, The standard model algebra (2017). arxiv.org/pdf/1702.04336

  128. J. Stolfi, Oriented Projective Geometry (Academic, New York, 1991)

    MATH  Google Scholar 

  129. I. Todorov, Clifford Algebras and Spinors. Bulg. J. Phys. 38, 3–28 (2011)

    MathSciNet  Google Scholar 

  130. G.F. Torres del Castillo, 3-D Spinors, Spin-Weighted Functions and Their Applications. Progress in Mathematical Physics, vol. 32 (Birkhäuser, Boston, 2003), 246 pp.

    Google Scholar 

  131. G. Trayling, W.E. Baylis, A geometric basis for the standard-model group. J. Phys. A Math. Gen. 34(15), 3309–3324 (2001)

    Article  MathSciNet  Google Scholar 

  132. L.W. Tu, An Introduction to Smooth Manifolds. Universitext, 2nd edn. (Springer, New York, 2011)

    Google Scholar 

  133. V.S. Varadarajan, Lie Groups, Lie Algebras, and Their Representations. Graduate Texts in Mathematics, vol. 102 (Springer, Cham, 1974)

    Google Scholar 

  134. V.S. Varadarajan, Supersymmetry for Mathematicians: An Introduction. Courant Lecture Notes, vol. 11 (American Mathematical Society, Providence, 2004)

    Google Scholar 

  135. J. Vaz Jr., R. da Rocha Jr., An Introduction to Clifford Algebras and Spinors (Oxford University Press, Oxford, 2016)

    Book  Google Scholar 

  136. A. Weil, Correspondence, by R. Lipschitz. Ann. Math. 69, 242–251 (1959). Reproduced in the second volume of A. Weil’s Œuvres Scientifiques, Collected Papers, 556–561

    Google Scholar 

  137. H. Weyl, Gruppentheorie und Quantenmechanik (Hirzel, Leipzig, 1928). Revised edition 1931, reprinted by Dover 1950

    Google Scholar 

  138. H. Weyl, The Classical Groups: Their Invariants and Representations (Princeton University Press, Princeton, 1939). 2nd edition, with supplement, 1953

    Google Scholar 

  139. S. Xambó, Escondidas sendas de la geometría proyectiva a los formalismos cuánticos, in El legado matemático de Juan Bautista Sancho Guimerá, ed. by D. Hernández-Ruipérez, A. Campillo (Real Sociedad Matemática Española & Ediciones Universidad de Salamanca, Salamanca, 2016), pp. 233–274. https://mat-web.upc.edu/people/sebastia.xambo/GA/2015-Xambo--EscondidasSendas-JBSG-in-memoriam.pdf

    Google Scholar 

  140. S. Xambó-Descamps, A Clifford View of Klein’s Geometry, 2009. Slides of the invited lecture delivered at the “International Conference on Didactics of Mathematics as a Mathematical Discipline” held 1–4 October 2009 in the University Madeira, Funchal, Madeira Island. https://mat-web.upc.edu/people/sebastia.xambo/K2/K2-Xambo.pdf

  141. S. Xambó-Descamps, From Leibniz’ characteristica geometria to contemporary Geometric Algebra. Qüaderns d’Història de l’Enginyeria 16(1), 109–141 (2017)

    MathSciNet  Google Scholar 

  142. S. Xambó-Descamps, J.M. Parra, Preface. Adv. Appl. Clifford Algebr. 27, 345–349 (2017)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Universitat Politècnica de Catalunya, Barcelona, Spain

    Sebastià Xambó-Descamps

Authors
  1. Sebastià Xambó-Descamps
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2018 The Author(s), under exclusive licence to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Xambó-Descamps, S. (2018). Postfaces. In: Real Spinorial Groups. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-030-00404-0_6

Download citation

Publish with us

Policies and ethics

Search

Navigation