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Books on the topic 'Quaternions space'

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Published: 9 November 2024

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1

Edmonds, James D. Relativistic reality: A modern view. Singapore: World Scientific, 1997.

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2

Krieg, Aloys. Modular Forms on Half-Spaces of Quaternions. Berlin, Heidelberg: Springer Berlin Heidelberg, 1985. http://dx.doi.org/10.1007/bfb0075946.

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3

Alpay, Daniel, Fabrizio Colombo, and Irene Sabadini. Quaternionic de Branges Spaces and Characteristic Operator Function. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-38312-1.

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4

Edmonds, J. D. Relativistic Reality: A Modern View. World Scientific Publishing Co Pte Ltd, 1997.

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5

Edmonds, J. D. Relativistic Reality: A Modern View. World Scientific Publishing Co Pte Ltd, 1997.

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6

Relativistic Reality: A Modern View (Knots and Everything, Vol 12). World Scientific Publishing Company, 1998.

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7

Lambek, Joachim. Six-Dimensional Lorentz Category. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198748991.003.0014.

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Joachim Lambek had a longstanding interest in the use of quaternions as a tool for explaining fundamental aspects of special relativity, dating from his days as a doctoral student to the end of his career. It is known (since the beginning of the twentieth century) that many areas of theoretical physics may be represented by quaternions with complex coefficients (so called “biquaternions”). This posthumous chapter illustrates how time may (or even should) be represented by three dimensions, so that space–time is represented by a six-dimensional Lorentz category (three space coordinates and three temporal coordinates).

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8

Spacecraft Modeling, Attitude Determination, and Control: Quaternion-Based Approach. Taylor & Francis Group, 2019.

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9

Yang, Yaguang. Spacecraft Modeling, Attitude Determination, and Control: Quaternion-Based Approach. Taylor & Francis Group, 2019.

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10

Yang, Yaguang. Spacecraft Modeling, Attitude Determination, and Control: Quaternion-Based Approach. Taylor & Francis Group, 2019.

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11

Yang, Yaguang. Spacecraft Modeling, Attitude Determination, and Control: Quaternion-Based Approach. Taylor & Francis Group, 2019.

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12

Yang, Yaguang. Spacecraft Modeling, Attitude Determination, and Control: Quaternion-Based Approach. Taylor & Francis Group, 2019.

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13

M¨uhlherr, Bernhard, Holger P. Petersson, and Richard M. Weiss. Linked Tori, II. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691166902.003.0006.

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This chapter proves several more results about weak isomorphisms between Moufang sets arising from quadratic forms and involutory sets. It first fixes a non-trivial anisotropic quadratic space Λ‎ = (K, L, q) before considering two proper anisotropic pseudo-quadratic spaces. It then describes a quaternion division algebra and its standard involution, a second quaternion division algebra and its standard involution, and an involutory set with a quaternion division algebra and its standard involution. It concludes with one more small observation regarding a pointed anisotropic quadratic space and shows that there is a unique multiplication on L that turns L into an integral domain with a multiplicative identity.

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14

Krieg, Aloys. Modular Forms on Half-Spaces of Quaternions. Springer London, Limited, 2006.

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15

Modular forms on half-spaces of quaternions. Berlin: Springer-Verlag, 1985.

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16

M¨uhlherr, Bernhard, Holger P. Petersson, and Richard M. Weiss. Linked Tori, I. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691166902.003.0005.

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This chapter investigates the consequences of the assumption that one Moufang set is weakly isomorphic to another. It first introduces some well-known facts about involutions which are assembled in a few lemmas, including those dealing with an involutory set, a biquaternion division algebra, and a quaternion division algebra with a standard involution. It then presents a notation for a non-trivial anisotropic quadratic space and another for an involutory set are presented, along with assumptions for a pointed anisotropic quadratic space and the standard involution of a quaternion. It also makes a number of propositions regarding the standard involution of a quaternion and a biquaternion. Results about weak isomorphisms between Moufang sets arising from involutory sets are given.

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17

Colombo, Fabrizio, Irene Sabadini, and Daniel Alpay. Quaternionic de Branges Spaces and Characteristic Operator Function. Springer, 2020.

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18

M¨uhlherr, Bernhard, Holger P. Petersson, and Richard M. Weiss. Semi-ramified Quadrangles of Type E6, E7 and E8. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691166902.003.0012.

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This chapter deals with the case that the building at infinity of the Bruhat-Tits building Ξ‎ is a Moufang semi-ramified quadrangle of type E⁶, E₇ and E₈. The basic proposition is that Ξ‎ is a semi-ramified quadrangle if δ‎Λ‎ = 1 and δ‎Ψ‎ = 2 holds. The chapter first considers the theorem supposing that ℓ = 6, that δ‎Λ‎ = 1 and δ‎Ψ‎ = 2, and that the Moufang residues R0 and R1 are not both indifferent. This is followed by cases ℓ = 7 and ℓ = 8 as well as theorems concerning an anisotropic pseudo-quadratic space, a quaternion division algebra, standard involution, a proper involutory set, and isotropic and anisotropic quadratic spaces.

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19

Zirnbauer, Martin R. Symmetry classes. Edited by Gernot Akemann, Jinho Baik, and Philippe Di Francesco. Oxford University Press, 2018. http://dx.doi.org/10.1093/oxfordhb/9780198744191.013.3.

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This article examines the notion of ‘symmetry class’, which expresses the relevance of symmetries as an organizational principle. In his 1962 paper The threefold way: algebraic structure of symmetry groups and ensembles in quantum mechanics, Dyson introduced the prime classification of random matrix ensembles based on a quantum mechanical setting with symmetries. He described three types of independent irreducible ensembles: complex Hermitian, real symmetric, and quaternion self-dual. This article first reviews Dyson’s threefold way from a modern perspective before considering a minimal extension of his setting to incorporate the physics of chiral Dirac fermions and disordered superconductors. In this minimally extended setting, Hilbert space is replaced by Fock space equipped with the anti-unitary operation of particle-hole conjugation, and symmetry classes are in one-to-one correspondence with the large families of Riemannian symmetric spaces.

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20

Yang, Yaguang. Spacecraft Modeling Attitude Determination and Control. Taylor & Francis Group, 2021.

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21

Ann, David. Quaternion That Provided Inspiration for Complex Spaces and Hamilton's Intuition. General Onologic Soft Corp., 2024.

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22

M¨uhlherr, Bernhard, Holger P. Petersson, and Richard M. Weiss. Residually Pseudo-Split Buildings. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691166902.003.0033.

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This chapter presents results about a residually pseudo-split Bruhat-Tits building Ξ‎L. It begins with a case for some quadratic space of type E⁶, E₇, and E₈ in order to identify an unramified extension such that the residue field is a pseudo-splitting field. It then considers a wild quaternion or octonion division algebra and the existence of an unramified quadratic extension L/K such that L is a splitting field of the quaternion division algebra. It also discusses the properties of an unramified extension L/K and shows that every exceptional Bruhat-Tits building is the fixed point building of a strictly semi-linear descent group of a residually pseudo-split building.

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23

M¨uhlherr, Bernhard, Holger P. Petersson, and Richard M. Weiss. Quadratic Forms of Type E6, E7 and E8. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691166902.003.0008.

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This chapter presents various results about quadratic forms of type E⁶, E₇, and E₈. It first recalls the definition of a quadratic space Λ‎ = (K, L, q) of type Eℓ for ℓ = 6, 7 or 8. If D₁, D₂, and D₃ are division algebras, a quadratic form of type E⁶ can be characterized as the anisotropic sum of two quadratic forms, one similar to the norm of a quaternion division algebra D over K and the other similar to the norm of a separable quadratic extension E/K such that E is a subalgebra of D over K. Also, there exist fields of arbitrary characteristic over which there exist quadratic forms of type E⁶, E₇, and E₈. The chapter also considers a number of propositions regarding quadratic spaces, including anisotropic quadratic spaces, and proves some more special properties of quadratic forms of type E₅, E⁶, E₇, and E₈.

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24

M¨uhlherr, Bernhard, Holger P. Petersson, and Richard M. Weiss. Existence. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691166902.003.0016.

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This chapter proves that Bruhat-Tits buildings exist. It begins with a few definitions and simple observations about quadratic forms, including a 1-fold Pfister form, followed by a discussion of the existence part of the Structure Theorem for complete discretely valued fields due to H. Hasse and F. K. Schmidt. It then considers the generic unramified cases; the generic semi-ramified cases, the generic ramified cases, the wild unramified cases, the wild semi-ramified cases, and the wild ramified cases. These cases range from a unique unramified quadratic space to an unramified separable quadratic extension, a tamely ramified division algebra, a ramified separable quadratic extension, and a unique unramified quaternion division algebra. The chapter also describes ramified quaternion division algebras D₁, D₂, and D₃ over K containing a common subfield E such that E/K is a ramified separable extension.

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25

Gantner, Jonathan. Operator Theory on One-Sided Quaternion Linear Spaces: Intrinsic $S$-Functional Calculus and Spectral Operators. American Mathematical Society, 2021.

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26

M¨uhlherr, Bernhard, Holger P. Petersson, and Richard M. Weiss. Totally Wild Quadratic Forms of Type E7. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691166902.003.0015.

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This chapter assumes that (K, L, q) is a totally wild quadratic space of type E₇. The goal is to prove the proposition that takes into account Λ‎ of type E₇, D as the quaternion division algebra over K whose image in Br(K) is the Clifford invariant of q, and the trace and trace map. The chapter also considers two other propositions: the first states that if the trace map is not equal to zero, then the Moufang residues R₀ and R₁ are not indifferent; the second states that if the trace map is equal to zero, then the Moufang residues R₀ and R₁ are both indifferent.

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27

M¨uhlherr, Bernhard, Holger P. Petersson, and Richard M. Weiss. Strictly Semi-linear Automorphisms. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691166902.003.0030.

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This chapter considers the action of a strictly semi-linear automorphism fixing a root on the corresponding root group. It begins with the hypothesis whereby Δ‎ is a Moufang spherical building and Π‎ is the Coxeter diagram of Δ‎; here the chapter fixes an apartment Σ‎ of Δ‎ and a root α‎ of Σ‎. The discussion then turns to a number of assumptions about an isomorphism of Moufang sets, anisotropic quadratic space, and root group sequence, followed by a lemma where E is an octonion division algebra with center F and norm N and D is a quaternion subalgebra of E. The chapter concludes with three versions of what is really one result about fixed points of non-linear automorphisms of the Moufang sets.

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28

M¨uhlherr, Bernhard, Holger P. Petersson, and Richard M. Weiss. Quadrangles of Type E6, E7 and E8: Summary. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691166902.003.0014.

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This chapter summarizes the different cases about Moufang quadrangles of type E⁶, E₇ and E₈. The first case is that the building at infinity of the Bruhat-Tits building Ξ‎ is an unramified quadrangle; the second, a semi-ramified quadrangle; and the third, a ramified quadrangle. The chapter considers a theorem that takes into account two root group sequences, both of which are either indifferent or the various dimensions, types, etc., are as indicated in exactly one of twenty-three cases. It also presents a number of propositions relating to a quaternion division algebra and a quadratic space of type Eℓ for ℓ = 6, 7 or 8. Finally, it emphasizes the fact that the quadrangles of type F₄ could have been overlooked in the classification of Moufang polygons.

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