scholarly journals A note on Clifford algebras and central division algebras with involution

1985 ◽  
Vol 26 (2) ◽  
pp. 171-176 ◽  
Author(s):  
D. W. Lewis
Keyword(s):  
Quadratic Form ◽  
Clifford Algebra ◽  
Division Algebra ◽  
Clifford Algebras ◽  
Matrix Ring ◽  
Division Algebras ◽  
Central Division

In this note we consider the question as to which central division algebras occur as the Clifford algebra of a quadratic form over a field. Non-commutative ones other than quaternion division algebras can occur and it is also the case that there are certain central division algebras D which, while not themselves occurring as a Clifford algebra, are such that some matrix ring over D does occur as a Clifford algebra. We also consider the further question as to which involutions on the division algebra can occur as one of two natural involutions on the Clifford algebra.

scholarly journals REPRESENTATIONS OF ALTERNATIVE CLIFFORD ALGEBRAS OF QUADRATIC FORMS

2014 ◽  
Vol 57 (3) ◽  
pp. 579-590 ◽  
Author(s):  
STACY MARIE MUSGRAVE
Keyword(s):  
Quadratic Form ◽  
Clifford Algebra ◽  
Algebraic Structure ◽  
Quadratic Forms ◽  
Clifford Algebras ◽  
Future Research ◽  
Octonion Algebras

AbstractThis work defines a new algebraic structure, to be called an alternative Clifford algebra associated to a given quadratic form. I explored its representations, particularly concentrating on connections to the well-understood octonion algebras. I finished by suggesting directions for future research.

scholarly journals Nicely semiramified division algebras over Henselian fields

2005 ◽  
Vol 2005 (4) ◽  
pp. 571-577 ◽  
Author(s):  
Karim Mounirh
Keyword(s):  
Division Algebra ◽  
Division Algebras ◽  
Central Division ◽  
Field Extensions ◽  
Finite Dimensional ◽  
Central Division Algebra ◽  
Radical Type ◽  
Maximal Subfield

This paper deals with the structure of nicely semiramified valued division algebras. We prove that any defectless finite-dimensional central division algebra over a Henselian fieldEwith an inertial maximal subfield and a totally ramified maximal subfield (not necessarily of radical type) (resp., split by inertial and totally ramified field extensions ofE) is nicely semiramified.

scholarly journals Projective Metric Geometry and Clifford Algebras

2021 ◽  
Vol 76 (4) ◽  
Author(s):  
Hans Havlicek
Keyword(s):  
Quadratic Form ◽  
Projective Space ◽  
Vector Space ◽  
Clifford Algebra ◽  
Clifford Algebras ◽  
Metric Geometry ◽  
Algebraic Description ◽  
Projective Metric ◽  
Point Set ◽  
Distinguished Group

AbstractEach vector space that is endowed with a quadratic form determines its Clifford algebra. This algebra, in turn, contains a distinguished group, known as the Lipschitz group. We show that only a quotient of this group remains meaningful in the context of projective metric geometry. This quotient of the Lipschitz group can be viewed as a point set in the projective space on the Clifford algebra and, under certain restrictions, leads to an algebraic description of so-called kinematic mappings.

Quadratic Forms of Type E6, E7 and E8

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Quadratic Form ◽  
Division Algebra ◽  
Quadratic Forms ◽  
Quadratic Extension ◽  
The Other ◽  
Quadratic Space ◽  
Division Algebras ◽  
Arbitrary Characteristic ◽  
Definition Of ◽  
Quaternion Division Algebra

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.

Octonion generalization of Pauli and Dirac matrices

2014 ◽  
Vol 12 (01) ◽  
pp. 1550007 ◽  
Author(s):  
B. C. Chanyal
Keyword(s):  
Division Algebra ◽  
Wave Equations ◽  
Clifford Algebras ◽  
Matrix Representation ◽  
Algebra Structure ◽  
Division Algebras ◽  
Octonion Algebra ◽  
Conserved Current ◽  
The Relationship

Starting with octonion algebra and its 4 × 4 matrix representation, we have made an attempt to write the extension of Pauli's matrices in terms of division algebra (octonion). The octonion generalization of Pauli's matrices shows the counterpart of Pauli's spin and isospin matrices. In this paper, we also have obtained the relationship between Clifford algebras and the division algebras, i.e. a relation between octonion basis elements with Dirac (gamma), Weyl and Majorana representations. The division algebra structure leads to nice representations of the corresponding Clifford algebras. We have made an attempt to investigate the octonion formulation of Dirac wave equations, conserved current and weak isospin in simple, compact, consistent and manifestly covariant manner.

scholarly journals Explicit isomorphisms of real Clifford algebras

2006 ◽  
Vol 2006 ◽  
pp. 1-13
Author(s):  
N. Değırmencı ◽  
Ş. Karapazar
Keyword(s):  
Quadratic Form ◽  
Clifford Algebra ◽  
Matrix Algebra ◽  
Clifford Algebras ◽  
Explicit Construction ◽  
The Other ◽  
Matrix Algebras ◽  
Direct Sum ◽  
Other Hand ◽  
Nondegenerate Quadratic Form

It is well known that the Clifford algebraClp,qassociated to a nondegenerate quadratic form onℝn (n=p+q)is isomorphic to a matrix algebraK(m)or direct sumK(m)⊕K(m)of matrix algebras, whereK=ℝ,ℂ,ℍ. On the other hand, there are no explicit expressions for these isomorphisms in literature. In this work, we give a method for the explicit construction of these isomorphisms.

scholarly journals Clifford division algebras and anisotropic quadratic forms: two counterexamples

1986 ◽  
Vol 28 (2) ◽  
pp. 227-228 ◽  
Author(s):  
P. Mammone ◽  
J. P. Tignol
Keyword(s):  
Quadratic Form ◽  
Tensor Product ◽  
Division Algebra ◽  
Quadratic Forms ◽  
Division Algebras ◽  
Quaternion Algebras ◽  
Image Position

In a recent paper [3], D. W. Lewis proposed the following conjecture. (The notation is the same as that in [2] and [3].)Conjecture. Let F be a field of characteristic not 2 and let a1, b1…, an, bn ∈ Fx. The tensor product of quaternion algebrasis a division algebra if and only if the quadratic form over Fis anisotropic.This equivalence indeed holds for n = 1 as is well known [2, Theorem 2.7], and Albert [1] (see also [4, §15.7]) has shown that it also holds for n = 2. The aim of this note is to provide counterexamples to both of the implications for n ≥ 3.

scholarly journals On the decomposition of a field as a tensor product

1979 ◽  
Vol 20 (2) ◽  
pp. 141-145 ◽  
Author(s):  
P. M. Cohn
Keyword(s):  
Tensor Product ◽  
Division Algebra ◽  
Prime Power ◽  
Tensor Products ◽  
Division Algebras ◽  
Commutative Field ◽  
Central Division ◽  
Central Division Algebra

The following two results in the theory of division algebras are well known and easily proved, for an arbitrary commutative field k (cf. for example [3, Chapter 10]).(i) The tensor product of two central division algebras over k of coprime degrees is again a division algebra.(ii) Every central division algebra over k is a tensor product of division algebras of prime power degrees.It is natural to ask whether corresponding results hold for commutative fields. The answers are not hard to find but (as far as I am aware) have not appeared in print before; since they throw some light on the nature of tensor products they seemed worth recording.

Existence

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Division Algebra ◽  
Quadratic Forms ◽  
Structure Theorem ◽  
Quadratic Extension ◽  
Quadratic Space ◽  
Division Algebras ◽  
Separable Extension ◽  
Valued Fields ◽  
Quaternion Division Algebra

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.

Series expansions for monogenic functions in Clifford algebras and their application

2020 ◽  
Vol 17 (3) ◽  
pp. 365-371
Author(s):  
Anatoliy Pogorui ◽  
Tamila Kolomiiets
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Vector Space ◽  
Clifford Algebra ◽  
Clifford Algebras ◽  
Monogenic Function ◽  
Second Order ◽  
Series Expansions ◽  
Monogenic Functions ◽  
Partial Differential

This paper deals with studying some properties of a monogenic function defined on a vector space with values in the Clifford algebra generated by the space. We provide some expansions of a monogenic function and consider its application to study solutions of second-order partial differential equations.

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