On the generalized Clifford algebra of a monic polynomial

Chapman, Adam; Kuo, Jung-Miao

doi:10.1016/j.laa.2014.12.030

Cited by 4 publications

(7 citation statements)

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“…The main theorem of [CK15] implies that there always exists a linear map φ : V → Mat r (k) for some r such that F v (φ(v)) = 0 for all v ∈ V . There is a natural non-commutative generalization of this problem.…”

Section: Questionsmentioning

confidence: 99%

“…Let V is a finite dimensional vector space and F (t) ∈ Sym • (V ∨ )[t] be monic and homogeneous. Given v ∈ V we can consider the image F v (t) of F (t) under the homomorphism Sym • (V ∨ )[t] → k[t] induced by v : V ∨ → k. The main theorem of [CK15] implies that there always exists a linear map φ : V → Mat r (k) for some r such that F v (φ(v)) = 0 for all v ∈ V . There is a natural non-commutative generalization of this problem.…”

Section: Questionsmentioning

confidence: 99%

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The characteristic polynomial of an algebra and representations

Kulkarni

Mustopa

Shipman

2017

Linear Algebra and its Applications

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Suppose that k is a field and let A be a finite dimensional, associative, unital k-algebra. Often one is interested in studying the finite-dimensional representations of A. Of course, a finite dimensional representation of A is simply a finite dimensional k-vector space M and a k-algebra homomorphism A → End k (M ). In this article we will not consider representations of algebras, but rather how to determine if a k-linear map φ : A → End k (M ) is actually a homomorphism. We restrict our attention to the case where A is a product of field extensions of k. If φ : A → End k (M ) is a representation then certainly, if a ∈ A satisfies a m = 1 then φ(a) m = id as well. Our first Theorem is a remarkable converse to this elementary observation.Let n > 2 be a natural number and assume that k has n primitive n th roots of unity. If φ(1 A ) = id and for each a ∈ A such that a n = 1 A we have φ(a) n = id, then φ is an algebra homomorphism.Consider the regular representation µ L : A → End k (A) of A on itself by left multiplication. For a ∈ A, let χ a (t) and χ a (t) be the characteristic and minimal polynomials of µ L (a), respectively. We note that χ a (a) = χ a (a) = 0 in A. Therefore if M is a finite dimensional left A module with structure map φ :The notion of assigning a characteristic polynomial to each element of an algebra and considering representations which are compatible with this assignment has appeared in [Pro87]. This idea has been applied to some problems in noncommutative geometry as well [LB03]. However, as far as we know the following related notion is new.Definition 1. Suppose that φ : A → End k (M ) is a k-linear map, where M is a finite dimensional k-vector space. We say that φ is a characteristic morphism if χ a (φ(a)) = 0 for all a ∈ A. We say that φ is minimal-characteristic if, moreover, χ a (φ(a)) = 0 for all a ∈ A.It is natural to ask whether or not the notions of characteristic morphism and minimal characteristic morphism are weaker than the notion of algebra morphism. Let us address minimal-characteristic morphisms first.Corollary. Assume that A = k d and that k has a full set of d th roots of unity. Then a minimalcharacteristic morphism φ : A → End k (M ) is an algebra morphism.Proof. First note that χ 1 (t) = t − 1. Hence

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Section: Questionsmentioning

confidence: 99%

Section: Questionsmentioning

confidence: 99%

The characteristic polynomial of an algebra and representations

Kulkarni

Mustopa

Shipman

2017

Linear Algebra and its Applications

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“…. , b s )I n " which is assumed initially in the axiom (17), is also compatible with matrix representation of the generalized Clifford algebras [24][25][26][27][28][29][30][31][32][33] associated with the r th degree homogeneous polynomials F(b 1 , b 2 , b 3 , . .…”

Section: The Basic Properties Of the Integral Domain Zmentioning

confidence: 99%

“…. , b s ) [28][29][30][31][32][33]. However, for some particular cases of the r th degree homogeneous forms F(b 1 , b 2 , b 3 , .…”

Section: The Basic Properties Of the Integral Domain Zmentioning

confidence: 99%

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A Unique Mathematical Derivation of the Fundamental Laws of Nature Based on a New Algebraic-Axiomatic (Matrix) Approach ‡

Zahedi

2017

Universe

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Abstract:In this article, as a new mathematical approach to origin of the laws of nature, using a new basic algebraic axiomatic (matrix) formalism based on the ring theory and Clifford algebras (presented in Section 2), "it is shown that certain mathematical forms of fundamental laws of nature, including laws governing the fundamental forces of nature (represented by a set of two definite classes of general covariant massive field equations, with new matrix formalisms), are derived uniquely from only a very few axioms." In agreement with the rational Lorentz group, it is also basically assumed that the components of relativistic energy-momentum can only take rational values. In essence, the main scheme of this new mathematical axiomatic approach to the fundamental laws of nature is as follows: First, based on the assumption of the rationality of D-momentum and by linearization (along with a parameterization procedure) of the Lorentz invariant energy-momentum quadratic relation, a unique set of Lorentz invariant systems of homogeneous linear equations (with matrix formalisms compatible with certain Clifford and symmetric algebras) is derived. Then by an initial quantization (followed by a basic procedure of minimal coupling to space-time geometry) of these determined systems of linear equations, a set of two classes of general covariant massive (tensor) field equations (with matrix formalisms compatible with certain Clifford, and Weyl algebras) is derived uniquely as well.Each class of the derived general covariant field equations also includes a definite form of torsion field appearing as the generator of the corresponding field' invariant mass. In addition, it is shown that the (1 + 3)-dimensional cases of two classes of derived field equations represent a new general covariant massive formalism of bispinor fields of spin-2, and spin-1 particles, respectively. In fact, these uniquely determined bispinor fields represent a unique set of new generalized massive forms of the laws governing the fundamental forces of nature, including the Einstein (gravitational), Maxwell (electromagnetic) and Yang-Mills (nuclear) field equations. Moreover, it is also shown that the (1 + 2)-dimensional cases of two classes of these field equations represent (asymptotically) a new general covariant massive formalism of bispinor fields of spin-3/2 and spin-1/2 particles, corresponding to the Dirac and Rarita-Schwinger equations.As a particular consequence, it is shown that a certain massive formalism of general relativity-with a definite form of torsion field appeared originally as the generator of gravitational field's invariant mass-is obtained only by first quantization (followed by a basic procedure of minimal coupling to space-time geometry) of a certain set of special relativistic algebraic matrix equations. It has been also proved that Lagrangian densities specified for the originally derived new massive forms of the Maxwell, Yang-Mills and Dirac field equations, are also gauge invariant, where the invariant mass of each field ...

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