Generalized Clifford Algebras Research Articles

Analysis on generalized Clifford algebras

Изучается вопрос, связанный с обобщенными алгебрами Клиффорда $\mathcal{C}_n(\underline{a})$, где $\underline{a}$ - ненулевой вектор. Если $\{e_1,…,e_n\}$ - ортонормированный базис, операция умножения определяется соотношениями \begin{align*} e_j^2=a_je_j-1, e_ie_j+e_je_i=a_ie_j+a_je_i, \end{align*} где $a_j=e_j\cdot\underline{a}$. Случай $\underline{a}=\underline{0}$ соответствует классической алгебре Клиффорда. Определяется оператор Дирака $D=\sum_je_j\partial_{x_j}$ и регулярные функции как его нулевое решение. Изучаются алгебраические свойства рассматриваемой алгебры. Доказываются основные формулы для оператора Дирака и изучаются свойства регулярных функций.

Monogenic Functions with Values in Generalized Clifford Algebras

Monogenic Functions with Values in Generalized Clifford Algebras

Monogenic functions taking values in generalized Clifford algebras

UDC 512.579Generalized Clifford algebras are constructed by various methods and have some applications in mathematics and physics.In this paper we introduce a new type of generalized Clifford algebra such that all components of a monogenic functionare solutions of an elliptic partial differential equation. One of our aims is to cover more partial differential equations inframework of Clifford analysis. We shall prove some Cauchy integral representation formulae for monogenic functions inthose cases.

Generalized Clifford Algebras Associated to Certain Partial Differential Equations

In the classical Clifford analysis the Laplace operator is factorized by the Cauchy–Riemann operator $$\Delta =\overline{D}D$$ . The consequence is all components of a monogenic function are harmonic functions. In more general situation, suppose that we are given a linear partial differential equation. We wish to find a generalized Clifford algebra such that all components of a generalized monogenic function taking values in that algebra satisfy the given partial differential equation. For instance, in order to represent biharmonic functions in the theory of plane elasticity, in 1934 L. Sobrero introduced a hypercomplex algebra which is generated by the imaginary e with the rule $$(e^2+1)^2=0$$ . In this paper we introduce an extension of the idea of L. Sobrero to construct some generalized Clifford algebras in order to cover more partial differential equations in higher dimensions.

How low can vacuum energy go when your fields are finite-dimensional?

According to the holographic bound, there is only a finite density of degrees of freedom in space when gravity is taken into account. Conventional quantum field theory does not conform to this bound, since in this framework, infinitely many degrees of freedom may be localized to any given region of space. In this paper, we explore the viewpoint that quantum field theory may emerge from an underlying theory that is locally finite-dimensional, and we construct a locally finite-dimensional version of a Klein–Gordon scalar field using generalized Clifford algebras. Demanding that the finite-dimensional field operators obey a suitable version of the canonical commutation relations makes this construction essentially unique. We then find that enforcing local finite dimensionality in a holographically consistent way leads to a huge suppression of the quantum contribution to vacuum energy, to the point that the theoretical prediction becomes plausibly consistent with observations.

A New View of Generalized Clifford Algebras

The main ingredient about generalized Clifford algebras (GCAs) in this paper is viewing them as algebras in certain symmetric linear categories of graded vector spaces. With this new view, we provide new derivations of some known results in a more direct manner. In particular, we prove the periodicity results on GCAs and propose a generalized Clifford process and show that any GCA can be obtained by repeating the process.

(q,σ,τ)-Differential Graded Algebras

We propose the notion of ( q , σ , τ ) -differential graded algebra, which generalizes the notions of ( σ , τ ) -differential graded algebra and q-differential graded algebra. We construct two examples of ( q , σ , τ ) -differential graded algebra, where the first one is constructed by means of the generalized Clifford algebra with two generators (reduced quantum plane), where we use a ( σ , τ ) -twisted graded q-commutator. In order to construct the second example, we introduce the notion of ( σ , τ ) -pre-cosimplicial algebra.

Superalgebraic structure of Lorentz transformations

We constructed superalgebraic representation of the algebra of second quantization of spinors. It is a fermionic analog of the Bargmann-Fock representation and is based on the use of the density of Grassmann variables in impulse space and their derivatives. We showed that Dirac matrices and Lorentz transformation generators can be expressed in terms of such densities. Lorentz transformations are considered as Bogolyubov transformations of creation and annihilation operators. We constructed superalgebraic form of the Dirac equation and the vacuum state vector. We showed that in the superalgebraic form of the complex Clifford algebra generators corresponding to Dirac gamma matrices are not equivalent. Clifford vector corresponding to diagonal matrix γ0 annihilates the vacuum, and the remaining ones give nonzero values. This means that there is asymmetric direction corresponding to the time axis.

Moyal deformations of Clifford gauge theories of gravity

A Moyal deformation of a Clifford [Formula: see text] Gauge Theory of (Conformal) Gravity is performed for canonical noncommutativity (constant [Formula: see text] parameters). In the very special case when one imposes certain constraints on the fields, there are [Formula: see text] first-order contributions in the [Formula: see text] parameters to the Moyal deformations of Clifford gauge theories of gravity. However, when one does [Formula: see text] impose constraints on the fields, there are first-order contributions in [Formula: see text] to the Moyal deformations in variance with the previous results obtained by other authors and based on different gauge groups. Despite that the generators of [Formula: see text] can be expressed in terms of the Clifford algebra generators this does [Formula: see text] imply that these algebras are isomorphic to the Clifford algebra. Therefore one should not expect identical results to those obtained by other authors. In particular, there are Moyal deformations of the Einstein–Hilbert gravitational action with a cosmological constant to first-order in [Formula: see text]. Finally, we provide a mechanism which furnishes a plausible cancellation of the huge vacuum energy density.

Generalized Clifford Algebras as Algebras in Suitable Symmetric Linear Gr-Categories

By viewing Clifford algebras as algebras in some suitable symmetric Gr-categories, Albuquerque and Majid were able to give a new derivation of some well known results about Clifford algebras and to generalize them. Along the same line, Bulacu observed that Clifford algebras are weak Hopf algebras in the aforementioned categories and obtained other interesting properties. The aim of this paper is to study generalized Clifford algebras in a similar manner and extend the results of Albuquerque, Majid and Bulacu to the generalized setting. In particular, by taking full advantage of the gauge transformations in symmetric linear Gr-categories, we derive the decomposition theorem and provide categorical weak Hopf structures for generalized Clifford algebras in a conceptual and simpler manner.

On the generalized Clifford algebra of a monic polynomial

In this paper we study the generalized Clifford algebra defined by Pappacena of a monic (with respect to the first variable) homogeneous polynomial Φ(Z,X1,…,Xn)=Zd−∑k=1dfk(X1,…,Xn)Zd−k of degree d in n+1 variables over some field F. We completely determine its structure in the following cases: n=2 and d=3 and either char(F)=3, f1=0 and f2(X1,X2)=eX1X2 for some e∈F, or char(F)≠3, f1(X1,X2)=rX2 and f2(X1,X2)=eX1X2+tX22 for some r,t,e∈F. Excluding a few exceptions, this algebra is an Azumaya algebra of rank nine whose center is the coordinate ring of an affine elliptic curve. We also discuss representations of arbitrary generalized Clifford algebras assuming the base field F is algebraically closed of characteristic zero.

Hyperbolic Polynomials and Generalized Clifford Algebras

We consider the problem of realizing hyperbolicity cones as spectrahedra, i.e. as linear slices of cones of positive semidefinite matrices. The generalized Lax conjecture states that this is always possible. We use generalized Clifford algebras for a new approach to the problem. Our main result is that if $$-1$$ - 1 is not a sum of hermitian squares in the Clifford algebra of a hyperbolic polynomial, then its hyperbolicity cone is spectrahedral. Our result also has computational applications, since this sufficient condition can be checked with a single semidefinite program.

Appell Bases for Monogenic Functions of Three Variables

Monogenic (or hyperholomorphic) functions are well known in general Clifford algebras but have been little studied in the particular case \({\mathbb{R}^{3} \rightarrow \mathbb{R}^{3}}\). We describe for this case the collection of all Appell systems: bases for the finite-dimensional spaces of monogenic homogeneous polynomials which respect the operator \({D = \partial_{0} - \vec{\partial}}\). We prove that no purely algebraic recursive formula (in a specific sense) exists for these Appell systems, in contrast to the existence of known constructions for \({\mathbb{R}^{3} \rightarrow \mathbb{R}^{4}}\) and \({\mathbb{R}^{4} \rightarrow \mathbb{R}^{4}}\). However, we give a simple recursive procedure for constructing Appell bases for \({\mathbb{R}^{3} \rightarrow \mathbb{R}^{3}}\) which uses the operation of integration of polynomials.

Clifford algebras of binary homogeneous forms

We study the generalized Clifford algebras associated to homogeneous binary forms of prime degree p, focusing on exponentiation forms of p-central spaces in division algebra.For a two-dimensional p-central space, we make the simplifying assumption that one basis element is a sum of two eigenvectors with respect to conjugation by the other. If the product of the eigenvalues is 1 then the Clifford algebra is a symbol Azumaya algebra of degree p, generalizing the theory developed for p=3. Furthermore, when p=5 and the product is not 1, we show that any quotient division algebra of the Clifford algebra is a cyclic algebra or a tensor product of two cyclic algebras, and every product of two cyclic algebras can be obtained as a quotient. Explicit presentation is given to the Clifford algebra when the form is diagonal.

Structure theory of central simple Z_d-graded algebras

This paper investigates the structure theory of Zd-central simple graded algebras and gives the complete decomposition into building block algebras. The results are also applied to generalized Clifford algebras, which are motivating examples of Zd-central simple graded algebras.

Pfaffian quartic surfaces and representations of Clifford algebras

Given a general ternary form f = f(x1, x2, x3) of degree 4 over an algebraically closed field of characteristic zero, we use the geometry of K3 surfaces and van den Bergh’s correspondence between representations of the generalized Clifford algebra Cf associated to f and Ulrich bundles on the surface Xf := {w = f(x1, x2, x3)} ⊆ P to construct a positive-dimensional family of 8-dimensional irreducible representations of Cf . The main part of our construction, which is of independent interest, uses recent work of Aprodu-Farkas on Green’s Conjecture together with a result of Basili on complete intersection curves in P to produce simple Ulrich bundles of rank 2 on a smooth quartic surface X ⊆ P with determinant OX(3). This implies that every smooth quartic surface in P is the zerolocus of a linear Pfaffian, strengthening a result of Beauville-Schreyer on general quartic surfaces. 2010 Mathematics Subject Classification: 14J60 (13C14, 16G30)

Generalized quadratic modules

In 1973 the japanese mathematician Kanzaki defined two categories of generalized quadratic modules over every commutative, associative and unital ring K. In both categories a generalized quadratic module (M, f, q) is provided with a quadratic form q and a linear form f. These categories give something new only when 2 is not invertible in K, and here they are studied when K is a field of characteristic 2. The first category is fit for the definition of generalized Clifford algebras Cl(M, f, q) where x2 = f(x)x + q(x) for all \({x\in M}\) ; all resulting Clifford algebras are described here. The second category is fit for the definition of nondegenerate objects, metabolic objects, orthogonal sums of objects, tensor products and extended Witt rings; we have managed to bring much information on the extended Witt ring We(K), and to propose an application.

On alphabetic presentations of Clifford algebras and their possible applications

In this paper, we address the problem of constructing a class of representations of Clifford algebras that can be named “alphabetic (re)presentations.” The Clifford algebra generators are expressed as m-letter words written with a three-character or a four-character alphabet. We formulate the problem of the alphabetic presentations, deriving the main properties and some general results. At the end, we briefly discuss the motivations of this work and outline some possible applications.

C-lattices and Decompositions of Generalized Clifford Algebras

In this note we introduce C-lattices to make use of them to provide a short and self-contained proof to the decomposition theorems of generalized Clifford algebras established by T. Y. Lam and T. L. Smith.

Algebra Forms with $$d^{N} = 0$$ on Quantum Plane. Generalized Clifford Algebra Approach

We construct a q-analog of exterior calculus with a differential d satisfying dN = 0, where N ≥ 2 and q is a primitive Nth root of unity, on a noncommutative space and introduce a notion of a q-differential k-form. A noncommutative space we consider is a reduced quantum plane. Our construction of a q-analog of exterior calculus is based on a generalized Clifford algebra with four generators and on a graded q-differential algebra. We study the structure of the algebra of q-differential forms on a reduced quantum plane and show that the first order calculus induced by the differential d is a coordinate calculus. The explicit formulae for partial derivatives of this first order calculus are found.