Advances in Applied Clifford Algebras最新文献

The mathematical tools of physics, based on group theory, are in permanent evolution. Major covariance groups are the orthogonal, unitary and symplectic groups. These groups are generally expressed in terms of real and complex matrices. Here we shall develop a new representation of the unitary symplectic groups USp(n) in terms of Clifford algebras constituted by tensor products of quaternion algebras called hyperquaternions. Concise expressions of the generators are obtained and a concrete example USp(4) is provided. Isomorphic quaternion matrix representations will also be used in the applications. The first application concerns classical mechanics. The Hamiltonian formalism, Poisson brackets and canonical transforms are related to the unitary symplectic groups. The 1D and 2D harmonic oscillators are examined within that framework. The second application concerns quantum mechanics. The Schrödinger and Heisenberg equations are derived in a new hyperquaternionic unitary symplectic way, the complex imaginary i being replaced by the quaternion k in phase space. The 1D and 2D quantum harmonic oscillators are treated within that formalism. Allowing a representation of both classical and quantum mechanics, it is hoped that the hyperquaternion algebras might deepen our mathematical comprehension of the foundational principles of physics.

A Jordanian deformation of the (N=2) supersymmetry algebra and its first-order noncommutative differential calculus is obtained from the q-deformed Hopf supersymmetry algebra via a singular limit of a linear transformation. We show that the Jordanian (N=2) supersymmetry algebra carries a Hopf superalgebra structure. It is enhanced to a twisted Hopf star superalgebra by four inequivalent families of (star )-structures. It is demonstrated that these star operations induce four types of (star )-involutions on the differential one-forms and partial derivatives. We introduce an appropriate Jordanian super-Hopf algebra that includes two even and two odd generators equipped with four different types of (star )-structures and its corresponding supergroup on the Hopf (N=2) supersymmetry algebra. It is shown that the noncommutative differential calculus over the Jordanian (N=2) supersymmetry algebra is left-covariant with respect to the Jordanian supergroup. The (star )-structures of the supergroup are (star )-preserving on the supersymmetry algebra only for zero values of their parameters.

Weighted Bergman projection operators on the Clifford monogenic Bergman spaces over the real ball of ({mathbb R}^n) have been already studied in a paper of Ren and Malonek. We extend and generalize their results by considering more general Bergman nonprojection operators and stating a necessary and sufficient condition for them to be bounded on weighted Lebesgue spaces. Sharp estimates for the weighted monogenic Bergman kernel are also given.

Let E(n) be the (2^{n+1})-dimensional Hopf algebra generated by anti-commuting elements (g,x_1, ldots , x_n), with g grouplike, each (x_i) skew-primitive, and (g^2=1), (x_i^2=0). In this article we prove that E(n)-coactions over a finite-dimensional algebra A are classified by tuples ((varphi , d_1, ldots , d_n)) consisting of an involution (varphi ) and a family ((d_i)_{i=1,ldots ,n}) of (varphi )-derivations satisfying appropriate conditions. Tuples of maps can be replaced by tuples of suitable elements ((c, u_1, ldots , u_n)), whenever A is a semisimple Clifford algebra.

In this paper, we establish a novel dual matrix representation for dual quaternion matrices, which forms the foundation for a fast and innovative dual structure-preserving algorithm for dual quaternion singular value decomposition (DQSVD). By leveraging the dual quaternion Householder transformation and exploiting the existing properties of dual quaternions, we design a structure-preserving algorithm. This algorithm has a remarkable advantage in that it can convert quaternion operations in the process of bidiagonalizing the dual quaternion matrix into a dual matrix during DQSVD into real operations. As a result, computational efficiency is significantly enhanced. To verify the effectiveness of our proposed algorithm, we present a series of numerical examples. In these examples, we construct the dual complex matrix representation of color images and apply the concept of the structure-preserving algorithm to the dual complex singular value decomposition (DCSVD). This has been successfully employed in the watermark design of color images.

In this paper, we introduce and study two classes of multiparameter Forelli–Rudin type operators from (L^{vec {p}}left( {mathcal {D}}right) ) to (L^{vec {q}}left( {mathcal {D}}right) ), especially on their boundedness, where (L^{vec {p}}left( {mathcal {D}}right) ) and (L^{vec {q}}left( {mathcal {D}}right) ) are both weighted Lebesgue spaces over the Cartesian product of two tubular domains (T_B), with mixed-norm and appropriate weights. We completely characterize the boundedness of these two operators when (1le vec {p}le vec {q}<infty ). Moreover, we provide the necessary and sufficient condition of the case that (vec {q}=(infty ,infty )). As an application, we obtain the boundedness of three common classes of integral operators, including the weighted multiparameter Bergman-type projection and the weighted multiparameter Berezin-type transform.

In this paper, starting from recently known scaled hypercomplexes ({mathbb {H}}_{t}), we define scaled hyperbolics ({mathbb {D}}_{t}) for scales (tin {mathbb {R}}). In particular, the (left( -1right) )-scaled hyperbolics ({mathbb {D}}_{-1}) is isomorphic to the complex field ({mathbb {C}}), the 0-scaled hyperbolics ({mathbb {D}}_{0}) is isomorphic to the dual numbers ({textbf{D}}), and the 1-scaled hyperbolics ({mathbb {D}}_{1}) is isomorphic to the classical hyperbolic numbers ({mathcal {D}}). For any fixed (tin {mathbb {R}}), initiated from the t-scaled hyperbolics ({mathbb {D}}_{t}), we construct the t-scaled-hyperbolic Clifford algebra ({mathscr {C}}_{t}=underrightarrow{textrm{lim}}{mathscr {C}}_{t,n}), where ({mathscr {C}}_{t,n}) are the n-th t-scaled-hyperbolic Clifford algebras for all (nin {mathbb {N}}cup left{ 0right} ), with ({mathscr {C}}_{t,0}={mathbb {R}}) and ({mathscr {C}}_{t,1}={mathbb {D}}_{t}), just like the classical Clifford algebra ({mathscr {C}}={mathscr {C}}_{-1}). To analyze this ({mathbb {R}})-algebra ({mathscr {C}}_{t}), we establish an operator algebra ({mathscr {M}}_{t}) (over ({mathbb {C}}), as usual), containing ({mathscr {C}}_{t}), and then construct a free-probabilistic structure (left( {mathscr {M}}_{t},tau _{t}right) ). From the operator theory, operator algebra and free probability on ({mathscr {M}}_{t}), we apply these analysis for studying ({mathscr {C}}_{t}.)

The last two decades, since the seminal work of Selig [18], has seen projective geometric algebra (PGA) gain popularity as a modern coordinate-free framework for doing classical Euclidean geometry and other Cayley-Klein geometries. This framework is based upon a degenerate Clifford algebra, and it is the purpose of this paper to delve deeper into its internal algebraic structure and extract meaningful information for the purposes of PGA. This includes exploiting the split extension structure to realise the natural decomposition of elements of this Clifford algebra into Euclidean and ideal parts. This leads to a beautiful demonstration of how Playfair’s axiom for affine geometry arises from the ambient degenerate quadratic space. The highlighted split extension property of the Clifford algebra also corresponds to a splitting of the group of units and the Lie algebra of bivectors. Central to these results is that the degenerate Clifford algebra ({{,textrm{Cl},}}(V)) is isomorphic to the twisted trivial extension ({{,textrm{Cl},}}(V/mathbb {F}{e_{0}})ltimes _alpha {{,textrm{Cl},}}(V/mathbb {F}{e_{0}})), where ({e_{0}}) is a degenerate vector and (alpha ) is the grade-involution.

In this paper, we introduce a notion of a probabilistic measure which takes values in t-scaled hyperbolic numbers for (tin mathbb {R}), with a system of axioms generalizing directly Kolmogorov’s axioms. i.e., we establish a suitable measure theory in the set (mathbb {D}_{t}) of all t-scaled hyperbolic numbers for arbitrarily fixed (tin mathbb {R}).

In this paper we study symmetry properties of the Hilbert transformation of the three real variables in the quaternion setting. In order to describe the symmetry properties we introduce the group (rtextrm{Spin}(3)+mathbb {R}^3) which is essentially an extension of the ax+b group. The study concludes that the Hilbert transformation has certain characteristic symmetry properties in terms of (rtextrm{Spin}(3)+mathbb {R}^3.) We first obtain the spinor representation of the group induced by one of (textrm{Spin}(2)) in (mathbb {H}). Then we decompose the natural representation of (rtextrm{Spin}(3)+mathbb {R}^3) into the direct sum of some two irreducible spinor representations, by which we characterize the Hilbert transformation in (mathbb {R}^3). Precisely, we show that a nontrivial operator is essentially the Hilbert transformation if and only if it is invariant under the action of the (rtextrm{Spin}(3)+mathbb {R}^3) group.