相关论文: Crystal planes and reciprocal space in Clifford ge…
Geometric Algebra and Calculus are mathematical languages encoding fundamental geometric relations that theories of physics seem to respect. We propose criteria given which statistics of expressions in geometric algebra are computable in…
We extend Kostant's results about $\mathfrak{g}$-invariants in the Clifford algebra $Cl(\mathfrak{g})$ of a complex semisimple Lie algebra $\mathfrak{g}$ to the relative case of $\mathfrak{k}$-invariants in the Clifford algebra…
We discuss $\mathcal{N}=1$ Klein and Klein-Conformal superspaces in $D=(2,2)$ space-time dimensions, realizing them in terms of their functor of points over the split composition algebra $\mathbb{C}_{s}$. We exploit the observation that…
Crystal structures and the Bloch theorem play a fundamental role in condensed matter physics. We extend the static crystal to the dynamic "space-time" crystal characterized by the general intertwined space-time periodicities in $D+1$…
Real Clifford algebras for arbitrary number of space and time dimensions as well as their representations in terms of spinors are reviewed and discussed. The Clifford algebras are classified in terms of isomorphic matrix algebras of real,…
Representations of $C^*$-algebras are realized on section spaces of holomorphic homogeneous vector bundles. The corresponding section spaces are investigated by means of a new notion of reproducing kernel, suitable for dealing with…
Clifford's geometric algebra has enjoyed phenomenal development over the last 60 years by mathematicians, theoretical physicists, engineers and computer scientists in robotics, artificial intelligence and data analysis, introducing a myriad…
An idea to present a classical Lie group of positive dimension by generators and relations sounds dubious, but happens to be fruitful. The isometry groups of classical geometries admit elegant and useful presentations by generators and…
Grassmannians are of fundamental importance in projective geometry, algebraic geometry, and representation theory. A vast literature has grown up utilizing using many different languages of higher mathematics, such as multilinear and tensor…
We introduce fractional flat space, described by a continuous geometry with constant non-integer Hausdorff and spectral dimensions. This is the analogue of Euclidean space, but with anomalous scaling and diffusion properties. The basic tool…
The last two decades, since the seminal work of Selig, 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…
We study application of the Clifford algebra and the Grassmann algebra to image recognitions in $(3+1)D$ using quaternions. Following S.L.Adler, we construct a quaternion-valued wave function model with fermions and bosons of equal degrees…
Here we briefly describe some topics along the lines of projective spaces and related geometric constructions connected to linear algebra, which provide fundamental examples in classical geometry and analysis.
We give a graphical calculus for a categorification of a Clifford algebra and its Fock space representation via differential graded categories. The categorical action is motivated by the gluing action between the contact categories of…
We define a structure of an algebra on the Lagrangian Floer cohomology of a Lagrangian submanifold over the quantum cohomology of the ambient symplectic manifold. The structure is analogous to the one defined by Biran-Cornea, but is…
The modern algebra concepts are used to construct tables of algebraic spinors related to Clifford algebra multivectors with real and complex coefficients. The following data computed by Mathematica are presented in form of tables for…
We present two geometric interpretations for complex multivectors and determinants: a little known one in terms of square roots of volumes, and a new one which uses fractions of volumes and allows graphical representations. The fraction…
The following topics are presented in these notes: Elements of Banach algebras, Banach algebras of the form $L^1(G)$, where $G$ is a locally compact group, spectrum of elements of Banach algebras, the spectral theory of compact operators on…
Of the various formalisms developed to treat relativistic phenomena, those based on Clifford's geometric algebra are especially well adapted for clear geometric interpretations and computational efficiency. Here we study relationships…
The Clifford algebra of a n-dimensional Euclidean vector space provides a general language comprising vectors, complex numbers, quaternions, Grassman algebra, Pauli and Dirac matrices. In this work, we present an introduction to the main…