The paper examines smoothness in graded skew Clifford algebras.
problem Smoothness of graded skew Clifford algebras.
method Investigation of differential smoothness.
result Results on the differential smoothness of graded skew Clifford algebras.
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48 results for “Clifford Algebra”
Extends Kostant's results to symmetric pairs in Clifford algebras.
problem Analyzing -invariants in Clifford algebras of symmetric pairs.
method Proves Cartan theorem, transgression theorem, Harish-Chandra isomorphism, and Clifford algebra conjecture for relative case.
result Establishes a relative transgression theorem and Harish-Chandra isomorphism for Clifford algebras.
We consider the diffeological version of the Clifford algebra of a (diffeological) finite-dimensional vector space; we start by commenting on the notion of a diffeological algebra (which is the expected analogue of the usual one) and that of a diffeological module (also an expected counterpart of the usual notion). Aft…
Classifies compact Clifford-Klein forms for specific Lie algebras.
problem Classifying compact Clifford-Klein forms for given Lie algebra structures.
method Using Onishchik's results on semisimple Lie algebras, the paper classifies forms for triples (g,h,l).
result New examples of reductive homogeneous spaces with non-standard compact Clifford-Klein forms.
This dissertation explores Clifford bundles and spinor fields in geometric and algebraic contexts.
problem Understanding spinor fields and their classification in geometric frameworks.
method Combines algebraic and geometric approaches to study Clifford structures on bundles and spinor fields.
result Identifies new spinor field classes in warped flux compactifications.
New proof of divisibility property for certain algebraic varieties.
problem Divisibility property for LQEL varieties.
method Construction of Clifford algebra representations to Severi varieties.
result New proof of Russo's Divisibility Property for LQEL varieties.
Non-trivial Clifford bundle from loop space tangent bundle.
problem Triviality obstruction of Clifford bundle on loop space.
method Constructing Clifford algebra bundle from loop space tangent bundle, showing non-triviality through Stiefel-Whitney and Pontrjagin classes.
result Clifford bundle is non-trivial, obstructed by manifold's Stiefel-Whitney and Pontrjagin classes.
Study on Lie algebras from Clifford modules, focusing on specific types.
problem Characterizing Lie algebras from Clifford modules and their graded structures.
method Analysis of pseudo -type Lie algebras and their representations through Clifford algebras.
result Different types of Lie algebras have varying possibilities of containing pseudo -type Lie algebras in their negative part.
Surveying isoparametric foliations and related geometric constructions.
problem Constructing isoparametric foliations using Clifford algebra.
method Representation theory of Clifford algebra and mean curvature flow.
result Investigation of geometric constructions related to isoparametric hypersurfaces.
Unified study of surfaces using Clifford algebras.
problem Classifying immersed surfaces in various manifolds.
method Using Clifford algebras to construct formalism for immersed bilegendrian surfaces.
result Full classifications of immersed bilegendrian surfaces in the unit tangent bundle of the 3-sphere.
Study cohomology rings of Grassmannians using Clifford algebras and symmetric spaces.
problem Understanding cohomology rings of Grassmannians over different fields.
method Explicit generators and relations for de Rham cohomology rings, filtered deformations related to Clifford algebras.
result Explicit generators and relations for the de Rham cohomology rings of Grassmannians.
We consider the diffeological pseudo-bundles of exterior algebras, and the Clifford action of the corresponding Clifford algebras, associated to a given finite-dimensional and locally trivial diffeological vector pseudo-bundle, as well as the behavior of the former three constructions (exterior algebra, Clifford action…
The paper extends matrix inequalities to various types of matrices.
problem Generalizing inequalities for different types of matrices.
method Extending known inequalities for real, complex, and quaternionic matrices.
result New inequalities for matrices in subspaces spanned by Clifford systems or algebras.
Paper connects geometric structures to algebra in high dimensions.
problem Understanding geometric structures in high dimensions.
method Relating minimal left ideals on Clifford algebras to geometric structures.
result Established a connection between algebraic and geometric properties.
Using representations of Clifford algebras we construct indecomposable singular Riemannian foliations on round spheres, most of which are non-homogeneous. This generalizes the construction of non-homogeneous isoparametric hypersurfaces due to by Ferus, Karcher and Munzner.
New method classifies special Vinberg cones of rank 4.
problem Classifying special Vinberg cones of rank 4.
method Using Clifford Nil-algebras and directed acyclic graphs.
result Explicit classification of rank 4 special Vinberg cones.
Paper analyzes -structures and their minimal left ideals.
problem Understanding -structures and their minimal left ideals.
method Identifying -structures with minimal left ideals via induced Kahler polynomial.
result Established link between -structures and minimal left ideals for Clifford algebras.
Novel CG-EGNNs learn equivariant functions from Clifford algebras.
problem Lack of equivariance in high-order graph neural networks.
method Integrates high-order local structures with Clifford algebras for equivariant learning.
result CG-EGNNs outperform previous methods on various benchmarks.
Paper describes invariants of slice regular functions' automorphism group.
problem Understanding invariants of slice regular functions' automorphism group.
method Analyzes automorphism group of slice regular functions over Clifford algebras.
result Describes invariants of the automorphism group of slice regular functions.
Study on curvature tensors, discovering new Osserman tensors.
problem Investigate properties of curvature tensors and their relations.
method Introduce quasi-Clifford curvature tensors and analyze their properties.
result Discovered an Osserman curvature tensor not satisfying the duality principle.
Extrinsically Symmetric Spaces, Submanifolds of Clifford Type and a Theorem of Harish-Chandramath.DG
The study proves a geometric result related to Harish-Chandra's theorem.
problem Understanding the relationship between symmetric submanifolds and Harish-Chandra's theorem.
method Analyzing maximal tori in Clifford tori within Euclidean spaces.
result A compact, intrinsically symmetric submanifold is extrinsically symmetric if and only if its maximal tori are Clifford tori.
The paper studies Majorana Fermions and their braid group representations.
problem Understanding the braid group representations associated with Majorana Fermions.
method Recalling and proving a general result about braid group representations associated with Clifford algebras, and comparing it with Ivanov braiding.
result Certain strings of Majorana operators give rise to extraspecial 2-groups and braiding representations of the Ivanov type.
Simplifies Riemann geometry computations without indices.
problem Complex computations in Riemann geometry.
method Uses forms with Clifford algebra values and a specific group action.
result Simplified computations in Riemann geometry.
Introduces a new geometric product for differential forms.
problem Constructing a Clifford algebra from differential forms.
method Introduces and revisits the Graf product with a new framework.
result Provides a structure for constructing a Clifford algebra.
The study examines differential smoothness in specific Artin-Schelter regular algebras of dimension 5.
problem Investigating the differential smoothness of Artin-Schelter regular algebras of dimension 5.
method Analyzing the relationship between the number of generators and Gelfand-Kirillov dimension to identify structural obstructions.
result Certain two- and four-generator AS-regular algebras of global dimension five fail to admit a differential calculus, while a five-generator graded Clifford algebra provides a positive example.
Geometric calculus introduced on pseudo-Riemannian manifolds without embedding.
problem Developing calculus on pseudo-Riemannian manifolds without embedding.
method Direct axiomatic approach to geometric calculus, paralleling general relativity.
result Full theory of differential calculus for vector, multivector, and tensor fields developed.
We construct families of differential graded algebras R and R \boxtimes R and give an algebraic formulation of the contact category of a disk through the differential graded category DGP(R) generated by some distinguished projective differential graded R-modules. The homology category H^0(DGP(R)) is a triangulated cate…
We construct two infinite families of algebraic minimal cones in . The first family consists of minimal cubics given explicitly in terms of the Clifford systems. We show that the classes of congruent minimal cubics are in one to one correspondence with those of geometrically equivalent Clifford systems. As a byp…
We provide explicit spinor representations for Clifford algebras.
problem Building explicit representations of Clifford algebras.
method Explicit construction of spinor modules and parallel spinor fields.
result Explicit spinor representations for all mixed signature Clifford algebras.
Study uses geometric algebra to analyze credit cycles, revealing dangerous feedback loops.
problem Understanding and predicting dangerous feedback loops in credit cycles.
method Represent economic states as multi-vectors in Clifford algebra, focusing on bivector elements for rotational coupling.
result Geometric relationship between unemployment and credit contraction shifts from simple correlation to dangerous rotational dynamics during crises.
A binary code for spinors simplifies calculations.
problem Efficiently encoding and manipulating spinors for computation.
method Binary encoding of spinors and Clifford multiplication using non-negative integers.
result Explicit descriptions of Lie algebras and automorphisms.
The algebraic and geometric properties of a novel generalization of Clifford's classical C4 point-circle configuration are analysed. A connection with the integrable quaternionic discrete Schwarzian Kadomtsev-Petviashvili equation is revealed.
Based on \cite{DH94}, we introduce a bijective correspondence between first order differential calculi and the graph structure of the symmetric lattice that allows one to encode completely the interconnection structure of the graph in the exterior derivative. As a result, we obtain the Grassmannian character of the lat…
Solves classification of compact Clifford-Klein forms for specific Lie groups.
problem Classification of standard compact Clifford-Klein forms of homogeneous spaces.
method Analysis of real Lie algebras and their subalgebras.
result Standard compact Clifford-Klein forms arise from specific Lie algebra triples.
No standard compact Clifford-Klein forms found for exceptional Lie groups.
problem Proving the non-existence of standard compact Clifford-Klein forms for exceptional Lie groups.
method Computer-aided approach, algorithmic methods for classifying semisimple subalgebras, and invariant calculations.
result Proves the non-existence of standard compact Clifford-Klein forms for homogeneous spaces of exceptional Lie groups.
The paper studies Clifford-Bianchi groups acting on hyperbolic spaces and their properties.
problem Understanding the actions of Clifford-Bianchi groups on hyperbolic spaces.
method Developed the abstract and computational theory for determining fundamental domains and generators for orders in low dimensions.
result Found that Clifford-Bianchi groups are arithmetic subgroups of SO(1, n+1) and their Möbius action.
In this paper, we continue the study of the existence problem of compact Clifford-Klein forms from a cohomological point of view, which was initiated by Kobayashi-Ono and extended by Benoist-Labourie and the author. We give an obstruction to the existence of compact Clifford-Klein forms by relating a natural homomorphi…
GKP codes connect quantum gates to algebraic curves, enabling fault-tolerant quantum computation.
problem Implementing fault-tolerant quantum computation in quantum harmonic oscillator systems.
method Exploring the topological and algebraic structure of GKP codes, showing how gates correspond to symplectic automorphisms and mapping class groups of surfaces.
result GKP Clifford gates are identified with symplectic automorphisms of GKP lattices and mapping class groups of surfaces, providing a topological interpretation of fault tolerance.
We give an account of the classical and integrable geometry of isothermic surfaces in arbitrary co-dimension. We show that the classical transformation theory of Darboux, Bianchi and Calapso goes through unchanged in arbitrary co-dimension as does the connection with the "curved flats" of Ferus and Pedit. Moreover, we …
Defines algebraic structures in Lagrangian Floer cohomology using differential forms.
problem Modeling algebraic structures in Lagrangian Floer cohomology.
method Defines two algebra structures using differential forms and a closed-open map, showing they coincide.
result Two algebra structures for the 2-dimensional Clifford torus coincide.
In this paper we show how to describe the general theory of a linear metric compatible connection with the theory of Clifford valued differential forms. This is done by realizing that for each spacetime point the algebra of Clifford bivectors is isomorphic to the Lie algebra of Sl(2,C). In that way the pullback of the …
We classify the algebraic curvature tensors which are both Osserman and complex Osserman in all but a finite number of exceptional dimensions.Information concerning the possible eigenvalue structures, which is provided by methods of algebraic topology, plays a central role in the analysis.
Consider a smooth map from a neighborhood of the origin in a real vector space to a neighborhood of the origin in a Euclidean space. Suppose that this map takes all germs of lines passing through the origin to germs of Euclidean circles, or lines, or a point. We prove that under some simple additional assumptions this …
Braiding Majorana Fermionscond-mat.str-el
In this paper we study a Clifford algebra generalization of the quaternions and its relationship with braid group representations related to Majorana fermions. The Fibonacci model for topological quantum computing is based on the fusion rules for a Majorana fermion. Majorana fermions can be seen not only in the structu…
Monogenic functions are basic to Clifford analysis. On Euclidean space they are defined as smooth functions with values in the corresponding Clifford algebra satisfying a certain system of first order differential equations, usually referred to as the Dirac equation. There are two equally natural extensions of these eq…
An almost Clifford and an almost Cliffordian manifold is a --structure based on the definition of Clifford algebras. An almost Clifford manifold based on $\mathcal O:= \cc l (s,t)$ is given by a reduction of the structure group to , where and . An…
Generative model designs highly designable proteins using geometric algebra.
problem Creating proteins with diverse and statistically accurate secondary structures.
method Introduced a geometric algebra flow matching model (FrameFlow) with Clifford Frame Attention (CFA) for protein backbone design.
result Achieved high designability, diversity, and novelty in protein backbone sampling.
Metric and Gauge Extensorsmath.DG
In this paper, the second in a series of eight we continue our development of the basic tools of the multivector and extensor calculus which are used in our formulation of the differential geometry of smooth manifolds of arbitrary topology . We introduce metric and gauge extensors, pseudo-orthogonal metric extensors, g…