Survey on algebraic minimal cones and nonassociative algebras.
problem Exploring algebraic minimal cones and nonassociative algebras.
method Recollections and recent developments in the field.
result Dedication to Vladimir Miklyukov's memory.
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48 results for “nonassociative algebras”
The purpose of these survey notes is to give a presentation of a classical theorem of Nomizu that relates the invariant affine connections on reductive homogeneous spaces and nonassociative algebras.
Many quantum groups and quantum spaces of interest can be obtained by cochain (but not cocycle) twist from their corresponding classical object. This failure of the cocycle condition implies a hidden nonassociativity in the noncommutative geometry already known to be visible at the level of differential forms. We exten…
Study on pre-Lie structures for semisimple Lie algebras over C.
problem Admissibility of pre-Lie structures in semisimple Lie algebras.
method Examined properties of anti-flexible algebras (AFAs), computed Lie-admissibility criteria, and provided examples.
result Explicit counterexample of an AFA admissible by sl(2, C).
The aim of this paper is to offer an overview of the most important applications of Jordan structures inside mathematics and also to physics, up-dated references being included. For a more detailed treatment of this topic see - especially - the recent book Iordanescu [364w], where sugestions for further developments ar…
Study of curvature tensor algebra with nonstandard multiplications.
problem Understanding curvature tensor algebra structures.
method Investigates orthogonally invariant commutative nonassociative multiplications on curvature tensors.
result Characterizes curvature tensor algebras in low dimensions and proves their simplicity in higher dimensions.
In (Phys. Rev. D 62, 081501, 2000) we proposed a unified approach to description of continuous and discrete spacetime based on nonassociative geometry and described nonassociative smooth and discrete de Sitter models. In our paper we give the description of nonassociative Friedmann-Robertson-Walker spacetime.
In the framework of nonassociative geometry (hep-th/0003238) a unified description of continuum and discrete spacetime is proposed. In our approach at the Planck scales the spacetime is described as a so-called "diodular discrete structure" which at large spacetime scales `looks like' a differentiable manifold. After a…
We show that the skew-symmetrized product on every Leibniz algebra E can be realized on a reductive complement to a subalgebra in a Lie algebra. As a consequence, we construct a nonassociative multiplication on E which, when E is a Lie algebra, is derived from the integrated adjoint representation. We apply this constr…
New geometric objects generalize Lie groupoids, with nontrivial tangent bundle properties.
problem Generalizing Lie groupoids to nonassociative structures.
method Introducing quasiloopoids and loopoids, proving properties of their tangent bundles, and reformulating discrete mechanics.
result Tangent bundles of loopoids are canonically loopoids, but cotangent bundles are not.
A nonassociative generalization of the principal fiber bundles with a smooth loop mapping on the fiber is presented. Our approach allows us to construct a new kind of gauge theories that involve higher ''nonassociative'' symmetries.
Equivalent categories of groups and risandles for 3-manifolds.
problem Constructing invariants of 3-manifolds.
method Introducing risandles and showing equivalence with groups, categorifying coloring invariants.
result Equivalence of fundamental groups and risandles for infinitely many 3-manifolds.
Study shows how holonomy and symplectic triple systems simplify computations in 3-Sasakian manifolds.
problem Understanding the holonomy groups in 3-Sasakian manifolds.
method Unified computation of holonomies using symplectic triple systems.
result Holonomy groups in homogeneous 3-Sasakian manifolds are smaller and can be described using symplectic triple systems.
A left order on a magma (e.g., semigroup) is a total order of its elements that is left invariant under the magma operation. A natural topology can be introduced on the set of all left orders of an arbitrary magma. We prove that this topological space is compact. Interesting examples of nonassociative magmas, whose spa…
New tools for studying Hsiang algebras discovered, linking them to known algebraic structures.
problem Classifying Hsiang algebras and understanding their properties.
method Introducing quasicomposition and tripling constructions to study Hsiang algebras.
result The triple of a quasicomposition algebra is an exceptional Hsiang algebra.
The study enumerates virtual quandles up to isomorphism.
problem Classifying virtual quandles up to isomorphism.
method Computer search and classification based on conjugacy class structures of rack automorphism groups.
result Classifications of virtual racks and quandles up to order 8.
Study of generalized Legendrian racks and their GL-structures.
problem Characterizing and classifying generalized Legendrian racks.
method Algebraic analysis, classification of racks and GL-racks, tensor products.
result Classification of several infinite families of GL-racks.
We study noncommutative bundles and Riemannian geometry at the semiclassical level of first order in a deformation parameter , using a functorial approach. The data for quantisation of the cotangent bundle is known to be a Poisson structure and Poisson preconnection and we now show that this data defines to a functo…
Three perspectives on quantizing magnetic Poisson structures are explored.
problem Quantization of magnetic Poisson structures with non-associativity.
method Deformation quantization, symplectic realization, geometric quantization using bundle gerbes.
result Comparison and contrast of different quantization approaches.
An affine hypersurface (AH) structure is a pair comprising a conformal structure and a projective structure such that for any torsion-free connection representing the projective structure the completely trace-free part of the covariant derivative of any metric representing the conformal structure is completely symmetri…
Abstract: Investigates octonion product deformations and related geometries.
problem Exploring geometries and deformations from the 7-sphere .
method Analyzing the spontaneous compactification and solutions of Lagrangian equations.
result Obtains a family of geometries including those with torsion and -structures.
This research tackles image classification with noise, proposing committees of CNNs.
problem Image classification with concurrent feature and label noise.
method Committees of Convolutional Neural Networks (CNNs) for MNIST, CIFAR-10, and CIFAR-100 datasets.
result Committees outperform single models in noisy conditions, especially on difficult datasets.
We investigate geometric aspects of double field theory (DFT) and its formulation as a doubled membrane sigma-model. Starting from the standard Courant algebroid over the phase space of an open membrane, we determine a splitting and a projection to a subbundle that sends the Courant algebroid operations to the correspo…
New smooth 2-group extensions from bundle gerbes on manifolds.
problem Classifying equivariant structures on bundle gerbes.
method Global approach to parallel transport, homotopy-coherent associated bundle construction.
result New models for the string group of compact simply-connected Lie groups.
New algebraic structure derived from Hopf algebra and Drinfel'd twist.
problem Developing a new algebraic structure from existing mathematical concepts.
method Extending -algebra to a Hopf algebra, twisting with Drinfel'd twist, and identifying Hopf morphisms and braided morphisms.
result Braided -algebra is derived from the process.
Study of cluster and skein algebras for surfaces, showing their connection.
problem Understanding algebraic structures of curve algebras on surfaces.
method Generalization and explicit definition of maps between cluster and skein algebras.
result Explicit maps between cluster and skein algebras, showing their close relationship.
Study on pseudo-Riemannian algebraic Ricci solitons in 4D Lie groups.
problem Investigating conditions for pseudo-Riemannian algebraic Ricci solitons on 4D Lie algebras.
method Analyzing the algebraic Ricci soliton equation for each 4D Lie algebra.
result Complete description of pseudo-Riemannian algebraic Ricci solitons in dimension four.
Characterizes Lie groups with specific structures and finds a correspondence between carrollian and galilean Lie algebras.
problem Understanding Lie groups with specific structures.
method Using structure theory of metric Lie algebras and defining new Lie algebras with skew-symmetric derivations.
result A canonical correspondence between carrollian and galilean Lie algebras mediated by bargmannian Lie algebras.
We extend the classical characterization of a finite-dimensional Lie algebra g in terms of its Maurer-Cartan algebra-the familiar differential graded algebra of alternating forms on g with values in the ground field, endowed with the standard Lie algebra cohomology operator-to sh Lie-Rinehart algebras. To this end, we …
The paper studies prolongations of Lie algebras associated with pseudo -type Lie algebras.
problem Investigating prolongations of Lie algebras associated with pseudo -type Lie algebras.
method Analyzing prolongations of associated fundamental graded Lie algebra and associated conformal pseudo-subriemannian fundamental graded Lie algebra.
result The prolongation of the associated conformal pseudo-subriemannian fundamental graded Lie algebra coincides with that of the associated fundamental graded Lie algebra under certain conditions.
A Lie-admissible algebra gives by anticommutativity a Lie algebra. In this work we study remarkable classes of Lie-admissible algebras such as Vinberg, PreLie algebras. We compute the corresponding binary quadratic operads and study their Koszul duality. Considering Lie algebras as Lie-admissible algebras we can define…
Study biderivations in complete Leibniz algebras, extending Lie algebra results.
problem Defining and studying biderivations in complete Leibniz algebras.
method Analyze biderivations according to two definitions, provide conditions for biderivations, and compare symmetric and skew-symmetric biderivations.
result Necessary and sufficient conditions for biderivations in Leibniz algebras are provided.
Universal enveloping algebras of Lie-Rinehart algebras: crossed products, connections, and curvaturemath.RA
The paper extends a theorem to Lie-Rinehart algebras and provides new decompositions of universal enveloping algebras.
problem Understanding universal enveloping algebras of Lie-Rinehart algebras.
method Extending a theorem to left Hopf algebroids and applying it to universal enveloping algebras of Lie-Rinehart algebras.
result Provides a crossed product decomposition of universal enveloping algebras for curved and flat connections.
The paper classifies Lie algebras with special operators.
problem Classifying 3D Lie algebras with regular semisimple algebraic Nijenhuis operators.
method Described all Nijenhuis eigenbases for each 3D Lie algebra.
result Different answers in real and complex cases, some Lie algebras admit operators, others do not.
Abstract defines and identifies a planar algebra with spin properties.
problem Identifying a specific planar algebra.
method Generators and relations defined, structure studied, identified with Jones' spin planar algebra.
result Identified a specific planar algebra with spin properties.
Symmetric spaces' connections form Lie admissible triple algebras.
problem Understanding the algebraic structure of symmetric spaces' connections.
method Analyzing the connection as a binary operator on tangent bundle sections, identifying Lie admissibility constraints.
result Connection algebra of symmetric spaces is a Lie admissible triple algebra.
The paper generalizes para-Kähler Lie algebras to k-para-Kähler Lie algebras and explores their structures.
problem Characterizing and understanding k-para-Kähler Lie algebras.
method Generalization of para-Kähler Lie algebras to k-para-Kähler Lie algebras, introduction of new structures, determination of Lie algebras.
result Determination of all k-symplectic Lie algebras of dimension (k+1) and six-dimensional 2-para-Kähler Lie algebras.
The paper investigates gradings of complex simple Lie algebras, focusing on -gradings and their algebraic structures.
problem Investigating the algebraic structure of -gradings of complex simple Lie algebras.
method Completely determining the possible reductive algebras and proving the uniqueness of a specific free nilpotent Lie algebra.
result The only free nilpotent Lie algebra of step 3 that appears as the negative part of a -grading is the usual -grading of the exceptional Lie algebra .
The pong algebramath.GT
New algebra pong algebra computed for knot Floer homology.
problem Computing A-infinity structure on knot Floer homology.
method Introduced differential graded algebra, pong algebra.
result Computed A-infinity structure on pong algebra's homology.
Study resolves conjecture linking two algebraic structures on surfaces.
problem Compatibility of skein and cluster algebra structures on surfaces.
method Established compatibility between skein and cluster algebras of surfaces.
result Cluster algebra of positive genus surfaces is not finitely generated.
Develops a bialgebra theory for post-Lie algebras using geometric interpretations and bilinear forms.
problem Characterizing and understanding post-Lie algebras and their associated structures.
method Utilizes Manin triples and generalized Hessian Lie groups to define and characterize post-Lie algebras with nondegenerate symmetric invariant bilinear forms.
result Establishes a bialgebra theory for post-Lie algebras via the Manin triple approach, including new algebraic structures like pp-post-Lie algebras.
Characterizes G2-structures on Lie algebras with non-trivial center.
problem Classifying Lie algebras with G2-structures.
method Analyzing Lie algebras with non-trivial center, using contactization and symplectic properties.
result Six unimodular Lie algebras with non-trivial center admit closed G2-structures.
Twilled L(ie-)R(inehart)-algebras generalize, in the Lie-Rinehart context, complex structures on smooth manifolds. An almost complex manifold determines an "almost twilled pre-LR algebra", which is a true twilled LR-algebra iff the almost complex structure is integrable. We characterize twilled LR structures in terms o…
This paper presents results on the framization of some knot algebras, defined by the authors. We explain the motivations of the concept of framization, coming from the Yokonuma--Hecke algebras, as well as recent results on the framization of the Temperley--Lieb algebra. Finally, we propose framizations for other knot a…
For finite dimensional real Lie algebras, we investigate the existence of an inner product having a basis comprised of geodesic elements. We give several existence and non-existence results in certain cases: unimodular solvable Lie algebras having an abelian nilradical, algebras having an abelian derived algebra, algeb…
Similarity algebra extends algebraic structures with quantitative bounds.
problem Exact algebraic structures with strict axioms.
method Framework for approximate algebraic and Lie structures with -estimates.
result Similarity structures converge to classical algebraic objects as .
Extends Loday-Quillen-Tsygan theorem to bornological Lie algebra homology.
problem Calculating Lie algebra homology of gauge algebras using cyclic homology.
method Extends proof to bornological Lie algebra homology of Fréchet and LF-algebras, prepares statements about homological algebra of topological vector spaces.
result Constructs a spectral sequence to calculate stable part of bornological Lie algebra homology of gauge algebras.
New algebra for twice-punctured torus curves.
problem Constructing a new algebra for skein theory.
method Using Heegaard dual of Iwahori--Hecke operator, Dehn twists are represented.
result Automorphisms correspond to Dehn twists on the twice-punctured torus.