Characterizes real left symmetric algebras with positive definite Koszul form and related Kähler-Einstein structures.
problem Characterizing real left symmetric algebras with positive definite Koszul form.
method Analyzes the properties of left multiplication operators and symmetric bilinear forms.
result Provides a complete characterization of real left symmetric algebras with positive definite Koszul form.
A general definition of a bimodule connection in noncommutative geometry has been recently proposed. For a given algebra this definition is compared with the ordinary definition of a connection on a left module over the associated enveloping algebra. The corresponding curvatures are also compared.
Redefined cord algebra using Morse Theory for knot invariants.
problem Defining a knot invariant using algebraic methods.
method Using Morse Theory to redefine the cord algebra.
result Proved the cord algebra is a knot invariant.
Discuss various definitions of vector fields on manifolds leading to Lie algebras.
problem Defining vector fields on convenient manifolds.
method Various definitions of vector fields and their equivalence.
result Equivalent definitions of vector fields on manifolds.
Paper defines multiplicities for quaternion eigenvalues without complex matrix concepts.
problem Defining multiplicities for quaternion eigenvalues without traditional matrix concepts.
method Introduces two definitions for algebraic and geometric multiplicities equivalent to classical definitions.
result Definitions are equivalent to classical ones and prove all properties easily.
New criteria for splitting definite 4-manifolds with cyclic groups.
problem Definite 4-manifolds with infinite cyclic fundamental groups.
method Two new criteria extending previous results, equivalent to algebraic representation conditions.
result Equivalence to algebraic representation conditions and production of new forms.
Proves algebraic cones for LCK manifolds with potential.
problem Characterizing algebraic cones for LCK manifolds.
method Analyzes LCK manifolds as complex submanifolds of Hopf manifolds and covers, proving algebraicity of the resulting cones.
result Affine algebraic structure on cones is independent of manifold choice.
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.
The paper defines and studies the geometric mean for tensors and its associated Riemannian geometry.
problem Defining and studying the geometric mean for tensors.
method Generalized geometric mean for tensors using T-product, verified properties, and investigated Riemannian manifold.
result Geometric mean of T-positive definite tensors is a unique solution of algebraic Riccati tensor equations and a midpoint of geodesics.
Algebraic definition classifies Markowitz markets simplifying portfolio optimization.
problem Complexity in Markowitz market classification.
method Algebraic definition and isomorphism classification.
result Classification shows little beyond portfolio optimization insights.
Study identifies specific subvarieties in translation surfaces with quadratic field.
problem Characterizing invariant subvarieties in translation surfaces with quadratic field.
method Analyzing algebraically primitive subvarieties in strata of translation surfaces.
result Only specific subvarieties identified: decagon, Weierstrass curves, etc.
The fundamental ideas of the definition of solvable and semisimple Bol algebras are given and some related theorems
New Lie algebras from knot homology.
problem Defining Lie algebras from knot homology.
method Using group homology, analogous to Goldman Lie algebra.
result Relations among new Lie algebras discussed.
Abstract: Review and definitions of generalised spin structures, their connections, and symmetry algebra.
problem Understanding and characterizing generalised spin structures and their properties.
method Definitions, basic notions, connections, covariant Lie derivative, covariant Cartan calculus, symmetry algebra.
result Characterization of homogeneous generalised spin structures.
We give a diagrammatic definition of Uq(sl2) when q is not a root of unity, including its Hopf algebra structure and its relationship with the Temperley-Lieb category.
We propose a new definition of so called Hamiltonian forms in n-plectic geometry and show that they have a non-trivial Lie infinity-algebra structure.
Unified determinants via a single equation.
problem Defining determinants with all known properties.
method Proposing a single equation implying all known properties of determinants.
result Unified definition of determinants with all properties.
These notes are devoted to the multiple generalization of a Lie algebra introduced by A.M.Vinogradov and M.M.Vinogradov. We compare definitions of such algebras in the usual and invariant case. Furthermore, we show that there are no simple n-ary Lie algebras of type (n−1,l) for l>0.
Defines null G-structures on Lorentzian manifolds and explores their symmetries.
problem Understanding geometric properties of null G-structures on Lorentzian manifolds.
method Definition and investigation of null G-structures, identification of induced geometries, algebra of diffeomorphisms.
result Interpolates between BMS and Lorentz symmetry algebras in some cases.
The submanifold quantum mechanics was opened by Jensen and Koppe (Ann. Phys. {\bf 63} (1971) 586-591) and has been studied for these three decades. This article gives its more algebraic definition and show what is the essential of the submanifold quantum mechanics from an algebraic viewpoint.
New approach to supervised learning in RKHS and vvRKHS using C∗-algebras.
problem Traditional supervised learning in RKHS and vvRKHS.
method Generalizing supervised learning to RKHM using C∗-algebras. result Constructing RKHMs with enhanced representation power.
New algebraic setup defines quantum link invariants.
problem Defining and controlling quantum link invariants.
method Quantum Schur--Weyl duality and variants.
result Global definitions of quantum polynomials.
We review the basic definition of a stack and apply it to the topological and smooth settings. We then address two subtleties of the theory: the correct definition of a ``stack over a stack'' and the distinction between small stacks (which are algebraic objects) and large stacks (which are generalized spaces).
Defines and classifies algebraic Schouten solitons in 3D Lorentzian Lie groups.
problem Classifying solitons in 3D Lorentzian Lie groups.
method Defined algebraic Schouten solitons and classified them for specific connections.
result Classified algebraic Schouten solitons for various connections on 3D Lorentzian Lie groups.
Paper introduces danceability index as a new bridge index definition.
problem Defining the bridge index in various mathematical contexts.
method Proves danceability index as equivalent to bridge index, extends to virtual knots.
result Danceability index is a new equivalent definition of the bridge index.
For a complex projective manifold Gromov-Witten invariants can be constructed either algebraically or symplectically. Using the versions of Gromov-Witten theory by Behrend and Fantechi on the algebraic side and by the author on the symplectic side, we prove that both points of view give the same results. A similar stat…
We give a survey on L^2-invariants such as L^2-Betti numbers and L^2-torsion taking an algebraic point of view. We discuss their basic definitions, properties and applications to problems arising in topology, geometry, group theory and K-theory.
Introduces Niebrzydowski algebras for trivalent spatial graphs and handles.
problem Counting and distinguishing trivalent spatial graphs and handlebody-links.
method Defines Niebrzydowski algebras with ternary operation and partially defined multiplication, motivated by Reidemeister moves.
result Niebrzydowski algebras can distinguish some trivalent spatial graphs and handlebody-links.
Defines formal vertex laws related to Lie conformal algebras.
problem No specific problem stated; focuses on definitions and proofs.
method Definitions and proofs of vertex/conformal versions of classical Lie theory results.
result Proves vertex/conformal versions of important Lie theory results.
Study of Pascal algebra matrices and their jet bundle map for vector bundles.
problem Defining and studying Pascal algebra matrices and their map on jet bundles.
method Identifying Pascal algebra matrices, showing generator well defines Pascal map, using it for intrinsic contact definition.
result Intrinsic definition of point-wise contact between Hermitian vector bundles using unitary equivalence of Pascal maps.
Kock [Bull. Austral. Math. Soc., 25 (1982), 357-386] has considered differential forms with values in a group in a context where neighborhood relations are available. By doing so, he has made it clear where the so-called Maurer-Cartan formula should come from. In this paper, while we retain the classical definition of …
It is shown that the non-trivial cocycles on simple Lie algebras may be used to introduce antisymmetric multibrackets which lead to higher-order Lie algebras, the definition of which is given. Their generalised Jacobi identities turn out to be satisfied by the antisymmetric tensors (or higher-order `structure constants…
On realizations of the Lie groups $ G_{2,\boldmath\scriptstyle{H}},F_{4,\boldmath\scriptstyle{H}},E_{6,\boldmath\scriptstyle{H}},E_{7,\boldmath\scriptstyle{H}},E_{8,\boldmath\scriptstyle{H}} $, second editionmath.DG The paper examines alternative definitions of exceptional Lie groups using quaternion numbers.
problem Defining exceptional Lie groups using quaternion numbers.
method Replacing Cayley algebra with quaternion numbers to define the groups.
result Determined the structure of the new Lie groups.
Conditions for exponentiating Lie algebras on complete locally convex spaces are established.
problem Conditions for exponentiating Lie algebras of linear operators on complete locally convex spaces.
method Focus on equicontinuous case, establishing necessary conditions for exponentiation to compact Lie groups.
result Necessary conditions for exponentiation to compact Lie groups are established.
Algebraic geometry replaces manifolds in differential geometry.
problem Eliminate the need for manifolds in differential geometry.
method Introduce algebraifolds and use commutative algebras with finitely generated projective module of derivations.
result General relativity can be formulated using algebraifolds.
For a spacelike 2-surface in spacetime, we propose a new definition of quasi-local angular momentum and quasi-local center of mass, as an element in the dual space of the Lie algebra of the Lorentz group. Together with previous defined quasi-local energy-momentum, this completes the definition of conserved quantities i…
Two elliptic Kashiwara-Vergne Lie algebras match.
problem Matching two elliptic Kashiwara-Vergne Lie algebras.
method Using mould theoretic techniques and graded formality isomorphisms.
result The two elliptic Kashiwara-Vergne Lie algebras coincide.
Real algebraic structures help classify overtwisted contact 3-spheres.
problem Classifying overtwisted contact structures on 3-spheres.
method Using real algebraic functions and open book decompositions.
result Most overtwisted contact structures are real algebraic.
We give an account of the construction of exterior differential systems based on the notion of tableaux over Lie algebras as developed in [Comm. Anal. Geom 14 (2006), 475-496; math.DG/0412169]. The definition of a tableau over a Lie algebra is revisited and extended in the light of the formalism of the Spencer cohomolo…
We give a new definition of the knot invariant associated to the Lie algebra su_{N+1}. The knot or link must be presented as the plat closure of a braid. The invariant is then a homological intersection pairing between two submanifolds of a configuration space of points in a disk. This generalizes previous work on the …
We extend the definition of algebraic entropy to endomorphisms of affine varieties. We calculate algebraic entropy of the action of elements of mapping class groups on various character varieties, and show that it is equal to a quantity we call the spectral radius, a generalization of the dilatation of a Pseudo-Anosov …
We study n-ary commutative superalgebras and L∞-algebras that possess a skew-symmetric invariant form, using the derived bracket formalism. This class of superalgebras includes for instance Lie algebras and their n-ary generalizations, commutative associative and Jordan algebras with an invariant form. We…
The paper solves a problem in constructing a bicategory of algebra bundles.
problem Defining a well-defined composition law for algebra bundles over a smooth manifold.
method Developed a complete solution for a bicategory of algebra bundles, addressing non-invertible bimodules and non-semisimple algebras.
result A complete solution to the problem of constructing a bicategory of algebra bundles.
On realizations of the complex Lie groups (F4,R)C,(E6,R)C,(E7,R)C,(E8,R)C and those compact real forms F4,R,E6,R,E7,R,E8,Rmath.DG The paper explores alternative definitions of complex Lie groups using real numbers.
problem Defining complex Lie groups using real numbers.
method Replacing Cayley algebra with real numbers to define groups.
result Determined the structure of complex and compact Lie groups.
Natural volume forms defined for pseudo-Finslerian manifolds with specific metrics.
problem Defining natural volume forms on pseudo-Finslerian manifolds with m-th root metrics. method Definitions depend on the parity of m, expressed in terms of Cayley hyperdeterminants. result Volume forms computation simplified by avoiding integration over the indicatrix.
New knot invariant from Kauffman states and algebraic modules.
problem Developing a new knot invariant related to Alexander polynomial.
method Defining a bigraded knot invariant using Kauffman states and differential graded algebras.
result The invariant's homology corresponds to the Alexander polynomial.
Formula derived for module ranks in 3-manifold character varieties.
problem Understanding essential surfaces in 3-manifolds.
method Algebraic geometry of SL(2,C)-character varieties and module structures.
result Formula for module ranks incorporating field of definition and Culler-Shalen norm.
As the third of our series of papers on differential geometry of microlinear Frolicher spaces, this paper is devoted to the Frolicher-Nijenhuis calculus of their named bracket. The main result is that the Frolicher-Nijenhuis bracket satisfies the graded Jacobi identity. It is also shown that the Lie derivation preserve…