New braided Frobenius algebras created from specific Hopf algebras.
problem Creating new algebraic structures from Hopf algebras.
method Heap operation and Yang-Baxter operator on tensor product.
result Heap operation induces a braiding compatible with Frobenius operations.
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.
Defines and characterizes operators on Lie ∞-algebras with respect to actions.
problem Characterizing operators on Lie ∞-algebras with respect to actions.
method Using higher derived brackets construction and Maurer-Cartan elements.
result Determines the Lie ∞-algebra controlling the deformation of operators.
The paper constructs new algebraic structures from Lie algebras and ternary Nambu-Lie algebras, leading to Yang-Baxter operators.
problem Constructing new algebraic structures from Lie algebras and ternary Nambu-Lie algebras.
method Using compositions of binary Lie algebras, 3-Lie algebras, and ternary Nambu-Lie algebras, the paper constructs ternary self-distributive objects and Yang-Baxter operators.
result The constructed Yang-Baxter operators are not gauge equivalent to the transposition operator and can be deformed to new solutions.
Determines algebra structure of complex differential forms operators.
problem Identifying the algebra structure of differential operators on complex-valued differential forms.
method Shows it is the universal enveloping algebra of a graded Lie algebra and determines its cohomology.
result Determines the cohomology of the graded Lie algebra with respect to various inner differentials.
Study BV operators on holomorphic polyvector fields on toric varieties.
problem Existence of BV operators in Gerstenhaber algebras.
method Analyzing BV operators on holomorphic polyvector fields on smooth compact toric varieties.
result Necessary and sufficient condition for BV operators existence.
Classifies 3D non-degenerate left-symmetric algebras.
problem Classifying left-symmetric algebras in 3D.
method Using Nijenhuis geometry and algebraic independence of coefficients in characteristic polynomial.
result Classification of differentially non-degenerate LSA in dimension 3.
This paper explores the relationship between Leibniz algebras and Nijenhuis operators.
problem Understanding the relationship between Leibniz algebras and Nijenhuis operators.
method Investigation of Nijenhuis operators on Leibniz algebras and classification of Leibniz bialgebras.
result Leibniz algebras are closely related to Nijenhuis operators, and triangular symplectic Leibniz bialgebras possess Nijenhuis operators.
We identify Melrose's suspended algebra of pseudodifferential operators with a subalgebra of the algebra of parametric pseudodifferential operators with parameter space R. For a general algebra of parametric pseudodifferential operators, where the parameter space may now be a cone Γ⊂Rp, we construct a uniq…
Novel duality theory for operator Frobenius algebras solves long-standing hydrodynamic integrable systems problem.
problem Long-standing Eisenhart-Stäckel problem for non-degenerate integrable systems.
method Introduce duality for operator Frobenius algebras and use mutual symmetry assumption.
result Construct new infinite-dimensional integrable systems of hydrodynamic type.
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 …
Examines algebraic conditions for positive sectional curvature in 4D and higher.
problem Determining when the sectional curvature of a Riemannian manifold is positive.
method Analyzes algebraic conditions for sectional positivity in 4D and higher dimensions.
result Characterizes a dense open subset of operators in 4D for sectional positivity.
The notion of pseudo-differential operators with coefficients in a continuous trace algebra over a manifold are introduced and their index theory is studied. The algebra of principal symbols in this calculus provides an abstract Poincaré dual to the continuous trace algebra. Index formulas for pseudo-differential opera…
Study describes index map for a specific algebra of pseudodifferential operators.
problem Understanding the K-theory index map for a particular algebra of pseudodifferential operators.
method Description of the K-theory index map associated with the continuous extension of the principal-symbol map.
result The index map takes values in K_0 of the commutator ideal E, which is isomorphic to Z^2.
Haantjes algebras help in diagonalizing operators on manifolds.
problem Diagonalizing operators on differentiable manifolds.
method Introducing Haantjes algebra, a family of operator fields with vanishing Haantjes torsion and compatibility conditions.
result Simultaneous diagonalization of operators in local coordinates or block-diagonal form in general cases.
Study YB operators and their deformations, finding integrable and nontrivial cases.
problem Understanding deformations of Yang-Baxter operators and their integrability.
method Relating deformations to Lie algebra deformations, analyzing cohomology groups.
result Existence of integrable YB deformations and nontrivial cases not arising from SD deformations.
We establish a correspondence between Young diagrams and differential operators of infinitely many variables. These operators form a commutative associative algebra isomorphic to the algebra of the conjugated classes of finite permutations of the set of natural numbers. The Schur functions form a complete system of com…
New theory allows simultaneous block-diagonalization of commuting operator fields.
problem Normal forms of operator fields.
method Generalized Nijenhuis torsions and generalized Haantjes algebra.
result Simultaneous block-diagonalization of commuting operator fields.
In this paper we get a necessary and sufficient condition for the Ricci operator of a solvable metric Lie algebra to have at least two negative eigenvalues. In particular, this condition implies that the Ricci operator of every non-unimodular solvable metric Lie algebra or every non-abelian nilpotent metric Lie algebra…
This paper explores A-infinity structures in contact categories and strand algebras.
problem Understanding A-infinity structures in contact categories and strand algebras.
method Explicit constructions and properties of A-infinity operations are established.
result Conditions for the vanishing and nonvanishing of A-infinity operations are derived.
New tools for analyzing Kähler manifolds, proving operator algebra and asymptotic kernel.
problem Analyzing Berezin-Toeplitz operators on Kähler manifolds.
method Introducing new tools for analytic microlocal analysis.
result Space of analytic Berezin-Toeplitz operators is an algebra.
Study Nijenhuis operators on homogeneous spaces related to C*-algebras.
problem Characterize Nijenhuis operators on homogeneous spaces of C*-algebras.
method Analyze vector bundle maps induced by admissible operators on C*-algebras.
result Identify conditions for vector bundle maps to be Nijenhuis operators.
Defines a new algebra for singular foliations, extending Schwartz kernels.
problem Extending Schwartz kernel operators to singular foliations.
method Defines convolution algebra of transverse distributions, proves representation as operators on spaces of functions.
result Generalizes Schwartz kernel operators to singular foliations.
This paper deals with sheaves of differential operators on noncommutative algebras. The sheaves are defined by quotienting a the tensor algebra of vector fields (suitably deformed by a covariant derivative) to ensure zero curvature. As an example we can obtain enveloping algebra like relations for Hopf algebras with di…
Researchers prove Lie algebras of differential operators and Grothendieck constructions coincide.
problem Understanding derivations and Lie algebras of vector bundles.
method Proving Lie algebras coincide through differential operators and Grothendieck constructions.
result Lie algebras coincide up to an isomorphism.
Characterizes operations on contact manifold differential forms.
problem Understanding natural operations on contact manifold differential forms.
method Introduces algebraic operators and the exterior derivative to characterize operations.
result All natural operations are built from introduced algebraic operators and the exterior derivative.
Homotopy operators help describe structures in equivariant deformation problems.
problem Equivariant deformation problems in algebraic structures.
method Use homotopy operators for an L∞-algebra associated with the problem. result Smooth parametrization of the space of structures around a given one.
Poisson algebra is usually defined to be a commutative algebra together with a Lie bracket, and these operations are required to satisfy the Leibniz rule. We describe Poisson structures in terms of a single bilinear operation. This enables us to explore Poisson algebras in the realm of non-associative algebras. We stud…
Researchers extend pseudodifferential calculus on filtered manifolds using fixed point algebras.
problem Defining operators with varying orders on filtered manifolds.
method Using generalized fixed point algebras and nilpotent Lie groups, they construct a new calculus.
result They establish a new calculus that reflects the behavior of differential operators on filtered manifolds.
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.
We consider the geometry of second order linear operators acting on the commutative algebra of densities on a (super)manifold introduced in our previous work. In the conventional language, operators on the algebra of densities correspond to operator pencils. This algebra has a natural invariant scalar product. We consi…
We review "quantum" invariants of closed oriented 3-dimensional manifolds arising from operator algebras.
Study non-commutative function algebras using contact geometry.
problem Quantize non-commutative function algebras in several variables.
method Contact geometry and rational differential operators.
result Generalizes known constructions of classical equivariants.
Study Nijenhuis operators and their linearization problem using left-symmetric algebras.
problem Linearization of Nijenhuis operators.
method Study points of scalar type, use left-symmetric algebras, classify 2D algebras.
result Complete classification of 2D real left-symmetric algebras.
We associate to any Riemannian symmetric space (of finite or infinite dimension) a L∗-algebra, under the assumption that the curvature operator has a fixed sign. L∗-algebras are Lie algebras with a pleasant Hilbert space structure. The L∗-algebra that we construct is a complete local isomorphism invariant and …
Study of differential operators on von Neumann algebras with applications to index theory.
problem Index theory for differential operators on non-compact manifolds.
method Relative index and Callias-type index theorems for von Neumann indexes.
result Cobordism invariance of the index of Callias-type operators.
We discuss algebraic properties for the symbols of geometric first order differential operators on almost Hermitian manifolds and Kähler manifolds. Through study on the universal enveloping algebra and higher Casimir elements, we know algebraic relations for the symbols like the Clifford algebra. From the relations, we…
Operators on the ring of algebraically constructible functions are used to compute local obstructions for a four-dimensional semialgebraic set to be homeomorphic to a real algebraic set. The link operator and arithmetic operators yield 243−43 independent characteristic numbers mod 2, which generalize the Akbulut-K…
Constructs a homotopy Loday algebra from symplectic 2-manifolds.
problem Tackles the construction of algebraic structures from symplectic 2-manifolds.
method Uses higher derived brackets and Voronov's technique to construct a homotopy Loday algebra.
result Constructs a homotopy Loday algebra with a specific structure accommodating the Dorfman bracket.
We consider differential operators acting on densities of arbitrary weights on manifold M identifying pencils of such operators with operators on algebra of densities of all weights. This algebra can be identified with the special subalgebra of functions on extended manifold M^. On one hand there is a canonical…
We construct bundles of modules of vertex operator algebras, and prove the rigidity and vanishing theorem for the Dirac operator on loop space twisted by such bundles. This result generalizes many previous results.
In this paper, we prove the algebraic K-theory Novikov conjecture for group algebras over the ring of Schatten class operators. The main technical tool in the proof is an explicit construction of the Connes-Chern character.
Paper develops Lie theory for Rota-Baxter operators on Lie algebras and groups.
problem Cohomology of relative Rota-Baxter operators on Lie algebras and groups.
method Cohomology construction, infinitesimal deformations study, differentiation and integration of Rota-Baxter operators.
result Integration of Rota-Baxter operators on Lie groups and Lie algebras.
Constructs index for elliptic operators using rapidly decaying kernels.
problem Index of elliptic operators in Fréchet algebra.
method Uses heat operators and heat kernel asymptotics.
result Index can be represented by an idempotent involving heat operators.
Study on deformation cohomology for braided commutative structures.
problem Classifying and understanding deformations of braided commutative algebras.
method Extending Yang-Baxter Hochschild cohomology to braided commutative deformations.
result Classifies infinitesimal deformations of braided algebras that are braided commutative.
The paper describes algebraic operations on surface fundamental groups.
problem Turaev's loop operations on surface fundamental groups.
method Formal power series and Drinfeld associators.
result Algebraic description of Turaev's self-intersection map.
New operations on Khovanov homology refine knot invariants.
problem Understanding finer knot invariants through homological operations.
method Developed an algebra of homological operations on Khovanov homology and lifted it to integral Khovanov homology.
result Conjectured infinite algebras of homological operations and provided evidence.
We compute the equivariant cohomology Chern character of the index of elliptic operators along the leaves of the foliation of a flat bundle. The proof is based on the study of certain algebras of pseudodifferential operators and uses techniques for analizing noncommutative algebras similar to those developed in Algebra…