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.
Research decouples Lie algebroids using bicocycle double cross product theory.
problem Understanding decoupling and coupling phenomena in Lie algebroids.
method Bicocycle double cross product realization method.
result Unified product, double cross product, semi-direct product, and cocycle extension frameworks are instances of the general method.
Researchers create spectral triples for twisted crossed products using Kasparov's external product.
problem Constructing spectral triples for twisted crossed products.
method Using Kasparov's external product, the construction of spectral triples for twisted crossed products is achieved.
result The construction of spectral triples for twisted crossed products is possible under suitable assumptions.
Isomorphism found between filtered calculus and crossed products.
problem Tackles isomorphism in filtered calculus and crossed products.
method Uses natural R-action and structure result for C*-algebra of graded nilpotent Lie groups.
result Found isomorphism between kernel of tangent groupoid and crossed product.
Study relative commutants in von Neumann algebras using contraction notions.
problem Understanding relative commutants in group and tracial crossed product von Neumann algebras.
method Introducing contraction notions to study relative commutants.
result Results applied to negatively curved groups and SL(d, Z).
Let G be a finite group. Noncommutative geometry of unital G-algebras is studied. A geometric structure is determined by a spectral triple on the crossed product algebra associated with the group action. This structure is to be viewed as a representative of a noncommutative orbifold. Based on a study of classical o…
This work merges 3-anchored bundles into 3-Lie algebroids.
problem Combining two 3-anchored bundles into a unified structure.
method Develops algebraic framework for merging bundles with mutual actions and cocycle terms.
result Unified setting for 3-Lie algebroids and special cases.
We study differential operators, whose coefficients define noncommutative algebras. As algebra of coefficients, we consider crossed products, corresponding to action of a discrete group on a smooth manifold. We give index formulas for Euler, signature and Dirac operators twisted by projections over the crossed product.…
The fundamental group of a hyperbolic manifold acts on the limit set, giving rise to a cross-product C^* algebra. We construct nontrivial K-cycles for the cross-product algebra, thereby extending some results of Connes and Sullivan to higher dimensions. We also show how the Patterson-Sullivan measure on the limit set c…
The paper explores rigidity and proximality in dynamical systems, proving new results about C∗-algebras.
problem Understanding rigidity and proximality in dynamical systems and their algebraic counterparts.
method Analyzing crossed products of dynamical systems and their C∗-algebras, focusing on uniform rigidity and proximality. result Uniformly rigid systems are almost reflecting, and certain crossed products are reflecting.
New dynamics derived from Lie groups using 2nd order tangent groups.
problem Deriving dynamics on complex Lie groups.
method Using double cross product groups and 2nd order Euler-Lagrange equations.
result 2nd order Lagrangian dynamics on double cross product groups derived.
Analytic torsion form constructed for non-commutative spaces.
problem Constructing analytic torsion form on non-commutative spaces.
method Using fiber bundles, flat vector bundles, and crossed product algebras, we define a non-commutative deRham differential form.
result The constructed torsion form appears in a transgression formula and leads to an index formula.
The orbit decomposition is given under the automorphism group on the real split Jordan algebra of all hermitian matrices of order three corresponding to any real split composition algebra, or the automorphism group on the complexification, explicitly, in terms of the cross product of H. Freudenthal and the characterist…
Two constructions of Chern character for equivariant vector bundles in noncommutative geometry.
problem Chern character for equivariant vector bundles in noncommutative geometry.
method Two constructions using cyclic cohomology of crossed product algebras.
result Equivalence of two constructions under proper action.
Defines complex manifolds on commutative Banach algebras and studies continuous families of compact complex manifolds.
problem Defines complex manifolds on commutative Banach algebras.
method Introduces a new manifold structure on continuous cross sections of complex vector bundles.
result Finite-dimensional C(X)-manifolds can be constructed from continuous families of compact complex manifolds.
The paper constructs quasi-isomorphisms for cyclic homology of group actions.
problem Computing cyclic homology and periodic cyclic homology of group actions.
method Explicit quasi-isomorphisms for crossed-product algebras and locally convex algebras.
result New spectral sequences for cyclic homology of group actions.
Abstract sketches historical development of Lie brackets, crossed modules, and Lie-Rinehart algebras.
problem Characterizing and understanding the relationships between Lie brackets, crossed modules, and Lie-Rinehart algebras.
method Historical review and combinatorial group theory considerations.
result The mutual relationship between Lie-Rinehart algebras and Lie brackets, and the historical development of these concepts.
Let g be a Leibniz algebra and E a vector space containing g as a subspace. All Leibniz algebra structures on E containing g as a subalgebra are explicitly described and classified by two non-abelian cohomological type objects: ${\mathcal H}{\mathcal L}^{2}_{\mathfrak{g}} \, (…
This paper studies curve multiplication in a specific algebraic structure.
problem Multiplication in the Kauffman bracket skein algebra of the thickened four-holed sphere.
method Developed an algorithm and explicit formula for curve multiplication, conjectured existence of a positive basis.
result Quasi-polynomial growth of the algorithm with respect to the number of crossings.
The paper studies coarse quotients and Roe algebras for group actions.
problem Understanding group actions on metric spaces and their algebraic properties.
method Defining coarse quotients and using large scale connectedness, the paper explores relations between Roe algebras and automorphism groups.
result There is a ∗-isomorphism between the maximal Roe algebras of X and XG under certain conditions. Study recovers C*-algebra from fields of Toeplitz algebras on specific groups.
problem Recovering C*-algebra from fields of Toeplitz algebras on specific groups.
method Using continuous fields of Toeplitz algebras and a crossed product.
result Algebra of principal symbols can be recovered from fields of Toeplitz algebras.
Proves magnetic helicity and cross-helicity are essential invariants in 3D magnetohydrodynamics.
problem Identifying essential invariants in 3D magnetohydrodynamics.
method Proves that Casimirs are functions of magnetic helicity and cross-helicity.
result Magnetic helicity and cross-helicity are the only independent regular integral invariants.
This paper provides a construction of a quantum statistical mechanical system associated to knots in the 3-sphere and cyclic branched coverings of the 3-sphere, which is an analog, in the sense of arithmetic topology, of the Bost-Connes system, with knots replacing primes, and cyclic branched coverings of the 3-sphere …
We prove that if M is a CW-complex, then the homotopy type of the skeletal filtration of M does not depend on the cell decomposition of M up to wedge products with n-disks Dn, when the later are given their natural CW-decomposition with unique cells of order 0, (n−1) and n; a result resembling J.H.C. Whi…
Algorithm finds positive braids with specific crossing matrices.
problem Finding positive braids with given crossing matrices.
method Finite algorithm using decomposition of braids.
result Algorithm determines existence of positive braids with specific matrices.
Constructs an analytic index for infinite dimensional manifolds with LT-action.
problem Analyzing infinite dimensional manifolds with LT-action. method Defines an analytic LT-equivariant index using a Hilbert space, Dirac operator, and crossed product. result Justifies the analytic index in terms of noncommutative geometry.
We show that any dimension 6 nearly Kähler (or nearly para-Kähler) geometry arises as a projective manifold equipped with a G2(∗) holonomy reduction. In the converse direction we show that if a projective manifold is equipped with a parallel 7-dimensional cross product on its standard tractor bundle …
Kasparov defined a distinguished K-homology fundamental class, so called the Dirac element. We prove a localization formula for the Dirac element in K-homology of crossed product of C^{*}-algebras. Then we define the quantization of Hamiltonian G-spaces as a push-forward of the Dirac element. With this, we develop a K-…
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.
The Conway potential function (CPF) for colored links is a convenient version of the multi-variable Alexander-Conway polynomial. We give a skein characterization of CPF, much simpler than the one by Murakami. In particular, Conway's `smoothing of crossings' is not in the axioms. The proof uses a reduction scheme in a t…
Defines cross product for m vectors in n-dimensional spaces.
problem No universal definition for cross product in high-dimensional spaces.
method Defines cross product for m vectors in n-dimensional spaces with any metric matrices.
result Cross product length represents m-dimensional volume, components represent volume directions.
New representations of Lie algebras via monoidal category actions.
problem Constructing representations of Lie algebras using monoidal categories.
method Using crossed homomorphisms and monoidal categories to generate representations.
result Established new bifunctor for weak and admissible representations of Lie-Rinehart algebras.
This paper explores topological aspects of index theory for infinite-dimensional manifolds.
problem Formulating index theory for infinite-dimensional manifolds with LT-actions.
method Introducing RKK-theory and constructing assembly maps for proper LT-spaces.
result Formulation of infinite-dimensional Poincaré duality and assembly maps.
Develops a universal Hermitian projective calculus for complex hyperbolic two-space
problem Complex hyperbolic geometry
method Algebraic invariant calculus
result Denominator-cleared identities for various geometric quantities
Let F be an incompressible, meridionally incompressible and not boundary-parallel surface with boundary in the complement of an algebraic tangle (B,T). Then F separates the strings of T in B and the boundary slope of F is uniquely determined by (B,T) and hence we can define the slope of the algebraic tang…
Here we study the deformations of associative submanifolds inside a G_2 manifold M^7 with a calibration 3-form φ. A choice of 2-plane field Λon M (which always exits) splits the tangent bundle of M as a direct sum of a 3-dimensional associate bundle and a complex 4-plane bundle TM= E\oplus V, and this helps us to relat…
Study boundary actions of CAT(0) spaces and their C∗-algebras.
problem Investigate boundary actions of CAT(0) spaces and their associated C∗-algebras. method Topological dynamics and C∗-algebras, focusing on actions of specific groups and their properties. result Established (strongly) pure infiniteness results for reduced crossed product C∗-algebras of boundary actions. New Boolean algebra method shows knot unknotting number is (c+1)/2.
problem Finding the minimum number of region crossing changes to unknot a knot.
method Boolean algebra applied to region crossing changes.
result Region unknotting number is (c+1)/2 for any knot with crossing number c.
Investigates adjustments on Lie group crossed modules for gauge theory.
problem Existence and classification of adjustments on crossed modules of Lie groups.
method Differentiation/integration correspondence with infinitesimal adjustments; Lie algebra techniques.
result Infinitesimal adjustments exist if and only if the Kassel-Loday class lies in the image of the Chern-Weil homomorphism.
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.
In this article, we give a geometric proof of the classification of complex vector cross product due to Lee-Leung.
We consider a smooth groupoid of the form Σ\rtimesΓwhere Σis a Riemann surface and Γa discrete pseudogroup acting on Σby local conformal diffeomorphisms. After defining a K-cycle on the crossed product C_0(Σ)\rtimesΓgeneralising the classical Dolbeault complex, we compute its Chern character in cyclic cohomology, using…
Study Killing forms on negatively curved manifolds, introducing generalized vector cross products.
problem Understanding Killing forms on negatively curved manifolds.
method Introduced generalized vector cross products and characterized SU(3) structures.
result Killing 3-forms on negatively curved manifolds vanish for dimensions greater than or equal to 4.
Study shows crossing numbers for algebraic knots can differ by arbitrarily large amounts.
problem Comparing two crossing number definitions for algebraic knots.
method Analyzed Hopf fibration and complex singularities to compare crossing numbers.
result Difference between crossing numbers can be arbitrarily large.
New algebras and maps defined in knot Floer homology for trivalent vertices.
problem Categorification of knot Floer homology for trivalent vertices.
method Definition of new algebras, local bimodules, and bimodule maps in bordered knot Floer homology.
result Categorification of representations of U_q(gl(1|1)^-).
Integrable dynamics explained via geometric maps and cluster algebras.
problem Integrable dynamics in projective geometry.
method Twisted triple crossing diagram maps and cluster integrable systems.
result Cross-ratio dynamics described by geometric R-matrices. In this paper we first state the classification of the prolongations of complex free fundamental graded Lie algebras. Next we introduce the notion of free pseudo-product fundamental graded Lie algebras and study the prolongations of complex free pseudo-product fundamental graded Lie algebras. Furthermore we investigate…
Develops index theory for infinite-dimensional manifolds with loop group actions using KK-theory.
problem Establish an index theory for infinite-dimensional manifolds acted upon by loop groups.
method Constructs objects in KK-theory and defines new cycles and indices.
result Defines a KK-theoretical index and compares it with the analytic index.