arXiv research

The paper extends a theorem to Lie-Rinehart algebras and provides new decompositions of universal enveloping algebras.

Research decouples Lie algebroids using bicocycle double cross product theory.

Researchers create spectral triples for twisted crossed products using Kasparov's external product.

Isomorphism found between filtered calculus and crossed products.

Study relative commutants in von Neumann algebras using contraction notions.

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.

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.

We observe that the iterated tangent group of a Lie group may be realized as a double cross product of the 2nd order tangent group, with the Lie algebra of the base Lie group. Based on this observation, we derive the 2nd order Euler-Lagrange equations on the 2nd order tangent group from the 1st order Euler-Lagrange equ…

In this paper we present the construction of explicit quasi-isomorphisms that compute the cyclic homology and periodic cyclic homology of crossed-product algebras associated with (discrete) group actions. In the first part we deal with algebraic crossed-products associated with group actions on unital algebras over any…

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.

Abstract sketches historical development of Lie brackets, crossed modules, and Lie-Rinehart algebras.

Based on the presentation of the Kauffman bracket skein module of the torus given by the third author in previous work, Charles D. Frohman and Răzvan Gelca established a complete description of the multiplicative operation leading to a famous product-to-sum formula. In this paper, we study the multiplicative structure …

Let $\mathfrak{g}$ be a Leibniz algebra and $E$ a vector space containing $\mathfrak{g}$ as a subspace. All Leibniz algebra structures on $E$ containing $\mathfrak{g}$ as a subalgebra are explicitly described and classified by two non-abelian cohomological type objects: ${\mathcal H}{\mathcal L}^{2}_{\mathfrak{g}} \, (…

An n-dimensional complex manifold is a manifold by biholomorphic mappings between open sets of the finite direct product of the complex number field. On the other hand, when A is a commutative Banach algebra, Lorch gave a definition that an A-valued function on an open set of A is holomorphic. The definition of a holom…

For a discrete metric space (or more generally a large scale space) $X$ and an action of a group $G$ on $X$ by coarse equivalences, we define a type of coarse quotient space $X_G$, which agrees up to coarse equivalence with the orbit space $X/G$ when $G$ is finite. We then restrict our attention to what we call coarsel…

Study recovers C*-algebra from fields of Toeplitz algebras on specific groups.

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 $D^n$, 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…

The crossing matrix of a braid on $N$ strands is the $N\times N$ integer matrix with zero diagonal whose $i,j$ entry is the algebraic number (positive minus negative) of crossings by strand $i$ over strand $j$ . When restricted to the subgroup of pure braids, this defines a homomorphism onto the additive subgroup of $N…

We prove that any regular Casimir in 3D magnetohydrodynamics is a function of the magnetic helicity and cross-helicity. In other words, these two helicities are the only independent regular integral invariants of the coadjoint action of the MHD group $\text{SDiff}(M)\ltimes\mathfrak X^*(M)$, which is the semidirect pro…

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-…

Given a fiber bundle $Z \to M \to B$ and a flat vector bundle $E \to M$ with a compatible action of a discrete group $G$, and regarding $B / G$ as the non-commutative space corresponding to the crossed product algebra, we construct an analytic torsion form as a non-commutative deRham differential form. We show that our…

We show that any dimension $6$ nearly Kähler (or nearly para-Kähler) geometry arises as a projective manifold equipped with a $\textrm{G}_2^{(*)}$ 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 …

Study Nijenhuis operators on homogeneous spaces related to C*-algebras.

Defines cross product for m vectors in n-dimensional spaces.

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…

This paper explores topological aspects of index theory for infinite-dimensional manifolds.

Develops a universal Hermitian projective calculus for complex hyperbolic two-space

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…

Using crossed homomorphisms, we show that the category of weak representations (resp. admissible representations) of Lie-Rinehart algebras (resp. Leibniz pairs) is a left module category over the monoidal category of representations of Lie algebras. In particular, the corresponding bifunctor of monoidal categories is e…

New Boolean algebra method shows knot unknotting number is (c+1)/2.

Investigates adjustments on Lie group crossed modules for gauge theory.

Study boundary actions of CAT(0) spaces and their $C^*$-algebras.

In this article, we give a geometric proof of the classification of complex vector cross product due to Lee-Leung.

New tools for studying Hsiang algebras discovered, linking them to known algebraic structures.

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 shows crossing numbers for algebraic knots can differ by arbitrarily large amounts.

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…

Integrable dynamics explained via geometric maps and cluster algebras.

New algebras and maps defined in knot Floer homology for trivalent vertices.

The paper generalizes Kuperberg invariants using twisted Drinfeld doubles.

We prove that if $M$ is a CW-complex and $*$ is a 0-cell of $M$, then the crossed module $Π_2(M,M^1,*)$ does not depend on the cellular decomposition of $M$ up to free products with $Π_2(D^2,S^1,*)$, where $M^1$ is the 1-skeleton of $M$. From this it follows that if $G$ is a finite crossed module and $M$ is finite, the…

Paper constructs a HOMFLYPT-type invariant for pseudo links.