arXiv research

In this paper we compute the deformation theory of a special class of algebras, namely of Azumaya algebras on a manifold ($C^{\infty}$ or complex analytic).

Researchers create explicit representations for skein algebras of small surfaces, revealing their Azumaya loci.

Criteria for extending degree-2 Azumaya algebras with C2-actions over curves.

Study centers of generalized skein algebras, showing almost Azumaya properties.

Paper extends algebraic geometry results to hyperbolic link complements.

Study of quaternion Azumaya algebras in hyperbolic once-punctured torus bundles.

New topological criterion extends arithmetic invariants in hyperbolic 3-manifolds.

We lay down an elementary yet fundamental lemma concerning a finite algebraicness property of a smooth map from an Azumaya/matrix manifold with a fundamental module to a smooth manifold. This gives us a starting point to build a synthetic (synonymously, $C^{\infty}$-algebraic) symplectic geometry and calibrated geometr…

Study deformation invariants of projective surfaces using twisted sheaves and Azumaya algebras.

Constructs noncommutative spaces for D-branes on complex algebraic spaces.

Let $M$ be a compact 3-manifold and $Γ=π_1(M)$. Work of Thurston and Culler--Shalen established the $\mathrm{SL}_2(\mathbb{C})$ character variety $X(Γ)$ as fundamental tool in the study of the geometry and topology of $M$. This is particularly the case when $M$ is the exterior of a hyperbolic knot $K$ in $S^3$. The mai…

Survey of stated skein modules/algebras of 3-manifolds/surfaces.

We show that for complex analytic K3 surfaces any torsion class in H^2(X,O_X^*) comes from an Azumaya algebra. In other words, the Brauer group equals the cohomological Brauer group. For algebraic surfaces, such results go back to Grothendieck. In our situation, we use twistor spaces to deform a given analytic K3 surfa…

Given a bundle gerbe on a compact smooth manifold or, more generally, on a compact étale Lie groupoid $M$, we show that the corresponding category of gerbe modules, if it is non-trivial, is equivalent to the category of finitely generated projective modules over an Azumaya algebra on $M$. This result can be seen as an …

In this follow-up of our earlier two works D(11.1) (arXiv:1406.0929 [math.DG]) and D(11.2) (arXiv:1412.0771 [hep-th]) in the D-project, we study further the notion of a `differentiable map from an Azumaya/matrix manifold to a real manifold'. A conjecture is made that the notion of differentiable maps from Azumaya/matri…

We consider D-branes in string theory and address the issue of how to describe them mathematically as a fundamental object (as opposed to a solitonic object) of string theory in the realm in differential and symplectic geometry. The notion of continuous maps, $k$-times differentiable maps, and smooth maps from an Azuma…

Mathai, Melrose, and Singer introduced the notion of projective elliptic operators on manifolds equipped with an Azumaya bundle. In this note we compute the equivariant index of transversally elliptic operators that are the pullback of projective elliptic operators on the trivialization of the Azumaya bundle. It encomp…

In this Part II of D(11), we introduce new objects: super-$C^k$-schemes and Azumaya super-$C^k$-manifolds with a fundamental module (or, synonymously, matrix super-$C^k$-manifolds with a fundamental module), and extend the study in D(11.1) ([L-Y3], arXiv:1406.0929 [math.DG]) to define the notion of `differentiable maps…

In this sequel to works D(11.1) (arXiv:1406.0929 [math.DG]), D(11.2) (arXiv:1412.0771 [hep-th]), and D(11.3.1) (arXiv:1508.02347 [math.DG]), we re-examine --- and reformulate when in need --- several basic notions in super $C^{\infty}$-algebraic geometry as guided by the mathematical formulation of Ramond-Neveu-Schwarz…

The paper finds a special pants decomposition for certain surface group representations.

In this lecture I review how a matrix/Azumaya-type noncommutative geometry arises for D-branes in string theory and how such a geometry serves as an origin of the master nature of D-branes; and then highlight an abundance conjecture on D0-brane resolutions of singularities that is extracted and purified from a work of …

Sliced skein algebras and geometric Kauffman bracket study algebraic structures and their properties.

In this note the fractional analytic index, for a projective elliptic operator associated to an Azumaya bundle, of DG/0402329 is related to the equivariant index of Atiyah and Singer for an associated transversally elliptic operator.

As the necessary background to construct from the aspect of Grothendieck's Algebraic Geometry dynamical fermionic D3-branes along the line of Ramond-Neveu-Schwarz superstrings in string theory, three pieces of the building blocks are given in the current notes: (1) basic $C^\infty$-algebrogeometric foundations of $d=4$…

D-branes on noncommutative spaces mimic string theory, offering new insights into mirror symmetry.

An index theory for projective families of elliptic pseudodifferential operators is developed when the twisting, i.e. Dixmier-Douady, class is decomposable. One of the features of this special case is that the corresponding Azumaya bundle can be realized in terms of smoothing operators. The topological and the analytic…

In D(13.1) (arXiv:1606.08529 [hep-th]), we introduced an admissible condition on differentiable maps $\varphi: (X^{\!A\!z}, E;\nabla)\rightarrow Y$ from an Azumaya/matrix manifold $X^{\!A\!z}$ (with the fundamental module $E$) with a connection $\nabla$ on $E$ to a manifold $Y$ in order to resolve a pull-push issue in …

New algebraic structure derived from Hopf algebra and Drinfel'd twist.

Study of cluster and skein algebras for surfaces, showing their connection.

We introduce a new action $S_{standard}^{(ρ,h; Φ,g,B,C)}$ for D-branes that is to D-branes as the Polyakov action is to fundamental strings. This `standard action' is abstractly a non-Abelian gauged sigma model --- based on maps $\varphi: (X^{\!A\!z},E;\nabla)\rightarrow Y$ from an Azumaya/matrix manifold $X^{\!A\!z}$ …

Study on pseudo-Riemannian algebraic Ricci solitons in 4D Lie groups.

Characterizes Lie groups with specific structures and finds a correspondence between carrollian and galilean 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 …

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

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.

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

The paper classifies Lie algebras with special operators.

Symmetric spaces' connections form Lie admissible triple algebras.

The paper generalizes para-Kähler Lie algebras to k-para-Kähler Lie algebras and explores their structures.

The paper investigates gradings of complex simple Lie algebras, focusing on $|3|$-gradings and their algebraic structures.

New algebra pong algebra computed for knot Floer homology.

Study resolves conjecture linking two algebraic structures on surfaces.

Study on pre-Lie structures for semisimple Lie algebras over C.

Develops a bialgebra theory for post-Lie algebras using geometric interpretations and bilinear forms.

Characterizes G2-structures on Lie algebras with non-trivial center.

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…