arXiv research

Extends Morse-Novikov Homology to include differential graded coefficients and fibration structures.

Develops a chain-level model for Chas-Sullivan products using Morse theory with differential graded coefficients.

Transport functions for principal bundles and Morse homology with differential graded coefficients

New Morse theory for path homology with coefficients.

Obstruction theory for complex bigraded differential algebras.

Colored knot polynomials possess a peculiar Z-expansion in certain combinations of differentials, which depends on the representation. The coefficients of this expansion are functions of the three variables (A,q,t) and can be considered as new distinguished coordinates on the space of knot polynomials, analogous to the…

We rewrite the recently proposed differential expansion formula for HOMFLY polynomials of the knot $4_1$ in arbitrary rectangular representation $R=[r^s]$ as a sum over all Young sub-diagrams $λ$ of $R$ with extraordinary simple coefficients $D_{λ^{tr}}(r)\cdot D_λ(s)$ in front of the $Z$-factors. Somewhat miraculously…

Explicit Taylor series for the volume of tubes in Lie groups

In this paper we discuss the question of integrating differential graded Lie algebras (DGLA) to differential graded Lie groups (DGLG). We first recall the classical problem of integration in the context, and present the construction for (non-graded) differential Lie algebras. Then, we define the category of differentia…

Given a unital associatve graded algebra we construct the graded q-differential algebra by means of a graded q-commutator, where q is a primitive N-th root of unity. The N-th power (N>1) of the differential of this graded q-differential algebra is equal to zero. We use our approach to construct the graded q-differentia…

This is the second in a series of papers laying the foundations for a differential graded approach to derived differential geometry (and other geometries in characteristic zero). In this paper, we extend the classical notion of a dg-algebra to define, in particular, the notion of a differential graded algebra in the wo…

Defines Floer homology with DG coefficients for symplectic manifolds.

The paper examines smoothness in graded skew Clifford algebras.

For a simply connected solvable Lie group G with a cocompact discrete subgroup Γ, we consider the space of differential forms on the solvmanifold G/Γ with values in certain flat bundle so that this space has a structure of a differential graded algebra(DGA). We construct Sullivan's minimal model of this DGA. This resul…

This paper develops a theory of graded manifolds in differential geometry.

Constructs graded jet bundles for Z-graded manifolds and vector bundles.

Let $C$ be a differential graded coalgebra, $\barΩC$ the Adams cobar construction and $C^\vee$ the dual algebra. We prove that for a large class of coalgebras $C$ there is a natural isomorphism of Gerstenhaber algebras between the Hochschild cohomologies $HH^\ast (C^\vee, C ^\vee)$ and $HH^\ast (\barΩC ; \barΩC)$. Thi…

Determines algebra structure of complex differential forms operators.

Given a compact Kaehler manifold, we consider the complement U of a divisor with normal crossings and a unitary local system V on it. We consider a differential graded Lie algebra (DGLA) of forms with holomorphic logarithmic singularities and vanishing residues. We construct a spectral sequence corresponding to the ant…

We iterate Manolescu's unoriented skein exact triangle in knot Floer homology with coefficients in the field of rational functions over $\mathbb{Z}/2\mathbb{Z}$. The result is a spectral sequence which converges to a stabilized version of delta-graded knot Floer homology. The $(E_2,d_2)$ page of this spectral sequence …

We introduce the concept of $N$-differential graded algebras (N-dga), and study the moduli space of deformations of the differential of a N-dga. We prove that it is controlled by what we call the N-Maurer-Cartan equation.

Constructs a new graded variety from algebraic data.

In this work, differential geometry of the Z$_3$-graded quantum superplane is constructed. The corresponding quantum Lie superalgebra and its Hopf algebra structure are obtained.

New dg-algebras link graph colorings to sheaves.

For a smooth family F of admissible elliptic pseudodifferential operators with differential form coefficients associated to a geometric fibration of manifolds M--> B we show that there is a natural zeta-form z(F,s) and zeta-determinant- form det(F) in the de-Rham algebra of smooth differential forms, generalizing the c…

This paper proves equivalence between derived manifolds and differential graded manifolds.

Differential geometry of the quantum Lie superalgebra of the extended quantum superplane and its Z$_2$-graded Hopf algebra structure is obtained. Its Z$_2$-graded dual Hopf algebra is also given.

This paper aims at setting out the basics of $\mathbb{Z}$-graded manifolds theory. We introduce $\mathbb{Z}$-graded manifolds from local models and give some of their properties. The requirement to work with a completed graded symmetric algebra to define functions is made clear. Moreover, we define vector fields and ex…

We show that the category of Lie triple systems is equivalent to the category of Lie algebras graded by Z/(2Z) such that the odd component generates the algbera and the second graded cohomology group coefficients in any trivial module is zero. As a corollary we obtain an analogous result for symmetric spaces and Lie gr…

Paper traces origins of graded Lie brackets theory.

We define the notions of trace, determinant and, more generally, Berezinian of matrices over a (Z_2)^n graded commutative associative algebra. The applications include a new approach to the classical theory of matrices with coefficients in a Clifford algebra, in particular of quaternionic matrices. In a special case, w…

The geometry of graded principal bundles is discussed in the framework of graded manifold theory of Kostant-Berezin-Leites. In particular, we prove that a graded principal bundle is globally trivial if and only if it admits a global graded section and, further, that the sheaf of vertical derivations on such a bundle co…

New algebra pong algebra computed for knot Floer homology.

We construct families of differential graded algebras R and R \boxtimes R and give an algebraic formulation of the contact category of a disk through the differential graded category DGP(R) generated by some distinguished projective differential graded R-modules. The homology category H^0(DGP(R)) is a triangulated cate…

Given a graded $E_1$-module over an $E_2$-algebra in spaces, we construct an augmented semi-simplicial space up to higher coherent homotopy over it, called its canonical resolution, whose graded connectivity yields homological stability for the graded pieces of the module with respect to constant and abelian coefficien…

Study graded coverings for supermanifolds, proving their universal properties.

A formula connects two algebraic structures derived from a category.

New calculus framework for vector bundles with metrics.

The derived bracket of a Maurer-Cartan element in a differential graded Lie algebra (DGLA) is well-known to define a differential graded Leibniz algebra. It is also well-known that a Lie infinity morphism between DGLAs maps a Maurer-Cartan element to a Maurer-Cartan element. Given a Lie-infinity morphism, a Maurer-elem…

Geometric structures on $\mathbb N Q$-manifolds, i.e.~non-negatively graded manifolds with an homological vector field, encode non-graded geometric data on Lie algebroids and their higher analogues. A particularly relevant class of structures consists of vector bundle valued differential forms. Symplectic forms, contac…

We prove that the isomorphism type of a large class of groups (containing finite groups, countable Artinian groups and mapping class groups of certain surfaces, among others) is determined by the set of differential graded $\mathbb Q$-algebras on which these groups act faithfully.

This is the second part in a series of two papers. The $k$-Dirac complex is a complex of differential operators which are natural to a particular $|2|$-graded parabolic geometry. In this paper we will consider the $k$-Dirac complex over a homogeneous space of the parabolic geometry and as a first result, we will prove …

Categorifies Jones polynomial for odd primes.

Let X be a pseudomanifold. In this text, we use a simplicial blow-up to define a cochain complex whose cohomology with coefficients in a field, is isomorphic to the intersection cohomology of X, introduced by M. Goresky and R. MacPherson. We do it simplicially in the setting of a filtered version of face sets, also cal…

In this work, the Z$_3$-graded differential geometry of the quantum plane is constructed. The corresponding quantum Lie algebra and its Hopf algebra structure are obtained. The dual algebra, i.e. universal enveloping algebra of the quantum plane is explicitly constructed and an isomorphism between the quantum Lie algeb…

Three new types of graded Lie groups are constructed and analyzed.

Normal forms for Q-structures on graded manifolds explained.

We elaborate on the recent observation that evolution for twist knots simplifies when described in terms of triangular evolution matrix ${\cal B}$, not just its eigenvalues $Λ$, and provide a universal formula for ${\cal B}$, applicable to arbitrary rectangular representation $R=[r^s]$. This expression is in terms of s…