Related papers: Birman-Ko-Lee Left Canonical Form and its Applicat…

Associated to all quasi-split Satake diagrams of type ADE and even spherical coweights $\mu$, we introduce the shifted iYangians ${}^\imath Y_\mu$ and establish their PBW bases. We construct the iGKLO representations of ${}^\imath Y_\mu$,…

Given a compact quantizable pseudo-K\"ahler manifold $(M,\omega)$ of constant signature, there exists a Hermitian line bundle $(L,h)$ over $M$ with curvature $-2\pi i\,\omega$. We shall show that the asymptotic expansion of the Bergman…

In this paper, we present a canonical quantization of Lie bialgebra structures on the formal power series $\mathfrak{d}[\![t]\!]$ with coefficients in the cotangent Lie algebra $\mathfrak{d} = T^*\mathfrak{g} = \mathfrak{g} \ltimes…

We construct a noncommutative Cartan calculus on any braided commutative algebra and study its applications in noncommutative geometry. The braided Lie derivative, insertion and de Rham differential are introduced and related via graded…

We explore properties of braids such as their fractional Dehn twist coefficients, right-veeringness, and quasipositivity, in relation to the transverse invariant from Khovanov homology defined by Plamenevskaya for their closures, which are…

We consider a pair of smooth manifolds, which are the counterparts in the even-dimensional and odd-dimensional cases. They are separately an almost complex manifold with Norden metric and an almost contact manifolds with B-metric,…

We prove some necessary conditions for a link to be either concordant to a quasi-positive link, quasi-positive, positive, or the closure of a positive braid. The main applications of our results are a characterisation of positive links with…

Defined on Birman-Ko-Lee monoids, the rotating normal form has strong connections with the Dehornoy's braid ordering. It can be seen as a process for selecting between all the representative words of a Birman-Ko-Lee braid a particular one,…

Many well studied knots can be realized as positive braid knots where the braid word contains a positive full twist; we say that such knots are twist positive. Some important families of knots are twist positive, including torus knots,…

We derive a relative version of the slicing Bennequin inequalities for cobordant Legendrian knots, and review a few proofs of the result.

We consider relatively prime integer numbers $m$ and $n$ such that each solvable group of order $mn$ has a normal subgroup of order $m$. We prove that each brace of size $mn$ is a semidirect product of a brace of size $m$ and a brace of…

In this paper, we focus on semiprime skew left braces provided by semidirect products. We show that if a semidirect product $B_1\rtimes B_2$ is semiprime and $B_1$ is Artinian, then $B_1$ must be semiprime. Moreover, we prove that the…

Using the Huynh and Le quantum determinant description of the colored Jones polynomial, we construct a new combinatorial description of the colored Jones polynomial in terms of walks along a braid. We then use this description to show that…

The Krein transform is the real counterpart of the Cayley transform and gives a one-to-one correspondence between the positive relations and symmetric contractions. It is treated with a slight variation of the usual one, resulting in an…

Structures of Lie algebras, Lie coalgebras, Lie bialgebras and Lie quasibialgebras are presented as solutions of Maurer-Cartan equations on corresponding governing differential graded Lie algebras using the big bracket construction of…

We obtain explicit formulas for the semi-classical twists deforming the coalgebraic structure of U(sl(3)) and U(sl(4)). In rank 2 and 3 the corresponding universal R-matrices quantize the boundary r-matrices of Cremmer-Gervais type defining…

We use monoidal category methods to study the noncommutative geometry of nonassociative algebras obtained by a Drinfeld-type cochain twist. These are the so-called quasialgebras and include the octonions as braided-commutative but…

Light-front coordinates offer a scenario in which a constituent approximation of hadron structure can emerge from QCD. This requires cutoffs that violate Lorentz covariance and gauge invariance, and a new renormalization group formalism…

A many variable $q$-calculus is introduced using the formalism of braided covector algebras. Its properties when certain of its deformation parameters are roots of unity are discussed in detail, and related to fractional supersymmetry. The…

We show that any exceptional non-trivial Dehn surgery on a hyperbolic two-bridge knot yields a 3-manifold whose fundamental group is left-orderable. This gives a new supporting evidence for a conjecture of Boyer, Gordon and Watson.