Related papers: Skein algebras and quantized Coulomb branches
For an unpunctured marked surface $\Sigma$, we consider a skein algebra $\mathscr{S}_{\mathfrak{sl}_{3},\Sigma}^{q}$ consisting of $\mathfrak{sl}_3$-webs on $\Sigma$ with the boundary skein relations at marked points. We construct a quantum…
We give a survey of some old and new results about the stated skein modules/algebras of 3-manifolds/surfaces. For generic quantum parameter, we discuss the splitting homomorphism for the 3-manifold case, general structures of the stated…
We develop the theory of ``branch algebras'', which are infinite-dimensional associative algebras that are isomorphic, up to taking subrings of finite codimension, to a matrix ring over themselves. The main examples come from groups acting…
The quantum cohomology algebra of a projective manifold X is the cohomology H(X,Q) endowed with a different algebra structure, which takes into account the geometry of rational curves in X. We show that this algebra takes a remarkably…
The present article is a continuation of QA/1303.4046, where we discussed the classification of quantum groups with quasi-classical limit $\mathfrak{g}$ and introduced a theory of Belavin-Drinfeld cohomology associated to any…
To any graded Frobenius algebra A we associate a sequence of graded Frobenius algebras A^[n] in such a way that for any smooth projective surface X with trivial canonical divisor there is a canonical isomorphism of rings between (H*X)^[n]…
We introduce a parametrized version of scissors congruence $K$-theory of manifolds with tangential structure, which includes a topologized version of the scissors congruence $K$-theory of oriented manifolds as a special case. We examine the…
We study the algebraic and geometric properties of stated skein algebras of surfaces with punctured boundary. We prove that the skein algebra of the bigon is isomorphic to the quantum group ${\mathcal O}_{q^2}(\mathrm{SL}(2))$ providing a…
The paper deals with braided Clifford algebras, understood as Chevalley-Kahler deformations of braided exterior algebras. It is shown that Clifford algebras based on involutive braids can be naturally endowed with a braided quantum group…
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 R\u{a}zvan Gelca established a complete description of the multiplicative operation leading to a…
The connection between braided Hopf algebra structure and the quantum group covariance of deformed oscillators is constructed explicitly. In this context we provide deformations of the Hopf algebra of functions on SU(1,1). Quantum subgroups…
We study the derived Hall algebra of the partially wrapped Fukaya category of a surface. We give an explicit description of the Hall algebra for the disk with m marked intervals and we give a conjectural description of the Hall algebras of…
We present a string theoretic approach to surface counting in local Calabi--Yau fourfolds via supersymmetric localization in topologically twisted four-dimensional gauge theories. This approach is based on a spectral correspondence between…
In earlier work, we constructed invariants of irreducible representations of the Kauffman skein algebra of a surface. We introduce here an inverse construction, which to a set of possible invariants associates an irreducible representation…
Let $X\subset \P^5$ be a smooth cubic fourfold. A well known conjecture asserts that $X$ is rational if and only if there an Hodge theoretically associated K3 surface $S$. The surface $S$ can be associated to $X$ in two other different…
Quillen's algebraic K-theory is reconstructed via Voevodsky's algebraic cobordism. More precisely, for a ground field k the algebraic cobordism P^1-spectrum MGL of Voevodsky is considered as a commutative P^1-ring spectrum. There is a…
Foams are surfaces with branch lines at which three sheets merge. They have been used in the categorification of sl(3) quantum knot invariants and also in physics. The 2D-TQFT of surfaces, on the other hand, is classified by means of…
We construct finite dimensional representations of the Kauffman bracket skein algebra of the one-punctured torus and four-punctured sphere at all roots of unity. The representations are given by explicit formulas. They all have dimensions…
The skein algebra of an oriented $3$-manifold is a classical limit of the Kauffman bracket skein module and gives the coordinate ring of the $SL_2(\mathbb{C})$-character variety. In this paper we determine the quotient of a polynomial ring…
We introduce a Lie algebra associated with a non-orientable surface, which is an analogue for the Goldman Lie algebra of an oriented surface. As an application, we deduce an explicit formula of the Dehn twist along an annulus simple closed…