Related papers: Okounkov's conjecture via BPS Lie algebras

Continuum Kac-Moody algebras have been recently introduced by the authors and O. Schiffmann. These are Lie algebras governed by a continuum root system, which can be realized as uncountable colimits of Borcherds-Kac-Moody algebras. In this…

We prove the Novikov conjecture on oriented Cheeger spaces whose fundamental group satisfies the strong Novikov conjecture. A Cheeger space is a stratified pseudomanifold admitting, through a choice of ideal boundary conditions, an L2-de…

We give an explicit description of the BPS Lie algebra of any quiver with zero potential, by relating the perverse filtration on the cohomological Hall algebra with certain limit conditions on polynomials. Our results also give a partial…

Quiver varieties have recently appeared in various different areas of Mathematics such as representation theory of Kac-Moody algebras and quantum groups, instantons on 4-manifolds, and resolutions Kleinian singularities. In this paper, we…

Let $A$ be a Koszul Calabi-Yau algebra. We show that there exists an isomorphism of Batalin-Vilkovisky algebras between the Hochschild cohomology ring of $A$ and that of its Koszul dual algebra $A^!$. This confirms (a generalization of) a…

In this thesis we discuss various classical problems in enumerative geometry. We are focused on ideas and methods which can be used explicitly for practical computations. Our approach is based on studying the limits of elliptic stable…

We establish an embedding of the quantum enveloping algebra of a symmetric generalized Kac--Moody algebra into a localized Hall algebra of $\mathbb Z_2$-graded complexes of representations of a quiver with (possible) loops. To overcome…

In this note, we give a description of the graded Lie algebra of double derivations of a path algebra as a graded version of the necklace Lie algebra equipped with the Kontsevich bracket. Furthermore, we formally introduce the notion of…

This survey is mostly concerned with unstable analogues of the Lichtenbaum-Quillen Conjecture. The Lichtenbaum-Quillen Conjecture (now implied by the Voevodsky-Rost Theorem) attempts to describe the algebraic K-theory of rings of integers…

We calculate the deformed and non-deformed cohomological Hall algebra (CoHA) of the preprojective algebra for the case of cyclic quivers by studying the Kontsevich-Soibelman CoHA and using tools from cohomological Donaldson-Thomas theory.…

Let $X$ be a projective variety of dimension $n$ over an algebraically closed field of arbitrary characteristic and let $A, B, C$ be nef divisors on $X$. We show that for any integer $1\leq k\leq n-1$, $$ (B^k\cdot A^{n-k})\cdot (A^k\cdot…

This paper is a continuation of the series of papers "Quantization of Lie bialgebras (QLB) I-V". We show that the image of a Kac-Moody Lie bialgebra with the standard quasitriangular structure under the quantization functor defined in…

This paper aims at a geometric realization of the Yangian of non-simply laced type in terms of quiver with potentials. For every quiver with symmetrizer, there is an extended quiver with superpotential, whose Jacobian algebra is the…

We show that if $g_\Gamma$ is the quantum tangent space (or quantum Lie algebra in the sense of Woronowicz) of a bicovariant first order differential calculus over a coquasitriangular Hopf algebra $(A,r)$, then a certain extension of it is…

Let $(\mathfrak{g}, \bullet)$ be a real left symmetric algebra, and $(\mathfrak{g}^-, [\;,\;])$ the corresponding Lie algebra. We denote by $L$ the left multiplication operator associated with the product $\bullet$. The symmetric bilinear…

We introduce affinizations and deformations of the BPS Lie algebra associated to a tripled quiver with potential, and use them to precisely determine the $T$-equivariant cohomological Hall algebra $\mathcal{H}_{\mathbb{A}^2}^T$ of compactly…

In this thesis we use the Beauville-Bogomolov decomposition to compute the LLV algebra of smooth projective complex varieties admitting a holomorphic symplectic form, generalizing known results from hyperk\"ahler and abelian varieties.…

An involutive Lie bialgebra induces a Batalin-Vilkovisky operator on its exterior algebra. We introduce a graded generalization of the necklace Lie bialgebra, which depends on a choice of a quiver $Q$. We relate the resulting…

In this paper we analyze the structure of some subalgebras of quantized enveloping algebras corresponding to unipotent and solvable subgroups of a simple Lie group G. These algebras have the non--commutative structure of iterated algebras…

Twisted Lie algebroid cohomologies, i.e. with values in representations, are shown to be Lie algebroid homotopy-invariant. Several important classes of examples are discussed. As an application, a generalized version of the Poincar\'e lemma…