Related papers: Symplectic groupoid and cluster algebras

Using the notion of compatibility between Poisson brackets and cluster structures in the coordinate rings of simple Lie groups, Gekhtman Shapiro and Vainshtein conjectured a correspondence between the two. Poisson Lie groups are classified…

We study skew-symmetrizable cluster algebras $\mathcal{A}$ associated with unpunctured surfaces $\tilde{\mathbf{S}}$ endowed with an orientation-preserving involution $\sigma$. We give a geometric realization of such cluster algebras by…

We present a general framework for constructing polynomial integrable systems on linearizations of Poisson varieties that admit log-canonical systems. Our construction is in particular applicable to Poisson varieties with compatible cluster…

Branching of symplectic groups is not multiplicity-free. We describe a new approach to resolving these multiplicities that is based on studying the associated branching algebra $B$. The algebra $B$ is a graded algebra whose components…

For each valued quiver $Q$ of Dynkin type, we construct a valued ice quiver $\Delta_Q^2$. Let $G$ be a simple connected Lie group with Dynkin diagram the underlying valued graph of $Q$. The upper cluster algebra of $\Delta_Q^2$ is graded by…

There is a natural action of the braid group on the symmetric matrices with units on the diagonal, appearing in various fields as Singularity Theory, Frobenius Manifolds or Isomonodromic deformations of certain classes of linear…

We give a self-contained algebraic description of a formal symplectic groupoid over a Poisson manifold M. To each natural star product on M we then associate a canonical formal symplectic groupoid over M. Finally, we construct a unique…

We continue the study of multiple cluster structures in the rings of regular functions on $GL_n$, $SL_n$ and $\operatorname{Mat}_n$ that are compatible with Poisson-Lie and Poisson-homogeneous structures. According to our initial…

We develop a general theory of cluster categories, applying to a 2-Calabi-Yau extriangulated category $\mathcal{C}$ and cluster-tilting subcategory $\mathcal{T}$ satisfying only mild finiteness conditions. We show that the structure theory…

In this article we use the cluster structure on the Grassmannian and the combinatorics of plabic graphs to exhibit a new aspect of mirror symmetry for Grassmannians in terms of polytopes. For our $A$-model, we consider the Grassmannian…

Geiss-Leclerc-Schroer defined the cluster algebra structure on the coordinate ring $C[N(w)]$ of the unipotent subgroup, associated with a Weyl group element $w$ and they proved cluster monomials are contained in Lusztig's dual semicanonical…

In the paper we introduce a notion of the Bruck-Reilly $\lambda$-polycyclic extension of a monoid $S$ with a homomorphism $\theta$ which is an analogue of the Bruck-Reilly extension of a monoid $S$. We describe idempotens of the semigroup…

The Marsden-Weinstein-Meyer symplectic reduction has an analogous version for cosymplectic manifolds. In this paper we extend this cosymplectic reduction to the context of groupoids. Moreover, we prove how in the case of an algebroid…

Let $S$ be a surface, $G$ a simply-connected classical group, and $G'$ the associated adjoint form of the group. We show that the spaces of moduli spaces of framed local systems $\X_{G',S}$ and $\A_{G,S}$, which were constructed by Fock and…

Let $G= GL_n(\mathbb{C})$, the general linear group over the complex numbers, and let $B$ be the set of invertible upper triangular matrices in $G$. Let $\mathfrak{b}=\text{Lie}(B)$. For $\mu:T^*(\mathfrak{b}\times \mathbb{C}^n)\rightarrow…

In this paper we show that the Poisson analogue of the Noether's Problem has a positive solution for essentially all finite symplectic reflection groups - the analogue of complex reflection groups in the symplectic world. Our proofs are…

Let $\lambda: \tilde{G}\to G$ be the non-trivial double covering of the symplectic group $G=Sp(V,\omega)$ of the symplectic vector space $(V,\omega)$ by the metaplectic group $\tilde{G}=Mp(V,\omega).$ In this case, $\lambda$ is also a…

Let $G$ be a connected semisimple Lie group. There are two natural duality constructions that assign to it the Langlands dual group $G^\vee$ and the Poisson-Lie dual group $G^*$. The main result of this paper is the following relation…

Let G be a connected reductive group defined over an algebraically closed field k of characteristic p > 0. The purpose of this paper is two-fold. First, when p is a good prime, we give a new proof of the ``order formula'' of D. Testerman…

We generalize a new class of cluster type mutations for which exchange transformations are given by reciprocal polynomials. In the case of second-order polynomials of the form $x+2\cos{\pi/n_o}+x^{-1}$ these transformations are related to…