Related papers: Cluster algebra structures on Poisson nilpotent al…
In our previous paper we have discussed Poisson properties of cluster algebras of geometric type for the case of a nondegenerate matrix of transition exponents. In this paper we consider the case of a general matrix of transition exponents.…
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…
In this paper, we study the invariant theory of quadratic Poisson algebras. Let G be a finite group of the graded Poisson automorphisms of a quadratic Poisson algebra A. When the Poisson bracket of A is skew-symmetric, a Poisson version of…
We show that the quantum coordinate ring of the unipotent subgroup N(w) of a symmetric Kac-Moody group G associated with a Weyl group element w has the structure of a quantum cluster algebra. This quantum cluster structure arises naturally…
We introduce (quantum) twist automorphisms for upper cluster algebras and cluster Poisson algebras with coefficients. Our constructions generalize the twist automorphisms for quantum unipotent cells. We study their existence and their…
In this paper we study a quadratic Poisson algebra structure on the space of bilinear forms on $C^{N}$ with the property that for any $n,m\in N$ such that $n m =N$, the restriction of the Poisson algebra to the space of bilinear forms with…
In previous work, the authors introduced the ozone group of an associative algebra as the subgroup of automorphisms which fix the center pointwise. The authors studied PI skew polynomial algebras, using the ozone group to understand their…
It is proved that on nilmanifolds with abelian complex structure, there exists a canonically constructed non-trivial holomorphic Poisson structure. We identify the necessary and sufficient condition for its associated cohomology to be…
It is known that many (upper) cluster algebras possess different kinds of good bases which contain the cluster monomials and are parametrized by the tropical points of cluster Poisson varieties. For a large class of upper cluster algebras…
Our motivation is to build a systematic method in order to investigate the structure of cluster algebras of geometric type. The method is given through the notion of mixing-type sub-seeds, the theory of seed homomorphisms and the view-point…
Given a Lie group G whose Lie algebra is endowed with a nondegenerate invariant symmetric bilinear form, we construct a Poisson algebra of continuous functions on a certain open subspace R of the space of representations in G of the…
We develop a structure theory for transposed Poisson algebras over fields of characteristic different from two. In particular, we prove that every finite-dimensional transposed Poisson algebra over an algebraically closed field decomposes…
The dual Lie bialgebra of a certain ``quasitriangular'' Lie bialgebra structure on the Heisenberg Lie algebra determines a (non-compact) Poisson--Lie group G. The compatible Poisson bracket on G is non-linear, but it can still be realized…
We present a class of Poisson structures on trivial extension algebras which generalize some known structures induced by Poisson modules. We show that there exists a one-to-one correspondence between such a class of Poisson structures and…
We give a uniform geometric realization for the cluster algebra of an arbitrary finite type with principal coefficients at an arbitrary acyclic seed. This algebra is realized as the coordinate ring of a certain reduced double Bruhat cell in…
We construct explicitly a class of coboundary Poisson-Lie structures on the group of formal diffeomorphisms of ${\Bbb R}^n$. Equivalently, these give rise to a class of coboundary triangular Lie bialgebra structures on the Lie algebra $W_n$…
The article gives a ring theoretic perspective on cluster algebras. Gei{\ss}-Leclerc-Schr\"oer prove that all cluster variables in a cluster algebra are irreducible elements. Furthermore, they provide two necessary conditions for a cluster…
We study the dual ${\rm G}^\ast$ of a standard semisimple Poisson-Lie group ${\rm G}$ from a perspective of cluster theory. We show that the coordinate ring $\mathcal{O}({\rm G}^\ast)$ can be naturally embedded into a cluster Poisson…
In this note, we initiate the systematic study of the Lie algebra structure of the necklace Lie algebra n of a free algebra in 2d variables. We begin by giving a description of n as an sp(2d)-module. Specializing to d = 1, we decompose n…
In this paper, we develop a construction of Poisson $n$-Lie algebras arising from $n$-Lie algebras of Jacobians and establish conditions under which this construction yields a Poisson $n$-Lie algebra. We also formulate a general conjecture…