BMW algebras of simply laced type

Non-associative Frobenius algebras for simply laced Chevalley groups

We provide an explicit construction for a class of commutative, non-associative algebras for each of the simple Chevalley groups of simply laced type. Moreover, we equip these algebras with an associating bilinear form, which turns them into Frobenius algebras. This class includes a 3876-dimensional algebra on which the Chevalley group of type E8 acts by automorphisms. We also prove that these algebras admit the structure of (axial) decomposition algebras.

Cluster algebras and preprojective algebras : the non simply-laced case

We generalize to the non simply-laced case results of Geiß, Leclerc and Schröer about the cluster structure of the coordinate ring of the maximal unipotent subgroups of simple Lie groups. In this way, cluster structures in the non simply-laced case can be seen as projections of cluster structures in the simply-laced case. This allows us to prove that cluster monomials are linearly independent in the non simply-laced case.

Symmetries of the Simply-Laced Quantum Connections and Quantisation of Quiver Varieties

We will exhibit a group of symmetries of the simply-laced quantum connections, generalising the quantum/Howe duality relating KZ and the Casimir connection. These symmetries arise as a quantisation of the classical symmetries of the simply-laced isomonodromy systems, which in turn generalise the Harnad duality. The quantisation of the classical symmetries involves constructing the quantum Hamiltonian reduction of the representation variety of any simply-laced quiver, both in filtered and in deformation quantisation.

An approach to non simply laced cluster algebras

Let $Δ$ be an oriented valued graph equipped with a group of admissible automorphisms satisfying a certain stability condition. We prove that the (coefficient-free) cluster algebra $\mathcal A(Δ/G)$ associated to the valued quotient graph $Δ/G$ is a subalgebra of the quotient $π(\mathcal A(Δ))$ of the cluster algebra associated to $Δ$ by the action of $G$. When $Δ$ is a Dynkin diagram, we prove that $\mathcal A(Δ/G)$ and $π(\mathcal A(Δ))$ coincide. As an example of application, we prove that affine valued graphs are mutation-finite, giving an alternative proof to a result of Seven.

Constructing simply laced Lie algebras from extremal elements

For any finite graph Gamma and any field K of characteristic unequal to 2 we construct an algebraic variety X over K whose K-points parameterise K-Lie algebras generated by extremal elements, corresponding to the vertices of the graph, with prescribed commutation relations, corresponding to the non-edges. After that, we study the case where Gamma is a connected, simply laced Dynkin diagram of finite or affine type. We prove that X is then an affine space, and that all points in an open dense subset of X parameterise Lie algebras isomorphic to a single fixed Lie algebra. If Gamma is of affine t…

简单连通量子仿射代数上的实单模与簇代数的范畴化

Real simple modules over simply-laced quantum affine algebras and categorifications of cluster algebras

本文引入简单连通量子仿射代数模范畴的子范畴，证明其量子Grothendieck环具有量子簇代数结构，并据此分类了实单模，提出了相关猜想。