For each simply-laced Dynkin graph $Δ$ we realize the simple complex Lie algebra of type $Δ$ as a quotient algebra of the complex degenerate composition Lie algebra $L(A)_{1}^{\mathbb{C}}$ of a domestic canonical algebra $A$ of type $Δ$ by some ideal $I$ of $L(A)_{1}^{\mathbb{C}}$ that is defined via the Hall algebra of $A$, and give an explicit form of $I$. Moreover, we show that each root space of $L(A)_{1}^{\mathbb{C}}/I$ has a basis given by the coset of an indecomposable $A$-module $M$ with root easily computed by the dimension vector of $M$.
We introduce the concept of a prime band in a string algebra $Λ$ and use it to associate to $Λ$ its finite bridge quiver. Then we introduce a new technique of `recursive systems' for showing that a graph map between finite dimensional string modules lies in its stable radical. Further we study two classes of non-domestic string algebras in terms of some connectedness properties of its bridge quiver. `Meta-$\bigcup$-cyclic' string algebras constitute the first class that is essentially characterized by the statement that each finite string is a substring of a band. Extending this class we have …
We show that the order type of the simplest version of a hammock for string algebras lies in the class of finite description linear orders--the smallest class of linear orders containing $\mathbf 0$, $\mathbf 1$, and that is closed under isomorphisms, finite order sum, anti-lexicographic product with $ω$ and $ω^*$, and shuffle of finite subsets--using condensation (localization) of linear orders as a tool. We also introduce two finite subsets of the set of bands and use them to describe the location of left $\mathbb N$-strings in the completion of hammocks.
An explicit formula for the canonical bilinear form on the Grothendieck ring of the Lie supergroup $GL(n,m)$ is given. As an application we get an algorithm for the decomposition Euler characters in terms of characters of irreducible modules in the category of partially polynomial representations.
Kang, Kashiwara, Kim and Oh have proved that cluster monomials lie in the dual canonical basis, under a symmetric type assumption. This involves constructing a monoidal categorification of a quantum cluster algebra using representations of KLR algebras. We use a folding technique to generalise their results to all Lie types.
In a previous paper we introduced the notion of a D-Lie algebra $\tilde{L}$. A D-Lie algebra $\tilde{L}$ is an $A/k$-Lie-Rinehart algebra with a right $A$-module structure and a canonical central element $D$ satisfying several conditions. We used this notion to define the universal enveloping algebra of the category of $\tilde{L}$-connections and to define the cohomology and homology of an arbitrary connection. In this note we introduce the canonical quotient $L$ of a D-Lie algebra $\tilde{L}$ and use this to classify D-Lie algebras where $L$ is projective as $A$-module. We define for any 2-co…
