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We determine all values of the parameters for which the cell modules form a standard system, for a class of cellular diagram algebras including partition, Brauer, walled Brauer, Temperley-Lieb and Jones algebras. For this, we develop and apply a general theory of algebras with Borelic pairs. The theory is also applied to give new uniform proofs of the cellular and quasi-hereditary properties of the diagram algebras and to construct quasi-hereditary 1-covers, in the sense of Rouquier, with exact Borel subalgebras, in the sense of Konig. Another application of the theory leads to a proof that Auslander-Dlab-Ringel algebras admit exact Borel subalgebras.
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The $n$-slice algebra is introduced as a generalization of path algebra in higher dimensional representation theory. In this paper, we give a classification of $n$-slice algebras via their $(n+1)$-preprojective algebras and the trivial extensions of their quadratic duals. One can always relate tame $n$-slice algebras to the McKay quiver of a finite subgroup of $mathrm{GL}(n+1, mathbb C)$. In the case of $n=2$, we describe the relations for the $2$-slice algebras related to the McKay quiver of finite Abelian subgroups of $mathrm{SL}(3, mathbb C)$ and of the finite subgroups obtained from embedding $mathrm{SL}(2, mathbb C)$ into $mathrm{SL}(3,mathbb C)$.
The objective of the present paper is to give a survey of recent progress on applications of the approaches of Ringel-Hall type algebras to quantum groups and cluster algebras via various forms of Greens formula. In this paper, three forms of Greens formula are highlighted, (1) the original form of Greens formula cite{Green}cite{RingelGreen}, (2) the degeneration form of Greens formula cite{DXX} and (3) the projective form of Greens formula cite{XX2007a} i.e. Green formula with a $bbc^{*}$-action.
We study periodicity and twisted periodicity of the trivial extension algebra $T(A)$ of a finite-dimensional algebra $A$. We prove that (twisted) periodicity of the trivial extension is equivalent to $A$ being (twisted) fractionally Calabi--Yau. Moreover, twisted periodicity of $T(A)$ is equivalent to the $d$-representation-finiteness of the $r$-fold trivial extension algebra $T_r(A)$ for some positive integers $r$ and $d$. These results allow us to construct a large number of new examples of periodic as well as fractionally Calabi--Yau algebras, and give answers to several open questions.
In cite{CK2005} and cite{Hubery2005}, the authors proved the cluster multiplication theorems for finite type and affine type. We generalize their results and prove the cluster multiplication theorem for arbitrary type by using the properties of 2--Calabi--Yau (Auslander--Reiten formula) and high order associativity.
Let $k$ be an algebraically closed field of characteristic $p > 0$ and let $G$ be a connected reductive algebraic group over $k$. Under some standard hypothesis on $G$, we give a direct approach to the finite $W$-algebra $U(mathfrak g,e)$ associated to a nilpotent element $e in mathfrak g = operatorname{Lie} G$. We prove a PBW theorem and deduce a number of consequences, then move on to define and study the $p$-centre of $U(mathfrak g,e)$, which allows us to define reduced finite $W$-algebras $U_eta(mathfrak g,e)$ and we verify that they coincide with those previously appearing in the work of Premet. Finally, we prove a modular version of Skryabins equivalence of categories, generalizing recent work of the second author.
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