Categorification and the quantum Grassmannian

In \cite{JKS} we gave an (additive) categorification of Grassmannian cluster algebras, using the category \CM(A) of Cohen-Macaulay modules for a certain Gorenstein order $A$. In this paper, using a cluster tilting object in the same category \CM(A), we construct a compatible pair $(B, L)$, which is the data needed to define a quantum cluster algebra. We show that when $(B, L)$ is defined from a cluster tilting object with rank 1 summands, this quantum cluster algebra is (generically) isomorphic to the corresponding quantum Grassmannian

Exceptional representations of a double quiver of type A, and Richardson elements in seaweed Lie algebras

In this paper, we study the set of $\Delta$-filtered modules of quasi-hereditary algebras arising from quotients of the double of quivers of type $A$. Our main result is that for any fixed $\Delta$-dimension vector, there is a unique (up to isomorphism) exceptional $\Delta$-filtered module. We then apply this result to show that there is always an open adjoint orbit in the nilpotent radical of a seaweed Lie algebra in \mathrm{gl}_{n}(\field), thus answering positively in this \mathrm{gl}_{n}(\field) case to a question raised independently by Michel Duflo and Dmitri Panyushev. An example of a seaweed Lie algebra in a simple Lie algebra of type $E_{8}$ not admitting an open orbit in its nilpotent radical is given

Sliding Mode Robustness Control Strategy for Shearer Height Adjusting System

This paper firstly established mathematical model of height adjusting hydro cylinder of the shearer, as well as the state space equation of the shearer height adjusting system. Secondly we designed a shearer automatic height adjusting controller adopting the sliding mode robustness control strategy. The height adjusting controller includes the sliding mode surface switching function based on Ackermann formula, as well as sliding mode control function with the improved butterworth filter. Then simulation of the height adjustment controller shows that the sliding mode robustness control solves buffeting of typical controller, and achieves automatic control for the rolling drum of the shearer. DOI : http://dx.doi.org/10.11591/telkomnika.v12i2.373