On solvability of the first Hochschild cohomology of a finite-dimensional algebra

For an arbitrary finite-dimensional algebra $A$, we introduce a general approach to determining when its first Hochschild cohomology ${\rm HH}^1(A)$, considered as a Lie algebra, is solvable. If $A$ is moreover of tame or finite representation type, we are able to describe ${\rm HH}^1(A)$ as the direct sum of a solvable Lie algebra and a sum of copies of $\mathfrak{sl}_2$. We proceed to determine the exact number of such copies, and give an explicit formula for this number in terms of certain chains of Kronecker subquivers of the quiver of $A$. As a corollary, we obtain a precise answer to a question posed by Chaparro, Schroll and Solotar

A reduction theorem for tau -rigid modules

We prove a theorem which gives a bijection between the support τ -tilting modules over a given finite-dimensional algebra A and the support τ -tilting modules over A / I, where I is the ideal generated by the intersection of the center of A and the radical of A. This bijection is both explicit and well-behaved. We give various corollaries of this, with a particular focus on blocks of group rings of finite groups. In particular we show that there are τ -tilting-finite wild blocks with more than one simple module. We then go on to classify all support τ -tilting modules for all algebras of dihedral, semidihedral and quaternion type, as defined by Erdmann, which include all tame blocks of group rings. Note that since these algebras are symmetric, this is the same as classifying all basic two-term tilting complexes, and it turns out that a tame block has at most 32 different basic two-term tilting complexes. We do this by using the aforementioned reduction theorem, which reduces the problem to ten different algebras only depending on the ground field k, all of which happen to be string algebras. To deal with these ten algebras we give a combinatorial classification of all τ -rigid modules over (not necessarily symmetric) string algebras

Unacylated ghrelin prevents mitochondrial dysfunction in a model of ischemia/reperfusion liver injury

Ischemia/reperfusion (I/R) injury is a common cause of liver dysfunction during hepatectomy, liver transplantation procedures and in generalized shock. Although effort has been dedicated to rescuing tissue damage in these clinical settings, there is still an urgent need for an effective treatment to protect the liver from the burden of I/R injury. In this study, we have investigated the potential clinical impact of unacylated-ghrelin (UnAG) in a liver I/R rat model. Particular attention has been paid to mitochondria. We demonstrate that UnAG was able to reduce the lag-phase time in response to ADP administration and increase oxygen consumption in ex vivo experiments using liver mitochondria recovered from rats subjected to I/R. Moreover, we found that UnAG rescued the expression of a key regulator of mitochondrial morphology and electron transport chain function; the optic atrophy 1 (Opa1) protein. Cytochrome c oxidase (COX), ATP synthase (complex V) activity and mitochondrial permeability transition pore (mPTP) opening were also affected by UnAG administration in vivo. An in vitro, hepatic I/R model was used to validate these data. We demonstrate that UnAG upregulates the expression of Cox subunit IV (CoxIV) and increases cellular ATP content. This results in Bcl-2 upregulation and protection against apoptosis. Opa1 silencing shows that Opa1 is crucial for a UnAG-induced increase in cellular ATP content, apoptosis resistance, Bcl-2 and CoxIV expression. Finally, we show that UnAG improves Opa1's interaction with MIC60 in the I/R setting, hinting at its role in cristae shape regulation. Our results demonstrate that UnAG administration rescues the intrinsic mitochondrial pathway triggered by I/R damage. Opa1's contribution in mediating this effect is also reported. This suggests that UnAG can interfere with mitochondrial dysfunction, via Opa1, in a preclinical liver I/R model. We therefore provide the rationale for exploiting UnAG as an alternative means to rescuing mitochondrial damage and organ dysfunction