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Take a quiver with $n$ vertices, and one arrow $i\to j$ for each pair $(i,j)$ of vertices, including those where $i=j$. Impose the following relations: all paths of length greater than two are zero …

I'll assume that $k$ is not the zero ring. If $u$ is not injective, then $\tilde{u}$ is not essentially surjective, since no faithful representation of $G_1$ is in the image. Suppose $u$ is injectiv …

Yes. The second condition says that the socle $S=\operatorname{soc}(V)$ of $V$ (i.e., the largest semisimple submodule) is one-dimensional, and the injective hull of $V$ is the same as the injective …

Yes, $A^G$ must have finite global dimension, as it has the same kind of triangular form as $A$. The Jacobson radical of $A$ is $$J(A)=\begin{pmatrix} 0 & M_{1,2} & \dots & M_{1,r} \\ 0 & 0 & …

Yes (and characteristic zero isn't necessary). Since $G$ is linearly reductive, $A$ is a direct sum $A^G\oplus B$ as $G$-modules, where $B$ is the sum of all non-trivial irreducible $G$-submodules of …

If $A$ is representation-infinite, then $\Gamma_A$ has more than one component, and so if its mesh category were equivalent to the category of indecomposable modules, the indecomposable modules could …

If $S$ is a finite dimensional algebra, and $M$ a finite dimensional $S$-bimodule, and if we construct the algebras $A=S\ltimes M$ and $B=S\ltimes DM$, then the trivial extension algebras $T(A)=A\ltim …

Here's one way of constructing counterexamples for finite groups. Suppose $M$ is a periodic $kG$-module with period $p$: i.e., the $p$th syzygy $\Omega^pM$ is isomorphic to $M$, but $\Omega M\not\con …

Suppose $U$ has this property. Let $V=FG$ and let $W$ be the direct sum of $|G|$ copies of the trivial representation. Then $V\otimes U$ is free of rank $\dim(U)$, $W\otimes U$ is a direct sum of $|G| …

I don't know exactly the relationship between the two definitions, but they're not the same, as the first one depends only on the class in $\operatorname{Ext}^n_\Lambda(C,A)$ but the second doesn't. F …

Probably there are no other examples. The paper Brenner, Sheila; Ringel, Claus Michael, Pathological modules over tame rings, J. Lond. Math. Soc., II. Ser. 14, 207-215 (1976). ZBL0356.16010. doesn' …

I don't know a classification, but another example is the group algebra (over a field $K$ of characteristic $2$) of the quaternion group $Q_8$ (or more generally any generalized quaternion group).

In the paper V.G. Kac, "Infinite root systems, representations of graphs and invariant theory", Invent. Math. 56, 57-92 (1980), Kac shows that the dimension vectors of indecomposable representatio …

As Geoff thought, the answer is contained in James and Kerber (it's Theorem 2.5.13 in "The Representation Theory of the Symmetric Group", Encyclopedia of Mathematics and its Applications vol. 16, 1981 …

There are examples of non-semisimple algebras with trivial Hochschild cohomology: for example, the path algebra $C$ of a quiver whose underlying graph is a tree. Also, for finite dimensional algebras …