Assume that K is an algebraically closed field. Let D be a complete discrete valuation domain with a unique maximal ideal p and residue field D/p ≌ K. We also assume that D is an algebra over the field K . We study subamalgam D-suborders ${\Lambda }^{•}$ (1.2) of tiled D-orders Λ (1.1). A simple criterion for a tame lattice type subamalgam D-order ${\Lambda }^{•}$ to be of polynomial growth is given in Theorem 1.5. Tame lattice type subamalgam D-orders ${\Lambda }^{•}$ of non-polynomial growth are completely described in Theorem 6.2 and Corollary...

We describe all finite-dimensional algebras A over an algebraically closed field for which the algebra $T_2\left(A\right)$ of 2×2 upper triangular matrices over A is of tame representation type. Moreover, the algebras A for which $T_2\left(A\right)$ is of polynomial growth (respectively, domestic, of finite representation type) are also characterized.

A criterion for tame prinjective type for a class of posets with zero-relations is given in terms of the associated prinjective Tits quadratic form and a list of hypercritical posets. A consequence of this result is that if ${\Lambda }^{•}$ is a three-partite subamalgam of a tiled order then it is of tame lattice type if and only if the reduced Tits quadratic form $q_{{\Lambda }^{•}}$ associated with ${\Lambda }^{•}$ in [26] is weakly non-negative. The result generalizes a criterion for tameness of such orders given by Simson [28] and gives an...

Let $𝕜$ be an algebraically closed field of characteristic $p\ne 2$, and let $Q_8$ be the quaternion group. We describe the structures of all simple modules over the quantum double $D\left(𝕜Q_8\right)$ of group algebra $𝕜Q_8$. Moreover, we investigate the tensor product decomposition rules of all simple $D\left(𝕜Q_8\right)$-modules. Finally, we describe the Grothendieck ring $G_0\left(D\left(𝕜Q_8\right)\right)$ by generators with relations.