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In this paper, we first show that for an acyclic gentle algebra A, the irreducible components of any moduli space of A-modules are products of projective spaces. Next, we show that the nice geometry of the moduli spaces of modules of an algebra does not imply the tameness of the representation type of the algebra in question. Finally, we place these results in the general context of moduli spaces of modules of Schur-tame algebras. More specifically, we show that for an arbitrary Schur-tame algebra A and theta-stable irreducible component C of a module variety of A-modules, the moduli space of theta-semi-stable points of C is either a point or a rational projective curve.
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We show that the irreducible components of any moduli space of semistable representations of a special biserial algebra are always isomorphic to products of projective spaces of various dimensions. This is done by showing that irreducible components of varieties of representations of special biserial algebras are isomorphic to irreducible components of products of varieties of circular complexes, and therefore normal, allowing us to apply recent results of the second and third authors on moduli spaces.
We determine the derived representation type of Nakayama algebras and prove that a derived tame Nakayama algebra without simple projective module is gentle or derived equivalent to some skewed-gentle algebra, and as a consequence, we determine its singularity category.
In this paper the authors investigate the $q$-Schur algebras of type B that were constructed earlier using coideal subalgebras for the quantum group of type A. The authors present a coordinate algebra type construction that allows us to realize these $q$-Schur algebras as the duals of the $d$th graded components of certain graded coalgebras. Under suitable conditions an isomorphism theorem is proved that demonstrates that the representation theory reduces to the $q$-Schur algebra of type A. This enables the authors to address the questions of cellularity, quasi-hereditariness and representation type of these algebras. Later it is shown that these algebras realize the $1$-faithful quasi hereditary covers of the Hecke algebras of type B. As a further consequence, the authors demonstrate that these algebras are Morita equivalent to Rouquiers finite-dimensional algebras that arise from the category ${mathcal O}$ for rational Cherednik algebras for the Weyl group of type B. In particular, we have introduced a Schur-type functor that identifies the type B Knizhnik-Zamolodchikov functor.
We introduce a Lie algebra $mathfrak{g}_{mathbf{Q}}(mathbf{m})$ and an associative algebra $mathcal{U}_{q,mathbf{Q}}(mathbf{m})$ associated with the Cartan data of $mathfrak{gl}_m$ which is separated into $r$ parts with respect to $mathbf{m}=(m_1, dots, m_r)$ such that $m_1+ dots + m_r =m$. We show that the Lie algebra $mathfrak{g}_{mathbf{Q}} (mathbf{m})$ is a filtered deformation of the current Lie algebra of $mathfrak{gl}_m$, and we can regard the algebra $mathcal{U}_{q, mathbf{Q}}(mathbf{m})$ as a $q$-analogue of $U(mathfrak{g}_{mathbf{Q}}(mathbf{m}))$. Then, we realize a cyclotomic $q$-Schur algebra as a quotient algebra of $mathcal{U}_{q, mathbf{Q}}(mathbf{m})$ under a certain mild condition. We also study the representation theory for $mathfrak{g}_{mathbf{Q}}(mathbf{m})$ and $mathcal{U}_{q,mathbf{Q}}(mathbf{m})$, and we apply them to the representations of the cyclotomic $q$-Schur algebras.
We give a necessary and sufficient condition on parameters for Ariki-Koike algebras (resp. cyclotomic q-Schur algebras) to be of finite representation type.
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