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Hello, This is Kaveh! Below are two questions that came out of my ongoing joint work with Charles Paquette and will be glad to receive your thoughts on. Let k be an algebraically closed field and A be a finite dimensional connected basic associative algebra which has multiplicative identity. Roughly speaking, A is called a tame algebra if for every positive integer d, almost all A-modules of length d can be parametrized in terms of only finitely many continuous 1-parameter families. By the celebrated Tame/Wild dichotomy of Drozd, A is either tame or wild. In [3], Crawley-Boevey gives an elegant characterization of tame algebras in terms of generic modules and their properties. We recall that an indecomposable A-module M is called generic if M is of infinite length, but of finite endolength (i.e, length of M as a module over End_A(M) is finite). Then, A is called generically tame if for each fixed integer d, there are only finitely many generic modules of endolength d. In [3], it is shown that A is tame if and only if it is generically tame.

As a counterpart of the aforementioned classical dichotomy theorem, more recently, in [1] Bodnarchuk-Drozd considered bricks and gave a similar dichotomy result with respect to their distribution. Recall that an A-module X is called a brick if the endomorphism algebra of X is a division ring. A brick of finite length is sometimes called a Schur representation, and that is the case exactly when the endomorphism algebra is the underlying ground field k. This modern dichotomy result of Bodnarchuk-Drozd has been further studied by Carroll-Chindris [2], where A is called Schur-tame if, roughly speaking, for every positive integer d, almost all Schur representations of length d can be parametrized in terms of only finitely many continuous 1-parameter families. For details, see "Moduli spaces of modules of Schur-tame algebras".

In our recent work (Mousavand-Paquette), we introduce the notion of generic brick, by which we mean those generic modules whose endomorphism algebra is a division ring. Furthermore, we say A is brick-continuous if there exists a 1-parameter family of bricks of the same length. For any biserial algebra A, we show that A is brick-continuous if and only it admits a generic brick. Furthermore, one can naturally define the brick-generically tame algebras as those algebras such that for each fixed integer d, there are only finitely many generic bricks of endolength d. Then, we are wondering whether the following hold in general:

• Question 1: Is it always true that for a tame algebra, A is brick-continuous if and only if A admits a generic brick?
  

• Question 2: Is it always true that A is brick-generically tame if and only if A is Schur tame?
  

References: [1] L. Bodnarchuk and Yu. Drozd. One class of wild but brick-tame matrix problems. J. Algebra, 323(10):3004–3019, 2010.

[2] A.T. Carroll, and C. Chindris. Moduli Spaces of Modules of Schur-Tame Algebras. Algebra and RepresentationTheory18, 961–976 (2015).

[3] W. Crawley-Boevey. Tame algebras and generic modules. Proc. London Math. Soc., 63 (1991), 241-265.