José Antonio de la Peña (Instituto de Matemáticas-UNAM, Mexico)
Degenerations of algebras and modules
Abstract:Let $A$ be a finite dimensional $k$-algebra.
We introduce some concepts on the geometry of representations and its applications.
(1) the representation type of the algebra (finite, tame or wild) can be read in the dimension of the module variety $mod_A(d)$ and the orbits $G(d) X$, where $G(d)$ if the group determining the iso-classes in $mod_A(d)$, $d \in \N$.
(2) Given two algebras $A$ and $B$, we say that $B$ is a degeneration of $A$ if there is an algebraic family $(A_z)_z \in Z$ such that $A_z$ is isomorphic to $A$ in a open and dense subset of $Z$ and $A_{z_0}$ is isomorphic to $B$ in some $z_0 \in Z$. If $B$ is a degeneration of $A$ then $\dim_k H^n(B) \ge \dim_k H^n(A)$, where $H^n$ denotes the Hochschild cohomology.
(3) Geiss degeneration theorem: a degeneration of a wild algebra is wild.
(4) Denote $q_A$ the (quadratic) Tits form of $A$, then:
(a) if $A$ is representation finite then $q_A$ is weakly positive (ie. $q_A(u) >0$ for $0 \ne u$ any vector with $0\le $ entries).
(b) if $A$ is tame the $q_A$ is weakly non-negative (ie. $q_A(u) \ge 0$ for any vector $u$ with $0\le $ entries).
(5) Bruestle-JAP-Skowronski theorem: let $A$ be a strongly simply connected algebra then $A$ is tame if and only if $q_A$ is wnn.
(6) We show that $A$ is strongly simply connected if and only if the trace of the Coxeter matrix satisfies $tr(\Phi_A)=-1$.
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