Representations of Algebras and Related Topics

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pp: 445–499

DOI: 10.4171/101-1/10

The Tits forms of tame algebras and their roots

José Antonio Peña[1] and Andrzej Skowroński[2]

(1) Universidad Nacional Autónoma de México, México, D.F., Mexico
(2) Nicolaus Copernicus University, Torun, Poland

We survey some old and recent results concerning properties of the Tits quadratic forms of triangular finite dimensional algebras of finite and tame representation type over an algebraically closed field, and realization of their roots as dimension vectors of indecomposable modules. In particular, we discuss when we may recover the representation type of a triangular algebra from the combinatorial properties of its Tits quadratic form.

Keywords: Integral quadratic form, weakly positive form, weakly nonnegative form, Tits form, Euler form, representation-finite algebra, tame algebra, wild algebra, strictly wild algebra, finite growth, polynomial growth, strongly simply connected algebra, Hochschild cohomology, tilted algebra, critical algebra, hypercritical algebra, pg-critical algebra, tubular algebra