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Let Λ be a tubular algebra over an arbitrary base field. We study the Grothendieck group $K_0\left(\Lambda \right)$, endowed with the Euler form, and its automorphism group $Aut\left(K_0\left(\Lambda \right)\right)$ on a purely K-theoretical level as in [7]. Our results serve as tools for classifying the separating tubular families of tubular algebras as in the example [5] and for determining the automorphism group $Aut\left(D^b\Lambda \right)$ of the derived category of Λ.

Canonical algebras, introduced by C. M. Ringel in 1984, play an important role in the representation theory of finite-dimensional algebras. They also feature in many other mathematical areas like function theory, 3-manifolds, singularity theory, commutative algebra, algebraic geometry and mathematical physics. We show that canonical algebras are characterized by a number of interesting extremal properties (among concealed-canonical algebras, that is, the endomorphism rings of tilting bundles on...