Title: Weighted Surface Algebras,
Abstract: A finite-dimensional algebra $A$ over an algebraically closed field $K$ is
called periodic if it is periodic under the action of the syzygy operator in
the category of $A-A-$ bimodules. The periodic algebras are self-injective and
occur naturally in the study of tame blocks of group algebras, actions of
finite groups on spheres, hypersurface singularities of finite Cohen-Macaulay
type, and Jacobian algebras of quivers with potentials. Recently, the tame
periodic algebras of polynomial growth have been classified and it is natural
to attempt to classify all tame periodic algebras. We introduce the weighted
surface algebras of triangulated surfaces with arbitrarily oriented triangles
and describe their basic properties. In particular, we prove that all these
algebras, except the singular tetrahedral algebras, are symmetric tame periodic
algebras of period $4$. Moreover, we describe the socle deformations of the
weighted surface algebras and prove that all these algebras are symmetric tame
periodic algebras of period $4$. The main results of the paper form an
important step towards a classification of all periodic symmetric tame algebras
of non-polynomial growth, and lead to a complete description of all algebras of
generalized quaternion type. Further, the orbit closures of the weighted
surface algebras (and their socle deformations) in the affine varieties of
associative $K$-algebra structures contain wide classes of tame symmetric
algebras related to algebras of dihedral and semidihedral types, which occur in
the study of blocks of group algebras with dihedral and semidihedral defect
groups.