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We study systems of linear and semilinear mappings considering them as representations of a directed graph $G$ with full and dashed arrows: a representation of $G$ is given by assigning to each vertex a complex vector space, to each full arrow a linear mapping, and to each dashed arrow a semilinear mapping of the corresponding vector spaces. We extend to such representations the classical theorems by Gabriel about quivers of finite type and by Nazarova, Donovan, and Freislich about quivers of tame types.
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V.I. Arnold [Russian Math. Surveys, 26 (no. 2), 1971, pp. 29-43] gave a miniversal deformation of matrices of linear operators; that is, a simple canonical form, to which not only a given square matrix A, but also the family of all matrices close to A, can be reduced by similarity transformations smoothly depending on the entries of matrices. We study miniversal deformations of quiver representations and obtain a miniversal deformation of matrices of chains of linear mappings.
We consider the problem of classifying oriented cycles of linear mappings $F^pto F^qtodotsto F^rto F^p$ over a field $F$ of complex or real numbers up to homeomorphisms in the spaces $F^p,F^q,dots,F^r$. We reduce it to the problem of classifying linear operators $F^nto F^n$ up to homeomorphism in $F^n$, which was studied by N.H. Kuiper and J.W. Robbin [Invent. Math. 19 (2) (1973) 83-106] and by other authors.
We give a method for constructing a regularizing decomposition of a matrix pencil, which is formulated in terms of the linear mappings. We prove that two pencils are topologically equivalent if and only if their regularizing decompositions coincide up to permutation of summands and their regular parts coincide up to homeomorphisms of their spaces.
Let k be an algebraically closed field of positive characteristic, and let W be the ring of infinite Witt vectors over k. Suppose G is a finite group and B is a block of kG of infinite tame representation type. We find all finitely generated kG-modules V that belong to B and whose endomorphism ring is isomorphic to k and determine the universal deformation ring R(G,V) for each of these modules.
We determine the derived representation type of Nakayama algebras and prove that a derived tame Nakayama algebra without simple projective module is gentle or derived equivalent to some skewed-gentle algebra, and as a consequence, we determine its singularity category.
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