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Register a new userTopological classification of oriented cycles of linear mappings
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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.
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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.
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 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.
Two sesquilinear forms $Phi:mathbb C^mtimesmathbb C^mto mathbb C$ and $Psi:mathbb C^ntimesmathbb C^nto mathbb C$ are called topologically equivalent if there exists a homeomorphism $varphi :mathbb C^mto mathbb C^n$ (i.e., a continuous bijection whose inverse is also a continuous bijection) such that $Phi(x,y)=Psi(varphi (x),varphi (y))$ for all $x,yin mathbb C^m$. R.A.Horn and V.V.Sergeichuk in 2006 constructed a regularizing decomposition of a square complex matrix $A$; that is, a direct sum $SAS^*=Roplus J_{n_1}oplusdotsoplus J_{n_p}$, in which $S$ and $R$ are nonsingular and each $J_{n_i}$ is the $n_i$-by-$n_i$ singular Jordan block. In this paper, we prove that $Phi$ and $Psi$ are topologically equivalent if and only if the regularizing decompositions of their matrices coincide up to permutation of the singular summands $J_{n_i}$ and replacement of $Rinmathbb C^{rtimes r}$ by a nonsingular matrix $Rinmathbb C^{rtimes r}$ such that $R$ and $R$ are the matrices of topologically equivalent forms. Analogous results for real and complex bilinear forms are also obtained.
Let $A$ be the locally unital algebra associated to a cyclotomic oriented Brauer category over an arbitrary algebraically closed field $Bbbk$ of characteristic $pge 0$. The category of locally finite dimensional representations of $A $ is used to give the tensor product categorification (in the general sense of Losev and Webster) for an integrable lowest weight with an integrable highest weight representation of the same level for the Lie algebra $mathfrak g$, where $mathfrak g$ is a direct sum of copies of $mathfrak {sl}_infty$ (resp., $ hat{mathfrak {sl}}_p$ ) if $p=0$ (resp., $p>0$). Such a result was expected in [3] when $Bbbk=mathbb C$ and proved previously by Brundan in [2] when the level is $1$.
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