Final Report Summary - LCFTDUAL (Logarithmic conformal field theory as a duality between Brauer-type algebras and quantum groups at roots of unity) The purpose of this project was to accomplish an important step forward in the study of Logarithmic conformal field theories, with a special emphasis on bulk or full field theories, by combining two approaches developed independently in the Saclay and Moscow schools. The first approach is based on the construction of explicit lattice regularizations, which are amenable to modern techniques from the field of quasi-hereditary and diagram algebras. The second approach is completely abstract, and uses vertex-operator algebras and W-symmetry in combination with factorizable ribbon Hopf algebras. In more concrete terms, the fundamental problem at which our project was aimed is the description and functorial correspondence of nonsemisimple braided monoidal categories associated with logarithmic CFTs and their vertex-operator algebras on the one hand, and with finite-dimensional modules over affine Hecke algebras (at roots of unity cases), on the other hand. An objective related to this problem involved an in-depth investigation of the duality between Brauer-type algebras and quantum groups at roots of unity -- analogues of the q-Schur-Weyl duality between ordinary Hecke algebras and quantized Lie algebras.During the project, we have established an important functorial correspondence between representation categories of the lattice algebras (TemperleyLieb and blob algebras) and the chiral algebras or Vertex Operator Aalgebras associated with Virasoro algebras at any central charge less than one. In this description, decompositions of quantum spin chains based on the lattice algebras or appropriate bimodules were studied. So, a complete picture for non-affine Temperley--Lieb modules at all roots of unity and their relations with Virasoro algebra representations was presented in several publications. Several types of periodic quantum spin chains based on affine Temperley-Lieb algebras were also studied. For understanding the continuum limit of spin chains and connection with Virasoro algebra representation theory, we used lattice discretizations (regularizations) of the Virasoro algebra generators in terms of (affine) Temperley-Lieb generators. Significant results obtained at the end of the project are around the following points:1) Several types of periodic quantum spin chains based on affine Temperley-Lieb algebras were fully described together with corresponding centralizers. For understanding the continuum limit of spin chains, we found lattice discretizations (regularizations) of the Virasoro algebra generators in terms of (affine) Temperley-Lieb generators. Using this algebraic approach, several chiral and bulk Logarithmic CFT were constructed in taking direct limits of the spin chains, including most interesting for applications bulk LCFT of zero central charge.2) New type of modules over Virasoro algebra were conjectured and a classification of Virasoro algebra modules was proposed based on the theory of tilting modules. Following our results on the classification of Virasoro algebra modules, we were able to propose a classification of spaces of boundary (conditions changing) quantum fields by the tilting Virasoro modules (and self-dual subquotients thereof) obtained as the inductive limits of the tiliting modules over the B-type Temperley-Lieb algebras.3) We found the centralizer of the small quantum sl(2) group at all roots of unity. It was identified with a proper extension of the Temperley-Lieb algebra. Using this result together with the Kazhdan-Lusztig duality, the finitizations (or lattice version) of the chiral triplet W-algebra modes were constructed on XXZ spin-chains.4) A logarithmic CFT model with central charge c=0 and modeling some important observables from the perocaltion problem was constructed as a scaling (direct) limit of quantum spin chains made of alternating tensor products of fundamental and anti-fundamental representations of super Lie algebra sl(2|1).Logarithmic CFTs are at the heart of several fundamental problems of theoretical physics, the most visible being the description of the transition between plateaux in the integer quantum Hall effect, and the solution of supertarget sigma models in the AdS/CFT correspondence. These problems are the subject of intense studies, and any step towards their solution would be a major progress. From a more fundamental side, recent developments in modern algebra have considerably improved our understanding of non semi-simple features. Bridging this with LCFTs was a key progress of this project. We believe that this progress will allow for the first time to gain a good understanding of the relation between boundary and bulk (non-chiral) properties of LCFTs, and answer such simple questions as ‘how many h= 0 fields are there in the model’ or ‘how many logarithmic partners does the stress energy tensor have’, which have eluded the experts so far.