Those court cases are from the 10th foreign convention on Representations of Algebras and similar subject matters (ICRA X) held on the Fields Institute. as well as the normal 'instructional' workshop previous the convention, there have been additionally workshops on 'Commutative Algebra, Algebraic Geometry and illustration Theory', 'Finite Dimensional Algebras, Algebraic teams and Lie Theory', and 'Quantum teams and corridor Algebras'. those workshops mirror the most recent advancements and the expanding curiosity in components which are heavily with regards to the illustration conception of finite dimensional associative algebras. even if those workshops have been prepared individually, their issues are strongly interrelated.The workshop on Commutative Algebra, Algebraic Geometry and illustration thought surveyed a variety of lately demonstrated connections, corresponding to these bearing on the class of vector bundles or Cohen-Macaulay modules over Noetherian jewelry, coherent sheaves on curves, or beliefs in Weyl algebras. additionally, equipment from algebraic geometry or commutative algebra when it comes to quiver representations and types of modules have been presented.The workshop on Finite Dimensional Algebras, Algebraic teams and Lie conception surveyed advancements in finite dimensional algebras and endless dimensional Lie thought, in particular because the components have interaction and will have destiny interactions. The workshop on Quantum teams and corridor Algebras handled different methods of utilizing the illustration conception of quivers (and species) as a way to build quantum teams, operating both over finite fields or over the advanced numbers. specifically, those court cases include a rather distinct define of using perverse sheaves as a way to receive canonical bases. The ebook is usually recommended for graduate scholars and researchers in algebra and geometry.