Let be a coalgebra over a field k. We introduce an operator Tr that takes a right quasi-finitely copresented -comodule M to a left quasi-finitely copresentedcomodule Tr M. If M is indecomposable not injective and Tr M is finite-dimensional over k, we prove the existence of an almost split sequence 0 → M → E → DTr M → 0 in the category of all right -comodules, where D = Hom k k . If is right semiperfect and the embedding of each simple right comodule S into its injective envelope I S has the property that the socle of I S /S is finite-dimensional, the above almost split sequence exists for each finite-dimensional M, and DTr M is also finite-dimensional.  2002 Elsevier Science (USA)

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We determine the regular representations, Gabriel quivers and representation type of all liftings of two-dimensional quantum linear spaces.Date: 10/17/08.

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