By Volodymyr Mazorchuk
The time period “categorification” was once brought by way of Louis Crane in 1995 and refers back to the strategy of changing set-theoretic notions by means of the corresponding category-theoretic analogues.
This textual content generally concentrates on algebraical elements of the speculation, provided within the old viewpoint, but in addition includes a number of topological purposes, specifically, an algebraic (or, extra accurately, representation-theoretical) method of categorification. It contains fifteen sections comparable to fifteen one-hour lectures given in the course of a grasp category at Aarhus college, Denmark in October 2010. There are a few routines accrued on the finish of the textual content and a slightly vast checklist of references. Video recordings of all (but one) lectures can be found from the grasp classification website.
The e-book presents an introductory review of the topic instead of a completely particular monograph. Emphasis is on definitions, examples and formulations of the implications. so much proofs are both in short defined or passed over. besides the fact that, whole proofs are available by way of monitoring references. it really is assumed that the reader understands the fundamentals of type thought, illustration conception, topology and Lie algebra.
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Extra resources for Lectures on Algebraic Categorification
Sample text
In O, respectively. For 2 hdom set P ´ 2W P . /. P /op . 4 BGG reciprocity and quasi-hereditary structure A module N 2 O is said to have a standard filtration or Verma flag if there is a filtration of N whose subquotients are Verma modules. /. 5 (BGG reciprocity). (a) Every projective module in O has a standard filtration. (b) If some N 2 O has a standard filtration, then for any 2 h the multiplicity ŒN W M. / of M. / as a subquotient of a standard filtration of N does not depend on the choice of such filtration.
0 (a) For every W -antidominant 2 W there is a unique indecomposable projective functor  ; such that  ; . / D P . /. (b) Every indecomposable projective functor from O to O 0 is isomorphic to  ; for some W -antidominant 2 W 0 . 2 implies that an indecomposable projective functor  is completely determined by its value  . / on the corresponding dominant Verma module . /. Moreover, as  . / is projective and projective modules form a basis of ŒO 0 (as O 0 , being quasi-hereditary, has finite global dimension), the functor  is already uniquely determined by ŒÂ .
8. For every 2 h there is a unique (up to isomorphism) indecomposable module T . r/ such that . / T . / and the cokernel of this inclusion admits a standard filtration. L For 2 hdom the module T ´ 2W T . / is called the characteristic tilting module. 9. The module T is ext-selforthogonal, has finite projective dimension and there is an exact sequence 0 ! P ! Q0 ! Q1 ! Qk ! T / for all i. The (opposite of the) endomorphism algebra of T is called the Ringel dual of B . The Ringel dual is defined for any quasi-hereditary algebra and is again a quasihereditary algebra.