Evgeny Feigin, Skoltech, spring 2021
Evgeny Feigin, Skoltech, spring 2021
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Exam problems, due date May 26.
May, 28: oral part of the exam. Please write down here (no later than 23:59, Wednesday, May, 26) the time of your availability.
Lectures
Lecture 1, 03.02 . lecture1_03.02 Quivers: definitions and examples. Representations of quivers: definitions and examples. Homomorphisms and isomorphisms of representations, irreducible and indecomposable representations of quivers. Krull-Schmidt theorem.
Lecture 2, 10.02. lecture2_10.02 Quotient representations for quivers, kernels and cokernels. Finite-dimensional representations of quivers as abelian category. Exact sequences, short exact sequences., split exact sequences. Examples of non split exact sequences. Sections and retractions, relation to splitting.
Lecture 3, 17.02. lecure3_17.02 Covariant and contravariant functors, functors of homomorphisms. Exact sequences and Hom functors, split exact sequences and Hom functors. Projective and injective modules. Paths, sinks, sources, oriented cycles. Simple modules S(i) and modules P(i) for quivers with no oriented cycles.
Lecture 4, 24.02. lecture4_24.02 Properties of the modules P(i) and I(i): projectivity and injectivity, description of homomorphisms spaces Hom(P(i),M). P(I) and I(i) are indecomposable. Hom spaces from P(i) to P(j), path algebra as an endomorphisms algebra of the direct sum of projectives.
Lecture 5, 03.03. lecture5_03.03 Two terms projective resolutions, explicit construction. Projective modules as direct summands of free modules. Complete classification of projective representations. Radical of P(i) as the maximal subrepresentation. Any subrepresentation of a projective representation is projective.
Lecture 6, 10.03. lecture6_10.03 Definition of Ext^1(M,N) via projective resolution, cokernels. Extensions, isomorphic extensions. Abelian group structure on the equivalence classes of extensions. An element of the space Ext^1(M,N) corresponding to an extension.
Lecture 7, 17.03. lecture7_17.03 Variety of representations with fixed dimension vector. The product of general linear groups group action. Description of the orbits of the action. Stabilizers and dimensions of orbits. Non split sequences and inequality on the dimensions of orbits. Quadratic form of a quiver.
Lecture 8, 24.03. lecture8_24.03 Dynkin and Eucledian quivers. A non Dynkin quiver has a Eulcedian quiver as a subquiver. Codimension of an orbit in the representation variety. Negative values of quadratic form of a quiver and infinite number of isoclasses of indecomposable representations.
Lecture 9. 31.03. lecture9_31.03 Kernel of the quadratic form for the Eucledian quivers. Positive definite quadratic forms and Dynkin quivers. Positive semi-definite quadratic forms and Eucledian quivers. Roots : real and imaginary, positive and negative. Dynkin quivers have finite number of roots. Positive roots for Dynkin quivers and indecomposable representations.
Lecture 10, lecture10_07.04 Gabriel's theorem: indecomposable representations and roots for Dynkin quivers. Unital associative algebras: left and right ideal, maximal ideals, radical. Path algebras: basic definitions.
Lwcture 11. lecture11_14.04 Radical of the path algebra of a quiver with no oriented cycles. Right and left modules, examples. Quiver representations and modules over the path algebras. Nakayama's lemma. Radical of a finite-dimensional algebra is nilpotent.
Lecture 12. lecture12_21.04 Idempotents, indecomposable (primitive) idempotents, orthogonal idemponents. Primitive idempotents in the path algebra. Decomposition of an algebra via orthogonal idempotents. Radicals and local algebras.
Lecture 13. lecture13_28.04 Local algebras and basic algebras. Admissible ideal, bound quiver algebras. Quiver attached to a basic algebra. Realization of a basic algebra in terms of bound quiver algebras.
Lecture 14. lecture14_12.05 Quiver Grassmannians: definition and examples. Nakajima quivers varieties: framed representations, doubled quivers, moment map, stability conditions.
Lecture 15. lecture15_19.05 Nakajima quiver Grassmannians: stability conditions in terms of subrepresentations of the double quiver, examples for the A_1 quiver. Universal quiver Grassmannians, projection to the product of classical Grassmannians.
Lecture 16. lecture16_26.05 Expected dimension of a quiver Grassmannian, rigid representations. Quotient construction of quiver Grassmannians. Tangent space as Hom space. Stratification of quiver Grassmannians by isoclasses of representations. Cotangent bundle to a partial flag variety as Nakajima quiver variety.
Problems
Problem 1. quivers1 due 09.02
Problem 2. quivers2 due 16.02
Problem 3. quivers3 due 23.02
Problem 4 quivers4 due 02.03
Problem 5 quivers5 due 09.03
Homework 1 hw1 due 16.03
Problem 6 quivers6 due 23.03
Problem 7 quivers7 due 14.04
Problem 8 quivers8 due 21.04
Problem 9 quivers9 due 28.04
Problem 10 quivers10 due 12.05
Problem 11 quivers11 due 19.05
Results results
Literature.
Ralf Schiffler, Quiver representations
W. W. Crawley-Boevey, Lectures on representations of quivers, https://www.math.uni-bielefeld.de/~wcrawley/
W. W. Crawley-Boevey, , Geometry of representations of algebras, https://www.math.uni-bielefeld.de/~wcrawley/
V.Ginzburg, Lectures on Nakajima’s quiver varieties
A. Kirillov, Jr., Quiver Representations and Quiver Varieties