Noncommutative Algebra Seminar

I`ll continue my talk on Hamiltonian reduction and quiver varieties. I`ll talk about the Hilbert scheme of points on the affine plane, Quiver varieties, HyperKahler reduction, Quantum Hamiltonian reduction, equivariant D-modules and quantum group hamiltonian reduction.

This time I`ll talk about equivariant D-modules, quantum Hamiltonian reduction, group valued moment maps and quantum group moment maps and quantum group Hamiltonian reduction. If time permits I might remark on microlocal differential operators and how they are related to the rational Cherednik algebra.

References

1. Pavel Etingof, Lectures on Calogero-Moser systems, arXiv:math/0606233.
2. Hiraku Nakajima, Lectures on Hilbert schemes of points on surfaces, American Mathematical Society, Providence, RI, 1999.
3. A.D.King, Moduli of representations of finite dimensional algebras, The Quarterly Journal of Mathematics 1994 45(4):515-530.

In this talk, I'll present a (fairly) complete proof of a remarkable theorem of Gabriel, which states that a quiver has only finitely many non-isomorphic indecomposable representations if, and only if, the underlying (unoriented) graph is a Dynkin diagram of type A,D, or E. (In particular, it doesn't matter how the graph is oriented into a quiver). Unlike Gabriel's original argument, the argument I will present (from [1] below, which I'll follow very closely) makes clear the connection to root systems and Weyl groups. I will also give a classification of "tame" quivers, which have not finitely many indecomposables but rather a discrete collection of one dimensional families of indecomposables (e.g. Jordan Form). The talk will be entirely elementary, and (not surprisingly) use many familiar methods and results from the classification of semi-simple f.d. lie algebras.

References

1. N Bernstein, I M Gel'fand, V A Ponomarev, COXETER FUNCTORS AND GABRIEL'S THEOREM, RUSS MATH SURV, 1973, 28 (2), 17-32.(this requires an MIT campus connection to access it)
2. Pavel Etingof and Students, Lectures and problems in representation theory, Clay Mathematics Research Academy.

For an abelian category satisfying some finiteness conditions I`ll introduce the Ringel-Hall algebra. I will also describe the Hopf algebra structure introduced by Green. In the case of ADE quivers I`ll relate this to the quantum Borel. I`ll also describe how Lusztig used this to construct canonical bases for quantum groups.

References

1. Olivier Schiffmann, Lectures on Hall algebras, arXiv:math/0611617.

I will introduce a moment map associated to the cotangent bundle of the space of representations of a quiver. I will explain how its fibers can be interpreted as spaces of representations of a certain quotient of the path algebra of the double quiver called deformed preprojective algebra. When the quiver is extended Dynkin I will explain how this construction is related to the Kleinian singularity C^2/G where G is a finite subgroup of SL(2, C).

References

1. W. Crawley-Boevey, Geometry of the moment map for representations of quivers. Compositio Math. 126 (2001), no. 3, 257--293. (also in Crawley-Boevey homepage http://www.amsta.leeds.ac.uk/~pmtwc/).
2. W. Crawley-Boevey,M.Holland, Noncommutative deformations of Kleinian singularities. Duke Math. J. 92 (1998), no. 3, 605--635.

The Koszulness of a quadratic algebra is probably the most fruitful property one can even state about it. The classical examples are the symmetric algebra S(V^*) and the exterior algebra Λ(V) for a finite-dimensional vector space V . Each Koszul algebra has a Koszul dual, in the example above S(V^*) and Λ(V) are Koszul dual.

Now let α be a quadratic Poisson bivector on V, one can regard it also as a quadratic Poisson bivector D(α) on the space V^*[1]. Therefore, the deformation quantizations S(V^*)_α and Λ(V)_D(α) are defined, they are graded quadratic algebras. The main result we are going to talk about is that these two algebras are Koszul dual as well. In other words, the Koszulness is preserved under the deformation quantization.

The suggested proof is not very elementary. We meet Kontsevich formality, differential graded categories and their A_∞ deformations, the AKSZ model of topological quantum field theory with many boundary conditions.... We deduce our result from the formality as dg Lie algebra of the Hochschild complex of some differential graded category, introduced recently by B. Keller. All these things are not supposed to be known and will be discussed in the talk.

The talk is based on the joint paper with P. Etingof, S. Loktev, and A. Oblomkov (in preparation). We present a "Lie algebra" construction of the spherical subalgebra in the wreath product symplectic reflection algebras of rank $N$ attached to a finite subgroup of $SL(2,\C)$ of type $D_4$, $E_6$, $E_7$, or $E_8$. Namely, the spherical subalgebra appears as a quantum Hamiltonian reduction of (some quotient of) the universal enveloping algebra of the direct sum of $sl_k$-s for some $k$. This gives a natural way to construct finite-dimensional representations of this spherical subalgebra.

I'll continue my talk on preprojective algebras. This time I'll consider deformed preprojective algebras for an extended Dynkin quiver Q. I will explain how this construction is related to the Kleinian singularity C^2/G, where G is the finite subgroup of SL(2, C) attached to the quiver Q by the McKay correspondence.

I will describe Kashiwara`s approach to crystal and global basis. These arise by taking q to 0 (in an appropriate sense) for the quantum group. I will discuss the construction and properties of these bases. If time permits I will also talk about Lusztig`s approach to canonical bases using perverse sheaves.

I will begin with the definition of rational Dunkl operators. I will define the rational Cherednik algebra for any complex reflection groups and will talk about some properties of this algebra. Then I will talk about the representation theory of the rational Cherednik algebra.

References

1. Pavel Etingof, Lectures on Calogero-Moser systems, arXiv:math/0606233.
2. Meinolf Geck, Gunter Malle, Reflection Groups. A Contribution to the Handbook of Algebra, arXiv:math/0311012.
3. Raphael Rouquier, Representations of rational Cherednik algebras, arXiv:math/0504600.

I will continue with the description of Kashiwara`s approach to crystal and global basis. I will discuss the construction and properties of these bases. If time permits I will also talk about Lusztig`s approach to canonical bases using perverse sheaves.

References

1. Masaki Kashiwara, On Crystal Bases.