# Crystals learning seminar

### From TorontoMathWiki

This seminar focuses on learning about crystals, which are nice combinatorial models for representations of Lie algebras. The first part of the seminar will focus on quantum groups and construction of crystal bases from this perspective (Kashiwara's "grand loop argument"). Later we will focus on combinatorial (Young tableaux, Littelmann path model) and geometric (quiver varieties, MV polytopes) constructions.

We will meet on Mondays from 3-5pm, in BA 6180, unless otherwise noted.

**Schedule of talks**

References for the talks will be added soon. For now, we recommend taking a look at the list of references for Peter Tingley's seminar on quantum groups from Spring 2011.

September 19: Overview of quantum groups. [Jaimie Thind]

- Abstract: A quantum group is a certain deformation of the universal enveloping algebra of a semisimple Lie algebra. In this talk we will review the presentation of a semisimple Lie algebra by generators and relations, define the quantum group associated to a semisimple Lie algebra, discuss some of its basic structures (in particular, its Hopf algebra structure), and look at examples.

- References: [HK] chapters 1 and 2, [J], [Kass]

September 26: More on quantum groups. [Iva Halacheva]

- Abstract: Let $g$ be a Kac-Moody algebra and $q\in\mathbb{C}$ not a root of unity. We introduce the category $\mathcal O^q_{\mathrm{int}}$ of integrable representations for the quantum group $U_q(g)$, and describe how it is a deformation of the category of integrable representations of $g$, in the sense that the latter is obtained as the "classical limit" $q\to 1$ of the former.

- References: [HK] chapter 3

October 3: Crystal bases, tensor products. Abstract crystals. [Chris Dodd]

- References: [HK] chapter 4

October 10: No talk due to Thanksgiving.

October 17: Existence and uniqueness of crystal bases. Global bases. [Stephen Morgan]

- References: [HK] chapters 5,6

October 24: Elementary approach to canonical bases. [Jaimie Thind]

- Reference: [Lusztig], section 1,2,3,8

October 31: Combinatorical constructions of crystals: Young tableaux, the Littelmann path model. [Bruce Fontaine]

- References: [Littelmann], [HK] chapter 7, [Kash2]

November 7: Quiver varieties I. [Oded Yacobi]

- Reference: [Savage] sections 1-4, [KS]

November 14: Quiver varieties II. [Sergey Arkhipov]

- Reference: [Savage] sections 1-4, [KS]

November 21: The affine Grassmannian and geometric Satake correspondence. [Brad Hannigan-Daley]

- Abstract: Given a reductive group $G$, the affine Grassmannian $Gr_G$ is a certain infinite-dimensional "variety" which can be thought of as a partial flag variety for the loop group of $G$. It has a natural stratification by finite-dimensional subvarieties labelled by the dominant coweights of $G$. At the same time, these coweights label the irreducible representations of the Langlands dual group $G^\vee$. This coincidence is a shadow of the geometric Satake correspondence, which gives an equivalence of pivotal categories between $\operatorname{Rep}(G^\vee)$ and the category of perverse sheaves on $Gr_G$ with respect to the aforementioned stratification. We will sketch a proof of this correspondence, along the way giving definitions (or at least some intuition) of the relevant objects.

- References: [Kam], [MV], [Ginz]

November 28: Crystal structure on MV cycles. [Brad Hannigan-Daley]

- Abstract: For $G$ a reductive algebraic group with Langlands dual $G^\vee$, the geometric Satake correspondence is an equivalence between the representation category of G∨ and the category of $G(\mathbb{C}((t)))$-equivariant perverse sheaves on $Gr_G$. One may then ask how the weight decomposition of representations looks on the perverse-sheaves side of things, and it turns out to be visible by way of certain subspaces of $Gr_G$ called semi-infinite orbits. In particular, the weight decomposition of an irreducible representation $V(\lambda)$ can be described in terms of subvarieties called Mirković-Vilonen (MV) cycles. Moreover, the corresponding set of MV cycles can be equipped with a crystal structure, thereby realizing the highest-weight crystal $B(\lambda)$.

- References: [BG], [BG]

December 5: MV polytope model for crystals. [Joel Kamnitzer]

- Abstract: I will explain the MV polytope model for crystals and how this model can be obtained from three different sources: Lusztig's elementary description of the canonical basis, the components of Lusztig quiver varieties, and the MV cycles in the affine Grassmannian.

December 12: Nakajima quiver varieties. [Sergey Arkhipov]

- References: [Ginz2]

**References:**

[BG] Baumann and Gaussent, http://front.math.ucdavis.edu/0606.5711

[BG] Braverman and Gaitsgory, http://front.math.ucdavis.edu/9909.5077

[HK] J. Hong and S.-J. Kang. *Introduction to quantum groups and crystal bases*. Graduate Studies in Mathematics, AMS.

[J] J.C. Jantzen. *Lectures on quantum groups*. Graduate Studies in Mathematics, AMS.

[Ginz] V. Ginzburg. Perverse sheaves on a loop group and Langlands duality. http://arxiv.org/abs/alg-geom/9511007

[Ginz2] V. Ginzburg. Lectures on Nakajima's quiver varieties. http://arxiv.org/PS_cache/arxiv/pdf/0905/0905.0686v2.pdf

[Kam] Kamnitzer, http://front.math.ucdavis.edu/0912.0569

[Kass] C. Kassel. *Quantum groups*. Springer GTM.

[Kash] M. Kashiwara. On crystal bases. http://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/on-crystal.pdf

[Kash2] M. Kashiwara. Similarity of Crystal Basis. *Lie Algebras and their Representations*. Contemporary Mathematics vol 194

[KS] M. Kashiwara and Y. Saito. Geometric construction of crystal bases. http://arxiv.org/abs/q-alg/9606009

[Littelmann] P. Littelmann, *Characters of Representations and Paths in $\mathfrak{h}_R*$*, http://www.mi.uni-koeln.de/~littelma/Edinburgh.pdf

[Lusztig] G. Lusztig, Canonical bases arising from quantized enveloping algebras

[MV] Mirkovic-Vilonen, http://front.math.ucdavis.edu/0401.5222

[Savage] A. Savage, http://front.math.ucdavis.edu/1003.5019