# Tensor product varieties and crystals. GL case

@article{Malkin2001TensorPV, title={Tensor product varieties and crystals. GL case}, author={Anton Malkin}, journal={arXiv: Algebraic Geometry}, year={2001} }

Let g be a simple simply laced Lie algebra. In this paper two families of varieties associated to the Dynkin graph of g are described: ``tensor product'' and ``multiplicity'' varieties. These varieties are closely related to Nakajima's quiver varieties and should play an important role in the geometric constructions of tensor products and intertwining operators. In particular it is shown that the set of irreducible components of a tensor product variety can be equipped with a structure of g…

## 13 Citations

### Comparison of quiver varieties, loop Grassmannians and nilpotent cones in type A

- MathematicsAdvances in Mathematics
- 2022

### Springer fibers and the Delta Conjecture at $t=0$

- Mathematics
- 2021

We introduce a family of varieties Yn,λ,s, which we call the ∆-Springer varieties, that generalize the type A Springer fibers. We give an explicit presentation of the cohomology ring H(Yn,λ,s) and…

### Heaps, crystals, and preprojective algebra modules

- Mathematics
- 2020

Fix a simply-laced semisimple Lie algebra. We study the crystal B(nλ), were λ is a dominant minuscule weight and n is a natural number. On one hand, B(nλ) can be realized combinatorially by height n…

### Existence of Affine Pavings for Varieties of Partial Flags Associated to Nilpotent Elements

- Mathematics
- 2013

The flag variety of a complex reductive linear algebraic group G is by definition the quotient G/B by a Borel subgroup. It can be regarded as the set of Borel subalgebras of Lie(G). Given a nilpotent…

### Cherednik algebras, W-algebras and the equivariant cohomology of the moduli space of instantons on A2

- Mathematics
- 2012

We construct a representation of the affine W-algebra of ${\mathfrak{g}}{\mathfrak{l}}_{r}$ on the equivariant homology space of the moduli space of Ur-instantons, and we identify the corresponding…

### Lectures on geometric realizations of crystals

- Mathematics
- 2010

These are notes for a lecture series given at the Fields Institute Summer School in Geometric Representation Theory and Extended Affine Lie Algebras, held at the University of Ottawa in June 2009. We…

### QUIVER VARIETIES AND BEILINSON-DRINFELD GRASSMANNIANS OF TYPE A

- Mathematics
- 2008

We construct Nakajima's quiver varieties of type A in terms of conjugacy classes of matrices and (non-Slodowy's) transverse slices naturally arising from affine Grassmannians. In full generality…

### Categorification of integrable representations of quantum groups

- Mathematics
- 2008

We categorify the highest weight integrable representations and their tensor products of a symmetric quantum Kac-Moody algebra. As a byproduct, we obtain a geometric realization of Lusztig’s…

## References

SHOWING 1-10 OF 28 REFERENCES

### Quivers, perverse sheaves, and quantized enveloping algebras

- Mathematics
- 1991

1. Preliminaries 2. A class of perverse sheaves on Ev 3. Multiplication 4. Restriction 5. Fourier-Deligne transform 6. Analysis of a sink 7. Multiplicative generators 8. Compatibility of…

### Geometric construction of crystal bases

- Mathematics
- 1996

We realize the crystal associated to the quantized enveloping algebras with a symmetric generalized Cartan matrix as a set of Lagrangian subvarieties of the cotangent bundle of the quiver variety. As…

### Quiver varieties and tensor products

- Mathematics
- 2001

Abstract.In this article, we give geometric constructions of tensor products in various categories using quiver varieties. More precisely, we introduce a lagrangian subvariety &?tilde; in a quiver…

### Canonical bases arising from quantized enveloping algebras

- Mathematics
- 1990

0.2. We are interested in the problem of constructing bases of U+ as a Q(v) vector space. One class of bases of U+ has been given in [DL]. We call them (or, rather, a slight modification of them, see…

### Crystalizing theq-analogue of universal enveloping algebras

- Mathematics
- 1990

For an irreducible representation of theq-analogue of a universal enveloping algebra, one can find a canonical base atq=0, named crystal base (conjectured in a general case and proven forAn, Bn, Cn…