ul 2 00 1 A category for the adjoint representation

The adjoint representation of a simple Lie algebra g admits a deformation into an irreducible representation R of the quantum group Uq(g). In this paper for a simply-laced g we realize R as the Grothendieck group of a particular abelian category C. There are exact functors from C to C which on the Grothendieck group act as the quantum group generators Eα, Fα, where α varies over simple roots. Various relations in the quantum group between products of Eα and Fα become functor isomorphisms. The adjoint representation R has a weight space decomposition as the direct sum of 1-dimensional vector spaces, one for each root of g, and the Cartan subalgebra. Mirroring this, we define C as the direct sum of copies of the category of graded vector spaces and the category of graded modules over the algebra A(Γ), naturally associated to the Dynkin diagram Γ of g. Change each edge of Γ into a pair of oriented edges, form the path algebra of this oriented graph, and quotient it out by the ideal generated by certain linear combinations of length 2 paths. A(Γ) is the resulting quotient algebra, and we name it the zigzag algebra of Γ. The Grothendieck group of the category of A(Γ)-modules is naturally identified with the weight lattice in the Cartan subalgebra of g. We introduce functors Eα and Fα lifting the generators Eα and Fα of Uq(g) and check that defining relations in the quantum group become isomorphisms of functors. We proceed to explore various properties of our categorification of the quantum group action on R. Among them is the adjointness of functors Eα and Fα, existence of several dualities in C and a braid group action in the derived category of A(Γ)-modules. We expect that not just the adjoint but any finite-dimensional irreducible representation L of the quantum group Uq(g), for a simple simply-laced Lie algebra g, admits a canonical realization as the Grothendieck group of an abelian category C(L). In this realization the Kashiwara-Lusztig basis in L should become the basis of indecomposable projective objects, the quantum group should act by exact functors and there should be a braid group action in the derived category of C(L). In short, all structures of the category C that we describe in this paper should also be present in categories C(L). Categories C(L) will be very close relatives of categories of coherent sheaves on Nakajima quiver varieties [Na] and categories of modules over cyclotomic Hecke algebras [A]. The work of Ariki [A], among other things, contains a categorification of all irreducible finite-dimensional representations of sln. His categories are made of blocks of the categories of modules over cyclotomic Hecke algebras for generic q. Ariki’s goals, which include a proof and generalizations of the Lascoux-Leclerc-Thibon conjecture [LLT], are quite different from ours. In particular, it has not been checked whether various fine structures of the category C, described in Section 4 of our paper and expected to hold in categories C(L), are present in Ariki’s categories. This work is intended to provide a simple model example of a ”perfect” categorification, with all structures visible in the representation L lifted to its categorification C(L). Another model example, a categorification of irreducible Uq(sl2) representations, will be treated in [Kh]. Our second goal is to draw the reader’s attention to the zigzag algebra A(Γ) of a graph Γ. Zigzag algebras have a variety of nice features, which we discuss in Sections 5 and 6: (i) A(Γ) is a trivial extension algebra and has a nondegenerate symmetric trace form; (ii) if Γ is a finite Dynkin diagram, then A(Γ) has finite type and there is a bijection between indecomposable representations of A(Γ) and roots of g;