1 99 9 TRIANGULATED HOPF CATEGORY n + SL ( 2 )

Crane and Frenkel proposed a notion of a Hopf category [2]. It was motivated by Lusztig’s approach to quantum groups – his theory of canonical bases. In particular, Lusztig obtains braided deformations Uqn+ of universal enveloping algebras Un+ for some nilpotent Lie algebras n+ together with canonical bases of these braided Hopf algebras [4, 5, 6]. The elements of the canonical basis are identified with isomorphism classes of simple perverse sheaves – certain objects of equivariant derived categories. They are contained in a semisimple abelian category of semisimple complexes. One of the proposals of Crane and Frenkel is to study this category rather than its Grothendieck ring Uqn+. Conjectural properties of this category were collected into a system of axioms of a Hopf category, equipped with functors of multiplication and comultiplication, isomorphisms of associativity, coassociativity and coherence which satisfy four equations [2]. The mathematical framework and some examples were provided by Neuchl [10]. Crane and Frenkel [2] gave an example of a Hopf category resembling the semisimple category encountered in Lusztig’s theory corresponding to one-dimensional Lie algebra n+ – nilpotent subalgebra of sl(2). We want to discuss an example of a related notion – triangulated Hopf category – the whole equivariant derived category equipped with multiplication and comultiplication functors and structure isomorphisms. In particular, it is a monoidal category. The new feature of coherence is that additive relations of [2] are replaced with distinguished triangles. This new structure does not induce a Hopf category structure of Crane and Frenkel on a subcategory of semi-simple complexes. The missing component is a consistent choice of splitting of splittable triangles. Verification of some of the consistency equations is still an open question. To give more details let us first recall the braided Hopf algebra H . As an algebra H is the polynomial algebra of one variable over R = Z[q, q]. More precisely, H ⊂ Q(q)[x] is the R-submodule spanned by the elements