A Categorification of Quantum sl_2

In this paper, we categorify the algebra U_q(sl_2) with the same approach as in [A. Lauda, Adv. Math. (2010), arXiv:math.QA/0803.3652; M. Khovanov, Comm. Algebra 11 (2001) 5033]. The algebra =_q(sl_2) is obtained from U_q(sl_2) by adjoining a collection of orthogonal idempotents 1_λ, λ∈P, in which P is the weight lattice of U_q(sl_2). Under such construction the algebra U is decomposed into a direct sum _(λ∈P) 1_λ,U1_λ. We set the collection of λ∈ P as the objects of the category u, 1-morphisms from λ to λ' are given by 1_λ,U1_λ, and 2-morphisms are constructed by some semilinear form defined on U. Hence we get a 2-category u from the algebra U_q(sl_2).

In this paper, we categorify the algebra Uq（sl2） with the same approach as in [A. Lauda, Adv. Math. （2010）, arXiv：math.QA/0803.3662; M. Khovanov, Comm. Algebra 11 （2001） 5033]. The algebra U =Uq（sl2） is obtained from Uq（sl2） by adjoining a collection of orthogonal idempotents 1λ,λ ∈ P, in which P is the weight lattice of Uq（sl2）. Under such construction the algebra U is decomposed into a direct sum λ∈p 1λ,U1λ. We set the collection of λ∈ P as the objects of the category U, 1-morphisms from λ to λ′ are given by 1λ,U1λ, and 2-morphisms are constructed by some semilinear form defined on U. Hence we get a 2-category u from the algebra Uq（sl2）.