A Categorification of Quantum widehat{sl_2}

In this paper, we categorify the algebraU_q(widehat{sl_2}) with the same approach as in [A. Lauda, Adv. Math. (2010), arXiv:math.QA/0803.3652;
M. Khovanov, Comm. Algebra11 (2001) 5033]. The algebra dot{U}=dot{U}_q(widehat{sl_2}) is obtained fromU_q(widehat{sl_2}) by adjoining a collection of orthogonal idempotents 1_λ,λ in P, in which P is the weight lattice ofU_q(widehat{sl_2}). Under such construction the algebraU is decomposed into a direct sum bigoplus_{lambdain P} 1_{lambda'}{U} 1_λ. We set the collection of λin P as the objects of the categoryU, 1-morphisms fromλ toλ' are given by 1_λ' }k U1_λ, and 2-morphisms are constructed by some semilinear form defined onU. Hence we get a 2-category {cal U} from the algebraU_q{(widehat{sl_2})}.