Categorification of the Vector Representation of U(so( 8,C ) )

To categorify the n-tensor products of vector representation of U (so (8, ) ) , some subcategoriees of Bernstein-Gelfand-Gelfand (BGG) category O of the general linear Lie algebra glnwere defined. The complexifications of their Grothendieck groups were used for categorifing base spaces of ntensor products of vector representation. And some projective endfunctors of BGG category O, which were used for categorifing the action of U (so (8, ) ) on n-tensor products of vector representation, were defined. It was got that hi (1≤i≤4) can be categorified by a pair of functors (H_i⁺, H_i^-) (1≤i≤4) , e_i, f_i (1≤i≤3) , can be categorified by ε_i, F_i (1≤i≤3) , e₄, f₄ can be categorified by a pair of functors (ε₄⁺, ε₄^-) , (F₄⁺, F₄^-) , respectively.