GLq(N)-COVARIANT BRAIDED DIFFERENTIAL BIALGEBRAS

We study a possibility to define the (braided) comultiplication for the GLq(N)covariant differential complexes on some quantum spaces. We discover such differential bialgebras (and Hopf algebras) on the bosonic and fermionic quantum hyperplanes (with additive coproduct) and on the braided matrix algebra BMq(N) with both multiplicative and additive coproducts. The latter case is related (for N = 2) to the q-Minkowski space and q-Poincare algebra. ∗ E-mail: [email protected] ⋄ E-mail: [email protected] 1. Throughout the recent development of differential calculus on quantum groups and quantum spaces, two principal and closely related concepts are readily seen. One of them, initiated by Woronowicz [1], is known as bicovariant differential calculus on quantum groups. Its characteristic feature is the covariance under the left and right “group shifts” ∆L and ∆R acting upon the differential complex in a consistent way. Brzezinski [2] has shown that this corresponds to existence of a differential bialgebra, i.e. a bialgebra structure with honest coproduct ∆ on the whole algebra of coordinate functions and their differentials. This allows one to treat all the subject using the standard Hopf-algebra technique. Another concept, introduced by Wess and Zumino [3] (see also [4]), proceeds from the requirement of covariance of the differential complex on a quantum space with respect to the coaction of some outer quantum group considered as a group of symmetry. In other words, the corresponding differential algebra must be a covariant comodule as well. In the present letter, we want to unify both concepts by formulating the following set of conditions to be satisfied by q-deformed differential calculi: α) associative algebra of generators and differential forms is respected by the (co)action of some quantum group; β) external differentiation d obeys d = 0 and the usual (graded) Leibnitz rule; γ) differential algebra admits a (braided) coproduct of the form [2] ∆(a) = a(1) ⊗ a(2) , ∆(da) = da(1) ⊗ a(2) + a(1) ⊗ da(2) . (1) Following these criteria, we obtain several examples of GLq(N)-covariant differential bialgebras: the braided matrix algebra BMq(N) (with additive and multiplicative coproducts) and also bosonic and fermionic quantum hyperplanes with additive coproduct. The first example seems to be of special interest because BMq(2) is presently considered as a candidate to the role of the q-Minkowski space [5]-[9]. 2. To formulate and study the quantum-group-covariant differential calculus, the R-matrix formalism [10] proved to be extremely convenient. Let us first consider the case of braided matrix algebra BMq(N) with the generators {1, u i j} (the latter form the N×N -matrix u) and relations R21u2R12u1 = u1R21u2R12 , (2) where R is the GLq(N) R-matrix [10, 11]. The multiplication rule (2) is invariant under adjoint coaction of GLq(N), uij → T i mS(T n j )⊗ u m n , or u → TuT −1 , (3) where T i j obey the relations R12T1T2 = T2T1R12 (4) and commute with umn . Eq.(2) (“reflection equations”) first appeared in the course of investigations of 2-dimensional integrable models on a half-line (see [12] and references therein). Further it was studied by Majid [13] within the general framework of braided algebras. From now on we prefer to use the notation a ⊗ 1 ≡ a, 1 ⊗ a ≡ a for any element a. The matrix notation will also be slightly modified [14] to simplify the relevant calculations: P12R12 ≡ R̂12 ≡ R , R −1 ≡ R , u1 ≡ u , u ′ 1 ≡ u ′ . Thus, the Hecke condition for R reads R−R = q − q ≡ λ , (5) whereas (2) becomes simply RuRu = uRuR . (6) Differential complex on BMq(N) is defined by (6) and RuRdu = duRuR , (7) RduRdu = −duRduR (8) (here and below we omit the wedge product symbol ∧ in the multiplication of differential forms). Of course, one could perfectly well use RuRdu = duRuR instead of (7): these possibilities are absolutely parallel. We should also note that an agreement of (7) with (6) (via the Leibnitz rule), and some other formulas below, rely heavily on the Hecke condition (5) that is specific to the GLq(N) case. Commutational relations (7),(8) have been found in the component form for N = 2 [6] in the context of the q-Poincare algebra, and then recast into the R-matrix form in [8]. Besides that, eq.(6) is known [13] to admit the multiplicative coproduct ∆(uij) = u i k ⊗ u k j , or ∆(u) = u⊗ u ≡ u u ′ , (9) provided the nontrivial braiding relations RuRu = uRu R (10) are used for commuting primed u-matrices with unprimed ones. Recall [15] that the braiding transformation Ψ : A⊗B → B ⊗A, where A and B are covariant comodules of a quantum group, is a map which commutes with the group coaction and, therefore, produces a covariant recipe for multiplying tensor products of generators: (1⊗ a) (b⊗ 1) ≡ a b = Ψ(a⊗ b) . For instance, eq.(10) is induced by the corresponding universal R-matrix through the (somewhat symbolic) relation Ψ(u ⊗ u) =< TuT ⊗ TuT ,R > . Now let us examine whether a map of the form (1) (see also [16, 17]), ∆(du) = du⊗ u+ u⊗ du ≡ du u + u du , (11) together with (9) yields a proper coproduct for the whole algebra (6)-(8). Our statement is that it really does. Moreover, two different sets of the braiding relations can be used here equally well: one based on (10),