Comment on the differential calculus on the quantum exterior plane

We give a two-parameter quantum deformation of the exterior plane and its differential calculus without the use of any R-matrix and relate it to the differential calculus with the R-matrix. We prove that there are two types of solutions of the Yang-Baxter equation whose symmetry group is GLp,q(2). We also give a two-parameter deformation of the fermionic oscillator algebra. Quantum groups are a generalization of the concept of groups. During the past few years, these new mathematical objects have found wide interest among theoretical physicsts and mathematicians. More precisely, the quantum group is an example of a Hopf algebra which is related to noncommutative geometry. After Wess-Zumino introduced the differential calculus on the quantum (hyper)plane, the quantum plane was generalized to the supersymmetric quantum (super)plane by some authors. In this letter we shall give a differential calculus on the quantum exterior plane whose symmetry group is GLp,q(2) and obtain two R-matrices. They are both solutions of the Yang-Baxter equation. Let us begin with the quantum exterior (dual) plane, defined as the polynomial ring generated by coordinates θ, φ which satisfy θφ+ pφθ = 0 (1) θ = 0 = φ (2) where p is a complex deformation parameter. To develop the differential calculus on the p-exterior plane in terms of the coordinates satisfying (1), (2), we shall make the following ansatz for the commutation relations of the coordinates with their differentials. Let the differentials of coordinates be denoted by Θ = dθ Φ = dφ. (3) In Ref. 1, Wess and Zumino have interpreted θ and φ as the differentials of coordinates of the quantum plane, respectively. Recall that the quantum plane is defined as the polynomial ring generated by coordinates x, y obeying the relation xy = qyx. As an alternative to this interpration, x and y can be identified with the differentials of θ and φ as in Ref. 6. The definition (3) is closely related to this approach. We assume that θΘ = AΘθ θΦ = F11Φθ + F12Θφ φΦ = BΦφ φΘ = F21Θφ+ F22Φθ. (4) Then we find, from (2), A = 1 B = 1. (5) The consistency condition d(θφ+ pφθ) = 0 2 (θφ+ pφθ)Θ = 0 (θφ+ pφθ)Φ = 0 (6) gives F22 + pF11 = 1 F21 + pF12 = p F12F22 = 0. (7) We define the exterior derivative d obeying the condition: d 2 = 0 (8) and the graded Leibniz rule d(fg) = (df)g + (−1)f(dg) (9) where f̂ = 0 for even variables and f̂ = 1 for odd variables. Applying the exterior differential d on second and fourth relations of eq.(4) and using (8), (9) one gets F11F21 = 1− F12 − F22. (10) We now assume that the commutation relation of differentials has the form ΘΦ = qΦΘ (11) where q is another complex deformation parameter. Then we have F11 = q(1− F12) F21 = q (1− F22). (12) The system (7), (10), (12) admits two solutions: Type I A = 1 F11 = q F21 = p B = 1 F12 = 0 F22 = 1− pq (13) and the relations (4) take the form θΘ = Θθ θΦ = qΦθ φΦ = Φφ φΘ = pΘφ+ (1− pq)Φθ. (14) Type II A = 1 F11 = p −1 F21 = q −1 B = 1 F12 = 1− p q F22 = 0 (15) and in this case the relations (4) take the form θΘ = Θθ θΦ = pΦθ − (1− pqΘφ φΦ = Φφ φΘ = qΘφ. (16) 3 To complete the differential geometric scheme we introduce derivatives of the quantum exterior plane in the standard way d = Θ∂θ + Φ∂φ. (17) Multiplying this expression from the right by θf and φf , respectively, and using the graded Leibniz rule for partial derivatives ∂i(fg) = (∂if)g + (−1) f(∂ig) (18) one finds ∂θθ = 1− θ∂θ − F12φ∂φ ∂θφ = −F21φ∂θ ∂φφ = 1− φ∂φ − F22θ∂θ ∂φθ = −F11θ∂φ. (19) These commutation relations, for type I, are ∂θθ = 1− θ∂θ ∂θφ = −pφ∂θ ∂φφ = 1− φ∂φ + (pq − 1)θ∂θ ∂φθ = −qθ∂φ (20a) and for type II, are ∂θθ = 1− θ∂θ + (p q − 1)φ∂φ ∂θφ = −q −1φ∂θ ∂φφ = 1− φ∂φ ∂φθ = −p −1θ∂φ. (20b) The commutation relations between the derivatives can be easily obtained by using that d = 0. So it follows that 0 = d = Θ∂ θ + ΦΘ(∂θ∂φ + q∂φ∂θ) + Φ ∂ φ which says that ∂θ∂φ + q∂φ∂θ = 0 ∂ 2 θ = 0 = ∂ 2 φ. (21) Finally to find the commutation rules between the differentials and derivatives we shall assume that they have the following form ∂θΘ = A11Θ∂θ + A12Φ∂φ ∂θΦ = A21Φ∂θ + A22Θ∂φ (22) ∂φΘ = B11Θ∂φ +B12Φ∂θ ∂φΦ = B21Φ∂φ +B22Θ∂θ. From the fact that ∂i(θ Θ) = δ jδ k l Θ k (23) 4 where ∂1 = ∂θ, θ 1 = θ, Θ = Θ, etc. we have A11 = 1 A22 = 0 B21 = 1 B12 = 0 F11A21 + F12A12 = 1 F21A12 + F22A21 = 0 F21B11 + F22B22 = 1 F11B22 + F12B11 = 0. (24) Thus there are two types of solutions. Type I ∂θΘ = Θ∂θ + (1− p −1q−1)Φ∂φ ∂θΦ = q −1Φ∂θ (25a) ∂φΘ = p −1Θ∂φ ∂φΦ = Φ∂φ and Type II ∂θΘ = Θ∂θ ∂θΦ = pΦ∂θ (25b) ∂φΘ = qΘ∂φ ∂φΦ = Φ∂φ + (1− pq)Θ∂θ. We now should compute the R-matrix for these two types of solutions satisfying the Yang-Baxter equations R12R13R23 = R23R13R12 R̂12R̂23R̂12 = R̂23R̂12R̂23 (26) where R̂ ij kl = R ji kl. (27) From the definition of R matrix for the Yang-Baxter equation θΘ = R klΘ θ (28) we have R = 