On q- and h-deformations of 3d-superspaces

Quantum de Rham complex on 𝔸h1|2

“…A two parameter first order differential calculus on the quantum superplane A 1|1 h that is covariant under the co-action of the quantum supergroup GL h,h ′ (1|1) is given in [2]. In the present paper, the first order differential calculus on the quantum superspace A 1|2 h , which is covariant under the co-action of the quantum supergroup GL h,h ′ (1|2) given in [4], will be introduced by following Wess-Zumino type differential calculus. This calculus is set up with the commutation relations between the generators of function rings O(A 1|2 h ) and their differentials as the generators of O(A 2|1 h ′ ).…”

“…In this section, we will briefly make mention of the Z 2 -graded Hopf algebra O(GL h,h ′ (1|2)) and (h, h ′ )-deformed quantum superspaces from [4], where h and h ′ are both odd (or Grassmann) numbers. It can be obtained (h, h ′ )-deformed commutation relations between the matrix entries of T from the standard FRT construction [6].…”

“…In the following theorem we assume that the odd matrix elements of T are anti-commutative with h and h ′ . Theorem 2.1 [4] The matrix entries of T satisfy the (h, h ′ )-deformed commu-tation relations…”

“…In this subsection, we introduce an h-and h ′ -deformations of the superspaces A 1|2 and A 2|1 given in [4]. Let K x, θ 1 , θ 2 be a free algebra with unit generated by x, θ 1 and θ 2 , where the coordinate x is even, the coordinates θ 1 and θ 2 are odd.…”

“…Remark 2.5 Although the commutation relations between the generators of exterior algebra given in the above definition seem to be obtained by taking −h ′ instead of h ′ in [4], due to the odd character of h ′ , the case in reality is not exactly so. An explanation about this case will be given in Sec.…”