
Introduction: Quantum groups (Drinfeld, 1986) have a rich mathematical structure (Klimyk and Schmüdgen, 1997; Majid, 1995) and possess concrete examples of noncommutative differential geometry (Connes, 1991) by introducing a consistent differential calculus on noncommutative spaces. The initiator works on covariant differential calculi on the quantum groups are due to Woronowicz (Woronowicz, 1987; Woronowicz, 1989). In that approach the quantum group is considered as the basic noncommutative space and the differential calculus on the group is obtained from the properties of the group. These works are very influential for the study of noncommutative differential calculi (see, for example, Giaquinto and Zhang, 1995; Heckenberger and Schmüdgen, 1998). Besides quantum vector spaces and quantum groups, covariant differential calculi have been investigated on quantum 2spheres (Podles, 1987), (Schirrmacher, Wess and Zumino, 1991). A theory of such calculi on the general quantum (super) spaces is still in progress. A quantum space is a space that quantum group acts with linear transformations and whose coordinates belong to a noncommutative associative algebra (Manin, 1988). By construing the exterior space of the quantum space as differentials of the coordinates, covariant differential calculus on the quantum space was developed in 1991 (Wess and Zumino, 1991). The natural extension of their scheme to \[{{\mathbb{Z}}_{2}}\]graded space or superspace (Manin, 1989) was introduced by Soni, 1990. The quantum (1+2)superspace, denoted by \[{{\mathbb{R}}_{q}}(12)\], is an example of a noncommutative (n+m)superspace. The elements of the (1+2)superspace \[{{\mathbb{R}}_{q}}(12)\] are supervectors generated by an even and two odd components. One defines the superspace \[{{\mathbb{R}}_{q}}(12)\] by dividing the superspace \[{{\mathbb{R}}_{q}}(12)\] of 3x1 vectors into two parts as \[{{\mathbb{R}}_{q}}(12)={{V}_{0}}\oplus {{V}_{1}}\] where the element of \[{{V}_{0}}\] is of the form \[{{(x,0,0)}^{t}}\] and grade 0 and the element of \[{{V}_{1}}\] is the form \[{{(0,{{\xi }_{1}},{{\xi }_{2}})}^{t}}\] and grade 1. The components of a supervector in \[{{\mathbb{R}}_{q}}(12)\] satisfy the following qcommutation relations (Manin, 1989)
\[x\cdot {{\xi }_{1}}=q{{\xi }_{1}}\cdot x,\text{ }x\cdot {{\xi }_{2}}=q{{\xi }_{2}}\cdot x,\text{ }{{\xi }_{1}}\cdot {{\xi }_{2}}={{q}^{1}}{{\xi }_{2}}\cdot {{\xi }_{1}},\text{ }{{\xi }_{i}}^{2}=0\]
where q is a non zero complex number. Oneparameter bicovariant differential calculus on the function algebra on \[{{\mathbb{R}}_{q}}(12)\] is studied by Celik, 2016. In our study, we construct two parameter differential calculi on the function algebra on \[{{\mathbb{R}}_{q}}(12)\] as well. These calculi are discussed from the covariance point of view, after introducing the corresponding two parameter quantum supergroup \[G{{L}_{p,}}_{q}(12)\]. Purpose: It is to construct twoparameter leftcovariant differential calculi on the function algebra on \[{{\mathbb{R}}_{q}}(12)\] with respect to superHopf algebra \[O(G{{L}_{p,}}_{q}(12))\]. Method: It is well known that in classical differential calculus, the functions commute with the differentials. From algebraic point of view, the space of 1forms is a free bimodule over the algebra of smooth functions generated by the first order differentials and the commutativity shows how its left and right structures are related to each other. Let the \[O({{\mathbb{R}}_{q}}(12))\]bimodule \[\Gamma \] be generated as a free right \[O({{\mathbb{R}}_{q}}(12))\]module by the differentials \[dx,d{{\xi }_{1}},d{{\xi }_{2}}\]. In general the coordinates will not commute with their differentials. Therefore, we assume that the possible commutation relations of the generators with their first order differentials are of the form
\[x\cdot dx={{P}_{1}}dx\cdot x+{{P}_{2}}d{{\xi }_{1}}\cdot {{\xi }_{1}}+{{P}_{3}}d{{\xi }_{1}}\cdot {{\xi }_{2}}+{{P}_{4}}d{{\xi }_{2}}\cdot {{\xi }_{1}}+{{P}_{5}}d{{\xi }_{2}}\cdot {{\xi }_{2}}\]
\[x\cdot d{{\xi }_{1}}={{A}_{1}}_{1}d{{\xi }_{1}}\cdot x+{{A}_{1}}_{2}dx\cdot {{\xi }_{1}}+{{A}_{1}}_{3}dx\cdot {{\xi }_{2}}+{{A}_{1}}_{4}d{{\xi }_{2}}\cdot x\] (1)
where the coefficients \[{{P}_{i}}\],\[{{A}_{ij}}\], etc. are possibly depend on p and/or q. That is, the left \[O({{\mathbb{R}}_{q}}(12))\]module structure on \[\Gamma \] should completely be defined by (1). To obtain nine crosscommutation relations between the elements of the set \[\left\{ x,{{\xi }_{1}},{{\xi }_{2}} \right\}\]and the elements of the set\[\left\{ dx,d{{\xi }_{1}},d{{\xi }_{2}} \right\}\], let us interpret the generators \[\theta ,y,z\] as the deformation of an algebra of differentials. In total, we have 41 indeterminate coefficients which are successively by the rest of the conditions. Findings: We can eliminate about half of those coefficients after applying the differential d to crosscommutation relations and comparing them with the relations of exterior plane. Next, we use compatibility with the left coaction of \[O(G{{L}_{p,}}_{q}(12))\]. This leaves one (free) parameter \[{{P}_{1}}\]. This parameter is fixed by checking associativity of the cubics, for instance, \[(x\cdot d{{\xi }_{1}})\wedge dx\] should be equal to \[x\cdot (d{{\xi }_{1}}\wedge dx)\]. We find that this parameter \[{{P}_{1}}\] should be either equal to p or to \[{{p}^{1}}\]. Conclusion: We see that, there exist two left covariant \[{{\mathbb{Z}}_{2}}\]graded first order differential calculi \[{{\Gamma }_{+}}\] and \[{{\Gamma }_{}}\] over the algebra \[O({{\mathbb{R}}_{q}}(12))\] with respect to the superHopf algebra \[O(G{{L}_{p,}}_{q}(12))\] such that the set \[\left\{ dx,d{{\xi }_{1}},d{{\xi }_{2}} \right\}\]is a free right \[O({{\mathbb{R}}_{q}}(12))\]module basis of \[{{\Gamma }_{\pm }}\]. The bimodule structures for these two calculi are described by the relations
\[{{(1)}^{\tau ({{x}_{i}})}}{{x}_{i}}\cdot dx_{i}^{{}}={{p}^{\pm 1}}\sum\limits_{k,l=1}^{3}{{{({{R}^{\pm 1}})}_{kl}}^{ij}}d{{x}_{k}}\cdot {{x}_{l}}\]
where \[\tau (u)\] denotes the grading of u.
Anahtar Kelimeler: Quantum Superspaces, Quantum Supegroup, SuperHopf Algebra, Differential Calculus
