Quantum sℓ(2) action on a divided-power quantum plane at even roots of unity

Abstract

We describe a nonstandard version of the quantum plane in which the basis is given by divided powers at an even root of unity q = e iπ/p . It can be regarded as an extension of the "nearly commutative" algebra C[X, Y ] with XY = (−1) p Y X by nilpotents. For this quantum plane, we construct a Wess-Zumino-type de Rham complex and find its decomposition into representations of the 2p 3 -dimensional quantum group Uqs (2) and its Lusztig extension U qs (2); we also define the quantum group action on the algebra of quantum differential operators on the quantum plane.

Loewy filtration and quantum de Rham cohomology over quantum divided power algebra

“…On the other hand, Hu [22] first defined the quantum divided power algebras A q (n) and the restricted quantum divided power subalgebras A q (n, 1) as u q (sl n )module algebras by defining the appropriate q-derivations, and thereby provided a realization model for some simple modules with highest weights (ℓ−1−s i )λ i−1 +s i λ i (0 ≤ s i < ℓ). Recently, Semikhatov [37] also exploited the divided-power quantum plane C q that is the rank 2 quantum divided power algebra A q (2) and its u q (sl 2 )module algebra realization to derive an explicit description of the indecomposable decompositions of (C q ) (np−1) and of the space of 1-forms (Ω 1 q ) (np−1) for the Wess-Zumino de Rham complex on C q (at q a 2p-th root of 1).…”

“…While, the indecomposable modules for the latter has been completely solved in different perspectives by many authors, like Chari-Premet [11], Suter [38], Xiao [39], etc. Recently, for the even order of root of unity case, Semikhatov [37] distinctly analyzed the submodules structure of the divided-power quantum plane for the Lusztig small quantum groupū q (sl 2 ) using a different way.…”

“…[3]). Section 4 is devoted to defining the q-differentials by using the q-derivations in [22], which are not the "differential calculus" in the sense of Woronowicz ([40]), as well as constructing the quantum de Rham complex Ω q (n) over A q (n) (see Propositions 4.2 & 4.4), which is different from the Wess-Zumino de Rham complex used in [32], [37] in the rank 1 case. For the quantum de Rham subcomplex Ω q (n, m), we give an interesting description of the corresponding quantum de Rham cohomologies (see Theorems 4.6 & 4.8).…”