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, the one in the basis of divided powers at an even root of unity $q=e^{i\pi /p}$. It can be regarded as an extension of the "nearly commutative" algebra $C[X,Y]$ with $XY=(-1{)}^{p}YX$ 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 $U_{q}sl(2)$ and its Lusztig extension; the quantum group action is also defined on the algebra of quantum differential operators on the quantum plane.

18 pages, amsart++, xy, times. V2: a reference and related comments added

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Cited by in corpus (1)

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