Classification of U_q(sl_2)-module algebra structures on the quantum plane

Abstract

A complete list of Uq(sl 2 )-module algebra structures on the quantum plane is produced and the (uncountable family of) isomorphism classes of these structures are described. The composition series of representations in question are computed. The classical limits of the Uq(sl 2 )-module algebra structures are discussed.

Hom-quantum groups III: Representations and module Hom-algebras

“…In this example, we consider Hom-type analogs of one such non-standard U q (sl 2 )-module-algebra structure on A 2|0 q . The reader is referred to [9] for a complete classification of U q (sl 2 )-module-algebra structures on the quantum plane. Following the setting of [9], we assume that 0 < q < 1.…”

“…The reader is referred to [9] for a complete classification of U q (sl 2 )-module-algebra structures on the quantum plane. Following the setting of [9], we assume that 0 < q < 1.…”

“…According to [9] (Theorem 9 and Proposition 18), there is a U q (sl 2 )-module-algebra structure ρ : U q (sl 2 ) ⊗ A 2|0 q → A 2|0 q on A 2|0 q determined by K ±1 (x m y n ) = q ±(m−2n) x m y n , E(x m y n ) = q 1−n [n] q x m y n−1 , and F (x m y n ) = q −m q 2m − q 2n q − q −1 x m y n+1 , (5.8.1)…”

“…In Example 5.7 we construct a multi-parameter family of U q (sl 2 ) α -module Hom-algebra structures on the Hom-quantum planes, generalizing the well-known U q (sl 2 )-module-algebra structure on the quantum plane [37] that is defined in terms of the quantum partial derivatives (5.7.3). In Example 5.8 we construct Hom-type analogs of a non-standard U q (sl 2 )-module-algebra structure on the quantum plane [9].…”

The Laurent Extension of Quantum Plane: a Complete List of U_q(sl₂)-Symmetries

“…The initial approach to considering different symmetries has been developed in the paper by S. Duplij and S. Sinel'shchikov [4]. The authors produced a complete list of U q (sl 2 )-symmetries on C q [x, y] and, in particular, demonstrated the existence of an uncountable collection of pairwise non-isomorphic such symmetries.…”