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We study certain $\mathrm{𝔰𝔩}(2,ℂ)$-actions associated to specific examples of branching of scalar generalized Verma modules for compatible pairs $(𝔤,𝔭)$, $({𝔤}^{\text{'}},{𝔭}^{\text{'}})$ of Lie algebras and their parabolic subalgebras.

Motivated by our attempts to construct an analogue of the Dirac operator in the setting of $U_q\left({\mathrm{𝔰𝔩}}_n\right)$, we write down explicitly the braided coproduct, antipode, and adjoint action for quantum algebra $U_q\left({\mathrm{𝔰𝔩}}_2\right)$. The braided adjoint action is seen to coincide with the ordinary quantum adjoint action, which also follows from the general results of S. Majid.

We propose a definition of a quantised ${\mathrm{𝔰𝔩}}_2$-differential algebra and show that the quantised exterior algebra (defined by Berenstein and Zwicknagl) and the quantised Clifford algebra (defined by the authors) of ${\mathrm{𝔰𝔩}}_2$ are natural examples of such algebras.