We propose a definition of a quantised -differential algebra and show that the quantised exterior algebra (defined by Berenstein and Zwicknagl) and the quantised Clifford algebra (defined by the authors) of are natural examples of such algebras.
Motivated by our attempts to construct an analogue of the Dirac operator in the setting of , we write down explicitly the braided coproduct, antipode, and adjoint action for quantum algebra . 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 study certain -actions associated to specific examples of branching of scalar generalized Verma modules for compatible pairs , of Lie algebras and their parabolic subalgebras.
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