In this paper, first we study infinitesimal deformations of a Lie algebra with a representation and introduce the notion of a Nijenhuis pair, which gives a trivial deformation of a Lie algebra with a representation. Then we introduce the notion of a Kupershmidt-(dual-)Nijenhuis structure on a Lie algebra with a representation, which is a generalization of the r − n structure (r-matrix-Nijenhuis structure) introduced by Ravanpak, Rezaei-Aghdam, and Haghighatdoost. We show that a Kupershmidt-(dual-)Nijenhuis structure gives rise to a hierarchy of Kupershmidt operators. Finally, we define a Rota-Baxter-Nijenhuis structure to be a Kupershmidt-Nijenhuis structure on a Lie algebra with respect to the adjoint representation and study the relation between Rota-Baxter-Nijenhuis structures and r-matrix-Nijenhuis structures.
All 62 monomial elements and 144 polynomial elements with one-dimensional support in the canonical basis B of the quantum group for type A4 have been determined in [5] and [4], respectively. It is conjectured in [4] that there are other polynomial elements in B with two- or three-dimensional support. In this paper, we compute 50 polynomial elements with two-dimensional support of different-exponent type and 62 polynomial elements with two-dimensional support of same-exponent type in the canonical basis B of the quantum group for type A4.
In this paper, we first prove that homogeneous spaces E 6 /A 4 and E 6 /A 1 admit Einstein metrics which are Ad(T × A 1 × A 4 )-invariant, and then show that they admit Non-Riemannian Einstein-Randers metrics.
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