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In this article, we study extensions of conformal modules over Lie conformal algebras W (a , b , r) with three parameters a, b, r ∈ C , which contain the Virasoro Lie conformal algebra Vir, W (b) and W (a , b) as subalgebras. We obtain the existence conditions of extensions of nontrivial conformal modules over W (a , b , r) and give a complete classification of extensions of finite irreducible conformal modules of W (a , b , r) . [ABSTRACT FROM AUTHOR]

In this paper, by studying the maximal good subspaces, we determine the dual Lie coalgebras of the centerless twisted Heisenberg–Virasoro algebra. Based on this, we construct the dual Lie bialgebras structures of the twisted Heisenberg–Virasoro type. As by-products, four new infinite dimensional Lie algebras are obtained. [ABSTRACT FROM AUTHOR]