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Let g be a (twisted or untwisted) affine Kac-Moody algebra, and μbe a diagram automorphism of g. In this paper, we give an explicit realization for the universal central extension ĝ[μ] of the twisted loop algebra of g with respect to μ, which provides a Moody-Rao-Yokonuma presentation for the algebra ĝ[μ] when μ, is non-transitive, and the presentation is indeed related to the quantization of twisted toroidal Lie algebras.

Let ⪯ be a compatible total order on the additive group ℤ2, and Lbe the rank two Heisenberg-Virasoro algebra. For any c= (c1, c2, c3, c4) ∈ℂ4, we define a ℤ2-graded Verma module M(c, ⪯) for L. A necessary and sufficient condition for M(c, ⪯) to be irreducible is provided. Moreover, the maximal ℤ2-graded submodules of M(c, ⪯) are characterized when M(c, ⪯) is reducible.