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In this paper, we give the explicit expressions of the differential operator representations of the exceptional Lie superalgebra D(2, 1;�). Based on these expressions, we construct free field realizations of the currents associated with D(2, 1;�) at an arbitrary level k.

Motivated by the recently discovered hidden symmetry of the type IIB Green–Schwarz superstring on certain background, the non-semisimple Kac–Moody twisted superalgebra gl(2)k is investigated by means of the vector coherent state method and boson-fermion realization. The free field realization of the twisted current superalgebra at general level k is constructed. The corresponding Conformal Field Theory (CFT) has zero central charge. According to the classification theory, this CFT is a nonunitary field theory. After projecting out a U(1) factor and an outer automorphism operator, we get the free field representation of psl(2)k, which is the algebra of gl(2)k modulo the 4-outer automorphism, the CFT has central charge -2.

We propose a rational quantum deformed nonlocal currents in the homogenous space $SU(2)_k/U(1)$, and in terms of it and a free boson field a representation for the Drinfeld currents of Yangian double at a general level $k=c$ is obtained. In the classical limit $\hbar \to 0$, the quantum nonlocal currents become $SU(2)_k$ parafermion, and the realization of Yangian double becomes the parafermion realization of $SU(2)_k$ current algebra., Latex, 9 pages

We derive a free boson representation of the Yangian double DY(slN) with arbitrary level k by the observation that there is a correspondence between the q-affine algebra and Yangian double associated with the same Cartan matrix. Vertex operator and screening currents cannot be obtained in the same way. From this point of view, the Yangian double with centre cannot be regarded as the degenerated case of the quantum affine algebra.

We express the vacuum expectation value of the SU(3)k parafermion fields by that of two bosons and SU(3)k current algebra. When k = 1,2,3, the SU(3)k current algebra becomes an inner product of a unitary representation, and T(z), W(3)(z) are equivalent to "quasi-self-adjoint" operators in this representation.