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Abstract
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Using the homogeneous G\"{o}del spacetimes we find some new solutions for the field equations of
bosonic string effective action up to first order in $\alpha'$ including both dilaton and axion fields.
We then discuss in detail the (non-)Abelian T-dualization of G\"{o}del string cosmologies via the Poisson-Lie (PL) T-duality approach.
In studying Abelian T-duality of the models we get seven dual models
in such a way that they are constructed by one-, two- and three-dimensional
Abelian Lie groups acting freely on the target space manifold.
The results of our study show that the Abelian T-dual models are, under some of the special conditions, self-dual;
moreover, by applying the usual rules of Abelian
T-duality without further corrections, we are still able to obtain two-loop solutions.
We also study the Abelian T-duality of G\"{o}del string cosmologies up to $\alpha'$-corrections by using
the T-duality rules at two-loop order derived by Kaloper and Meissner.
Afterwards, non-Abelian duals of the G\"{o}del spacetimes are constructed by two- and three-dimensional non-Abelian Lie groups
such as $A_2$, $A_2 \oplus A_1$ and $SL(2, \mathbb{R})$.
In this way, the PL self-duality of $AdS_3 \times \mathbb{R}$ space is discussed.
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