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In this paper we study minimal algebraic surfaces with $p_g=q=1,K^2=4$ andnonhyperelliptic Albanese fibrations of genus 4. We construct for the firsttime a family of such surfaces as complete intersections of type $(2,3)$ in a$\mathbb{P}^3$-bundle over an elliptic curve. For the surfaces we constructhere, the direct image of the canonical sheaf under the Albanese map isdecomposable (which is a topological invariant property).  Moreover we prove that, all minimal surfaces with $p_g=q=1,K^2=4$ andnonhyperelliptic Albanese fibrations of genus 4 such that the direct image ofthe canonical sheaf under the Albanese map is decomposable are contained in ourfamily. As a consequence, we show that these surfaces constitute a4-dimensional irreducible subset $\mathcal{M}$ of $\mathcal{M}_{1,1}^{4,4}$,the Gieseker moduli space of minimal surfaces with $p_g=q=1, K^2=g=4$.Moreover, the closure of $\mathcal{M}$ is an irreducible component of$\mathcal{M}_{1,1}^{4,4}$.

Algebraic surfaces with $p_g=q=1, K^2=4$ and nonhyperelliptic Albanese fibrations of genus 4