  We study symmetry breaking of solitons in the framework of a nonlinear
fractional Schr\"{o}dinger equation (NLFSE), characterized by its L\'{e}vy
index, with cubic nonlinearity and a symmetric double-well potential.
Asymmetric, symmetric, and antisymmetric soliton solutions are found, with
stable asymmetric soliton solutions emerging from unstable symmetric and
antisymmetric ones by way of symmetry-breaking bifurcations. Two different
bifurcation scenarios are possible. First, symmetric soliton solutions undergo
a symmetry-breaking bifurcation of the pitchfork type, which gives rise to a
branch of asymmetric solitons, under the action of the self-focusing
nonlinearity. Second, a family of asymmetric solutions branches off from
antisymmetric states in the case of self-defocusing nonlinearity through a
bifurcation of an inverted-pitchfork type. Systematic numerical analysis
demonstrates that increase of the L\'{e}vy index leads to shrinkage or
expansion of the symmetry-breaking region, depending on parameters of the
double-well potential. Stability of the soliton solutions is explored following
the variation of the L\'{e}vy index, and the results are confirmed by direct
numerical simulations.
