  Hot and dense quark matter with isospin and chiral imbalances is investigated
in the framework of the (3+1)-dimensional Nambu--Jona-Lasinio model (NJL) in
the large-$N_c$ limit ($N_c$ is the number of quark colors). Its phase
structure is considered in terms of baryon -- $\mu_B$, isospin -- $\mu_I$ and
chiral isospin -- $\mu_{I5}$ chemical potentials. It is shown in the paper that
(i) in the chiral limit there is a duality between chiral symmetry breaking
(CSB) and charged pion condensation (PC) phenomena. (ii) At the physical point,
i.e. at nonzero bare quark mass $m_0$, and temperature this duality relation is
only approximate, although rather accurate. (iii) We have shown that the chiral
isospin chemical potential $\mu_{I5}$ in dense quark matter generates charged
pion condensation both at zero and nonzero $m_0$, and at $\mu_{I5}\ne 0$ this
phase might be observed up to temperatures as high as 100 MeV. (iv)
Pseudo-critical temperature of the chiral crossover transition rises in the NJL
model with increasing $\mu_{I5}$. (v) It has been found an agreement between
particular sections of the phase diagram in the framework of NJL model and
corresponding ones in lattice QCD simulations. Two different plots from
different lattice simulations that are completely independent and are not
connected at the first sight are in reality dual to each other, it means that
lattice QCD simulations support the hypothesis that in real quark matter there
exists the (approximate) duality between CSB and charged PC. Moreover, we can
reverse the logic and we can predict the increase of pseudo-critical
temperature with chiral chemical potential, the much debated effect recently,
just by the duality notion, hence bolster confidence in this result (lattice
QCD showed this feature for unphysically large pion mass) and put it on the
considerably more solid ground.
