  We consider Schramm-Loewner evolutions (SLEs) with internal degrees of
freedom that are associated with representations of affine Lie algebras,
following group theoretical formulation of SLEs. We reconstruct the SLEs
considered by Bettelheim {\it et al.} [Phys. Rev. Lett. {\bf 95}, 251601
(2005)] and Alekseev {\it et al.} [Lett. Math. Phys. {\bf 97}, 243-261 (2011)]
in correlation function formulation. We also explicitly formulate stochastic
differential equations on internal degrees of freedom for Heisenberg algebras
and the affine $\mathfrak{sl}_{2}$. Our formulation enables us to find several
local martingales associated with SLEs with internal degrees of freedom from
computation on a representation of an affine Lie algebra. Indeed, we formulate
local martingales associated with SLEs with internal degrees of freedom
described by Heisenberg algebras and the affine $\mathfrak{sl}_{2}$. We also
find an affine $\mathfrak{sl}_{2}$ symmetry of a space of SLE local martingales
for the affine $\mathfrak{sl}_{2}$.
