  We present a lattice QCD calculation of the $\Delta(1232)$ matrix elements of
the axial-vector and pseudoscalar currents. The decomposition of these matrix
elements into the appropriate Lorentz invariant form factors is carried out and
the techniques to calculate the form factors are developed and tested using
quenched configurations. Results are obtained for 2+1 domain wall fermions and
within a hybrid scheme with domain wall valence and staggered sea quarks. Two
Goldberger-Treiman type relations connecting the axial to the pseudoscalar
effective couplings are derived. These and further relations based on the
pion-pole dominance hypothesis are examined using the lattice QCD results,
finding support for their validity. Utilizing lattice QCD results on the axial
charges of the nucleon and the $\Delta$, as well as the nucleon-to-$\Delta$
transition coupling constant, we perform a combined chiral fit to all three
quantities and study their pion mass dependence as the chiral limit is
approached.
