  Unitary transformations of a Hamiltonian generally induce interaction terms
beyond the particle rank present in the untransformed Hamiltonian that have to
be captured and included in a many-body calculation. In systems with
strangeness such as hypernuclei, the three-body terms induced by the
hyperon-nucleon interaction are strong, so their inclusion is crucial. We
present in detail a procedure for computing hyperon-nucleon-nucleon interaction
terms that are induced during a similarity renormalization group (SRG) flow.
The SRG is carried out in a basis spanned by antisymmetric harmonic-oscillator
states with respect to three-body Jacobi coordinates. We discuss basis
construction, antisymmetrization, numerical evaluation of the flow equations,
and separation of the genuine three-body terms. We then use the hypernuclear
no-core shell model with SRG-evolved Hamiltonians, addressing the sensitivity
of hypernuclear states and hyperon separation energies to changes in the
nucleonic Hamiltonian by example of $^7_\Lambda$Li, $^9_\Lambda$Be,
$^{11}_\Lambda$B, $^{13}_\Lambda$C, and the hyper-helium chain. We also present
a survey of the hyper-hydrogen chain, exploring the structure of hypernuclei
with extreme neutron-proton asymmetries.
