We report the results of high spatial and spectral resolution integral-field spectroscopy of the central ~3 x 3 arcsec^2 of the active galaxy NGC 1275 (Perseus A), based on observations with the Near-infrared Integral Field Spectrograph (NIFS) and the ALTAIR adaptive-optics system on the Gemini North telescope. The circum-nuclear disc in the inner R~50 pc of NGC 1275 is seen in both the H2 and [FeII] lines. The disc is interpreted as the outer part of a collisionally-excited turbulent accretion disc. The kinematic major axis of the disc at a position angle of 68 deg is oriented perpendicular to the radio jet. A streamer-like feature to the south-west of the disc, detected in H2 but not in [FeII], is discussed as one of possibly several molecular streamers, presumably falling into the nuclear region. Indications of an ionization structure within the disc are deduced from the HeI and Br gamma emission lines, which may partially originate from the inner portions of the accretion disc. The kinematics of these two lines agrees with the signature of the circum-nuclear disc, but both lines display a larger central velocity dispersion than the H2 line. The rovibrational H2 transitions from the core of NGC 1275 are indicative of thermal excitation caused by shocks and agree with excitation temperatures of ~1360 and ~4290 K for the lower- and higher-energy H2 transitions, respectively. The data suggest X-ray heating as the dominant excitation mechanism of [FeII] emission in the core, while fast shocks are a possible alternative. The [FeII] lines indicate an electron density of ~4000 cm^{-3}. The H2 disc is modelled using simulated NIFS data cubes of H2 emission from inclined discs in Keplerian rotation around a central mass. Assuming a disc inclination of 45 deg +/- 10 deg, the best-fitting models imply a central mass of (8^{+7}_{-2}) x 10^8 Msun. (abridged)
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