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A hyperluminous infrared galaxy (HLIRG or HyLIRG) is a galaxy emitting more than 1013 times solar luminosity (LSun) in the far infrared. Similar galaxy categories:
By definition, a HLIRG fits these definitions as well. A LIRG is generally a starburst galaxy and/or has an active galactic nucleus (AGN) or quasar.
I've also seen these thresholds specified as for the galaxy's full bolometric luminosity, sometimes merely stipulating that most of its emission is within the infrared or far infrared; I presume that these varying specifications don't greatly change the sets of qualifying galaxies.