Are actual chords being played in classical music?
Of course they are. But I sense that the title of your question doesn't really show what you're actually asking.
What is the left hand doing? Is it primarily arpeggiating chords, or is it doing something trickier?
This is the better question, but the answer isn't too helpful: the left hand can do whatever the the composer wants it to do.
Sometimes it just provides the harmonies by playing chords in the left hand, like in Beethoven's Piano Sonata Op. 10/2, first movement:
Other times it lends harmonic support by only playing one pitch in the left hand, as in Beethoven's Piano Sonata Op. 13, second movement:
Other times it arpeggiates the harmonies with what we call an Alberti bass, like in Mozart's Piano Sonata K545, first movement:
It can also just play a melodic line, causing the right hand to play the harmonic support, as in Liszt's Consolation No. 4:
Or why not multiple melodic lines at once, like in the first fugue from Bach's Well-Tempered Clavier? Here the melodic line in mm. 4--5 is added in in mm. 6--7 while the earlier line continues.
And then some composers just said screw it and put the whole damn piece in the left hand.