Thelonious Monk tune, Lu Lu's Back in Town, who is Lu Lu?

I saw a documentary about Monk that played that tune during the part of Monk's career when he was returning to NYC from his 1966 European tour, it seemed to suggest that Monk is Lu Lu, Is he? If not, who is Lu Lu?

thanks in advance.

Lulu's Back In Town is a 1935 song by Harry Warren and Al Dubin from thr movie Broadway Gondolier. The most famous recording is by Fats Waller. It's got a verse, but here's just the chorus:

Gotta get my old tuxedo pressed
Gotta sew a button on my vest
'Cause tonight I've gotta look my best
Lulu's back in town

Gotta get half a buck somewhere
Gotta shine my shoes and slick my hair
Gotta get myself a boutonniere
Lulu's back in town

You can tell all my pets
All my Harlem coquettes
Mister Otis regrets
That he won't be aroun'

You can tell the mailman not to call
I ain't comin' home until the fall
And I might not get back at all
Lulu's back in town