QUANTUM CALCULUS AND ALGEBRA WORKSHEET
S MAJID
Q1 Suppose that (dx)x = qx(dx) where q is a numerical parameter and assume the
product rule. Show that in this case
1 − qn
dxn = [n]q xn−1 dx, [n]q =
1−q
(the last expression is called a q-integer).
Q2 Deduce from Q1 that
df (x) = ∂q f (x)dx,
∂q f (x) =
f (qx) − f (x)
x(q − 1)
for any polynomial function f (x). The last expression is called the q-derivative.
Q3 Show that
q n−m [m]q + [n − m]q = [n]q .
[n] !
n
q
or zero if m > n.
Q4 Define [n]q ! = [n]q [n − 1]q !, [0]q ! = 1 and [ m
]q = [m]q ![n−m]
q!
Deduce from Q3 that
n−1
n−1
n
q n−m [
]q + [
]q = [ ]q
m−1
m
m
for all m < n.
Q5 Show that if BA = qAB then
(A + B)n =
n
X
n
[ ]q Am B n−m
m
m=0
(this is called the q-binomial theorem).
Q6 Find a function eq (x) so that
∂q eq (x) = eq (x)
(this is called the q-exponential) and use Q5 to show that if BA = qAB then
eq (A)eq (B) = eq (A + B).
Q7* Using the language and methods of braided algebra from my lecture, write
down and prove the braided version of the usual identity
a(bcb−1 )a−1 = (ab)c(ab)−1 .
(you will need the lemma from my lecture).
References
[1] On Space and Time, A Connes, M. Heller, S. Majid, R. Penrose, J. Polkinghorne and A.
Taylor, Cambridge University Press 2008
[2] A Quantum Groups Primer, S. Majid, Cambridge University Press 2002
1