Acta Polytechnica


doi:10.14311/AP.2014.54.0394
Acta Polytechnica 54(6):394–397, 2014 © Czech Technical University in Prague, 2014

available online at http://ojs.cvut.cz/ojs/index.php/ap

AUTOMORPHISMS OF ALGEBRAS AND ORTHOGONAL
POLYNOMIALS

Daniel Gromada, Severin Pošta∗

Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University
in Prague, Trojanova 13, CZ-120 00 Prague, Czech Republic

∗ corresponding author: severin.posta@fjfi.cvut.cz

Abstract. Suitable automorphisms together with a complete classification of representations of some
algebras can be used to generate some sets of orthogonal polynomials “at no cost”. This is also the case
for the nonstandard Klimyk-Gavrilik deformation U′q(so3), which is connected to q-Racah polynomials.

Keywords: orthogonal polynomials, algebra representation, automorphism.

1. Introduction
The connection of orthogonal polynomials with Lie al-
gebras has been known for a long time (see [1–4]. For
a nice introduction and a detailed historical survey see
[5] and references therein). Having available the classi-
fication of irreducible representations and making use
of some automorphisms of algebras, one can obtain
sets of orthogonal polynomials for free, without the
need to proving their properties manually. The same
is true for some of their q-analogs. We show this ap-
proach on a well-known example of the sl2 algebra and
Krawtchouk polynomials [5], and we then apply the
same procedure to the nonstandard Klimyk-Gavrilik
deformation U′q(so3), for which the complete classifi-
cation of its irreducible representations is known (see
[6–9]).

2. Lie algebra sl2
Let us first consider the Lie algebra sl2 of 2×2 complex
matrices with zero trace. It has the standard Chevalley
basis

e =
(

0 1
0 0

)
, f =

(
0 0
1 0

)
, h =

(
1 0
0 −1

)
.

These matrices satisfy the commutation relations

[h,e] = 2e, [h,f] = −2f, [e,f] = h,

where [x,y] = xy −yx.
Let ϕ be a finite-dimensional irreducible represen-

tation of sl2 acting on the space VN+1 of dimension
N + 1 with some fixed basis via the matrices

H = ϕ(h) =



N 0 0 · · · 0
0 N − 2 0 0
0 0 N − 4 0
...

...
...

0 0 0 · · · −N


 ,

ϕ(e) =




0 1 0 · · · 0
0 0 2 0
0 0 0

... 0
...

...
...

0 0 0 · · · 0


 ,

ϕ(f) =




0 0 0 · · · 0
N 0 0 0
0 N − 1 0 0
...

... ...
...

0 0 0 · · · 0


 .

Now let us define a matrix S as

S = ϕ(e) + ϕ(f) =




0 1 0 · · · 0
N 0 2 0
0 N − 1 0

... 0
...

... ...
...

0 0 0 · · · 0


 .

Making use of an automorphism σ sending

h → e + f, e →
1
2

(h−e + f), f →
1
2

(h + e−f)

and taking into account the classification of irreducible
representations of the sl2 algebra, we see that matrices
H and S form a so-called Leonard pair (a Leonard
pair is a pair of diagonalizable finite-dimensional linear
transformations, each of which acts in an irreducible
tridiagonal fashion on an eigenbasis for the other one).
Particularly, they have the same eigenvalues.
It follows that there exists a matrix P such that

S = PHP−1. We can consider the rows of P or the
columns of P−1 as coordinate vectors of a polynomial.
The proposition is that

Kj(k; 1/2,N) = [P−1]kj =
(
N

j

)−1
Pjk (1)

where Kn(x; p,N) is n-th Krawtchouk polynomial n =
0, 1, . . . ,N with parameters p ∈ (0, 1) and N ∈ N0.
The matrix P is defined by a similarity relation up

394

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vol. 54 no. 6/2014 Automorphisms of algebras and orthogonal polynomials

to a multiplicative constant. Equation 1 holds if we
choose P00 = 1.

The Krawtchouk polynomials are defined by means
of the hypergeometric function

Kn(x; p,N) = 2F1
(
−n,−x
−N

∣∣∣∣ 1p
)

=
N∑
j=0

(−n)j(−x)j
(−N)j

1
pjj!

where (a)j =
j−1∏
k=0

(a + k) and (a)0 = 1.

To show (1), we can use the similarity relations
P−1S = HP−1, i.e.,

[P−1S]kj = j[P−1]k,j−1 + (N − j)[P−1]k,j+1,
[HP−1]jk = (N − 2k)[P−1]kj,

construct a recurrence relation

j[P−1]k,j−1 − (N − 2k)[P−1]k,j
+ (N − j)[P−1]k,j+1 = 0 (2)

and compare it with general recurrence for Krawt-
chouk polynomials,

j(p− 1)Kj−1(x) −
(
j(p− 1) + (j −N)p + x

)
Kj(x)

+ (j −N)pKj+1(x) = 0 (3)

where Kn(x) = Kn(x; p,N) for fixed p and N. One
can show a similar result for the relation SP = PH.

This also proves the relation between P−1 and P∗,
which can be written as P−1 = P∗G−1, where Gjk =
δjk
(
N
j

)
. This means that P is orthogonal with respect

to the inner product defined by matrix G. Thus, P−1
is orthogonal and so the columns of P−1 (Krawtchouk
polynomials) are orthogonal with respect to the inner
product defining the orthogonality relation

N∑
j=1

(
N

j

)
Km(j)Kn(j) = hnδmn. (4)

The inner product is of course determined up to nor-
malization ((4) can be multiplied by the arbitrary se-
quence an). This result corresponds with the general
orthogonality relation for Krawtchouk polynomials
(see [10])

N∑
j=0

(
N

j

)
pj(1 −p)N−jKm(j)Kn(j)

=
(
N

n

)−1 (1 −p
p

)n
δmn.

Relation (4) can be derived, not just proven, using
the properties of the Leonard pair. We will make use
of the fact that S has N + 1 distinct eigenvalues and P
diagonalizes S, so the columns of P are eigenvectors

of S. If we find an inner product such that S is a
matrix of a self-adjoint operator with respect to this
product, then P will be orthogonal with respect to
this product.
We will try to find a diagonal matrix A =

diag(a0, . . . ,aN ), such that A−1SA is a Hermitian
matrix, and so it is a matrix of the self-adjoint oper-
ator in the case of a standard inner product. After
a change of basis we can see that S is a self-adjoint
operator in the case of the inner product defined by
the matrix G = A∗A = diag(w0, . . . ,wn). Thus, we
require

[A−1SA]j−1,j = [A−1SA]j,j−1,

which leads to the condition

|aj|2 = |aj−1|2
S̄j,j−1
Sj−1,j

= |aj−1|2
N − j + 1

j
.

This request is fulfilled if we choose

wj = |aj|2 =
j∏

k=1

N −k + 1
k

=
(
N

j

)
.

3. Algebra U ′q(so3)
Now let us consider from the same point of view
the algebra U′q(so3), a complex associative algebra
generated by three elements I1, I2 and I3 satisfying
the relations

q1/2I1I2 −q−1/2I2I1 = I3, (5)
q1/2I2I3 −q−1/2I3I2 = I1, (6)
q1/2I3I1 −q−1/2I1I3 = I2. (7)

Let us assume that q is not the root of unity and
define matrices ϕ(I1), ϕ(I2), ϕ(I3) by

[ϕ(I1)]j+1,j =
[2M − j]

q−M+j + qM−j
,

[ϕ(I1)]j−1,j = −
[j]

q−M+j + qM−j
,

[ϕ(I1)]jk = 0 for k 6= j ± 1,
[ϕ(I3)]jk = i[−M + j]δjk,

where M = N/2 and [ν] = (qν−q−ν)/(q−q−1). (The
matrix ϕ(I2) can be obtained from the third defining
relation (7).) Then the triple form is an irreducible so
called classical representation of U′q(so3) of dimension
N + 1.

The matrices ϕ(I1) and ϕ(I3) have the same eigen-
values, which follows from the classification of all irre-
ducible representations (there is one classical represen-
tation per dimension, see [9]) and from the existence
of a rotational automorphism which sends

I1 → I2, I2 → I3, I3 → I1.

Thus, we can construct matrix P such that

ϕ(I1) = Pϕ(I3)P−1.

395



Daniel Gromada, Severin Pošta Acta Polytechnica

We will show that matrix P corresponds to q-Racah
polynomials. The general q-Racah polynomials are
defined by means of hypergeometric series as

Rn
(
µ(x); α,β,γ,δ | q

)
= 4ϕ3

(
q−n,αβqn+1,q−x,γδqx+1

αq,βδq,γq

∣∣∣∣ q; q
)

=
∞∑
k=0

[q−n]k[αβqn+1]k[q−x]k[γδqx+1]k
[αq]k[βδq]k[γq]k

qk

[q]k
, (8)

where Rn(x; α,β,γ,δ | q) is n-th q-Racah polynomial
with parameters α, β, γ, δ, and with n = 0, 1, . . . ,N,
where N is a nonnegative integer,

µ(x) = q−x + γδqx+1,

[a]k =
k−1∏
j=0

(1 −aqj), [a]0 = 1.

The parameters must satisfy

αq = q−N or βδq = q−N or γq = q−N.

In the definition of basic hypergeometric orthogonal
polynomials it is usually assumed that q ∈ (0, 1).
However, in this calculation it is sufficient to assume
q ∈ R\{−1, 0, 1}.
The correspondence has the following form

−ijRj
(
µ(k); α,β,γ,δ | q

)
= [P−1]kj = w−1j P̄jk, (9)

where i is an imaginary unit. The weight sequence
and parameters are

wj =
[q−N ]j[−q−N ]j

[q]j[−q]j
1 + q−N+2j

(−q−N )j(1 + q−N )
,

α = β = −γ = −δ = iq
−N −1

2 . (10)

From now, we will again omit the parameters of
the polynomials and write only Rn(µ(x)) instead of
Rn(µ(x); α,β,γ,δ | q).
In order to prove (9), we construct the recurrence

relation and compare it to the general form of the
recurrence relation for q-Racah polynomials. The
equation P−1ϕ(I1) = ϕ(I3)P−1 gives us the relation

− (q−N+k −q−k)Rj
(
µ(k)

)
=
−q−N (1 −q2n)

1 + q−N+2n
Rj−1

(
µ(k)

)
+

1 −q−2N+2n

1 + q−N+2n
Rj+1

(
µ(k)

)
. (11)

We can see that this form corresponds to the general
recurrence (see [10])

− (1 −q−x)(1 −γδqx+1)Rn
(
µ(x)

)
= AnRn+1

(
µ(x)

)
− (An + Cn)Rn

(
µ(x)

)
+ CnRn−1

(
µ(x)

)
, (12)

where

An =
(1 −αqn+1)(1 −αβqn+1)

1 −αβq2n+1
(1 −βδqn+1)(1 −γqn+1)

1 −αβq2n+2
,

Cn =
q(1 −qn)(1 −βqn)(γ −αβqn)(δ −αqn)

(1 −αβq2n)(1 −αβq2n+1)
,

if the parameters are set up the way as in (10).
The way of deriving the weight sequence is similar

to the former case. We again try to find a diagonal
matrix A. However, there is no diagonal matrix that
transforms ϕ(I1) to a Hermitian matrix. Nevertheless,
we can transform ϕ(I1) to a symmetric matrix and
then show that the transformed matrix is normal. The
elements of A have to satisfy

a2j = a
2
j−1

[ϕ(I1)]j,j−1
[ϕ(I1)]j−1,j

= a2j−1
−(q2M−j+1 −q−2M+j−1)(q−M+j + qM−j)

(qj −q−j)(q−M+j−1 + qM−j+1)
.

The elements are determined up to a multiplicative
constant as a product

a2j =
j∏

k=1
−

(q2M−k+1 −q−2M+k−1)(q−M+k + qM−k)
(qk −q−k)(q−M+k−1 + qM−k+1)

=
j∏

k=1
q2M

(1 −q−4M+2k−2)(1 + q−2M+2k)
(1 −q2k)(1 + q−2M+2k−2)

=
j∏

k=1

(1 −q−N+k−1)(1 + q−N+k−1)(1 + q−N+2k)
q−N (1 −qk)(1 + qk)(1 + q−N+2k−2)

=
[q−N ]j[−q−N ]j

[q]j[−q]j
1 + q−N+2j

(q−N )j(1 + q−N )
.

If we assume q ∈ (0, 1) then for all k ≥ 1 the factor
1−q−N+k−1 is negative whereas the other factors are
positive. Therefore, |aj|2 = (−1)ja2j. It can be easily
seen that this holds for all q ∈ R\{−1, 0, 1} by similar
reasoning. Finally, we have |aj|2 = wj.
Now we just need to verify that B := A−1ϕ(I1)A

is normal using the fact that B is symmetric. Thus,
we have to verify

∑
BjlB̄kl =

∑
B̄jlBkl. We can just

show that for all indices j, k, l we have

BjlB̄kl = a−1j [ϕ(I1)]jlalā
−1
k [ϕ(I1)]klāl ∈ R.

Since ϕ(I1) is real and alāl = |al|2, we just have
to decide whether ajak is real for indices j, k, whose
difference is even (otherwise [ϕ(I1)]jl or [ϕ(I1)]kl is
zero due to its special form). Considering a2j is real
and alternates, we see that aj and ak are both real or
purely imaginary. Therefore, ajak ∈ R.

4. Conclusion
On the example of the algebra U′q(so3) we have shown
that the existence of a complete classification of rep-
resentations together with a suitable use of some au-
tomorphism can produce as a by-product a set of

396



vol. 54 no. 6/2014 Automorphisms of algebras and orthogonal polynomials

orthogonal polynomials (see also [11]). Because the
algebra U′q(so3) is a special case of Askey-Wilson al-
gebra AW(3), introduced by Zhedanov [12], it would
be nice to generalize this approach to this case (see
also [13–16]).

References
[1] W. Miller, Jr. Lie theory and difference equations. I. J
Math Anal Appl 28:383–399, 1969.

[2] T. H. Koornwinder. Krawtchouk polynomials, a
unification of two different group theoretic
interpretations. SIAM J Math Anal 13(6):1011–1023,
1982. doi:10.1137/0513072.

[3] P. Feinsilver. Lie algebras and recurrence relations. I.
Acta Appl Math 13(3):291–333, 1988.

[4] Y. I. Granovskĭı, A. S. Zhedanov. Orthogonal
polynomials on Lie algebras. Izv Vyssh Uchebn Zaved
Fiz 29(5):60–66, 1986.

[5] K. Nomura, P. Terwilliger. Krawtchouk polynomials,
the Lie algebra sl2, and Leonard pairs. Linear Algebra
Appl 437(1):345–375, 2012.
doi:10.1016/j.laa.2012.02.006.

[6] A. M. Gavrilik, N. Z. Iorgov. q-deformed algebras
Uq (son) and their representations. Methods Funct Anal
Topology 3(4):51–63, 1997.

[7] A. M. Gavrilik, N. Z. Iorgov, A. U. Klimyk.
Nonstandard deformation U ′q (son): the embedding
U ′q (son) ⊂ Uq (sln) and representations. In Symmetries
in science, X (Bregenz, 1997), pp. 121–133. Plenum,
New York, 1998.

[8] A. M. Gavrilik, N. Z. Iorgov. Representations of the
nonstandard algebras Uq (so(n)) and Uq (so(n − 1, 1)) in
Gel′ fand-Tsetlin basis. Ukraïn F̄ız Zh 43(6-7):791–797,
1998. International Symposium on Mathematical and
Theoretical Physics (Kyiv, 1997).

[9] M. Havlíček, S. Pošta. On the classification of
irreducible finite-dimensional representations of U ′q (so3)
algebra. J Math Phys 42(1):472–500, 2001.
doi:10.1063/1.1328078.

[10] R. Koekoek, R. F. Swarttouw. The askey-scheme of
hypergeometric orthogonal polynomials and its
q-analogue, 1996. arXiv:math/9602214.

[11] P. Terwilliger. Leonard pairs and the q-Racah
polynomials. Linear Algebra Appl 387:235–276, 2004.
doi:10.1016/j.laa.2004.02.014.

[12] A. S. Zhedanov. “Hidden symmetry” of
Askey-Wilson polynomials. Teoret Mat Fiz
89(2):190–204, 1991. doi:10.1007/BF01015906.

[13] P. Terwilliger. The universal Askey-Wilson algebra.
SIGMA Symmetry Integrability Geom Methods Appl
7:Paper 069, 24, 2011. doi:10.3842/SIGMA.2011.069.

[14] P. Terwilliger. The universal Askey-Wilson algebra
and the equitable presentation of Uq (sl2). SIGMA
Symmetry Integrability Geom Methods Appl 7:Paper
099, 26, 2011.

[15] P. Terwilliger, A. Žitnik. Distance-regular graphs of
q-Racah type and the universal Askey–Wilson algebra.
J Combin Theory Ser A 125:98–112, 2014.
doi:10.1016/j.jcta.2014.03.001.

[16] P. Terwilliger. The universal Askey-Wilson algebra
and DAHA of type (C ∨1 , C1). SIGMA Symmetry
Integrability Geom Methods Appl 9:Paper 047, 40, 2013.

397

http://dx.doi.org/10.1137/0513072
http://dx.doi.org/10.1016/j.laa.2012.02.006
http://dx.doi.org/10.1063/1.1328078
arXiv:math/9602214
http://dx.doi.org/10.1016/j.laa.2004.02.014
http://dx.doi.org/10.1007/BF01015906
http://dx.doi.org/10.3842/SIGMA.2011.069
http://dx.doi.org/10.1016/j.jcta.2014.03.001

	Acta Polytechnica 54(6):394–397, 2014
	1 Introduction
	2 Lie algebra sl2
	3 Algebra Uq'(so3)
	4 Conclusion
	References