ap-5-10.dvi


Acta Polytechnica Vol. 50 No. 5/2010

The Diamond Lemma and the PBW Property in Quantum Algebras

M. Havlíček, S. Pošta

Abstract

The use of the diamond lemma for proving various facts about the center of the algebra U ′q(so3) is demonstrated. The
approach presented here is successful in other cases of quantum algebras and superalgebras.

Keywords: quantum algebra, PBW property, center.

1 Introduction
A fundamental issue when examining quantum
groups or similar structures is to explore and clas-
sify all representations.
Usually one encounters two main cases which are

of differentdifficulty. When the deformationparame-
ter, in quantumgroups typically denoted by q, is not
a root of unity, i. e. when qn 
= 1 for all integers n,
the representationtheory is “the same”as in the clas-
sical case, that is in the case of enveloping algebras
of classical Lie algebras. When q is a root of unity,
it is not an easy task to classify finite dimensional
representations even in the lowest dimensions.
Various preparatory steps must be taken before

one can investigate representations of the consid-
ered algebra. Important supportingknowledgewhich
helps inmaking such a classification is detailed infor-
mation about the center of the considered algebra,
because irreducible representation operators which
belong to the center of the algebra (Casimir opera-
tors) are represented by a scalar multiple of the unit
operator.
TheHarish-Chandra homomorphism (1951) is an

isomorphism which maps the center Z(U(g)) of the
universal enveloping algebra U(g) of a semisimpleLie
algebra g to the elements S(h)W of the symmetric
algebra S(h) of a Cartan subalgebra h ⊂ g that are
invariant under the corresponding Weyl group W .
Let r be the rank of g, which is the dimension

of the Cartan subalgebra h. Coxeter observed that
S(h)W is a polynomial algebra in r variables. There-
fore, the center of the universal enveloping algebra
of a semisimple Lie algebra is a free polynomial ring.
Any Casimir operator is an arbitrary polynomial in
basic algebra invariants. The number and degrees of
these fundamental invariants are shown in Table 1.
In the case of standard Drinfeld-Jimbo quantum

groups,wehaveananalogyto thenon-deformedcase.
When deformation parameter q is not a root of unity,
a modified version of the Harish-Chandra homomor-
phism exists and the center is again isomorphic to
the ring of polynomials of the fundamental invariants
(see [1]).

Table 1: Degrees of the fundamental invariants

Ar 2,3,4, ..., n +1

Br 2,4,6, ...,2n

Cr 2,4,6, ...,2n

Dr n;2,4,6, ...,2n − 2
E6 2,5,6,8,9,12

E7 2,6,8,10,12,14,18

E8 2,8,12,14,18,20,24,30

F4 2,6,8,12

G2 2,6

For example, in the simplest case of the algebra
Uq(sl2), generated by four letters (generators) de-
noted by E, F , K, K−1 and relations

KK−1 = K−1K =1,

KEK−1 = q2E,

KF K−1 = q−2F,

[E, F ] ≡ EF − F E = [K]q ≡
K − K−1
q − q−1

,

the center is generated by the Casimir element

Cq = EF +
Kq−1 + K−1q
(q − q−1)2

(1)

(for the standard proof of this fact see [2], theo-
rem 45).
When q is a primitive root of unity, say qn = 1,

n > 1, qm 
=1 for m < n, the situation is muchmore
difficult. The center is typically much larger and the
central elements satisfy nontrivial polynomial rela-
tions [3]. In the case of Uq(sl2), there are four more
additional elements in the center, namely

Ep, F p, Kp, K−p,

where p = n if n is odd and p =
n

2
if n is even. These

five elements (together with (1)) are no longer alge-
braically independent. One can show by induction

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Acta Polytechnica Vol. 50 No. 5/2010

that

p−1∏
j=0

(Cq −(q − q−1)−2(Kqj+1+K−1q−j−1))= EpF p,

which implies

Cpq + γ1C
p−1
q + . . . + γp−1Cq +

(−1)p(q − q−1)−2p(Kp − K−p)= EpF p,

where γi are certain complex coefficients.
Quantum groups are not the only kind of quan-

tum deformations. There exist also other, non-
standard deformations. For example, q-deformation
U ′q(so3) of the universal enveloping algebra U(so3),
which does not coincide with the Drinfeld-Jimbo
quantum algebra Uq(so3) is constructed without us-
ing the Cartan subalgebra and roots by deforming
Serre-type relations directly. We substitute simply
2 → [2]q = q + q−1 in cubic defining relations of
U(so3). As a result we obtain a complex associative
algebrawithunity generatedby elements I21, I32 sat-
isfying the relations

I221I32 − (q + q
−1)I21I32I21 + I32I

2
21 = −I32,

I21I
2
32 − (q + q

−1)I32I21I32 + I
2
32I21 = −I21.

It can be shown that this is isomorphic to an algebra
generated by three generators I1, I2, I3 and relations
[5]

q
1
2 I1I2 − q−

1
2 I2I1 = I3,

q
1
2 I2I3 − q−

1
2 I3I2 = I1, (2)

q
1
2 I3I1 − q−

1
2 I1I3 = I2.

One can quickly explore the following Casimir ele-
ment, which belongs to the center of this algebra:

C = q2I21 + I
2
2 + q

2I23 − (q
5
2 − q

1
2)I1I2I3. (3)

Similarly as in the case of ordinary Hopf quantum
groups, one can expect that when q is not a root of
unity, this element generates the center of the alge-
bra U ′q(so3). However, there is no analogy of Harish-
Chandra homomorphismhere so onemust prove this
fact by other methods. As is shown below, these
methods areuseful even in themore complicated case
when q is a root of unity.

2 The diamond lemma
In 1978, M. Bergmann recalled a rather deep and
forgotten result of Newman from graph theory, often
called the diamond lemma. He showed its usefulness
for other fields of mathematics also, namely for the
theory of associative algebras. The original Newman
formulation was as follows (see [6]). Let G be an
oriented graph. Now suppose that

1)The orientedgraph G has the descending chain
condition. That is, all positively oriented paths in G
terminate, in other words, there are no circles in the
graph.
2) Whenever two edges, e and e′, proceed from

one vertex a of G, there exist positively oriented
paths p, p′ in G leading from the endpoints b, b′ of
these edges to a common vertex c. (This is often
called the “confluence” or “diamond” condition, see
fig. 1.)

Fig. 1: Diamond condition

Then one can show that every connected compo-
nent C of G has a unique minimal vertex mC. This
means that every maximal positively oriented path
beginning at a point of C will terminate at mC.
Let us now describe the version of the diamond

lemma in the theoryof associativealgebras. Let R be
an associative algebrawith unity over complex num-
bers, given by the finite set of generators X and the
set of relations

S = {Wσ = fσ|σ ∈ Σ},

where Wσ is monomial (the product of a finite num-
ber of generators from X) and fσ is a complex linear
combination of monomials. Let us have partial or-
dering < defined on monomials which satisfies three
conditions: it is invariantwith respect to multiplica-
tion, i. e. for each monomial A, B, B′, C we have

B < B′ =⇒ ABC < AB′C,

it is S-compatible (i. e. fσ is a linear combination
of elements being less than Wσ for each σ ∈ Σ)
and fulfils DCC (the descending chain condition,
which is the nonexistence of an infinite sequence
x1 > x2 > ...). Furthermore, let all monomialswhich
can be written as product ABC, where AB = Wσ
and BC = Wτ, B 
= 1, σ, τ ∈ Σ or in the form
ABC = Wσ and B = Wτ, σ 
= τ (these monomi-
als are called ambiguities) be reduced by the rela-
tions in S to a common value. Then the diamond
lemma states that all irreducible (i. e. completely
reduced, towhich one cannotapply any relation from
S) monomials form a basis of algebra R.

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Acta Polytechnica Vol. 50 No. 5/2010

The diamond lemma can be effectively used for
decidingvariouskinds ofproblems. The typicalprob-
lem, as mentioned in [6], is as follows. Let us have
an algebra with generators a, b, c and the relations

a2 = a, b2 = b, c2 = c, (4)

(a + b + c)2 = a + b + c. (5)

Nowtheproblem is to answer the questionwhether it
follows from these relations that ab =0. The second
relation (5) can be rewritten as

cb = −ab − ba − ac − ca − bc. (6)

Now we test whether (4), (5) and (6) imply a unique
canonical form of the elements of the considered al-
gebra. We must examine the following ambiguities:

a3, b3, c3, cb2, c2b.

The first three are trivial to reduce, and the fourth,
reduced in two possible ways, gives us the following:

c(bb)= cb = −ab − ba − ac − ca − bc

and

(cb)b = (−ab − ba − ac − ca − bc)b =
−ab2 − bab − acb − cab − bcb =
−ab − bab − a(−ab − ba − ac − ca − bc) −
cab − b(−ab − ba − ac − ca − bc)=
−ab − bab + a2b + aba + a2c +
aca + abc − cab + bab +
b2a + bac + bca + b2c =

−ab − bab + ab + aba + ac + aca +
abc − cab + bab + ba + bac + bca + bc =
aba + ac + aca + abc − cab +
ba + bac + bca + bc.

So we have

−ab − ba − ac − ca − bc =
aba + ac + aca + abc − cab + ba + bac + bca + bc.

This equality can be rewritten as

cab = aba +2ac + aca + abc + (7)

2ba + bac + bca +2bc + ab + ca.

Reducing the fifth ambiguity

c(cb)= (cc)b

leads to the same relation. Now what happens if we
add (7) to the list of relations? The ambiguities cb2

and c2b nowreduceautomatically toa commonvalue.
But two new ambiguities of higher degree arise:

c2ab, cab2.

We therefore test again: we have

(cc)ab = cab = aba +2ac + aca + abc +2ba +

bac + bca +2bc + ab + ca,

and, after some computation,

c(cab)= . . . = aba +2ac + aca + abc +2ba +

bac + bca +2bc + ab + ca.

The second ambiguity reduces to a common value,
too. The basis of the considered algebra consists of
all words consisting of letters a, b, c not containing
substrings a2, b2, c2, cab and cb. Therefore the word
ab is irreducible, hence nonzero.

3 PBW property

One of simple consequences of the diamond lemma
is the Poincaré-Birkhoff-Witt property of universal
enveloping algebras and its analogy in the case of
quantum deformations.
As a simple example, let us have an enveloping

algebra U(sl2) of Lie algebra sl2 which is given by
three generators E, F , H satisfying the relations

[E, F ] = EF − F E = H,
[H, E] = 2E, (8)

[H, F ] = −2F.

Wedefine the total ordering ≺ of the generators as it
is in the alphabet, i. e. E ≺ F ≺ H. Partial ordering
amongmonomials is defined in suchwaythat X < Y ,
when the length of X (the number of letters in the
product X) is less than the length of Y , orwhen X is
a permutation of the letters from Y , but has a lower
number of inverses (the monomial X = x1 . . . xs has
inverse (i, j), 1 ≤ i, j ≤ s, when i < j and xi � xj).
One can easily see that this partial ordering is com-
patiblewith relations (8), whichwepresent in amore
suitable form of “rewriting rules”:

F E → EF − H,
HE → EH +2E, (9)
HF → F H − 2F.

One can also easily check that the ordering fulfils
DCC. Simple computation gives us

(HF)E = (F H − 2F)E = F HE − 2F E =
F(EH +2E) − 2F E =
F EH =(EF − H)H = EF H − H2,

H(F E) = H(EF − H)= HEF − H2 =
(EH +2E)F − H2 =
EHF +2EF − H2 =
E(F H − 2F)+2EF − H2 =
EF H − 2EF +2EF − H2 =
EF H − H2.

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Acta Polytechnica Vol. 50 No. 5/2010

We see that the ambiguityHFE is reduced to a com-
mon value. One can easily list all irreducible mono-
mials. These are precisely

EαF β Hγ , α, β, γ ≥ 0.

The diamond lemma states that these monomials
form the basis of the algebra U(sl2) (as stated by
the well known PBW theorem).

4 Center of U(sl2)

As was said in the introduction, there is a standard
way to explore the structure of the center of the alge-
bra U(sl2). Let us find this structure now using the
diammond lemma. This procedure can then be gen-
eralized to other algebras where standard tools like
the Harish-Chandra homomorphism cannot be used.
Theproblem is tofindall elements X from U(sl2),

for which we have

[X, A] ≡ XA − AX =0 for all A ∈ U(sl2).

This condition is clearly equivalent to

[X, E] = [X, F ] = [X, H] = 0, (10)

that is, one can restrict oneself to making commu-
tations with generators only. Let us take a general
element of the form

X =
n∑

i,j,k=0

αi,j,kE
iF j Hk

for some small values of n, let us generally compute
commutators [X, E], [X, F ] a [X, H] and solve a sys-
tem of linear equations (10) for coefficients αi,j,k.
For n = 1 we get nothing (trivial zero solution

only).
For n =2 we get any scalar multiple of

X = H2 +4EF − 2H ≡ C.

For higher n we get elements of the form

αC + βC2,

then
αC + βC2 + γC3,

etc., where α, β, γ, . . . are arbitrary complex coeffi-
cients.
Of course this leads to the hypothesis that any

element X which commutes with E, F and H is of
the form p(C), where p is an arbitrary complex poly-
nomial.
The proof is based on the change of the original

basis {EαF β Hγ }. We add to generators E, F and

H another letter C and to the rewriting rules (9) we
add the following:

EF →
1
4
(C +2H − H2),

EC → CE,
F C → CF,
HC → CH.

The basis now consists of irreducible elements, i. e.

{CjEkHm|j, k, m ≥ 0} ∪ {Cj F lHm|j, l, m ≥ 0}.

We now take the general element as a linear combi-
nation

X =
n∑

j,k,m=0

βj,k,mC
j EkHm +

n∑
j,l,m=0

γj,l,mC
j F lHm.

When computing commutators [X, E], [X, F ] and
[X, H] one canmake use of the fact that, for example

[Cj EkHm, E] = Cj[EkHm, E] = Cj Ek[Hm, E]

etc. It is also clear (as opposed to the original case)
what it is sufficient to show: one must show that co-
efficients βj,k,m = 0 for (k, m) 
= (0,0), similarly for
γ’s. This can be seen from (10).

5 Center of U ′q(so3)

Letus apply the process introducedabove to the case
of the nonstandarddeformation U ′q(so3). First, let us
assume qn 
=1 for all n.
Using the first relation (2) as a definition for I3

and substituting into the second and third relations
we get two cubic relations

I2I
2
1 − (q + q

−1)I1I2I1 + I
2
1I2 = −I2,

I22I1 − (q + q
−1)I2I1I2 + I1I

2
2 = −I1.

It can be shown that U ′q(so3) is isomorphic to the al-
gebra with the generators I1, I2 satisfying the two
above relations. The casimir element (3) can be
rewritten to the form (we ommit scalar factor q)

C =(q+q−1)(I21+I
2
1I
2
2+I

2
2)+I2I1I2I1−[3]qI1I2I1I2.

We now construct a different basis of the algebra
U ′q(so3). We deal with the following rewriting rules:

I2I
2
1 → (q + q

−1)I1I2I1 − I21I2 − I2,
I22I1 → (q + q

−1)I2I1I2 − I1I22 − I1,
I2I1I2I1 → C +[3]qI1I2I1I2 −

(q + q−1)(I21 + I
2
1I
2
2 + I

2
2),

I2C → CI2,
I1C → CI1.

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Acta Polytechnica Vol. 50 No. 5/2010

It can easily be seen that precisely the elements

Cγ Iα1 (I2I1)
kI

β
2 , γ, α, β ≥ 0, k ∈ {0,1}

are irreducible, thus forming a new linear basis of the
algebra.
Now it is sufficient to take arbitrary element X of

the form

X =
∑

α,β,k

pα,β,k(C)I
α
1 (I2I1)

kI
β
2

where pα,β,k are arbitrary complex polynomials of
one variable, commute X with I1, I2 and by virtue
of its equality to zero to show that all polynomials
pα,β,k =0 with the only exception p0,0,0.
When q is a primitive root of unity, qn = 1, the

center of the algebra does not only consist of the or-
dinary Casimir element (3) but there are three more
elements having the form

Cn1 =

[n−12 ]∑
j=0

(
n − j

j

)
n

n − j
( i

q − q−1
)2j

I
n−2j
1

(and the same polynomial in I2 and I3 denoted by
Cn2, Cn3). It is not an easy task to show that these
elements really belong to the center of the algebra
(see [4]). After some transformation one can see that
Cnj, j =1,2,3 are actually Chebyshev polynomials.
It turns out that these four Casimir elements are

no longer polynomial independent. When one wants
to prove this fact it may come in handy to have ex-
plicit polynomial dependence for small values of n.
However, it turns out that it is practically impossi-
ble to obtain the relation between Casimir elements
by “brute force”, even for the simplest cases. For
example, for n =3 one can show that

C3 − qC2 − C231 − C
2
32− (11)

C233 +3(q + q
1
2)C31C32C33 = 0.

Note that C3 is of degree nine, so it is quite compli-
cated even to prove the given explicit relation.
The diamond lemma can be quite useful for ob-

taining relations of type (11) directly. We must only
construct a suitable set of rewriting rules and with
the help of it new advantageous basis of the algebra.
First we start with ordinary rewriting rules com-

ing from the commutation relations, namely

I2I1 → qI1I2 − q
1
2 I3,

I3I2 → qI2I3 − q
1
2 I1,

I3I1 → q−1I1I3 + q−
1
2 I2,

I1C → CI1,
I2C → CI2,
I3C → CI3.

The powers of the generators can be reduced using
Casimir elements Cnk, thereforewe add the relations

Ink → Cnk −
[n−12 ]∑
j=1

(
n − j

j

)
n

n − j
( i

q − q−1
)2j

I
n−2j
k ,

k =1,2,3.

Next we must ensure that Casimir element C does
not appear too many times. This is ensured by the
relation

ACB → A(q2I21 + I
2
2 + q

2I23 − (q
5
2 − q

1
2)I1I2I3)B,

where A, B are arbitrary monomials, and it comes
into play only if the count of letters Ik (k = 1,2,3)
and C in monomial ACB is greater or equal to n.
The last rule (or, better to say, set of rules)whichwe
do not present explicitly transforms

Cj W → Cj+1W̃ ,

where W is any product of generators Ik, k = 1,2,3
with the property that every generator I1, I2, I3 is
present in the product. Using the commutation re-
lations, W is transformed to the form I1I2I3W1 and
the product I1I2I3 is then converted to C using the
rule

I1I2I3 → (q
5
2 − q

1
2)−1(q2I21 + I

2
2 + q

2I23 − C).

These transformation rules now lead to the basis con-
taining the following elements:

Cαn1C
β
n2C

γ
n3C

δIk1 I
l
2I

m
3 ,

α, β, γ ≥ 0,0 ≤ δ ≤ n − 1,
0 ≤ k, l, m ≤ n − 1 − δ, klm =0.

If we want to find the explicit relation between
Casimir elements, we simply express Cn in the ba-
sis specified (using a finite number of rewriting rules
reducing Cn to canonical form).
In general, one can show that for odd n the ele-

ments C, Cn1, Cn2, Cn3 fulfil the relation

C2n1 + C
2
n2 + C

2
n3 +(q − q

−1)nCn1Cn2Cn3 =
n−1∏
k=0

(C + q [k]q [k +1]q).

The relation for even n as well as the proof of this
can be found in [7].

Acknowledgement

We acknowledge financial support from grant
201/10/1509 of the Czech Science Foundation and
MSM6840770039 of the Ministry of Education,
Youth, and Sports of the Czech Republic.

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Acta Polytechnica Vol. 50 No. 5/2010

References

[1] Lusztig, G: Introduction to Quantum Groups,
Birkhäuser, Boston, 1993.

[2] Klimyk, A. U., Schmüdgen, K.: Quantum groups
and their representations, Springer, Berlin, 1997.

[3] Tange, R.: The centre of quantum sln at root of
unity, J. Algebra 301 (2006) 425–445.

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

[5] Havlíček, M., Klimyk, A. U., Pošta, S.: Repre-
sentations of the cyclically symmetric q-deformed

algebra soq(3), J. Math. Phys. 40 (1999),
2135–2161.

[6] Bergmann, G. M.: Diamond Lemma for ring
theory, Adv. Math. 29 (1978), 178–218.

[7] Havlíček, M., Pošta, S.: Center of algebra
U ′q(so3), submitted to J. Math. Phys.

Prof. Ing. Miloslav Havlíček, DrSc.
E-mail: miloslav.havlicek@fjfi.cvut.cz
doc. Ing. Severin Pošta, Ph.D.
E-mail: severin.posta@fjfi.cvut.cz
Department of Mathematics
FNSPE, Czech Technical University in Prague
Trojanova 13, CZ-120 00 Prague

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