CUBO A Mathematical Journal

Vol.10, N
o
¯ 03, (171–194). October 2008

Rosso-Yamane Theorem on PBW Basis of Uq(AN )
∗

Yuqun Chen, Hongshan Shao

School of Mathematical Sciences, South China Normal University,

Guangzhou 510631, P. R. China

emails: yqchen@scnu.edu.cn, shaohongshan118@163.com

and

K.P. Shum

Department of Mathematics, The University of Hong Kong,

Pokfulam Road, Hong Kong, China (SAR)

email: kpshum@maths.hku.hk

ABSTRACT

Let Uq(AN ) be the Drinfeld-Jimbo quantum group of type AN . In this paper, by using

Gröbner-Shirshov bases, we give a simple (but not short) proof of the Rosso-Yamane

Theorem on PBW basis of Uq(AN ).

RESUMEN

Sea Uq(AN ) el grupo cuántico de Drinfel-Jimbo de tipo AN . En este art́ıculo, us-

ando bases de Gröbner-Shirshov damos una demonstración simple (pero no corta) del

Teorema de Rosso-Yamane sobre bases PBW de Uq(AN ).

Key words and phrases: Quantum group, Quantum enveloping algebra, Gröbner-Shirshov basis.

Math. Subj. Class.: 20G42, 16S15, 13P10.

∗Supported by the NNSF of China (No.10771077) and the NSF of Guangdong Province (No.06025062).



172 Yuqun Chen et al. CUBO
10, 3 (2008)

1 Introduction

Since any algebra (commutative, associative, Lie), as well as any module over an algebra, can be

presented by generators and defining relations, it is important to have a general method to deal

with these presentations. Such a method now exists and is called the Gröbner bases method (due

to B. Buchberger [18], [19]), or standand bases method (due to H. Hironaka [21]), or Gröbner-

Shirshov bases method (due to A. I. Shirshov [35]). The original Shirshov’s paper [35] is for Lie

algebra presentations, but it can be easily adopted for associative algebra presentations as well,

see L. A. Bokut [3] and G. Bergman [1].

Let, for example, L = Lie(X|[xixj ] −
∑

αkij xk, i > j, xi, xj , xk ∈ X) be a Lie algebra

over a field (or a commutative ring) k presented by a k-basis X and the multiplication table.

Then S = {[xixj ] −
∑

αkij xk| i > j, xi, xj , xk ∈ X} is a Gröbner-Shirshov basis (subset) of the

free Lie algebra Lie(X) over k. On the other hand, the universal enveloping algebra U (L) =

k〈X|xixj − xjxi −
∑

αkij xk, i > j, xi, xj , xk ∈ X〉 is the associative algebra presented by the same

set X and the defining relations S(−) (we rewrite S using [xy] = xy − yx). There is a general

but not difficult result that for any S ⊂ Lie(X), S is a Gröbner-Shirshov basis in the sense of

Lie algebras if and only if S(−) ⊂ k〈X〉 is a Gröbner-Shirshov basis in the sense of associative

algebras (see, for example, [9] and [7]). This means that in our case, S(−) is a Gröbner-Shirshov

basis (subset) in k〈X〉. By Composition-Diamond lemma (see below), the S-irreducible words on

X, Irr(S) = {xi1 . . . xik , i1 ≤ . . . ≤ ik, k ≥ 0} form a k-basis of U (L). This is a conceptional

proof of the PBW-Theorem by using Gröbner-Shirshov bases theory.

There are many results on Gröbner-Shirshov bases for associative and Lie algebras, as well as

for semigroups, groups, conformal algebras, dialgebras, and so on, see, for example, surveys [14],

[15], [25] and [8]. Let us mention those for simple Lie algebras and Lie superalgebras via Serre’s

presentations ([10], [11], [12], [13], [9]), for modules over simple Lie algebras and Iwahori-Hecke

algebras ([23], [24], [25]), for Kac-Moody algebras of types A
(1)

n , B
(1)

n , C
(1)

n , D
(1)

n ([31], [32], [33]),

for Coxeter groups ([17]), for braid groups via Artin-Burau, Artin-Garside and Briman-Ko-Lee

presentations ([4], [5] and [6]).

Drinfeld-Jimbo ([20], [22]) presentations for quantized enveloping algebras Uq(g), where g is

a semisimple Lie algebra, are a natural source of associative presentations. M. Rosso [34] and I.

Yamane [36] found the PBW-basis of Uq(AN ). G. Lusztig [29] and [30], and M. Kashiwara [26]

and [27] found the bases of Uq(g) for any semisimple algebra g, as well as for their representations.

Their approach work equally well for quantized enveloping algebras associated with arbitrary sym-

metrizable Cartan matrix, not just those corresponding to finite dimensional Lie algebras. V. K.

Kharchenko [28] found the approach to linear bases of quantized enveloping algebras via the so

called character Hopf algebras.

It the paper [16], Gröbner-Shirshov bases approach was applied to study Uq(g) for any sym-

metrizable Cartan matrix. Using this approach, they got a new proof of the triangular decomposi-



CUBO
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Rosso-Yamane Theorem on PBW Basis of Uq(AN ) 173

tion of Uq(g) (see, for example, Jantzen [37]). For Uq(AN ), it was proved by Bokut and Malcolmson

[16] that the Jimbo relations (see [36]) of U +q (AN ) constitute a Gröbner-Shirshov basis of U
+

q (AN )

in Jimbo generators xij , 1 ≤ i, j ≤ N + 1 (see below).

In this paper, we give an elementary proof that Jimbo relations S is a Gröbner-Shirshov basis

of U +q (AN ). For such a purpose, we just check all possible compositions of polynomials from S and

proved that all them can be resolved. Also in §1 in this paper, we are giving necessary definitions

and Composition-Diamond lemma following Shirshov [35].

2 Preliminaries

We first cite some concepts and results from the literature which are related to the Gröbner-

Shirshov bases for associative algebras.

Let k be a field, k〈X〉 the free associative algebra over k generated by X and X∗ the free

monoid generated by X, where the empty word is the identity which is denoted by 1. For a word

w ∈ X∗, we denote the length of w by deg(w). Let X∗ be a well ordered set. Let f ∈ k〈X〉 with

the leading word f̄ . Then we call f monic if f̄ has coefficient 1.

Definition 2.1. ([35], see also [2], [3]) Let f and g be two monic polynomials in k〈X〉 and < a

well ordering on X∗. Then, there are two kinds of compositions:

(i) If w is a word such that w = f̄ b = aḡ for some a, b ∈ X∗ with deg(f̄ )+deg(ḡ) >deg(w),

then the polynomial (f, g)w = f b − ag is called the intersection composition of f and g with respect

to w.

(ii) If w = f̄ = aḡb for some a, b ∈ X∗, then the polynomial (f, g)w = f − agb is called the

inclusion composition of f and g with respect to w.

Definition 2.2. ([2], [3], cf. [35]) Let S ⊂ k〈X〉 such that every s ∈ S is monic. Then the

composition (f, g)w is called trivial modulo (S, w) if (f, g)w =
∑

αiaisibi, where each αi ∈ k,

ai, bi ∈ X
∗, si ∈ S and aisibi < w. If this is the case, then we write

(f, g)w ≡ 0 mod(S, w).

In general, for p, q ∈ k〈X〉, we write p ≡ q mod(S, w) which means that p − q =
∑

αiaisibi,

where each αi ∈ k, ai, bi ∈ X
∗, si ∈ S and aisibi < w.

Definition 2.3. ([2], [3], cf. [35]) We call the set S with respect to the well ordering < a

Gröbner-Shirshov set (basis) in k〈X〉 if any composition of polynomials in S is trivial modulo S.

If a subset S of k〈X〉 is not a Gröbner-Shirshov basis, then we can add to S all nontrivial

compositions of polynomials of S, and by continuing this process (maybe infinitely) many times,

we eventually obtain a Gröbner-Shirshov basis Sc. Such a process is called the Shirshov algorithm.



174 Yuqun Chen et al. CUBO
10, 3 (2008)

A well ordering > on X∗ is called a monomial order if it is compatible with the multiplication

of words, that is, for u, v ∈ X∗, we have

u > v ⇒ w1uw2 > w1vw2, f or all w1, w2 ∈ X
∗.

A standard example of monomial order on X∗ is the deg-lex order to compare two words first by

degree and then lexicographically, where X is a well ordered set.

The following lemma was first proved by Shirshov [35] for free Lie algebras (with deg-lex order)

in 1962 (see also Bokut [2]). In 1976, Bokut [3] specialized the approach of Shirshov to associative

algebras (see also Bergman [1]). For the case of commutative polynomials, this lemma is known

as the Buchberger’s Theorem in [18] and [19].

Lemma 2.4. (Composition-Diamond Lemma) Let k be a field, k〈X|S〉 = k〈X〉/Id(S) and <

a monomial order on X∗, where Id(S) is the ideal of k〈X〉 generated by S. Then the following

statements are equivalent:

(i) S is a Gröbner-Shirshov basis.

(ii) f ∈ Id(S) ⇒ f̄ = as̄b for some s ∈ S and a, b ∈ X∗.

(iii) Irr(S) = {u ∈ X∗|u 6= as̄b, s ∈ S, a, b ∈ X∗} is a basis of the algebra k〈X|S〉. �

3 Rosso-Yamane theorem on PBW basis of Uq(AN )

Let k be a field, A = (aij ) an integral symmetrizable N × N Cartan matrix so that aii = 2,

aij ≤ 0 (i 6= j) and there exists a diagonal matrix D with diagonal entries di which are nonzero

integers such that the product DA is symmetric. Let q be a nonzero element of k such that q4di 6= 1

for each i. Then the quantum enveloping algebra is (see [20], [22])

Uq(A) = k〈X ∪ H ∪ Y |S
+ ∪ K ∪ T ∪ S−〉,

where



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Rosso-Yamane Theorem on PBW Basis of Uq(AN ) 175

X = {xi},

H = {h±1
i

},

Y = {yi},

S+ = {

1−aij
∑

ν=0

(−1)ν

(

1 − aij

ν

)

t

x
1−aij −ν

i
xj x

ν
i , where i 6= j, t = q

2di},

S− = {

1−aij
∑

ν=0

(−1)ν

(

1 − aij

ν

)

t

y
1−aij −ν

i
yj y

ν
i , where i 6= j, t = q

2di},

K = {hihj − hj hi, hih
−1

i
− 1, h−1

i
hi − 1, xjh

±1

i
− q∓diaij h±1xj , h

±1

i
yj − yj h

±1},

T = {xiyj − yjxi − δij
h2i − h

−2

i

q2di − q−2di
} and

(

m

n

)

t

=







n
∏

i=1

t
m−i+1

−t
i−m−1

ti−t−i
m > n > 0,

1 n = 0 or m = n.

Let

A = AN =



















2 −1 0 · · · 0

−1 2 −1 · · · 0

0 −1 2 · · · 0

· · · · ·

0 0 0 · · · 2



















and q8 6= 1.

It is reminded that in this case, the diagonal matrix D is identity.

We introduce some new variables defined by Jimbo (see [36]) which generate Uq(AN ):

˜X = {xij , 1 ≤ i < j ≤ N + 1},

where

xij =

{

xi j = i + 1,

qxi,j−1xj−1,j − q
−1xj−1,j xi,j−1 j > i + 1.

We now order the set ˜X in the following way.

xmn > xij ⇐⇒ (m, n) >lex (i, j).



176 Yuqun Chen et al. CUBO
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Let us recall from Yamane [36] the following notation:

C1 = {((i, j), (m, n))|i = m < j < n},

C2 = {((i, j), (m, n))|i < m < n < j},

C3 = {((i, j), (m, n))|i < m < j = n},

C4 = {((i, j), (m, n))|i < m < j < n},

C5 = {((i, j), (m, n))|i < j = m < n},

C6 = {((i, j), (m, n))|i < j < m < n}.

Let the set ˜S+ consist of Jimbo relations:

xmnxij − q
−2xij xmn ((i, j), (m, n)) ∈ C1 ∪ C3,

xmnxij − xij xmn ((i, j), (m, n)) ∈ C2 ∪ C6,

xmnxij − xij xmn + (q
2 − q−2)xinxmj ((i, j), (m, n)) ∈ C4,

xmnxij − q
2xij xmn + qxin ((i, j), (m, n)) ∈ C5.

It is easily seen that U +q (AN ) = k〈
˜X|˜S+〉.

The following theorem is from [16].

Theorem 3.1. ([16] Theorem 4.1) Let the notation be as before. Then, with the deg-lex order on
˜X∗, ˜S+ is a Gröbner-Shirshov basis for k〈 ˜X|˜S+〉 = U +q (AN ).

Proof. We will prove that all compositions in ˜S+ are trivial modulo ˜S+. We consider the

following cases.

Case 1. f = xmnxij − q
−2xij xmn, g = xij xkl − q

−2xklxij , w = xmnxij xkl.

In the case, we have

(f, g)w = −q
−2

xij xmnxkl + q
−2

xmnxklxij .

There are four subcases to consider.

((i, j), (m, n)) ∈ C1 ((i, j), (m, n)) ∈ C3

((k, l), (i, j)) ∈ C1 1.1. ((k, l), (m, n)) ∈ C1 1.3. ((k, l), (m, n)) ∈ C4, C5 or C6

((k, l), (i, j)) ∈ C3 1.2. ((k, l), (m, n)) ∈ C4 1.4. ((k, l), (m, n)) ∈ C3

1.1. ((i, j), (m, n)) ∈ C1, ((k, l), (i, j)) ∈ C1 and ((k, l), (m, n)) ∈ C1.

Then, we have

(f, g)w ≡ −q
−4xij xklxmn + q

−4xklxmnxij

≡ −q−6xklxij xmn + q
−6

xklxij xmn

≡ 0.



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Rosso-Yamane Theorem on PBW Basis of Uq(AN ) 177

1.2. ((i, j), (m, n)) ∈ C1, ((k, l), (i, j)) ∈ C3 and ((k, l), (m, n)) ∈ C4.

Then, we have (i, j) = (m, l), ((k, n), (i, j)) ∈ C2 and

(f, g)w ≡ −q
−2xij [xklxmn − (q

2 − q−2)xknxml] + q
−2[xklxmn − (q

2 − q−2)xknxml]xij

≡ −q−4xklxij xmn + q
−2(q2 − q−2)xknxij xml

+q−4xklxij xmn − q
−2(q2 − q−2)xknxij xml

≡ 0.

1.3. ((i, j), (m, n)) ∈ C3, ((k, l), (i, j)) ∈ C1 and ((k, l), (m, n)) ∈ C4, C5 or C6.

1.3.1. If ((k, l), (m, n)) ∈ C4 (m < l), then (k, n) = (i, j), ((i, j), (m, l)) ∈ C2 and

(f, g)w ≡ −q
−2

xij [xklxmn − (q
2 − q−2)xknxml] + q

−2[xklxmn − (q
2 − q−2)xknxml]xij

≡ −q−4xklxij xmn + q
−2(q2 − q−2)xij xknxml + q

−4
xklxij xmn

−q−2(q2 − q−2)xknxij xml

≡ 0.

1.3.2. If ((k, l), (m, n)) ∈ C5 (m = l), then (k, n) = (i, j) and

(f, g)w ≡ −q
−2xij (q

2xklxmn − qxkn) + q
−2(q2xklxmn − qxkn)xij

≡ −xij xklxmn + q
−1xij xkn + xklxmnxij − q

−1xknxij

≡ −q−2xklxij xmn + q
−2xklxij xmn

≡ 0.

1.3.3. If ((k, l), (m, n)) ∈ C6 (m > l), then

(f, g)w ≡ −q
−2

xij xklxmn + q
−2

xklxmnxij

≡ −q−4xklxij xmn + q
−4xklxij xmn

≡ 0.

1.4. ((i, j), (m, n)) ∈ C3, ((k, l), (i, j)) ∈ C3 and ((k, l), (m, n)) ∈ C3.

This case is similar to 1.1.

Case 2. f = xmnxij − q
−2xij xmn, g = xij xkl − xklxij , w = xmnxij xkl.

In the case, we have

(f, g)w = −q
−2xij xmnxkl + xmnxklxij .

There are also four subcases to consider.

((i, j), (m, n)) ∈ C1 ((i, j), (m, n)) ∈ C3

((k, l), (i, j)) ∈ C2 2.1. ((k, l), (m, n)) ∈ C2, C3 or C4 2.3. ((k, l), (m, n)) ∈ C2

((k, l), (i, j)) ∈ C6 2.2. ((k, l), (m, n)) ∈ C6 2.4. ((k, l), (m, n)) ∈ C6



178 Yuqun Chen et al. CUBO
10, 3 (2008)

2.1. ((i, j), (m, n)) ∈ C1, ((k, l), (i, j)) ∈ C2 and ((k, l), (m, n)) ∈ C2, C3 or C4.

2.1.1. If ((k, l), (m, n)) ∈ C2 (n < l), then

(f, g)w ≡ −q
−2xij xklxmn + xklxmnxij

≡ −q−2xklxij xmn + q
−2xklxij xmn

≡ 0.

2.1.2. If ((k, l), (m, n)) ∈ C3 (n = l), then

(f, g)w ≡ −q
−4

xij xklxmn + q
−2

xklxmnxij

≡ −q−4xklxij xmn + q
−4

xklxij xmn

≡ 0.

2.1.3. If ((k, l), (m, n)) ∈ C4 (n > l), then ((k, n), (i, j)) ∈ C2, ((i, j), (m, l)) ∈ C1 and

(f, g)w ≡ −q
−2xij [xklxmn − (q

2 − q−2)xknxml] + [xklxmn − (q
2 − q−2)xknxml]xij

≡ −q−2xklxij xmn + q
−2(q2 − q−2)xknxij xml + q

−2xklxij xmn

−q−2(q2 − q−2)xknxij xml

≡ 0.

For the cases 2.2, 2.3 and 2.4, the proofs are similar to 2.1.1.

Case 3. f = xmnxij − q
−2xij xmn, g = xij xkl − xklxij + (q

2 − q−2)xkj xil, w = xmnxij xkl.

In the case, we have

(f, g)w = −q
−2

xij xmnxkl + xmnxklxij − (q
2 − q−2)xmnxkj xil.

There are two subcases to consider.

((i, j), (m, n)) ∈ C1 ((i, j), (m, n)) ∈ C3

3.1. 3.2.

((k, l), (i, j)) ∈ C4 ((k, l), (m, n)), ((k, j), (m, n)) ∈ C4 ((k, l), (m, n)) ∈ C4, C5 or C6

((k, j), (m, n)) ∈ C3

3.1. ((i, j), (m, n)) ∈ C1, ((k, l), (i, j)) ∈ C4 and (k, l), (m, n)), ((k, j), (m, n)) ∈ C4.

Then, we have ((k, n), (i, j)) ∈ C2, ((i, l), (m, n)) ∈ C1, ((i, l), (m, j)) ∈ C1, ((m, l), (i, j))



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Rosso-Yamane Theorem on PBW Basis of Uq(AN ) 179

∈ C1 and

(f, g)w ≡ −q
−2xij [xklxmn − (q

2 − q−2)xknxml] + [xklxmn − (q
2 − q−2)xknxml]xij

−(q2 − q−2)[xkj xmn − (q
2 − q−2)xknxmj ]xil

≡ −q−2[xklxij − (q
2 − q−2)xkj xil]xmn + q

−2(q2 − q−2)xknxij xml + q
−2xklxij xmn

−(q2 − q−2)xknxmlxij − q
−2(q2 − q−2)xkj xilxmn + q

−2(q2 − q−2)2xknxilxmj

≡ q−4(q2 − q−2)xknxmlxij − (q
2 − q−2)xknxmlxij + q

−2(q2 − q−2)xknxmlxij

≡ 0.

3.2. ((i, j), (m, n)) ∈ C3, ((k, l), (i, j)) ∈ C4, (k, l), (m, n)) ∈ C4, C5 or C6 and ((k, j), (m, n)) ∈

C3.

3.2.1. If ((k, l), (m, n)) ∈ C4 (l > m) and ((k, j), (m, n)) ∈ C3, then ((k, n), (i, j)) ∈ C3, ((i, j), (m, l)) ∈

C2, ((i, l), (m, n)) ∈ C4 and

(f, g)w ≡ −q
−2xij [xklxmn − (q

2 − q−2)xknxml] + [xklxmn − (q
2 − q−2)xknxml]xij

−q−2(q2 − q−2)xkj xmnxil

≡ −q−2[xklxij − (q
2 − q−2)xkj xil]xmn + q

−4(q2 − q−2)xknxij xml + q
−2xklxij xmn

−(q2 − q−2)xknxij xml − q
−2(q2 − q−2)xkj [xilxmn − (q

2 − q−2)xinxml]

≡ 0.

3.2.2. If ((k, l), (m, n)) ∈ C5 (l = m) and ((k, j), (m, n)) ∈ C3, then ((k, l), (i, j)) ∈ C4, ((k, n), (i, j)) ∈

C3, ((i, l), (m, n)) ∈ C5 and

(f, g)w ≡ −q
−2xij (q

2xklxmn − qxkn) + (q
2xklxmn − qxkn)xij − q

−2(q2 − q−2)xkj xmnxil

≡ −[xklxij − (q
2 − q−2)xkj xil]xmn + q

−3
xknxij + xklxij xmn − qxknxij

−q−2(q2 − q−2)xkj [q
2
xilxmn − qxin]

≡ q−3xknxij − qxknxij + q
−1(q2 − q−2)xknxij

≡ 0.

3.2.3. If ((k, l), (m, n)) ∈ C6 (l < m) and ((k, j), (m, n)) ∈ C3, then ((i, l), (m, n)) ∈ C6 and

(f, g)w ≡ −q
−2xij xklxmn + xklxmnxij − q

−2(q2 − q−2)xkj xmnxil

≡ −q−2[xklxij − (q
2 − q−2)xkj xil]xmn + q

−2xklxij xmn − q
−2(q2 − q−2)xkj xilxmn

≡ 0.

Case 4. f = xmnxij − q
−2xij xmn, g = xij xkl − q

2xklxij + qxkj , w = xmnxij xkl.

In the case, we have

(f, g)w = −q
−2

xij xmnxkl + q
2
xmnxklxij − qxmnxkj .



180 Yuqun Chen et al. CUBO
10, 3 (2008)

There are two subcases to consider.

((i, j), (m, n)) ∈ C1 ((i, j), (m, n)) ∈ C3

4.1. 4.2.

((k, l), (i, j)) ∈ C5 ((k, l), (m, n)) ∈ C5 ((k, l), (m, n)) ∈ C6

((k, j), (m, n)) ∈ C4 ((k, j), (m, n)) ∈ C3

4.1. ((i, j), (m, n)) ∈ C1, ((k, l), (i, j)) ∈ C5, ((k, l), (m, n)) ∈ C5 and ((k, j), (m, n)) ∈ C4.

Then, we have ((k, n), (i, j)) ∈ C2 (m = i) and

(f, g)w ≡ −q
−2

xij (q
2
xklxmn − qxkn) + q

2(q2xklxmn − qxkn)xij

−q[xkj xmn − (q
2 − q−2)xknxmj ]

≡ −(q2xklxij − qxkj )xmn + q
−1xknxij + q

2xklxij xmn

−q3xknxij − qxkj xmn + q(q
2 − q−2)xknxmj

≡ 0.

4.2. ((i, j), (m, n)) ∈ C3, ((k, l), (i, j)) ∈ C5, ((k, l), (m, n)) ∈ C6 and ((k, j), (m, n)) ∈ C3.

Then, we have

(f, g)w ≡ −q
−2xij xklxmn + q

2xklxmnxij − q
−1xkj xmn

≡ −q−2(q2xklxij − qxkj )xmn + xklxij xmn − q
−1xkj xmn

≡ 0.

Case 5. f = xmnxij − xij xmn, g = xij xkl − q
−2xklxij , w = xmnxij xkl.

In the case, we have

(f, g)w = −xij xmnxkl + q
−2xmnxklxij .

There are four subcases to consider.

((i, j), (m, n)) ∈ C2 ((i, j), (m, n)) ∈ C6

((k, l), (i, j)) ∈ C1 5.1. ((k, l), (m, n)) ∈ C2, C3, C4, C5 or C6 5.3. ((k, l), (m, n)) ∈ C6

((k, l), (i, j)) ∈ C3 5.2. ((k, l), (m, n)) ∈ C2 5.4. ((k, l), (m, n)) ∈ C6

5.1. ((i, j), (m, n)) ∈ C2, ((k, l), (i, j)) ∈ C1, and ((k, l), (m, n)) ∈ C2, C3, C4, C5 or C6.

5.1.1. If ((k, l), (m, n)) ∈ C2 (l > n), then we have ((k, l), (i, j)) ∈ C1 and

(f, g)w ≡ −xij xklxmn + q
−2xklxmnxij

≡ −q−2xklxij xmn + q
−2

xklxij xmn

≡ 0.



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5.1.2. If ((k, l), (m, n)) ∈ C3 (l = n), then

(f, g)w ≡ −q
−2xij xklxmn + q

−4xklxmnxij

≡ −q−4xklxij xmn + q
−4xklxij xmn

≡ 0.

5.1.3. If ((k, l), (m, n)) ∈ C4 (m < l < n), then we have ((k, l), (i, j)) ∈ C1, ((k, n), (i, j)) ∈

C1, ((i, j), (m, l)) ∈ C2 and

(f, g)w ≡ −xij [xklxmn − (q
2 − q−2)xknxml] + q

−2[xklxmn − (q
2 − q−2)xknxml]xij

≡ −xij xklxmn + (q
2 − q−2)xij xknxml + q

−2
xklxmnxij − q

−2(q2 − q−2)xknxmlxij

≡ −q−2xklxij xmn + q
−2(q2 − q−2)xknxij xml + q

−2
xklxij xmn

−q−2(q2 − q−2)xknxij xml

≡ 0.

5.1.4. If ((k, l), (m, n)) ∈ C5 (m = l), then we have ((k, n), (i, j)) ∈ C1 and

(f, g)w ≡ −xij (q
2
xklxmn − qxkn) + q

−2(q2xklxmn − qxkn)xij

≡ −q2xij xklxmn + qxij xkn + xklxmnxij − q
−1xknxij

≡ −xklxij xmn + q
−1xknxij + xklxij xmn − q

−1xknxij

≡ 0.

5.1.5. If ((k, l), (m, n)) ∈ C6 (l < m), the proof is similar to 5.1.1.

For the cases of 5.2, 5.3 and 5.4, the proofs are also similar to 5.1.1.

Case 6. f = xmnxij − xij xmn, g = xij xkl − xklxij , w = xmnxij xkl.

In the case, we have

(f, g)w = −xij xmnxkl + xmnxklxij .

There are four subcases to consider.

((i, j), (m, n)) ∈ C2 ((i, j), (m, n)) ∈ C6

((k, l), (i, j)) ∈ C2 6.1. ((k, l), (m, n)) ∈ C2 6.3. ((k, l), (m, n)) ∈ C2, C3, C4, C5 or C6

((k, l), (i, j)) ∈ C6 6.2. ((k, l), (m, n)) ∈ C6 6.4. ((k, l), (m, n)) ∈ C6

6.1. ((i, j), (m, n)) ∈ C2, ((k, l), (i, j)) ∈ C2 and ((k, l), (m, n)) ∈ C2.

Then, we have

(f, g)w ≡ −xij xklxmn + xklxmnxij

≡ −xklxij xmn + xklxij xmn

≡ 0.



182 Yuqun Chen et al. CUBO
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6.2. ((i, j), (m, n)) ∈ C2, ((k, l), (i, j)) ∈ C6 and ((k, l), (m, n)) ∈ C6.

This case is similar to 6.1.

6.3. ((i, j), (m, n)) ∈ C6, ((k, l), (i, j)) ∈ C2 and ((k, l), (m, n)) ∈ C2, , C3, C4, C5 or C6.

6.3.1. If ((k, l), (m, n)) ∈ C2 (l > n), the proof is similar to 6.1.

6.3.2. If ((k, l), (m, n)) ∈ C3 (l = n), then

(f, g)w ≡ −q
−2xij xklxmn + q

−2xklxmnxij

≡ −q−2xklxij xmn + q
−2xklxij xmn

≡ 0.

6.3.3. If ((k, l), (m, n)) ∈ C4 (m < l < n), then we have ((k, n), (i, j)) ∈ C2, ((i, j), (m, n)) ∈

C6 and

(f, g)w ≡ −xij [xklxmn − (q
2 − q−2)xknxml] + [xklxmn − (q

2 − q−2)xknxml]xij

≡ −xij xklxmn + (q
2 − q−2)xij xknxml + xklxmnxij − (q

2 − q−2)xknxmlxij

≡ −xklxij xmn + (q
2 − q−2)xknxij xml + xklxij xmn − (q

2 − q−2)xknxij xml

≡ 0.

6.3.4. If ((k, l), (m, n)) ∈ C5 (m = l), then we have ((k, n), (i, j)) ∈ C2 and

(f, g)w ≡ −xij (q
2
xklxmn − qxkn) + (q

2
xklxmn − qxkn)xij

≡ −q2xij xklxmn + qxij xkn + q
2
xklxmnxij − qxknxij

≡ −q2xklxij xmn + qxknxij + q
2xklxij xmn − qxknxij

≡ 0.

6.3.5. If ((k, l), (m, n)) ∈ C6 (l < m), the proof is similar to 6.1.

6.4. ((i, j), (m, n)) ∈ C6, ((k, l), (i, j)) ∈ C6 and ((k, l), (m, n)) ∈ C6.

This case is also similar to 6.1.

Case 7. f = xmnxij − xij xmn, g = xij xkl − xklxij + (q
2 − q−2)xkj xil, w = xmnxij xkl.

In the case, we have

(f, g)w = −xij xmnxkl + xmnxklxij − (q
2 − q−2)xmnxkj xil.

There are two subcases to consider.

((i, j), (m, n)) ∈ C2 ((i, j), (m, n)) ∈ C6

7.1. 7.2.

((k, l), (i, j)) ∈ C4 ((k, l), (m, n)) ∈ C2, C3, C4, C5 or C6 ((k, l), (m, n)),

((k, j), (m, n)) ∈ C2 ((k, j), (m, n)) ∈ C6



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7.1. ((i, j), (m, n)) ∈ C2, ((k, l), (i, j)) ∈ C4, ((k, l), (m, n)) ∈ C2, C3, C4, C5 or C6 and

((k, j), (m, n)) ∈ C2.

7.1.1. If ((k, l), (m, n)) ∈ C2 (n < l) and ((k, j), (m, n)) ∈ C2, then we have ((i, l), (m, n)) ∈ C2

and

(f, g)w ≡ −xij xklxmn + xklxmnxij − (q
2 − q−2)xkj xmnxil

≡ −[xklxij − (q
2 − q−2)xkj xil]xmn + xklxij xmn − (q

2 − q−2)xkj xilxmn

≡ 0.

7.1.2. If ((k, l), (m, n)) ∈ C3 (n = l) and ((k, j), (m, n)) ∈ C2, then ((i, l), (m, n)) ∈ C3 and

(f, g)w ≡ −q
−2

xij xklxmn + q
−2

xklxmnxij − (q
2 − q−2)xkj xmnxil

≡ −q−2[xklxij − (q
2 − q−2)xkj xil]xmn + q

−2
xklxij xmn − q

−2(q2 − q−2)xkj xilxmn

≡ 0.

7.1.3. If ((k, l), (m, n)) ∈ C4 (m < l < n) and ((k, j), (m, n)) ∈ C2, then we obtain

((k, n), (i, j)) ∈ C4, ((i, j), (m, l)) ∈ C2, ((i, l), (m, n)) ∈ C4 and

(f, g)w ≡ −xij [xklxmn − (q
2 − q−2)xknxml] + [xklxmn − (q

2 − q−2)xknxml]xij

−(q2 − q−2)xkj xmnxil

≡ −xij xklxmn + (q
2 − q−2)xij xknxml + xklxmnxij − (q

2 − q−2)xknxmlxij

−(q2 − q−2)xkj [xilxmn − (q
2 − q−2)xinxml]

≡ −[xklxij − (q
2 − q−2)xkj xil]xmn + (q

2 − q−2)[xknxij − (q
2 − q−2)xkj xin]xml

+xklxij xmn − (q
2 − q−2)xknxij xml

≡ 0.

7.1.4. If ((k, l), (m, n)) ∈ C5 (m = l) and ((k, j), (m, n)) ∈ C2, then ((k, n), (i, j)) ∈ C4, ((i, l), (m, n)) ∈

C5 and

(f, g)w ≡ −xij (q
2xklxmn − qxkn) + (q

2xklxmn − qxkn)xij − (q
2 − q−2)xkj xmnxil

≡ −q2xij xklxmn + qxij xkn + q
2xklxmnxij − qxknxij

−(q2 − q−2)xkj (q
2xilxmn − qxin)

≡ −q2[xklxij − (q
2 − q−2)xkj xil]xmn + q[xknxij − (q

2 − q−2)xkj xin]

+q2xklxij xmn − qxknxij − q
2(q2 − q−2)xkj xilxmn + q(q

2 − q−2)xkj xin

≡ 0.

7.1.5. If ((k, l), (m, n)) ∈ C6 (l < m) and ((k, j), (m, n)) ∈ C2, then ((i, l), (m, n)) ∈ C6. This

case is similar to 7.1.1.



184 Yuqun Chen et al. CUBO
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7.2. ((i, j), (m, n)) ∈ C6, ((k, l), (i, j)) ∈ C4, ((k, l), (m, n)), ((k, j), (m, n)) ∈ C6.

This case is also similar to 7.1.1.

Case 8. f = xmnxij − xij xmn, g = xij xkl − q
2xklxij + qxkj , w = xmnxij xkl.

In the case, we have

(f, g)w = −xij xmnxkl + q
2xmnxklxij + qxmnxkj .

There are two subcases to consider.

((i, j), (m, n)) ∈ C2 ((i, j), (m, n)) ∈ C6

8.1. 8.2.

((k, l), (i, j)) ∈ C5 ((k, l), (m, n)) ∈ C6 ((k, l), (m, n)), ((k, j), (m, n)) ∈ C6

((k, j), (m, n)) ∈ C2

8.1. ((i, j), (m, n)) ∈ C2, ((k, l), (i, j)) ∈ C5, ((k, l), (m, n)) ∈ C6 and ((k, j), (m, n)) ∈ C2.

Then, we have

(f, g)w ≡ −xij xklxmn + q
2xklxmnxij + qxkj xmn

≡ −(q2xklxij − qxkj )xmn + q
2xklxij xmn + qxkj xmn

≡ 0.

8.2. ((i, j), (m, n)) ∈ C6, ((k, l), (i, j)) ∈ C5, ((k, l), (m, n)), ((k, j), (m, n)) ∈ C6.

This case is similar to 8.1.

Case 9. f = xmnxij − xij xmn + (q
2 − q−2)xinxmj , g = xij xkl − q

−2xklxij , w = xmnxij xkl.

In the case, we have

(f, g)w = −xij xmnxkl + (q
2 − q−2)xinxmj xkl + q

−2xmnxklxij .

There are two subcases to consider.

((i, j), (m, n)) ∈ C4

((k, l), (i, j)) ∈ C1 9.1. ((k, l), (m, n)), ((k, l), (m, j)) ∈ C4, C5 or C6

((k, l), (i, j)) ∈ C3 9.2. ((k, l), (m, n)) ∈ C4 ((k, l), (m, j)) ∈ C3

9.1. ((i, j), (m, n)) ∈ C4, ((k, l), (i, j)) ∈ C1 and ((k, l), (m, n)), ((k, l), (m, j)) ∈ C4, C5 or

C6.



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9.1.1. If ((k, l), (m, n)), ((k, l), (m, j)) ∈ C4 (l > m), then we have ((i, j), (k, n)) ∈ C1, ((k, n), (m, l)) ∈

C2, ((k, j), (i, n)) ∈ C1, ((k, l), (i, n)) ∈ C1, ((i, j), (m, l)) ∈ C2 and

(f, g)w ≡ −xij [xklxmn − (q
2 − q−2)xknxml] + (q

2 − q−2)xin[xklxmj − (q
2 − q−2)xkj xml]

+q−2[xklxmn − (q
2 − q−2)xknxml]xij

≡ −xij xklxmn + (q
2 − q−2)xij xknxml + (q

2 − q−2)xinxklxmj

−(q2 − q−2)2xinxkj xml + q
−2xklxmnxij − q

−2(q2 − q−2)xknxmlxij

≡ −q−2xklxij xmn + (q
2 − q−2)xij xknxml + q

−2(q2 − q−2)xklxinxmj

−q−2(q2 − q−2)2xkj xinxml + q
−2xkl[xij xmn − (q

2 − q−2)xinxmj ]

−q−2(q2 − q−2)xknxij xml

≡ (q2 − q−2)xij xknxml − q
−2(q2 − q−2)2xkj xinxml − q

−4(q2 − q−2)xij xknxml

≡ 0.

9.1.2. If ((k, l), (m, n)), ((k, l), (m, j)) ∈ C5 (l = m), then we have ((i, j), (k, n)) ∈ C1, ((k, l), (i, n), ((k, j), (i, n)) ∈

C1 and

(f, g)w ≡ −xij (q
2xklxmn − qxkn) + (q

2 − q−2)xin(q
2xklxmj − qxkj )

+q−2(q2xklxmn − qxkn)xij

≡ −q2xij xklxmn + qxij xkn + q
2(q2 − q−2)xinxklxmj − q(q

2 − q−2)xinxkj

+xklxmnxij − q
−1xknxij

≡ −xklxij xmn + qxij xkn + (q
2 − q−2)xklxinxmj − q

−1(q2 − q−2)xkj xin

+xkl[xij xmn − (q
2 − q−2)xinxmj ] − q

−3
xij xkn

≡ qxij xkn − qxkj xin + q
−3

xkj xin − q
−3

xij xkn

≡ 0.

9.1.3. If ((k, l), (m, n)), ((k, l), (m, j)) ∈ C6 (l < m), then we have ((k, l), (i, n)) ∈ C1 and

(f, g)w ≡ −xij xklxmn − (q
2 − q−2)xinxklxmj + q

−2
xklxmnxij

≡ −q−2xklxij xmn − q
−2(q2 − q−2)xklxinxmj + q

−2xkl[xij xmn − (q
2 − q−2)xinxmj ]

≡ 0.

9.2. ((i, j), (m, n)) ∈ C4, ((k, l), (i, j)) ∈ C3, ((k, l), (m, n)) ∈ C4 and ((k, l), (m, j)) ∈ C3.



186 Yuqun Chen et al. CUBO
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Then, we have ((k, n), (i, j)) ∈ C2, ((k, l), (i, n)) ∈ C4, ((i, j), (m, l)) ∈ C3 and

(f, g)w ≡ −xij [xklxmn − (q
2 − q−2)xknxml] + q

−2(q2 − q−2)xinxklxmj

+q−2[xklxmn − (q
2 − q−2)xknxml]xij

≡ −xij xklxmn + (q
2 − q−2)xij xknxml + q

−2(q2 − q−2)xinxklxmj + q
−2

xklxmnxij

−q−2(q2 − q−2)xknxmlxij

≡ −q−2xklxij xmn + (q
2 − q−2)xknxij xml + q

−2(q2 − q−2)[xklxin

−(q2 − q−2)xknxil]xmj + q
−2xkl[xij xmn − (q

2 − q−2)xinxmj ]

−q−4(q2 − q−2)xknxij xml

≡ (q2 − q−2)xknxij xml − q
−2(q2 − q−2)xknxilxmj − q

−4(q2 − q−2)xknxij xml

≡ 0.

Case 10. f = xmnxij − xij xmn + (q
2 − q−2)xinxmj , g = xij xkl − xklxij , w = xmnxij xkl.

In the case, we have

(f, g)w = −xij xmnxkl + (q
2 − q−2)xinxmj xkl + xmnxklxij .

There are two subcases to consider.

((i, j), (m, n)) ∈ C4

((k, l), (i, j)) ∈ C2 10.1. ((k, l), (m, n)) ∈ C2, C3 or C4 ((k, l), (m, j)) ∈ C2

((k, l), (i, j)) ∈ C6 10.2. ((k, l), (m, n)), ((k, l), (m, j)) ∈ C6

10.1. ((i, j), (m, n)) ∈ C4, ((k, l), (i, j)) ∈ C2, ((k, l), (m, n)) ∈ C2, C3 or C4 and ((k, l), (m, j)) ∈

C2.

10.1.1. If ((k, l), (m, n)) ∈ C2 (l > n), then we have ((k, l), (i, n)) ∈ C2 and

(f, g)w ≡ −xij xklxmn + (q
2 − q−2)xinxklxmj + xklxmnxij

≡ −xklxij xmn + (q
2 − q−2)xklxinxmj + xkl[xij xmn − (q

2 − q−2)xinxmj ]

≡ 0.

10.1.2. If ((k, l), (m, n)) ∈ C3 (l = n), then we have ((k, l), (i, n)) ∈ C3 and

(f, g)w ≡ −q
−2xij xklxmn + (q

2 − q−2)xinxklxmj + q
−2xklxmnxij

≡ −q−2xklxij xmn + q
−2(q2 − q−2)xklxinxmj + q

−2
xkl[xij xmn − (q

2 − q−2)xinxmj ]

≡ 0.



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10.1.3. If ((k, l), (m, n)) ∈ C4 (l < n), then we have ((k, n), (i, j)) ∈ C2, ((k, l), (i, n)) ∈

C4, ((i, j), (m, l)) ∈ C4 and

(f, g)w ≡ −xij [xklxmn − (q
2 − q−2)xknxml] + (q

2 − q−2)xinxklxmj

+[xklxmn − (q
2 − q−2)xknxml]xij

≡ −xij xklxmn + (q
2 − q−2)xij xknxml + (q

2 − q−2)xinxklxmj

+xklxmnxij − (q
2 − q−2)xknxmlxij

≡ −xklxij xmn + (q
2 − q−2)xknxij xml + (q

2 − q−2)[xklxin − (q
2 − q−2)xknxil]xmj

+xkl[xij xmn − (q
2 − q−2)xinxmj ] − (q

2 − q−2)xkn[xij xml − (q
2 − q−2)xilxmj ]

≡ 0.

10.2. ((i, j), (m, n)) ∈ C4, ((k, l), (i, j)) ∈ C6, ((k, l), (m, n)), (k, l), (m, j)) ∈ C6.

This case is similar to 10.1.

Case 11. f = xmnxij −xij xmn + (q
2 −q−2)xinxmj , g = xij xkl −xklxij + (q

2 −q−2)xkj xil, w =

xmnxij xkl.

In the case, we have

(f, g)w = −xij xmnxkl + (q
2 − q−2)xinxmj xkl + xmnxklxij − (q

2 − q−2)xmnxkj xil,

with
((i, j), (m, n)) ∈ C4

((k, l), (i, j)) ∈ C4 ((k, l), (m, n)), ((k, l), (m, j)) ∈ C4, C5 or C6

11.1. If ((k, l), (m, n)), ((k, l), (m, j)) ∈ C4 (l > m), then we have ((k, n), (i, j)) ∈ C2, ((k, l),

(i, n)) ∈ C4, ((k, j), (i, n)) ∈ C4, ((i, j), (m, l)) ∈ C2, ((i, l), (m, n)) ∈ C4,

((i, l), (m, j)) ∈ C4 and

(f, g)w ≡ −xij [xklxmn − (q
2 − q−2)xknxml] + (q

2 − q−2)xin[xklxmj − (q
2 − q−2)xkj xml]

+[xklxmn − (q
2 − q−2)xknxml]xij − (q

2 − q−2)[xkj xmn − (q
2 − q−2)xknxmj ]xil

≡ −xij xklxmn + (q
2 − q−2)xij xknxml + (q

2 − q−2)xinxklxmj

−(q2 − q−2)xinxkj xml + xklxmnxij − (q
2 − q−2)xknxmlxij

−(q2 − q−2)xkj xmnxil + (q
2 − q−2)2xknxmj xil

≡ −[xklxij − (q
2 − q−2)xkj xil]xmn + (q

2 − q−2)xknxij xml + (q
2 − q−2)[xkj xin

−(q2 − q−2)xknxil]xmj − (q
2 − q−2)[xkj xin − (q

2 − q−2)xknxij ]xml

+xkl[xij xmn − (q
2 − q−2)xinxmj ] − (q

2 − q−2)xknxij xml

−(q2 − q−2)xkj [xilxmn − (q
2 − q−2)xinxml]

+(q2 − q−2)2xkn[xilxmj − (q
2 − q−2)xij xml]

≡ 0.



188 Yuqun Chen et al. CUBO
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11.2. If ((k, l), (m, n)), ((k, l), (m, j)) ∈ C5 (l = m), then we have ((k, n), (i, j)) ∈ C2, ((k, l),

(i, n)) ∈ C4, ((k, j), (i, n)) ∈ C4, ((i, l), (m, n)) ∈ C5, ((i, l), (m, j)) ∈ C5 and

(f, g)w ≡ −xij (q
2xklxmn − qxkn) + (q

2 − q−2)xin(q
2xklxmj − qxkj ) + (q

2xklxmn − qxkn)xij

−(q2 − q−2)[xkj xmn − (q
2 − q−2)xknxmj ]xil

≡ −q2xij xklxmn + qxij xkn + q
2(q2 − q−2)xinxklxmj − q(q

2 − q−2)xinxkj

+q2xklxmnxij − qxknxij − (q
2 − q−2)xkj xmnxil + (q

2 − q−2)2xknxmj xil

≡ −q2[xklxij − (q
2 − q−2)xkj xil]xmn + qxknxij + q

2(q2 − q−2)[xklxin

−(q2 − q−2)xknxil]xmj − q(q
2 − q−2)[xkj xin − (q

2 − q−2)xknxij ]

+q2xkl[xij xmn − (q
2 − q−2)xinxmj ] − qxknxij

−(q2 − q−2)xkj [q
2xilxmn − qxin] + (q

2 − q−2)2xkn[q
2xilxmj − qxij ]

≡ 0.

11.3. If ((k, l), (m, n)), ((k, l), (m, j)) ∈ C6 (l < m), then ((k, j), (m, n)) ∈ C4, ((k, l), (i, n)) ∈

C4, ((i, l), (m, n)), ((i, l), (m, j)) ∈ C6 and

(f, g)w ≡ −xij xklxmn + (q
2 − q−2)xinxklxmj + xklxmnxij

−(q2 − q−2)[xkj xmn − (q
2 − q−2)xknxmj ]xil

≡ −[xklxij − (q
2 − q−2)xkj xil]xmn + (q

2 − q−2)[xklxin − (q
2 − q−2)xknxil]xmj

+xkl[xij xmn − (q
2 − q−2)xinxmj ] − (q

2 − q−2)xkj xilxmn + (q
2 − q−2)2xknxilxmj

≡ 0.

Case 12. f = xmnxij −xij xmn+(q
2−q−2)xinxmj , g = xij xkl−q

2xklxij +qxkj , w = xmnxij xkl,

with
((i, j), (m, n)) ∈ C4

((k, l), (i, j)) ∈ C5 ((k, l), (m, n)), ((k, l), (m, j)) ∈ C6

((k, j), (m, n)) ∈ C4 ((k, l), (i, n)) ∈ C5

In the case, we can deduce that

(f, g)w = −xij xmnxkl + (q
2 − q−2)xinxmj xkl + q

2
xmnxklxij − qxmnxkj

≡ −xij xklxmn + (q
2 − q−2)xinxklxmj + q

2xkj xmnxij

−q[xkj xmn − (q
2 − q−2)xknxmj ]

≡ −(q2xklxij − qxkj )xmn + (q
2 − q−2)(q2xklxin − qxkn)xmj

+q2xkl[xij xmn − (q
2 − q−2)xinxmj ] − qxkj xmn + q(q

2 − q−2)xknxmj

≡ −q2xklxij xmn + qxkj xmn + q
2(q2 − q−2)xklxinxmj − q(q

2 − q−2)xknxmj

+q2xklxij xmn − q
2(q2 − q−2)xklxinxmj − qxkj xmn + q(q

2 − q−2)xknxmj

≡ 0.



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Rosso-Yamane Theorem on PBW Basis of Uq(AN ) 189

Case 13. f = xmnxij − q
2xij xmn + qxin, g = xij xkl − q

−2xklxij , w = xmnxij xkl.

In the case, we have

(f, g)w = −q
2xij xmnxkl + qxinxkl + q

−2xmnxklxij .

There are two subcases to consider.

((i, j), (m, n)) ∈ C5

((k, l), (i, j)) ∈ C1 13.1. ((k, l), (m, n)) ∈ C6 ((k, l), (i, n)) ∈ C1

((k, l), (i, j)) ∈ C3 13.2. ((k, l), (m, n)) ∈ C5 ((k, l), (i, n)) ∈ C4

13.1. ((i, j), (m, n)) ∈ C5, ((k, l), (i, j)) ∈ C1, ((k, l), (m, n)) ∈ C6 and ((k, l), (i, n)) ∈ C1.

Then, we have

(f, g)w = −q
2xij xklxmn + q

−1xklxin + q
−2xklxmnxij

≡ −xklxij xmn + q
−1xklxin + q

−2xkl(q
2xij xmn − qxin)

≡ −xklxij xmn + q
−1xklxin + xklxij xmn − q

−1xklxin

≡ 0.

13.2. ((i, j), (m, n)) ∈ C5, ((k, l), (i, j)) ∈ C3, ((k, l), (m, n)) ∈ C5 and ((k, l), (i, n)) ∈ C4.

Then, we have ((k, n), (i, j)) ∈ C2 and

(f, g)w ≡ −q
2
xij (q

2
xklxmn − qxkn) + q[xklxin − (q

2 − q−2)xknxil]

−q−2(q2xklxmn − qxkn)xij

≡ −q4xij xklxmn + q
3xij xkn + qxklxin − q(q

2 − q−2)xknxil + xklxmnxij

−q−1xknxij

≡ −q2xklxij xmn + q
3xknxij + qxklxin − q

3xknxil

+q−1xknxil + q
2xklxij xmn − qxklxin − q

−1xknxij

≡ 0.

Case 14. f = xmnxij − q
2xij xmn + qxin, g = xij xkl − xklxij , w = xmnxij xkl.

In the case, we have

(f, g)w = −q
2
xij xmnxkl + qxinxkl + xmnxklxij .

There are two subcases to consider.

((i, j), (m, n)) ∈ C5

((k, l), (i, j)) ∈ C2 14.1. ((k, l), (m, n)), ((k, l), (i, n)) ∈ C2, C3 or C4

((k, l), (i, j)) ∈ C6 14.2. ((k, l), (m, n)), ((k, l), (i, n)) ∈ C6



190 Yuqun Chen et al. CUBO
10, 3 (2008)

14.1. ((i, j), (m, n)) ∈ C5, ((k, l), (i, j)) ∈ C2 and ((k, l), (m, n)), ((k, l), (i, n)) ∈ C2, C3 or C4.

14.1.1. If ((k, l), (m, n)) and ((k, l), (i, n)) ∈ C2 (l > n), then

(f, g)w = −q
2xij xklxmn + qxklxin + xklxmnxij

≡ −q2xklxij xmn + qxklxin + xkl(q
2xij xmn − qxin)

≡ 0.

14.1.2. If ((k, l), (m, n)) and ((k, l), (i, n)) ∈ C3 (l = n), then

(f, g)w = −xij xklxmn + q
−1xklxin + q

−2xklxmnxij

≡ −xklxij xmn + q
−1xklxin + xklxij xmn − q

−1xklxin

≡ 0.

14.1.3. If ((k, l), (m, n)), ((k, l), (i, n)) ∈ C4 (l < n), then we have ((k, n), (i, j)) ∈ C2, ((i, j), (m, l)) ∈

C5 and

(f, g)w ≡ −q
2xij [xklxmn − (q

2 − q−2)xknxml] + q[xklxin − (q
2 − q−2)xknxil]

+[xklxmn − (q
2 − q−2)xknxml]xij

≡ −q2xij xklxmn + q
2(q2 − q−2)xij xknxml + qxklxin − q(q

2 − q−2)xknxil

+xklxmnxij − (q
2 − q−2)xknxmlxij

≡ −q2xklxij xmn + q
2(q2 − q−2)xknxij xml + qxklxin − q(q

2 − q−2)xknxil

+xkn(q
2xij xmn − qxin) − (q

2 − q−2)xkn(q
2xij xmn − qxil)

≡ 0.

14.2. ((i, j), (m, n)) ∈ C5, ((k, l), (i, j)) ∈ C6 and ((k, l), (m, n)), ((k, l), (i, n)) ∈ C6.

This case is similar to 14.1.1.

Case 15. f = xmnxij −q
2xij xmn +qxin, g = xij xkl −xklxij +(q

2 −q−2)xkj xil, w = xmnxij xkl,

with

((i, j), (m, n)) ∈ C5

((k, l), (i, j)) ∈ C4 ((k, l), (m, n)) ∈ C6 ((k, l), (i, n)) ∈ C4

((k, j), (m, n)) ∈ C5 ((i, l), (m, n)) ∈ C6



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Rosso-Yamane Theorem on PBW Basis of Uq(AN ) 191

Then, we have

(f, g)w = −q
2
xij xmnxkl + qxinxkl + xmnxklxij − (q

2 − q−2)xmnxkj xil

≡ −q2xij xklxmn + q[xklxin − (q
2 − q−2)xknxil] + xklxmnxij

−(q2 − q−2)(q2xkj xmn − qxkn)xil

≡ −q2[xklxij − (q
2 − q−2)xkj xil]xmn + qxklxin − q(q

2 − q−2)xknxil

+xkl(q
2xij xmn − qxin) − q

2(q2 − q−2)xkj xmnxil + q(q
2 − q−2)xknxil

≡ −q2xklxij xmn + q
2(q2 − q−2)xkj xilxmn + qxklxin − q(q

2 − q−2)xknxil

+q2xklxij xmn − qxklxin − q
2(q2 − q−2)xkj xilxmn + q(q

2 − q−2)xknxil

≡ 0.

Case 16. f = xmnxij − q
2xij xmn + qxin, g = xij xkl − q

2xklxij + qxkj , w = xmnxij xkl, with

((i, j), (m, n)) ∈ C5

((k, l), (i, j)) ∈ C5 ((k, l), (m, n)) ∈ C6 ((k, l), (i, n)), ((k, j), (m, n)) ∈ C5

In the case, we have

(f, g)w = −q
2xij xmnxkl + qxinxkl + q

2xmnxklxij − qxmnxkj

≡ −q2xij xklxmn + q(q
2xklxin − qxkn) + q

2xklxmnxij − q(q
2xkj xmn − qxkn)

≡ −q2(q2xklxij − qxkj )xmn + q
3xklxin − q

2xkn + q
2xkl(q

2xij xmn − qxin)

−q3xkj xmn + q
2xkn

≡ 0.

Thus, ˜S+ is a Gröbner-Shirshov basis. This completes the proof of Theorem 3.1. �

Similarly, with the deg-lex order on ˜Y ∗, ˜S− is a Gröbner-Shirshov basis for U −q (AN ) =

k〈˜Y |˜S−〉.

We now use the same notation as before. Order the generators by: xi > xj , hi > h
−1

i
> hj >

h
−1

j
, yi > yj if i > j, and xi > h

±1

j
> ym for all i, j, m. Then we obtain a well ordering (deg-lex)

on ˜X ∪ H ∪ ˜Y . Thus, by Theorem 3.1, we re-obtain the following theorem in [16].

Theorem 3.2. ([16] Theorem 2.7) Let the notation be as before. Then with the deg-lex order on

{ ˜X ∪ H ∪ ˜Y }∗, ˜S+ ∪ T ∪ K ∪ ˜S− is a Gröbner-Shirshov basis for Uq(AN ) = k〈 ˜X ∪ H ∪ ˜Y |˜S
+ ∪

T ∪ K ∪ ˜S−〉.



192 Yuqun Chen et al. CUBO
10, 3 (2008)

Acknowledgement: The authors would like to express their deepest gratitude to Professor L. A.

Bokut for his kind guidance, useful discussions and enthusiastic encouragement.

Received: June 2008. Revised: August 2008.

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