Alternative factorization of the reduced R matrix

From Remark4.17, one can use the Cartan involution and replace the first factorei⊗1 in the reducedRmatrix with the embedding by theFipaths. Then the embeddingι^w⊗ι induces a very simple factorization of the reducedRmatrix, where

gb(e⁻_i )=gb(f_i¹^,−) . . .gb(f_iⁿⁱ^,−), gb(e⁺_i )=gb(f_iⁿⁱ^,+) . . .gb(f_i¹^,+), and hence by Corollary9.6,

Fig. 51 The flipping of triangleμR₂ of the basic quivers in typeG₂, before changing the index back to standard form. The basic quivers are stacked according to Fig.42

Corollary 10.8 Under the embeddingι^w⊗ι, the reduced R matrix factorizes as R=R4·R3·R2·R1,

and recall that^opmeans multiplying from the right.

The embeddingι^w⊗ιcorresponds to a new quiverZg, which is another amalga-mation of the two quiversDg, where the nodes{f_iⁿⁱ}of the first quiver are glued to {fⁿ_iⁱ}of the second quiver instead (see Fig.52). Then one can describe for every type ofgthe mutation sequence giving the flip of triangulations onZgeasily:

Q D Q Q Q

C B

Q Q Q Q

Fig. 52 Half-Dehn twist of the quiverZg

Proposition 10.9 Let

P_i^Q^Q := P_F^Q_i −P_F^Q_i

be the concatenation of the Fi-paths in the corresponding subquivers ofZg. Then the mutation sequence giving the flip of triangulation is

μR1 = {P1−→P2−→. . .PN},

where as before if imis the k-th appearance of the root index i from the right ofi, then Pm=P_i^Q^Q[k,ni].

Whenicorresponds to the Coxeter element of the Weyl group, w0 =wc^h^/², this coincides with the mutation sequence of the flip of triangulations (where two quivers mirrored to each other are glued) described in [35] in the classical type. Hence this construction generalizes those of [35], and at the same time provides a representation theoretic meaning of the sequences giving the flip of triangulations described there.

Although the description of the R matrix factorization is very nice, we see that after 4 flips it does not return to the original quiver, but rather a mirror image with all the arrows flipped. A full Dehn twist, however, return us to the original configuration.

If Conjecture8.5is true, which gives a quiver mutation equivalence betweenι and ι^w (with Dynkin involution), this will relate such nice presentation of theR matrix factorization to the canonical one found in the main theorem.

11 Proof of Theorem9.5

LetRdenote the right hand side of (9.8). The strategy is to show thatKRalso gives the braiding relations (9.3) as well. First of all, we have

Ad_K(1⊗ei+ei⊗K_i)=Ki ⊗ei +ei ⊗1, (11.1) Ad_K(fi⊗1+Ki⊗fi)=fi ⊗K_i+1⊗fi, (11.2)

Ad_K(Ki)=Ki⊗Ki. (11.3)

Hence in order to prove the braiding relations, it suffices to show

R(ei)=(1⊗ei+ei⊗K_i), (11.4) R(fi)=(fi⊗1+Ki⊗fi), (11.5)

R(K i)=Ki⊗Ki, (11.6)

where the last one is trivial. We begin with several Lemmas:

Lemma 11.1 For anysl₂triple(e,f,K,K), and any self-adjoint element X , we have Adg_b(e⊗X)(f⊗1+K⊗X)=f⊗1+K⊗X. (11.7) Proof This is a well-known result by considering the formal power series expansion ofgb(recall that we restrict ourselves to the compact case, but it holds for the non-compact case as well). Recall

gb(u)=E x p_q−2

− u q−q⁻¹

n≥0

(−1)ⁿq¹²ⁿ⁽ⁿ⁻¹⁾uⁿ (qⁿ−q⁻ⁿ) . . . (q−q⁻¹), and that we have

eⁿf−feⁿ=(qⁿ−q⁻ⁿ)(qⁿ⁻¹K−q¹⁻ⁿK)eⁿ⁻¹. Hence we can work out

gb(e⊗X)(f⊗1+K⊗X)−(f⊗1+K⊗X)gb(e⊗X)

⎛

⎝

n≥0

(−1)ⁿq¹²ⁿ⁽ⁿ⁻¹⁾eⁿ

(qⁿ−q⁻ⁿ) . . . (q−q⁻¹)⊗Xⁿ

⎞

⎠(f⊗1+K⊗X)

−(f⊗1+K ⊗X)

n≥0

(−1)ⁿq¹²ⁿ⁽ⁿ⁻¹⁾eⁿ

(qⁿ−q⁻ⁿ) . . . (q−q⁻¹)⊗Xⁿ

=(f⊗1)

n≥0

(−1)ⁿq¹²ⁿ⁽ⁿ⁻¹⁾eⁿ

(qⁿ−q⁻ⁿ) . . . (q−q⁻¹)⊗Xⁿ

Proof We observe that by Lemma5.2, Ad_g

wheneverY^l_jcomes afterY_i^k⁻¹and beforeY_i^kin the decomposition (9.8).

Lemma 11.3 For the reduced wordi=(i1, . . . ,iN)∈R, if iN =i , then KR(ei)=^op(ei)KR.

Proof Note that if iN = i, then X^±_N = X_f±ni i

,ei = X_fni

i + X_fni

i ,e⁰_i and Ki =

X_fni

i ,e⁰_i,f_i⁻ⁿⁱ. We have

X_f−ni i

X_e⁰

i =q_i²X_e⁰

iX_f−ni i

Hence Ad_g

b(e_i⊗X⁻_N)(1⊗ei+ei⊗Ki)

= Adg_b(e_i⊗X

f−ni i

)(1⊗X_fni

i +1⊗X_fni

i ,e⁰_i +ei ⊗Ki)

=1⊗X_fni

i +(1⊗X_fni

i ,e⁰_i +ei⊗Ki)(1+qiei⊗X_f−ni

X_e0 i)⁻¹

=1⊗X_fni

i +1⊗X_fni

i ,e⁰_i(1+qieβi ⊗X_f−ni

i )(1+qiei⊗X_f−ni

X_e⁰

i)⁻¹

=1⊗ei.

One then check directly that 1⊗ei commutes with all the factorsej ⊗X_k^±for every j,k, except the last termei ⊗X⁺_N, where we have the reverse of the above:

Ad_g

b(e_i⊗X⁺_N)(1⊗ei)=1⊗ei +ei⊗K_i,

and hence

KR(ei)=K(1⊗ei+ei⊗K_i)R=^op(ei)KR

as required.

In general, we use the fact that the decomposition ofRis invariant under the change of wordsM. Letfi :=!f_i^k^,±denote the representation offiusing the mutated cluster variables"Xi :=M(Xi)under the change of wordsM(cf. Sect.7)

Lemma 11.4 We have the following identities:

(1) For the change of wordsM:(. . .i j i. . .)←→(. . .j i j. . .)we have gb(e1⊗f₁^k⁺¹^,±)gb(e2⊗f₂^l^,±)gb(e1⊗f₁^k^,±)=gb(e2⊗f₂^l⁺¹^,±) gb(e1⊗f₁^k^,±)gb(e2⊗f₂^l^,±)

forv(i,k) < v(j,l) < v(i,k+1).

(2) For the change of wordsM : (. . .i j i j. . .) ←→ (. . .j i j i. . .)where i is short and j is long, we have

gbs(ei⊗f_i^k⁺¹^,±)gb(ej ⊗f_j^l⁺¹^,±)gbs(ei⊗f_i^k^,±)gb(ej ⊗f_j^l^,±)

=gb(ej ⊗f_j^l⁺¹^,±)gbs(ei ⊗f_i^k⁺¹^,±)gb(ej⊗f_j^l^,±)gbs(ei ⊗f_i^k^,±) forv(j,l) < v(i,k) < v(j,l+1) < v(i,k+1).

Proof We will prove the+case, while the−case is similar.

Proof of (1)In the simply-laced case, recall that we have f_i^k⁺¹^,+f_i^k^,+=q²f_i^k^,+f_i^k⁺¹^,+, f_i^k^,+f_j^l^,+=q⁻¹f_j^l^,+f_i^k^,+. Hence

[ej ⊗f_j^l^,+,ei⊗f_i^k^,+]

q−q⁻¹ =ejei⊗f_j^l^,+f_i^k^,+−eiej ⊗f_i^k^,+f_j^l^,+

=ejei⊗ −q⁻¹ejei

q−q⁻¹ ⊗f_j^l^,+f_i^k^,+

=ei j ⊗q⁻¹^/²f_j^l^,+f_i^k^,+

whereei j =Ti(ej)is given by the Lusztig’s isomorphism.

Hence using (A.9), we have

gb(ei⊗f_i^k⁺¹^,+)gb(ej⊗f_j^l^,+)gb(ei⊗f_i^k^,+)

=gb(ei⊗f_i^k⁺¹^,+)gb(ei ⊗f_i^k^,+)gb(ei j ⊗q⁻¹^/²f_j^l^,+f_i^k^,+)gb(ej ⊗f_j^l^,+)

=gb(ei⊗(f_i^k⁺¹^,++f_i^k^,+))gb(ei j⊗q⁻¹^/²f_j^l^,+f_i^k^,+)gb(ej⊗f_j^l^,+).

Similarly, we have

gb(ej⊗f^l_j⁺¹^,+)gb(ei ⊗f_i^k^,+)gb(ej ⊗f^l_j^,+)

=gb(ei ⊗f_i^k^,+)gb(ei j ⊗q⁻¹^/²f^l_j⁺¹^,+f_i^k^,+)gb(ej⊗f^l_j⁺¹^,+)gb(ej⊗f^l_j^,+)

=gb(ei ⊗f_i^k^,+)gb(ei j ⊗q⁻¹^/²f^l_j⁺¹^,+f_i^k^,+)gb(ej⊗(f^l_j⁺¹^,++f^l_j^,+)).

If we write down the quantum cluster variables as

f_i^k =X1, f_i^k⁺¹=X1,2, f^l_j =X3, f_i^k ="X1, fj

l ="X3, fj l+1

=X3,4,

then we have

X1=X1(1+q X2),

X3=X3(1+q X⁻₂¹)⁻¹,

X4=X₂⁻¹, and one can see that

f_i^k⁺¹^,++f_i^k^,+=f_i^k^,+, f_j^l^,+f_i^k^,+=f^l_j⁺¹^,+f_i^k^,+,

f_j^l^,+=f^l_j⁺¹^,++f^l_j^,+

as required.

Proof of (2)We havef_i^k^,+f_j^l^,+=q⁻¹f_j^l^,+f_i^k^,+wheneverv(j,l) < v(i,k). Let v=ei⊗f_i^k^,+,

u =ej⊗f_j^l^,+, c

[2]qs

= [u, v]

q−q⁻¹ =q¹^/²ejei −q⁻¹^/²eiej

q−q⁻¹ ⊗q⁻¹^/²f_j^l^,+f_i^k^,+=eY ⊗q⁻¹^/²f_j^l^,+f_i^k^,+, d =q_s⁻¹cv−qsvc

q−q⁻¹ = eYei−eieY

qs −qs⁻¹

⊗q⁻¹f_j^l^,+(f_i^k^,+)²=eX⊗q⁻¹f_j^l^,+(f_i^k^,+)²,

whereeX :=Ti(ej)andeY :=TiTj(ei)are given by the Lusztig’s isomorphism. We have

eYeX =qeXeY, eXei =qeieX, ejeY =qeYej, [ej,eX]

q−q⁻¹ =e²_Y,

and henceu,c,d, vsatisfies the condition for (A.10). Applying (A.10) repeatedly and rearranging, we have (we underline the terms to be transformed):

g_b_s(e_i⊗f_i^k⁺¹^,+)g_b(e_j⊗f_j^l⁺¹^,+)g_b_s(e_i⊗f_i^k^,+)g_b(e_j⊗f_j^l^,+)

=₍A.10)gb(ej⊗f_j^l+1,+)gbs(eY⊗q^−1/2f_j^l+1,+f_i^k+1,+)gb(eX⊗q⁻¹f_j^l+1,+(f_i^k+1,+)²)

×g_b_s(e_i⊗f_i^k⁺¹^,+)

gb(ej⊗f_j^l,+)gbs(eY⊗q^−1/2f_j^l,+f_i^k,+)gb(eX⊗q⁻¹f_j^l,+(f_i^k,+)²)gbs(ei⊗f_i^k,+)

=₍A.10)gb(ej⊗f_j^l⁺¹^,+)gbs(eY⊗q⁻¹^/²f_j^l⁺¹^,+f_i^k⁺¹^,+)

×gb(eX⊗q⁻¹f_j^l⁺¹^,+(f_i^k⁺¹^,+)²)gb(ej⊗f_j^l^,+) gbs(eY⊗q^−1/2f_j^l,+f_i^k+1,+)gb(eX⊗q⁻¹f_j^l,+(f_i^k+1,+)²)

g_b_s(e_i⊗f_i^k⁺¹^,+)g_b_s(e_Y⊗q⁻¹^/²f_j^l^,+f_i^k^,+)g_b(e_X⊗q⁻¹f_j^l^,+(f_i^k^,+)²)g_b_s(e_i⊗f_i^k^,+)

=₍A.7)gb(ej⊗f_j^l+1,+)gbs(eY⊗q^−1/2f_j^l+1,+f_i^k+1,+)gb(ej⊗f_j^l,+)

×gb(e²Y⊗f_j^l⁺¹^,+f^l_j^,+(f_i^k⁺¹^,+)²)

gb(eX⊗q⁻¹f_j^l+1,+(f_i^k+1,+)²)gbs(eY⊗q^−1/2f_j^l,+f_i^k^+1,+)gb(eX⊗q⁻¹f_j^l,+(f_i^k+1,+)²) g_b_s(e_Y⊗q⁻¹^/²f_j^l^,+f_i^k^,+)g_b_s(e_X⊗f_j^l^,+f_i^k^,+f_i^k⁺¹^,+)

gbs(ei⊗f_i^k+1,+)gb(eX⊗q⁻¹f_j^l,+(f_i^k,+)²)gbs(ei⊗f_i^k,+)

=₍A.8)g_b(e_j⊗(f_j^l⁺¹^,++f_j^l^,+))

gbs(eY⊗q^−1/2f_j^l+1,+f_i^k+1,+)gb(e²_Y⊗f_j^l+1,+f_j^l,+(f_i^k+1,+)²)gbs(eY⊗q^−1/2f_j^l,+f_i^k+1,+) g_b_s(e_Y⊗q⁻¹^/²f_j^l^,+f_i^k^,+)g_b(e_X⊗q⁻¹f_j^l⁺¹^,+(f_i^k⁺¹^,+)²)

gb(eX⊗q⁻¹f_j^l,+(f_i^k+1,+)²)gbs(eX⊗f_j^l,+f_i^k,+f_i^k+1,+)gb(eX⊗q⁻¹f_j^l,+(f_i^k,+)²)

g_b_s(e_i⊗(f_i^k⁺¹^,++f_i^k^,+))

=₍A.12)g_b(e_j⊗(f^l_j⁺¹^,++f_j^l^,+))

×gbs(eY⊗q^−1/2(f_j^l+1,++f_j^l,+)f_i^k+1,+)gbs(eY⊗q^−1/2f_j^l,+f_i^k,+)

g_b(e_X⊗q⁻¹f_j^l⁺¹^,+(f_i^k⁺¹^,+)²)g_b(e_X⊗q⁻¹f_j^l^,+(f_i^k⁺¹^,++f_i^k^,+)²)g_b_s(e_i⊗(f_i^k⁺¹^,++f_i^k^,+))

=₍A.8)gb(ej⊗(f_j^l+1,++f_j^l,+))gbs(eY⊗q^−1/2(f_j^l+1,++f_j^l,+)f_i^k+1,++q^−1/2f_j^l,+f_i^k,+) gb(eX⊗q⁻¹f_j^l⁺¹^,+(f_i^k⁺¹^,+)²+q⁻¹f_j^l^,+(f_i^k⁺¹^,++f_i^k^,+)²)gbs(ei⊗(f_i^k⁺¹^,++f_i^k^,+)), where in the last line, we observe that the termsq² commute, hence we can apply (A.12).

On the other hand, by applying (A.10) once, we have

gb(ej⊗f₂^l⁺¹^,+)gb_s(ei⊗f₁^k⁺¹^,+)gb(ej⊗f₂^l^,+)gb_s(ei⊗f₁^k^,+)

=gb(ej⊗(f₂^l⁺¹^,++f₂^l^,+))gb_s(eY ⊗q⁻¹^/²f₂^l^,+f₁^k⁺¹^,+) gb(eX⊗q⁻¹f₂^l^,+(f₁^k⁺¹^,+)²)gb_s(ei⊗(f₁^k⁺¹^,++f₁^k^,+)).

To compare, again we write out the quantum cluster variables as

f_i^k^,+=X1, f_i^k⁺¹^,+=X1,2, f_j^l^,+=X3, f_j^l⁺¹^,+=X3,4, f_i^k^,+=X1, f_i^k⁺¹^,+=X1,2, f^l_j^,+=X3, f^l_j⁺¹^,+=X3,4. Recall that we need to do mutation three times according to Sect.7.2, which gives at the end

X1=D₂⁻¹X1,2,4,

X2=X⁻₂_,¹₄D1, X3=D₁⁻¹X3D3, X4=D₃⁻¹X4, where

D1=(1+qsX2)(1+q_s³X2)+q X₂2,4, D2=(1+qsX2+qsX2,4),

D3=(1+qsX2+qsX2,4)(1+q_s³X2+q_s³X2,4).

Now we can check directly that

f_j^l⁺¹^,++f_j^l^,+=f₂^l⁺¹^,++f₂^l^,+, f_j^l⁺¹^,++f_j^l^,+)f_i^k⁺¹^,++q⁻¹^/²f_j^l^,+f_i^k^,+=f₂^l^,+f₁^k⁺¹^,+, f_j^l⁺¹^,+(f_i^k⁺¹^,+)²+f_j^l^,+(f_i^k⁺¹^,++f_i^k^,+)²=f₂^l^,+(f₁^k⁺¹^,+)²,

f_i^k⁺¹^,++f_i^k^,+=f₁^k⁺¹^,++f₁^k^,+

and this completes the proof.

Remark 11.5 In typeG2, using the mutation sequence that gives the half-Dehn twist from Sect.10.1.5, one can conjugate the representation of(e2)by (9.8) and check the braiding relation directly. Using the fact that the standard form of the universal R matrix is invariant under the change of words, we conclude that the analogue of Lemma11.4also holds in typeG2.

Proof of Theorem9.5 First it is obvious thatKandRcommute with both(Ki)and (K_i)by direct calculation.

As a consequence of Lemma11.2, we have

KR(fi)=K(fi⊗1+K⊗fi)R=^op(fi)KR (11.9) as required.

As a consequence of Lemma11.4, we can choose freely the reduced wordiwith any choice of index on the right ofi, and by Lemma11.3, we obtain

KR(e i)=^op(ei)KR

for every root indexi, thus completing the proof of the braiding relations.

Finally, recall that by the construction of the positive representationsP_λ, one can choose appropriate discrete parametersλand restrict it to give any irreducible highest weight finite dimensional representations ofUq(g)[21]. ThenKRsatisfies the braiding (9.3) on every finite dimensional representations ofUq(g), and as a formal power series it has constant term equals 1, hence we conclude thatKRequals the universalRmatrix.

Acknowledgements I would like to thank Gus Schrader and Alexander Shapiro for stimulating discussions who inspired the constructions carried out in this work. I would also like to thank Masahito Yamazaki and Rei Inoue for valuable comments. This work is supported by JSPS KAKENHI Grant Numbers JP16K17571 and Top Global University Project, MEXT, Japan.

A Quantum dilogarithm identities

The compact quantum dilogarithm function is defined to be the infinite product ^q(x)=

∞ r=0

(1+q^2r⁺¹x)⁻¹, (A.1)

which is well defined for 0 <q < 1. In the split real case, whereq = e^π^{i b}² with 0 <b <1, the infinite product is not so well-behaved. To treat this case, the non-compact quantum dilogarithmgb(x)is composed of two commuting copies, associated to the so-calledFaddeev’s modular double, of the compact quantum dilogarithm^q(x) [6,8]. It is a meromorphic function that can be represented as an integral expression:

gb(x):=exp 1

R+i0

x^{i b}^t

sinh(πbt)sinh(πb⁻¹t) dt

, (A.2)

such that by functional calculus, it is a unitary operator whenxis positive self-adjoint, and there is ab-duality:

gb(x)=g_b−1(x^b¹²). (A.3) In this paper however, we are only interested in the formal algebraic calculation, hence one may consider only the compact part and think about the correspondence in terms of formal power series

gb(x)∼^q(x)⁻¹= ∞ r=0

(1+q^2r⁺¹x)=E x p_q−2

− u q−q⁻¹

, (A.4)

where

E x pq(x):=

k≥0

x^k

(k)q!, (A.5)

(k)q:= 1−q^k

1−q . (A.6)

In particular, we can rewrite the identities of E x pq(x) derived in [31] for the quantum dilogarithm functiongb(x)that are needed in this paper. In particular, by

writing in this way, the argument ofgb(x)are all manifestly positive self-adjoint so that the identities are well-defined in the split real setting.

We will be interested in two types of identities: the pentagon equation (PE) and the quantum exponential relation (QE), together with their generalizations.

Simply-laced caseLetu, vbe self-adjoint variables. Ifuv=q²vu, then we have the pentagon equation and the quantum exponential relation:

(P E): gb(v)gb(u)=gb(u)gb(q⁻¹uv)gb(v), (A.7)

(Q E): gb(u+v)=gb(u)gb(v). (A.8)

Let againu, vbe self-adjoint and

c:= [u, v]

q−q⁻¹, such that

uc=q²cu, cv=q²vc.

Then we have the generalized pentagon equation:

(P E): gb(v)gb(u)=gb(u)gb(c)gb(v). (A.9) in which (A.7) is a special case.

Doubly-laced caseIn the doubly-laced case we haveqs =q¹^/². Letu, vbe self-adjoint variables, and let

c:= [u, v]

qs−qs⁻¹

, d:= q_s⁻¹cv−qsvc q−q⁻¹ , such that

uc=q²cu, cd =q²dc, dv=q²vd, q⁻¹ud−qdu q−q⁻¹ = c²

[2]²qs

We have

(P E): gbs(v)gb(u)=gb(u)gbs

c [2]qs

gb(d)gbs(v), (A.10) (Q E): gbs(c+v)=gbs(c)gb([2]qsd)gbs(v). (A.11) In particular ifuv=q²vuand substituteu →quv⁻¹/[2]q_s, we have:

gbs(u+v)=gbs(u)gb(q⁻¹uv)gbs(v), (A.12)

gb((u+v)²)=gb(u²)gb_s(q⁻¹^/²uv)gb(v²). (A.13) These two relations are related by theb-duality (A.3).

Triply-laced caseFor completeness we also translate the typeG2identity of [31] to gb(x), which becomes more natural looking.

Letqs =q¹^/³, and letu, vbe self-adjoint. Define c:= q_s⁻¹uv−qsvu

q_s²−qs⁻²

, d:= q_s⁻²cv−q_s²vc

qs−qs⁻¹

, d:= q_s⁻²uc−q_s²cu

qs−qs⁻¹

such that these relations are satisfied:

ud=q²du, dc=q²cd, cd=q²dc, dv=q²vd, c²= q⁻¹ud−qdu

q−q⁻¹ , c²= q⁻¹dv−qvd

q−q⁻¹ , c³= q⁻²dd−q²dd q−q⁻¹ . Then we have

(Q E): gb_s(u+v)=gb_s(u)gb(d)gb_s(c)gb(d)gb_s(v). (A.14) In particular ifuv=q²vu =q_s⁶vu, we have

gbs(u+v)=gbs(u)gb(q⁻²u²v)gbs(q⁻¹uv)gb(q⁻²uv²)gbs(v), (A.15) gb((u+v)³)=gb(u³)gbs(q⁻²u²v)gb(q⁻³u³v³)gbs(q⁻²uv²)gb(v³), (A.16) which are related by theb-duality (A.3).

On the other hand, lete1,e2be the generators ofUq(gG₂)withe1long ande2short, and letζ1, ζ2be positive variables satisfyingζ1ζ2=q⁻¹ζ2ζ1. Let the non-simple root generators be

eW :=T1(e2)= [e2,e1]_q³/2 s

q_s³−qs⁻³

, eX :=T1T2(e1)= [eY,eW]_q−1/2

qs−qs⁻¹

, eY :=T1T2T1(e2)= [e2,eW]_q¹/2

q_s²−qs⁻²

, eZ :=T1T2T1T2(e1)= [e2,eY]_q−1/2

qs−qs⁻¹

Then we have (PE):

gb_s(e2⊗ζ2)gb(e1⊗ζ1)

=gb(e1⊗ζ1)gb_s(eW ⊗q¹^/²ζ1ζ2)gb(eX⊗q³ζ1²ζ2³)gb_s

(eY ⊗qζ1ζ2²)gb(eZ ⊗q³^/²ζ1ζ2³)gb_s(e2⊗ζ2). (A.17)

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