量子群と q 差分量子 Weyl 群双有理作用

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量子群と q _差分量子 Weyl _{群双有理作用}

Quantum groups and quantized q -diﬀerence birational Weyl group actions

黒木玄

(

_東北大学

)

Gen KUROKI (Tohoku University, Japan)

24 September 2010

日本数学会

2010

_{年度秋季総合分科会} 名古屋大学大学院多元数理科学研究科

2010

_年

9

_月

22–25

_日

(24 September 2010, Version 1.7)

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Quantum birational Weyl group action

GCM [a

_ij

] d

_i

a

_ij

= d

_j

a

_ji

, ⟨ h

_i

, α

_i

⟩ = a

_ij

, α

^∨_i

= d

⁻_i ¹

α

_i

Ore domain q-Serre: ∑

k

( − 1)

^k

[

₁₋_a_ij

k

]

q_i

e

¹_i⁻^a^ij⁻^k

e

_j

e

^k_i

= 0 (i ̸ = j ) A

^q

= ⟨ e

_i

, a

_i

⟩

ⁱ_∈^I

Parameters a

_i

= q

^αⁱ

: a

_i

e

_j

= e

_j

a

_i

↓

Field of fractions K

^q

= Q( A

^q

) = { ab

⁻¹

| a, b ∈ A

^q

, b ̸ = 0 }

↓

e

ⁿ_i

e

_j

e

⁻_i ⁿ

is an n-independent rational function of q

_iⁿ

↓ ↓ q

_iⁿ

7→ a

_i

= q

_i^α^∨ⁱ

(q

_i

= q

^dⁱ

) Weyl group action s

_i

(e

_j

) = e

^α_i ^∨ⁱ

e

_j

e

^α_i ^∨ⁱ

on K

^q

s

_i

(a

_j

) = a

⁻_i ^a^ij

a

_j

( ⇐⇒ s

_i

(α

_j

) = α

_j

− a

_ij

α

_i

) Both q-diﬀerence analogue and canonical quantization of

the birational Weyl group action given by Noumi-Yamada math/0012028.

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Explicit formulae for s

_i

(e

_j

) = e

^α_i ^∨ⁱ

e

_j

e

^α_i ^∨ⁱ

• [x, y]

_q

:= xy − qyx, q(k) := q

_i^2k+a^ij

(i ̸ = j )

• Deﬁne (ad

_q

e

_i

)

^k

(e

_j

) for k = 0, 1, 2, . . . by

(ad

_q

e

_i

)

^k

(e

_j

) = [e

_i

, [ · · · , [e

_i

, [e

_i

, e

_j

]

_q(0)

]

_q(1)

· · · ]

_q(k₋₂₎

]

_q(k₋₁₎

.

• Then (ad

_q

e

_i

)

^k

(e

_j

) =

∑

k ν=0

( − 1)

^ν

q

_i^ν^(k⁻^1+a^ij⁾

[ k

ν ]

q_i

e

^k_i ⁻^ν

e

_j

e

^ν_i

.

• q-Serre relations ⇐⇒ (ad

_q

e

_i

)

^k

(e

_j

) = 0 if i ̸ = j and k > − a

_ij

.

• s

_i

(e

_j

) =

 

 



 

 

e

_i

(i = j ),

−a_ij

∑

k=0

q

_i^(k+a^ij^)(α^∨ⁱ ⁻^k)

[ α

^∨_i

k

]

q_i

(ad

_q

e

_i

)

^k

(e

_j

)e

⁻_i ^k

(i ̸ = j ).

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Quantum geometric crystal structure on K

^q

GCM [a

_ij

] d

_i

a

_ij

= d

_j

a

_ji

, ⟨ h

_i

, α

_i

⟩ = a

_ij

, α

^∨_i

= d

⁻_i ¹

α

_i

Ore domain q-Serre: ∑

k

( − 1)

^k

[

₁₋_a_ij

k

]

q_i

e

¹_i⁻^a^ij⁻^k

e

_j

e

^k_i

= 0 (i ̸ = j ) A

^q

= ⟨ e

_i

, a

_i

⟩

ⁱ_∈^I

Parameters a

_i

= q

^αⁱ

: a

_i

e

_j

= e

_j

a

_i

↓

Field of fractions K

^q

= Q( A

^q

) = { ab

⁻¹

| a, b ∈ A

^q

, b ̸ = 0 }

↓

e

ⁿ_i

e

_j

e

⁻_i ⁿ

is an n-independent rational function of q

_iⁿ

↓ ↓ q

_iⁿ

7→ t

Quantum geometric e

^t_i

(e

_j

) = e

ⁿ_i

e

_j

e

⁻_i ⁿ

|

^q_iⁿ_7→^t

crystal str. on K

^q

e

^t_i

(a

_j

) = t

⁻^a^ij

a

_j

The Verma relations for e

^t_i

= ⇒ Weyl group action s

_i

= e

^a_i ⁱ

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Generalization

q-Serre relations ∑

k

( − 1)

^k

[

₁₋_a_ij

k

]

q_i

e

¹_i⁻^a^ij⁻^k

e

_j

e

^k_i

= 0 (i ̸ = j )

↓

Verma relations e

^k_i

e

^k+l_j

e

^l_i

= e

^l_j

e

^k+l_i

e

^k_j

if a

_ij

a

_ji

= 1, etc.

↓

Assumption e

ⁿ_i

xe

⁻_i ⁿ

is a rational fucntion of q

_iⁿ

.

↓ ↓ q

_iⁿ

7→ t

Quantum e

^t_i

(x) = e

ⁿ_i

xe

⁻_i ⁿ

|

^q_iⁿ_7→^t

geomtric crystal e

^t_i

(a

_j

) = t

⁻^a^ij

a

_j

(action on parameters)

↓

Weyl group action s

_i

= e

^α_i ^∨ⁱ

(a

_i

= q

_i^α^∨ⁱ

)

Actions of the lattice parts of aﬃne Weyl groups

→ q-diﬀerence quantum Painlev´ e systems

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Quantum Schubert cell

Reduced expression of w ∈ W : w = s

_i₁

· · · s

_i_N

, i = (i

₁

, . . . , i

_N

).

↓

A

ⁱ

= ⟨ x

_ν

, a

_i

⟩ , K

ⁱ

= Q( A

^q

) = { ab

⁻¹

| a, b ∈ A

ⁱ

, b ̸ = 0 }

Deﬁning relations: x

_ν

x

_µ

= q

_µν

x

_µ

x

_ν

(µ < ν ), a

_i

∈ center.

q

_µν

:= q

^b^iµiν

, b

_ij

:= d

_i

a

_ij

.

↓

(x

₁

, . . . , x

_N

) 7→ e

_q_i

1

(x

₁

F

_i₁

) · · · e

_q

iN

(x

_N

F

_i_N

)

is quantization of a positive structure of a Schubert cell.

↓ e

_i

:= ∑

i_ν=i

x

_ν

{ q-Serre relations for e

_i

,

e

ⁿ_i

x

_ν

e

⁻_i ⁿ

is a rational function of q

_iⁿ

.

↓

Quantum geometric crystal structure on K

ⁱ

.

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Explicit formulae for e

^t_i

(x

_ν

)

• X := ∑

i_µ=i, µ<ν

x

_µ

, Y := ∑

i_µ=i, µ>ν

x

_µ

..

• Then e

_i

= X + δ

_i_ν_i

x

_ν

+ Y .

• If i

_ν

= i, then

e

^t_i

(x

_ν

) = t

⁻²

x

_ν

1 + q

_i²

(x

_ν

+ q

_i²

Y )X

⁻¹

1 + q

_i²

t

⁻²

(x

_ν

+ q

_i²

Y )X

⁻¹

1 + q

_i²

Y (X + x

_ν

)

⁻¹

1 + q

_i²

t

⁻²

Y (X + x

_ν

)

⁻¹

,

• If i

_ν

̸ = i, then − a

_ii_ν

= 0 and

e

^t_i

(x

_ν

) = t

⁻^a^iiν

x

_ν

−

∏

a_iiν k=1

1 + q

_i⁻^2(k⁻¹⁾

t

⁻²

Y X

⁻¹

1 + q

_i⁻^2(k⁻¹⁾

Y X

⁻¹

.

(8)

Remarks

• All the expressions for e

^t_i

(x

_ν

) are subtraction-free.

K

ⁱ

depends only on w ∈ W up to canonical positive isomorphisms.

↓

generalization

Various quantum positive geometric crystals

↓

classical limit

Various positive geometric crystals

↓

ultra-discretization

Various crystals

• φ

_i

and ε

_i

.

◦ e

^t_i

(e

_i

) = e

_i

, e

^t_i

(q

^αⁱ

) = t

⁻²

q

^αⁱ

.

◦ ε

_i

:= const.q

^αⁱ

e

_i

, φ

_i

:= const.q

⁻^αⁱ

e

_i

. (ε

_i

= q

^2αⁱ

φ

_i

)

◦ Then e

^t_i

(φ

_i

) = t

²

φ

_i

, e

^t_i

(ε

_i

) = t

⁻²

ε

_i

.

◦ quantum t

⁻²

, q

^2αⁱ

←→ classical “c”, “α

_i

”

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Files

• Old version of this ﬁle −→

http://www.math.tohoku.ac.jp/˜kuroki/LaTeX/20100924 Nagoya.pdf

• Quantum M -matrix for A

_∞

case −→ § 1.6 of

http://www.math.tohoku.ac.jp/˜kuroki/LaTeX/20100630 Osaka.pdf

• Quantization of the birational action of W (A

⁽¹⁾_m₋₁

) × W (A

⁽¹⁾_n₋₁

) given by Kajiwara-Noumi-Yamada nlin/0106029 for mutually prime m, n

→ http://www.math.tohoku.ac.jp/˜kuroki/LaTeX/20100630 WxW.pdf

• Theory of quantum geometric crystals −→ in preparation

For more details see the following pages.

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Symmetrizable GCM and root datum

• Let A = [a

_ij

]

_i,j_∈_I

be a symmetrizable GCM:

◦ a

_ii

= 2, a

_ij

5 0 (i ̸ = j ), a

_ij

= 0 ⇐⇒ a

_ji

= 0;

◦ d

_i

a

_ij

= d

_j

a

_ji

, d

_i

∈ Z

^>0

.

• Let ( ⟨ , ⟩ : Q

^∨

× P → Z , { h

_i

}

ⁱ_∈^I

⊂ Q

^∨

, { α

_i

}

ⁱ_∈^I

⊂ P ) be a root datum:

• ﬁnitely generated free Z -modules Q

^∨

, P and perfect bilinear pairing ⟨ , ⟩ : Q

^∨

× P → Z .

◦ { h

_i

}

ⁱ_∈^I

⊂ Q

^∨

is called a set of simple coroots.

Q

^∨

is called a coroot lattice.

◦ { α

_i

}

ⁱ_∈^I

⊂ P is called a set of simple roots.

P is called a weight lattice.

◦ ⟨ h

_i

, α

_j

⟩ = a

_ij

.

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The group algebra F [q

^P

] of the weight lattice P

• Base ﬁeld F := Q (q).

• F [q

^P

] := ⊕

λ∈P

F q

^λ

, q

^λ

q

^µ

= q

^λ+µ

(λ, µ ∈ P ).

• [x]

_q

:= q

^x

− q

⁻^x

q − q

⁻¹

, [k]

_q

! := [1]

_q

[2]

_q

· · · [k ]

_q

(k ∈ Z

₌0

).

• [ x k

]

q

:= [x]

_q

[x − 1]

_q

· · · [x − k + 1]

_q

[k ]

_q

! (q-binomial coeﬃcients).

• q

_i

:= q

^dⁱ

, α

^∨_i

:= d

⁻_i ¹

α

_i

(= a simple coroot).

Remark. q

^±^dⁱ^α^∨ⁱ

= q

^±^αⁱ

∈ F [q

^P

] = ⇒

[ α

^∨_i

k

]

q_i

∈ F [q

^P

].

(12)

Quantum algebra A

^q

= ⟨ q

^λ

, e

_i

| λ ∈ P, i ∈ I ⟩

Assumptions.

(1) A

^q,0

is an associative algebra over F generated by e

_i

̸ = 0 (i ∈ I ).

(2) q-Serre relations:

1−a_ij

∑

k=0

( − 1)

^k

[ 1 − a

_ij

k

]

q_i

e

¹_i⁻^a^ij⁻^k

e

_j

e

^k_i

= 0 (i ̸ = j ).

(3) A

^q

:= F [q

^P

] ⊗

_F

A

^q,0

is an Ore domain.

Identiﬁcation. q

^λ

= q

^λ

⊗ 1 ∈ A

^q

, e

_i

= 1 ⊗ e

_i

∈ A

^q

. Remark. q

^λ

e

_i

= e

_i

q

^λ

in A

^q

.

• Q( A

^q

) := (the quotient skew ﬁeld of A

^q

) = { as

⁻¹

| a, s ∈ A

^q

, s ̸ = 0 } . Example. The root datum is of ﬁnite or aﬃne type

= ⇒ A

^q,0

= U

_q

(n

₊

) satisﬁes all the assumptions above.

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Iterated adjoint by e

_i

• Assume i ̸ = j .

• [x, y]

_q

:= xy − qyx, q(k) := q

_i^2k+a^ij

• Deﬁne (ad

_q

e

_i

)

^k

(e

_j

) for k = 0, 1, 2, . . . by (ad

_q

e

_i

)

⁰

(e

_j

) = e

_j

,

(ad

_q

e

_i

)

¹

(e

_j

) = [e

_i

, e

_j

]

_q(0)

,

(ad

_q

e

_i

)

²

(e

_j

) = [e

_i

, [e

_i

, e

_j

]

_q(0)

]

_q(1)

, . . .,

(ad

_q

e

_i

)

^k

(e

_j

) = [e

_i

, [ · · · , [e

_i

, [e

_i

, e

_j

]

_q(0)

]

_q(1)

· · · ]

_q(k₋₂₎

]

_q(k₋₁₎

.

• Then (ad

_q

e

_i

)

^k

(e

_j

) =

∑

k ν=0

( − 1)

^ν

q

_i^ν^(k⁻^1+a^ij⁾

[ k

ν ]

q_i

e

^k_i ⁻^ν

e

_j

e

^ν_i

.

• q-Serre relations ⇐⇒ (ad

_q

e

_i

)

^k

(e

_j

) = 0 if i ̸ = j and k > − a

_ij

.

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Conjugation by powers of e

_i

• For n = 0, 1, 2, . . .,

e

ⁿ_i

e

_j

e

⁻_i ⁿ

=

 

 



 

 

e

_i

(i = j ),

−a_ij

∑

k=0

q

_i^(k+a^ij⁾⁽ⁿ⁻^k)

[ n

k ]

q_i

(ad

_q

e

_i

)

^k

(e

_j

)e

⁻_i ^k

(i ̸ = j ).

• Deﬁne e

^α_i ^∨ⁱ

e

_j

e

⁻_i ^α^∨ⁱ

∈ Q( A

^q

) by

e

^α_i ^∨ⁱ

e

_j

e

⁻_i ^α^∨ⁱ

=

 

 



 

 

e

_i

(i = j ),

−a_ij

∑

k=0

q

_i^(k+a^ij^)(α^∨ⁱ ⁻^k)

[ α

^∨_i

k

]

q_i

(ad

_q

e

_i

)

^k

(e

_j

)e

⁻_i ^k

(i ̸ = j ).

• x 7→ e

ⁿ_i

xe

⁻_i ⁿ

is an algebra automorphism of Q( A

^q

)

= ⇒ e

_j

7→ e

^α_i ^∨ⁱ

e

_j

e

⁻_i ^α^∨ⁱ

is uniquely extended to an alg. autom. of Q( A

^q

).

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Qauntized birational Weyl group action

Theorem 1. The algebra automorphim s

_i

of Q( A

^q

) can be deﬁned by s

_i

(e

_j

) = e

^α_i ^∨ⁱ

e

_j

e

⁻_i ^α^∨ⁱ

(i ∈ I ), s

_i

(q

^λ

) = q

^λ^−⟨^hⁱ^,λ^⟩^αⁱ

(λ ∈ P ).

Then { s

_i

}

ⁱ_∈^I

satisﬁes the deﬁning relations of the Weyl group W :

s

_i

s

_j

= s

_j

s

_i

(a

_ij

a

_ji

= 0), s

_i

s

_j

s

_i

= s

_j

s

_i

s

_j

(a

_ij

a

_ji

= 1), s

_i

s

_j

s

_i

s

_j

= s

_j

s

_i

s

_j

s

_i

(a

_ij

a

_ji

= 2),

s

_i

s

_j

s

_i

s

_j

s

_i

s

_j

= s

_j

s

_i

s

_j

s

_i

s

_j

s

_i

(a

_ij

a

_ji

= 3), s

²_i

= 1.

Thus we obtain the action of the Weyl group W on Q( A

^q

).

Remark. This is a q-diﬀerence version of quantization of the birational

Weyl group action given by Noumi-Yamada math/0012028.

(16)

The Verma relations of { e

_i

}

ⁱ_∈^I

• (Lusztig’s book (1993)) q-Serre relations of { e

_i

}

ⁱ_∈^I

implies

◦ (a

_ij

, a

_ji

) = (0, 0) = ⇒ e

^k_i

e

^l_j

= e

^l_j

e

^k_i

,

◦ (a

_ij

, a

_ji

) = ( − 1, − 1) = ⇒ e

^k_i

e

^k+l_j

e

^l_i

= e

^l_j

e

^k+l_i

e

^k_j

,

◦ (a

_ij

, a

_ji

) = ( − 1, − 2) = ⇒ e

^k_i

e

^2k+l_j

e

^k+l_i

e

^l_j

= e

^l_j

e

^k+l_i

e

^2k+l_j

e

^k_i

,

◦ (a

_ij

, a

_ji

) = ( − 1, − 3)

= ⇒ e

^k_i

e

^3k+l_j

e

^2k+l_i

e

^3k+2l_j

e

^k+l_i

e

^l_j

= e

^l_j

e

^k+l_i

e

^3k+2l_j

e

^2k+l_i

e

^3k+l_j

e

^k_i

. These relations are called the Verma relations.

• The Verma relations

= ⇒ { s

_i

}

ⁱ_∈^I

satisﬁes the deﬁning relation of the Weyl group.

• For details see arXiv:0808.2604.

(17)

Quantum geometric crystal structure on A

^q

• The algebra homomorphism e

^t_i

: Q( A

^q

) → Q( A

^q

)(t) is deﬁned by e

^t_i

(e

_j

) = e

ⁿ_i

e

_j

e

⁻_i ⁿ

q_iⁿ7→t

(j ∈ I ), e

^t_i

(q

^λ

) = t

^−⟨^hⁱ^,λ^⟩

q

^λ

(λ ∈ P ).

• Then e

¹_i

= id

_Q(_A_q₎

, e

^t_i¹

e

^t_i²

= e

^t_i¹^t²

: Q( A

^q

) → Q( A

^q

)(t

₁

, t

₂

).

• Furthermore { e

^t_i

}

ⁱ_∈^I

satisﬁes the Verma relations:

◦ (a

_ij

, a

_ji

) = (0, 0) = ⇒ e

^t_i¹

e

^t_j²

= e

^t_j²

e

^t_i¹

,

◦ (a

_ij

, a

_ji

) = ( − 1, − 1) = ⇒ e

^t_i¹

e

^t_j¹^t²

e

^t_i²

= e

^t_j²

e

^t_i¹^t²

e

^t_j¹

,

◦ (a

_ij

, a

_ji

) = ( − 1, − 2) = ⇒ e

^t_i¹

e

^t_j¹^t²

e

^t¹^t

2 2

i

e

^t_j²

= e

^t_j²

e

^t¹^t

2 2

i

e

^t_j¹^t²

e

^t_i¹

,

◦ (a

_ij

, a

_ji

) = ( − 1, − 3)

= ⇒ e

^t_i¹

e

^t_j¹^t²

e

^t

2 1t³₂

i

e

^t¹^t

2 2

j

e

^t¹^t

3 2

i

e

^t_j²

= e

^t_j²

e

^t¹^t

3 2

i

e

^t¹^t

2 2

j

e

^t

2 1t³₂

i

e

^t_j¹^t²

e

^t_i¹

.

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Deﬁnition of quantum geometric crystal

Deﬁnition. ( K , { e

^t_i

}

ⁱ_∈^I

) is called a quantum geometric crystal if it satisﬁes the following conditions:

◦ K is a skew ﬁeld.

◦ e

^t_i

is an algebra homomorphism K → K (t).

◦ e

^t_i

is regular at t = 1. e

¹_i

= id

_K

, e

^t_i¹

e

^t_i²

= e

^t_i¹^t²

.

◦ { e

^t_i

}

ⁱ_∈^I

satisﬁes the Verma relations.

◦ F [q

^P

] is a subalgebra of the center of K .

◦ e

^t_i

is regular at t = q

^λ

for any λ ∈ P .

◦ e

^t_i

(q

^λ

) = t

^−⟨^hⁱ^,λ^⟩

q

^λ

for λ ∈ P .

Remark. For the classical case, see Berenstein-Kazhdan math/9912105.

Proposition 2. (Q( A

^q

), { e

^t_i

}

ⁱ_∈^I

) is a quantum geometric crystal.

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Weyl group action on a quantum geometric crystal

Proposition 3.

Let ( K , { e

^t_i

}

ⁱ_∈^I

) be a quantum geometric crystal.

Put a

_i

= q

^αⁱ

= q

_i^α^∨ⁱ

and s

_i

(x) = e

^a_i ⁱ

(x) for i ∈ I , x ∈ K . Then s

_i

is an algebra automorphism of K with

s

_i

(q

^λ

) = q

^λ^−⟨^hⁱ^,λ^⟩^αⁱ

= q

^sⁱ^(λ)

for λ ∈ P .

Moreover { s

_i

}

ⁱ_∈^I

satisﬁes the deﬁning relations of the Weyl group W and hence generates the action of W on K .

• Propositions 2 and 3 = ⇒ Theorem 1.

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Quantum Schubert cell

• b

_ij

:= d

_i

a

_ij

. Then b

_ji

= b

_ij

and q

^b^ij

= q

_i^a^ij

.

• i := (i

₁

, i

₂

, . . . , i

_N

) ∈ I

^N

.

• A

^i,0

:= the associative algebra over F = Q (q) generated by { x

_ν

}

^N_ν₌₁

with deﬁning relations: x

_ν

x

_µ

= q

^b^iµiν

x

_µ

x

_ν

(µ < ν ).

• A

ⁱ

:= F [q

^P

] ⊗

_F

A

^i,0

= ⟨ q

^λ

, x

_ν

| λ ∈ P, 1 5 ν 5 N ⟩ . (Identiﬁcation. q

^λ

⊗ 1 = q

^λ

, 1 ⊗ x

_ν

= x

_ν

)

• Then A

ⁱ

is an Ore domain.

• If w = s

_i₁

s

_i₂

· · · s

_i_N

is a reduced expression of w ∈ W ,

then Q( A

^0,i

) depends only on w (Berenstein q-alg/9605016)

and is the rational function ﬁeld of a quantum Schubert cell.

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Quantum geometric crystal structure on A

ⁱ

• e

_i

:= ∑

i_ν=i

x

_ν

. Then { e

_i

}

ⁱ_∈^I

satisﬁes the q-Serre relations.

• Assume { i

_ν

| ν = 1. . . . , N } = I . ( ←− inessential assumtion) Then e

_i

̸ = 0 for all i ∈ I .

Theorem 4. (quant. geom. crys. str. on A

ⁱ

)

The algebra hom. e

^t_i

: Q( A

ⁱ

) → Q( A

ⁱ

)(t) can be deﬁned by e

^t_i

(x

_ν

) = e

ⁿ_i

x

_ν

e

⁻_i ⁿ

q_iⁿ7→t

, e

_i

(q

^λ

) = t

^−⟨^hⁱ^,λ^⟩

q

^λ

. Then (Q( A

ⁱ

), { e

_i

}

ⁱ_∈^I

) is a quantum geometric crystal.

Remark. An induction on n = 0, 1, 2, . . . proves that

e

ⁿ_i

x

_ν

e

⁻_i ⁿ

is an n-independent rational function of q

_iⁿ

.

Explicit formulae −→ Next page

(22)

Explicit formulae for e

^t_i

(x

_ν

) and their positivity

• X := ∑

i_µ=i, µ<ν

x

_µ

, Y := ∑

i_µ=i, µ>ν

x

_µ

..

• Then e

_i

= X + δ

_i_ν_i

x

_ν

+ Y .

• If i

_ν

= i, then

e

^t_i

(x

_ν

) = t

⁻²

x

_ν

1 + q

_i²

(x

_ν

+ q

_i²

Y )X

⁻¹

1 + q

_i²

t

⁻²

(x

_ν

+ q

_i²

Y )X

⁻¹

1 + q

_i²

Y (X + x

_ν

)

⁻¹

1 + q

_i²

t

⁻²

Y (X + x

_ν

)

⁻¹

,

• If i

_ν

̸ = i, then − a

_ii_ν

= 0 and

e

^t_i

(x

_ν

) = t

⁻^a^iiν

x

_ν

−

∏

a_iiν k=1

1 + q

_i⁻^2(k⁻¹⁾

t

⁻²

Y X

⁻¹

1 + q

_i⁻^2(k⁻¹⁾

Y X

⁻¹

.

Positivity. All the formulae for e

^t_i

(x

_ν

) are subtraction-free.