- Email: [email protected]
Kawashima's relations for interpolated multiple zeta values
Journal of Algebra 447 (2016) 424–431
Contents lists available at ScienceDirect
Journal of Algebra www.elsevier.com/locate/jalgebra
Kawashima’s relations for interpolated multiple zeta values Tatsushi Tanaka a,∗ , Noriko Wakabayashi b a
Department of Mathematics, Faculty of Science, Kyoto Sangyo University, Motoyama Kamigamo, Kita-ku, Kyoto-city, Kyoto 603-8555, Japan b College of Science and Engineering, Ritsumeikan University, 1-1-1, Nojihigashi, Kusatsu-city, Shiga 525-8577, Japan
a r t i c l e
i n f o
Article history: Received 4 October 2014 Available online xxxx Communicated by Michel Van den Bergh Keywords: t-MZVs Harmonic products t-Kawashima relations Cyclic sum formula
a b s t r a c t Recently, Yamamoto introduced polynomials in one variable t which interpolates multiple zeta and zeta-star values (t-MZVs for short), provided new prospects on two-one conjecture of Ohno and Zudilin and proved the cyclic sum formula for t-MZVs. In this paper, we establish a generalization of Kawashima’s relations (t-Kawashima relations) for t-MZVs. We prove the cyclic sum formula for t-MZVs using a type of derivation operator, together with the t-Kawashima relations. © 2015 Elsevier Inc. All rights reserved.
1. Introduction Multiple zeta and zeta-star values are defined, for positive integers k1 , . . . , kl with k1 ≥ 2, by convergent series ζ(k1 , . . . , kl ) =
k1 m1 >···>ml >0 m1
1 · · · mkl l
, ζ (k1 , . . . , kl ) =
mk11 m1 ≥···≥ml ≥1
* Corresponding author. E-mail addresses: [email protected] (T. Tanaka), [email protected] (N. Wakabayashi). http://dx.doi.org/10.1016/j.jalgebra.2015.09.015 0021-8693/© 2015 Elsevier Inc. All rights reserved.
1 · · · mkl l
,
T. Tanaka, N. Wakabayashi / Journal of Algebra 447 (2016) 424–431
425
abbreviated as MZVs and MZSVs, respectively. Decomposing the region of the sum of the defining series of MZSVs into 2n−1 regions by separating each ‘≥’ into ‘>’ and ‘=’, we immediately find the (symbolic) invertible linear transformation between MZVs and MZSVs. For example, ζ (k1 , k2 ) = ζ(k1 , k2 ) + ζ(k1 + k2 ), ζ (k1 , k2 , k3 ) = ζ(k1 , k2 , k3 ) + ζ(k1 + k2 , k3 ) + ζ(k1 , k2 + k3 ) + ζ(k1 + k2 + k3 ). Let t be a variable. In [11] Yamamoto introduced the following interpolation polynomial of MZVs and MZSVs. ζ t (k1 , . . . , kl ) =
tl−dep(p) ζ(p) (∈ R[t]),
(1)
p
where dep(p) is the depth (length) of p and p runs over all indices of the form p = (k1 · · · kl ) in which each is filled by the comma, or the plus +. We call this polynomial t-MZV. Note that ζ 0 = ζ (MZV) and ζ 1 = ζ (MZSV). In this paper, we discuss certain quadratic formula which contains the cyclic sum formula for t-MZVs. In Section 2, we show the quadratic formula for t-MZVs as follows. Notations are described in next two sections. Theorem 1. For any m ≥ 1 and any v, w ∈ Ht y, we have
t
t
Zt (ϕt (v) (−tx + y)p−1 y)Zt (ϕt (w) (−tx + y)q−1 y)
p+q=m p,q≥1 t
t
= −Zt (ϕt (v ∗ w) (−tx + y)m−1 y). When t = 0 and t = 1, these formulas for MZVs and MZSVs are called Kawashima relations which first appeared in [5] (also see [8,9]). Kawashima relations are of importance since those for MZVs are expected to give whole relations among MZVs as well as extended double shuffle relations [4,7] or associator relations [1]. As an application, the following cyclic sum formula for t-MZVs is again proved. (The original appearance of the cyclic sum formula for MZVs and MZSVs are proved respectively in [3] and [6].)
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Theorem 2. (See [11, Theorem 5.4].) Let k1 , . . . , kl be positive integers and not all 1. Put k = k1 + · · · + kl . Then we have l k i −1
ζ t (ki − j + 1, ki+1 , . . . , kl , k1 , . . . , ki−1 , j)
i=1 j=1
= (1 − t)
l
ζ t (ki + 1, ki+1 , . . . , kl , k1 , . . . , ki−1 ) + kζ(k + 1)tl .
i=1
This theorem is proved in Section 3. 2. Kawashima’s relations for t-MZVs 2.1. Algebraic setup We begin with the algebraic setup introduced for example in [2,11]. Let Ht = Q[t]x, y denote the non-commutative polynomial algebra over Q[t] in two indeterminates x and y, and let H1t and H0t denote the subalgebras Q[t] + Ht y and Q[t] + xHt y, respectively. We denote H0 (= Qx, y), H10 , H00 simply by H, H1 , H0 , respectively. Put zk = xk−1 y for k ≥ 1. We define the Q[t]-linear map Zt : H0t → R[t] by Zt (1) = 1 and Zt (zk1 · · · zkl ) = ζ t (k1 , . . . , kl ). The Q[t]-linear map St : H1t → H1t is defined by St (wy) = σt (w)y (w ∈ Ht ), where σt is an automorphism on Ht characterized by σt (x) = x, σt (y) = tx + y. This operator St is nothing but the operator S t introduced in [11, Definition 3.1]. Then we find that St is bijective (in fact St−1 = S−t ) and Zt = Z0 St .
(2)
ϕt = −St−1 ϕSt ,
(3)
We let
where ϕ denotes the automorphism on H characterized by ϕ(x) = x + y, ϕ(y) = −y. Note that ϕ0 = −ϕ.
T. Tanaka, N. Wakabayashi / Journal of Algebra 447 (2016) 424–431
427
As is introduced in [2], the harmonic product ∗ on H1 is defined, for positive integers p, q and v, w ∈ H1 , recursively by 1 ∗ w = w ∗ 1 = w, zp v ∗ zq w = zp (v ∗ zq w) + zq (zp v ∗ w) + zp+q (v ∗ w) t
and the Q-bilinearity. We also define the t-harmonic product ∗ on H1t by t
v ∗ w = St−1 (St (v) ∗ St (w)) (v, w ∈ H1t ),
(4)
where the harmonic product ∗ on the right hand side is regarded as a Q[t]-bilinear map. t
(In [11, Definition 3.7], the product ∗ is defined recursively as above for the harmonic t
product ∗. The property (4) we adopt as the definition of ∗ here is proved afterwards.) Because of (2), (4) and the fact that the map Z0 is a homomorphism with respect to the harmonic product ∗, we find that the map Zt is a homomorphism with respect to the t harmonic product ∗, i.e., t
Zt (v ∗ w) = Zt (v)Zt (w) (v, w ∈ H0t ). The product on H1 is defined by zp v zq w = zp+q (v ∗ w) t
for v, w ∈ H1 . Using , we define the product on H1t by t
v w = St−1 (St (v) St (w)) (v, w ∈ H1t ),
(5)
where, as above, the harmonic product ∗ and hence the product on the right hand side are regarded as Q[t]-bilinear maps. Remark 3. It is off the subjects, but we notice that the product represents the rule of the harmonic product formula for truncated multiple zeta values ζm (k1 , . . . , kl ) =
k1 m+1=m1 >···>ml >0 m1
1 · · · mkl l t
for any positive integer m. Therefore we also notice that the product represents the rule of the harmonic product formula for an interpolation of truncated multiple zeta values t ζm (k1 , . . . , kl ) =
p
which is defined as well as (1).
tl−dep(p) ζm (p)
428
T. Tanaka, N. Wakabayashi / Journal of Algebra 447 (2016) 424–431
2.2. Proof of Theorem 1 Now we show that Kawashima’s relations for t-MZVs (Theorem 1) follows straightforwardly from the case t = 0 which is a result of [5]. When t = 0, we have
Z0 (ϕ(v) y p )Z0 (ϕ(w) y q ) = Z0 (ϕ(v ∗ w) y m )
p+q=m p,q≥1
for any positive integer m and any v, w ∈ Hy. By (2), (3) and (5),
LHS =
Zt St−1 (St ϕt St−1 (v) y p )Zt St−1 (St ϕt St−1 (w) y q )
p+q=m p,q≥1
=
t
t
Zt (ϕt St−1 (v) St−1 (y p ))Zt (ϕt St−1 (w) St−1 (y q )).
p+q=m p,q≥1
By (2), (3), (4) and (5), RHS = −Zt St−1 (St ϕt St−1 (v ∗ w) y m ) t
t
= −Zt (ϕt (St−1 (v) ∗ St−1 (w)) St−1 (y m )). Hence we conclude Theorem 1. 3. Cyclic sum formula 3.1. Algebraic setup ⊗(n+1)
Let n be a positive integer. We denote an action of Ht on Ht defined by
by “”, which is
a (w1 ⊗ · · · ⊗ wn+1 ) = w1 ⊗ · · · ⊗ wn ⊗ awn+1 , (w1 ⊗ · · · ⊗ wn+1 ) b = w1 b ⊗ w2 ⊗ · · · ⊗ wn+1 . ⊗(n+1)
The action is a Ht -bimodule structure on Ht ⊗(n+1) the Q[t]-linear map Cn,t : Ht → Ht by
. For a positive integer n, we define
Cn,t (x) = −Cn,t (y) = x ⊗ ((1 − t)x + y)⊗(n−1) ⊗ y and the Leibniz rule Cn,t (vw) = Cn,t (v) σt−1 (w) + σt−1 (v) Cn,t (w)
T. Tanaka, N. Wakabayashi / Journal of Algebra 447 (2016) 424–431
⊗(n+1)
for any v, w ∈ Ht . Let Mn : Ht
429
→ Ht denote the multiplication map, i.e.,
Mn (w1 ⊗ · · · ⊗ wn+1 ) = w1 · · · wn+1 , and let ρn,t = Mn Cn,t . We also denote by Hˇ1t a subvector space of H1t generated by words of H1t except for powers of y. 3.2. Proof of Theorem 2 By the definition, we find that ρn,0 = St ρn,t .
(6)
According to [10, Proposition 2.5], we have ρn,0 (Hˇ10 ) ⊂ ϕ(Hy ∗ Hy) y. By (3), (4), (5) and (6), ρn,t (Hˇ10 ) ⊂ St−1 (ϕ(Hy ∗ Hy) y) t
= St−1 ϕ(Hy ∗ Hy) y t
= −ϕt (St−1 (Hy ∗ Hy)) y t
t
= −ϕt (St−1 (Hy) ∗ St−1 (Hy)) y. This shows t t ρn,t (Hˇ1t ) ⊂ ϕt (Ht y ∗ Ht y) y.
Therefore Theorem 1 for m = 1 implies the following proposition. Proposition 4. For any positive integer n, we have ρn,t (Hˇ1t ) ⊂ ker Zt . If n = 1, we calculate C1,t (σt (zk1 · · · zkl )) = C1,t (xk1 −1 (tx + y) · · · xkl −1 (tx + y)) =
l k i −1
σt−1 (xk1 −1 (tx + y) · · · xki−1 −1 (tx + y)xj−1 ) C1,t (x)
i=1 j=1
σt−1 (xki −j−1 (tx + y) · · · xkl −1 (tx + y))
T. Tanaka, N. Wakabayashi / Journal of Algebra 447 (2016) 424–431
430
+
l
σt−1 (xk1 −1 (tx + y) · · · xki −1 (tx + y)) C1,t (tx + y)
i=1
σt−1 (xki+1 −1 (tx + y) · · · xkl −1 (tx + y)) =
l k i −1
xki −j yxki+1 −1 y · · · xkl −1 y ⊗ xk1 −1 y · · · xki−1 −1 yxj−1 y
i=1 j=1
+ (t − 1)
l
x · xki+1 −1 y · · · xkl −1 y ⊗ xk1 −1 y · · · xki −1 y,
i=1
C1,t (xk ) =
k
xi ⊗ xk−i y.
i=1
Hence we have ρ1,t (σt (zk1 · · · zkl ) − tl xk ) =
l k i −1
zki −j+1 zki+1 · · · zkl zk1 · · · zki−1 zj
i=1 j=1
+ (t − 1)
l
zki +1 zki+1 · · · zkl zk1 · · · zki−1 − ktl zk+1 .
i=1
When k1 + · · · + kl = k with kq > 1 for some q, σt (zk1 · · · zkl ) − tl xk is an element in Hˇ1t . Therefore we have Theorem 2 by Proposition 4. Acknowledgments The first author is supported by Kyoto Sangyo University Research Grants E1405. The second author is supported by the Japan Society for the Promotion of Science, Grant-in-Aid for Young Scientists (B) 15K17523. References [1] V.G. Drinfel’d, On quasitriangular quasi-Hopf algebras and a group closely connected with Gal(Q/Q), Leningrad Math. J. 2 (1991) 829–860. [2] M. Hoffman, The algebra of multiple harmonic series, J. Algebra 194 (1997) 477–495. [3] M. Hoffman, Y. Ohno, Relations of multiple zeta values and their algebraic expression, J. Algebra 262 (2003) 332–347. [4] K. Ihara, M. Kaneko, D. Zagier, Derivation and double shuffle relations for multiple zeta values, Compos. Math. 142 (2006) 307–338. [5] G. Kawashima, A class of relations among multiple zeta values, J. Number Theory 129 (4) (2009) 755–788. [6] Y. Ohno, N. Wakabayashi, Cyclic sum of multiple zeta values, Acta Arith. 123 (2006) 289–295. [7] G. Racinet, Doubles melanges des polylogarithmes multiples aux racines de l’unite, Publ. Math. Inst. Hautes Études Sci. 95 (2002) 185–231.
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[8] T. Tanaka, On the quasi-derivation relation for multiple zeta values, J. Number Theory 129 (9) (2009) 2021–2034. [9] T. Tanaka, Algebraic interpretation of Kawashima relation for multiple zeta values and its applications, RIMS Kôkyûroku Bessatsu B19 (2010) 117–134. [10] T. Tanaka, N. Wakabayashi, An algebraic proof of the cyclic sum formula for multiple zeta values, J. Algebra 323 (2010) 766–778. [11] S. Yamamoto, Interpolation of multiple zeta and zeta-star values, J. Algebra 385 (2013) 102–114.
Contents lists available at ScienceDirect
Journal of Algebra www.elsevier.com/locate/jalgebra
Kawashima’s relations for interpolated multiple zeta values Tatsushi Tanaka a,∗ , Noriko Wakabayashi b a
Department of Mathematics, Faculty of Science, Kyoto Sangyo University, Motoyama Kamigamo, Kita-ku, Kyoto-city, Kyoto 603-8555, Japan b College of Science and Engineering, Ritsumeikan University, 1-1-1, Nojihigashi, Kusatsu-city, Shiga 525-8577, Japan
a r t i c l e
i n f o
Article history: Received 4 October 2014 Available online xxxx Communicated by Michel Van den Bergh Keywords: t-MZVs Harmonic products t-Kawashima relations Cyclic sum formula
a b s t r a c t Recently, Yamamoto introduced polynomials in one variable t which interpolates multiple zeta and zeta-star values (t-MZVs for short), provided new prospects on two-one conjecture of Ohno and Zudilin and proved the cyclic sum formula for t-MZVs. In this paper, we establish a generalization of Kawashima’s relations (t-Kawashima relations) for t-MZVs. We prove the cyclic sum formula for t-MZVs using a type of derivation operator, together with the t-Kawashima relations. © 2015 Elsevier Inc. All rights reserved.
1. Introduction Multiple zeta and zeta-star values are defined, for positive integers k1 , . . . , kl with k1 ≥ 2, by convergent series ζ(k1 , . . . , kl ) =
k1 m1 >···>ml >0 m1
1 · · · mkl l
, ζ (k1 , . . . , kl ) =
mk11 m1 ≥···≥ml ≥1
* Corresponding author. E-mail addresses: [email protected] (T. Tanaka), [email protected] (N. Wakabayashi). http://dx.doi.org/10.1016/j.jalgebra.2015.09.015 0021-8693/© 2015 Elsevier Inc. All rights reserved.
1 · · · mkl l
,
T. Tanaka, N. Wakabayashi / Journal of Algebra 447 (2016) 424–431
425
abbreviated as MZVs and MZSVs, respectively. Decomposing the region of the sum of the defining series of MZSVs into 2n−1 regions by separating each ‘≥’ into ‘>’ and ‘=’, we immediately find the (symbolic) invertible linear transformation between MZVs and MZSVs. For example, ζ (k1 , k2 ) = ζ(k1 , k2 ) + ζ(k1 + k2 ), ζ (k1 , k2 , k3 ) = ζ(k1 , k2 , k3 ) + ζ(k1 + k2 , k3 ) + ζ(k1 , k2 + k3 ) + ζ(k1 + k2 + k3 ). Let t be a variable. In [11] Yamamoto introduced the following interpolation polynomial of MZVs and MZSVs. ζ t (k1 , . . . , kl ) =
tl−dep(p) ζ(p) (∈ R[t]),
(1)
p
where dep(p) is the depth (length) of p and p runs over all indices of the form p = (k1 · · · kl ) in which each is filled by the comma, or the plus +. We call this polynomial t-MZV. Note that ζ 0 = ζ (MZV) and ζ 1 = ζ (MZSV). In this paper, we discuss certain quadratic formula which contains the cyclic sum formula for t-MZVs. In Section 2, we show the quadratic formula for t-MZVs as follows. Notations are described in next two sections. Theorem 1. For any m ≥ 1 and any v, w ∈ Ht y, we have
t
t
Zt (ϕt (v) (−tx + y)p−1 y)Zt (ϕt (w) (−tx + y)q−1 y)
p+q=m p,q≥1 t
t
= −Zt (ϕt (v ∗ w) (−tx + y)m−1 y). When t = 0 and t = 1, these formulas for MZVs and MZSVs are called Kawashima relations which first appeared in [5] (also see [8,9]). Kawashima relations are of importance since those for MZVs are expected to give whole relations among MZVs as well as extended double shuffle relations [4,7] or associator relations [1]. As an application, the following cyclic sum formula for t-MZVs is again proved. (The original appearance of the cyclic sum formula for MZVs and MZSVs are proved respectively in [3] and [6].)
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T. Tanaka, N. Wakabayashi / Journal of Algebra 447 (2016) 424–431
Theorem 2. (See [11, Theorem 5.4].) Let k1 , . . . , kl be positive integers and not all 1. Put k = k1 + · · · + kl . Then we have l k i −1
ζ t (ki − j + 1, ki+1 , . . . , kl , k1 , . . . , ki−1 , j)
i=1 j=1
= (1 − t)
l
ζ t (ki + 1, ki+1 , . . . , kl , k1 , . . . , ki−1 ) + kζ(k + 1)tl .
i=1
This theorem is proved in Section 3. 2. Kawashima’s relations for t-MZVs 2.1. Algebraic setup We begin with the algebraic setup introduced for example in [2,11]. Let Ht = Q[t]x, y denote the non-commutative polynomial algebra over Q[t] in two indeterminates x and y, and let H1t and H0t denote the subalgebras Q[t] + Ht y and Q[t] + xHt y, respectively. We denote H0 (= Qx, y), H10 , H00 simply by H, H1 , H0 , respectively. Put zk = xk−1 y for k ≥ 1. We define the Q[t]-linear map Zt : H0t → R[t] by Zt (1) = 1 and Zt (zk1 · · · zkl ) = ζ t (k1 , . . . , kl ). The Q[t]-linear map St : H1t → H1t is defined by St (wy) = σt (w)y (w ∈ Ht ), where σt is an automorphism on Ht characterized by σt (x) = x, σt (y) = tx + y. This operator St is nothing but the operator S t introduced in [11, Definition 3.1]. Then we find that St is bijective (in fact St−1 = S−t ) and Zt = Z0 St .
(2)
ϕt = −St−1 ϕSt ,
(3)
We let
where ϕ denotes the automorphism on H characterized by ϕ(x) = x + y, ϕ(y) = −y. Note that ϕ0 = −ϕ.
T. Tanaka, N. Wakabayashi / Journal of Algebra 447 (2016) 424–431
427
As is introduced in [2], the harmonic product ∗ on H1 is defined, for positive integers p, q and v, w ∈ H1 , recursively by 1 ∗ w = w ∗ 1 = w, zp v ∗ zq w = zp (v ∗ zq w) + zq (zp v ∗ w) + zp+q (v ∗ w) t
and the Q-bilinearity. We also define the t-harmonic product ∗ on H1t by t
v ∗ w = St−1 (St (v) ∗ St (w)) (v, w ∈ H1t ),
(4)
where the harmonic product ∗ on the right hand side is regarded as a Q[t]-bilinear map. t
(In [11, Definition 3.7], the product ∗ is defined recursively as above for the harmonic t
product ∗. The property (4) we adopt as the definition of ∗ here is proved afterwards.) Because of (2), (4) and the fact that the map Z0 is a homomorphism with respect to the harmonic product ∗, we find that the map Zt is a homomorphism with respect to the t harmonic product ∗, i.e., t
Zt (v ∗ w) = Zt (v)Zt (w) (v, w ∈ H0t ). The product on H1 is defined by zp v zq w = zp+q (v ∗ w) t
for v, w ∈ H1 . Using , we define the product on H1t by t
v w = St−1 (St (v) St (w)) (v, w ∈ H1t ),
(5)
where, as above, the harmonic product ∗ and hence the product on the right hand side are regarded as Q[t]-bilinear maps. Remark 3. It is off the subjects, but we notice that the product represents the rule of the harmonic product formula for truncated multiple zeta values ζm (k1 , . . . , kl ) =
k1 m+1=m1 >···>ml >0 m1
1 · · · mkl l t
for any positive integer m. Therefore we also notice that the product represents the rule of the harmonic product formula for an interpolation of truncated multiple zeta values t ζm (k1 , . . . , kl ) =
p
which is defined as well as (1).
tl−dep(p) ζm (p)
428
T. Tanaka, N. Wakabayashi / Journal of Algebra 447 (2016) 424–431
2.2. Proof of Theorem 1 Now we show that Kawashima’s relations for t-MZVs (Theorem 1) follows straightforwardly from the case t = 0 which is a result of [5]. When t = 0, we have
Z0 (ϕ(v) y p )Z0 (ϕ(w) y q ) = Z0 (ϕ(v ∗ w) y m )
p+q=m p,q≥1
for any positive integer m and any v, w ∈ Hy. By (2), (3) and (5),
LHS =
Zt St−1 (St ϕt St−1 (v) y p )Zt St−1 (St ϕt St−1 (w) y q )
p+q=m p,q≥1
=
t
t
Zt (ϕt St−1 (v) St−1 (y p ))Zt (ϕt St−1 (w) St−1 (y q )).
p+q=m p,q≥1
By (2), (3), (4) and (5), RHS = −Zt St−1 (St ϕt St−1 (v ∗ w) y m ) t
t
= −Zt (ϕt (St−1 (v) ∗ St−1 (w)) St−1 (y m )). Hence we conclude Theorem 1. 3. Cyclic sum formula 3.1. Algebraic setup ⊗(n+1)
Let n be a positive integer. We denote an action of Ht on Ht defined by
by “”, which is
a (w1 ⊗ · · · ⊗ wn+1 ) = w1 ⊗ · · · ⊗ wn ⊗ awn+1 , (w1 ⊗ · · · ⊗ wn+1 ) b = w1 b ⊗ w2 ⊗ · · · ⊗ wn+1 . ⊗(n+1)
The action is a Ht -bimodule structure on Ht ⊗(n+1) the Q[t]-linear map Cn,t : Ht → Ht by
. For a positive integer n, we define
Cn,t (x) = −Cn,t (y) = x ⊗ ((1 − t)x + y)⊗(n−1) ⊗ y and the Leibniz rule Cn,t (vw) = Cn,t (v) σt−1 (w) + σt−1 (v) Cn,t (w)
T. Tanaka, N. Wakabayashi / Journal of Algebra 447 (2016) 424–431
⊗(n+1)
for any v, w ∈ Ht . Let Mn : Ht
429
→ Ht denote the multiplication map, i.e.,
Mn (w1 ⊗ · · · ⊗ wn+1 ) = w1 · · · wn+1 , and let ρn,t = Mn Cn,t . We also denote by Hˇ1t a subvector space of H1t generated by words of H1t except for powers of y. 3.2. Proof of Theorem 2 By the definition, we find that ρn,0 = St ρn,t .
(6)
According to [10, Proposition 2.5], we have ρn,0 (Hˇ10 ) ⊂ ϕ(Hy ∗ Hy) y. By (3), (4), (5) and (6), ρn,t (Hˇ10 ) ⊂ St−1 (ϕ(Hy ∗ Hy) y) t
= St−1 ϕ(Hy ∗ Hy) y t
= −ϕt (St−1 (Hy ∗ Hy)) y t
t
= −ϕt (St−1 (Hy) ∗ St−1 (Hy)) y. This shows t t ρn,t (Hˇ1t ) ⊂ ϕt (Ht y ∗ Ht y) y.
Therefore Theorem 1 for m = 1 implies the following proposition. Proposition 4. For any positive integer n, we have ρn,t (Hˇ1t ) ⊂ ker Zt . If n = 1, we calculate C1,t (σt (zk1 · · · zkl )) = C1,t (xk1 −1 (tx + y) · · · xkl −1 (tx + y)) =
l k i −1
σt−1 (xk1 −1 (tx + y) · · · xki−1 −1 (tx + y)xj−1 ) C1,t (x)
i=1 j=1
σt−1 (xki −j−1 (tx + y) · · · xkl −1 (tx + y))
T. Tanaka, N. Wakabayashi / Journal of Algebra 447 (2016) 424–431
430
+
l
σt−1 (xk1 −1 (tx + y) · · · xki −1 (tx + y)) C1,t (tx + y)
i=1
σt−1 (xki+1 −1 (tx + y) · · · xkl −1 (tx + y)) =
l k i −1
xki −j yxki+1 −1 y · · · xkl −1 y ⊗ xk1 −1 y · · · xki−1 −1 yxj−1 y
i=1 j=1
+ (t − 1)
l
x · xki+1 −1 y · · · xkl −1 y ⊗ xk1 −1 y · · · xki −1 y,
i=1
C1,t (xk ) =
k
xi ⊗ xk−i y.
i=1
Hence we have ρ1,t (σt (zk1 · · · zkl ) − tl xk ) =
l k i −1
zki −j+1 zki+1 · · · zkl zk1 · · · zki−1 zj
i=1 j=1
+ (t − 1)
l
zki +1 zki+1 · · · zkl zk1 · · · zki−1 − ktl zk+1 .
i=1
When k1 + · · · + kl = k with kq > 1 for some q, σt (zk1 · · · zkl ) − tl xk is an element in Hˇ1t . Therefore we have Theorem 2 by Proposition 4. Acknowledgments The first author is supported by Kyoto Sangyo University Research Grants E1405. The second author is supported by the Japan Society for the Promotion of Science, Grant-in-Aid for Young Scientists (B) 15K17523. References [1] V.G. Drinfel’d, On quasitriangular quasi-Hopf algebras and a group closely connected with Gal(Q/Q), Leningrad Math. J. 2 (1991) 829–860. [2] M. Hoffman, The algebra of multiple harmonic series, J. Algebra 194 (1997) 477–495. [3] M. Hoffman, Y. Ohno, Relations of multiple zeta values and their algebraic expression, J. Algebra 262 (2003) 332–347. [4] K. Ihara, M. Kaneko, D. Zagier, Derivation and double shuffle relations for multiple zeta values, Compos. Math. 142 (2006) 307–338. [5] G. Kawashima, A class of relations among multiple zeta values, J. Number Theory 129 (4) (2009) 755–788. [6] Y. Ohno, N. Wakabayashi, Cyclic sum of multiple zeta values, Acta Arith. 123 (2006) 289–295. [7] G. Racinet, Doubles melanges des polylogarithmes multiples aux racines de l’unite, Publ. Math. Inst. Hautes Études Sci. 95 (2002) 185–231.
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