Note on Dilogarithm Identities from Nilpotent Double Affine Hecke Algebras
Recently Cherednik and Feigin [arXiv:1209.1978] obtained several Rogers-Ramanujan type identities via the nilpotent double affine Hecke algebras (Nil-DAHA). These identities further led to a series of dilogarithm identities, some of which are known, while some are left conjectural. We confirm and ex...
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| citation_txt | Note on Dilogarithm Identities from Nilpotent Double Affine Hecke Algebras / T. Nakanishi // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 9 назв. — англ. |
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| description | Recently Cherednik and Feigin [arXiv:1209.1978] obtained several Rogers-Ramanujan type identities via the nilpotent double affine Hecke algebras (Nil-DAHA). These identities further led to a series of dilogarithm identities, some of which are known, while some are left conjectural. We confirm and explain all of them by showing the connection with Y-systems associated with (untwisted and twisted) quantum affine Kac-Moody algebras.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 8 (2012), 104, 5 pages
Note on Dilogarithm Identities
from Nilpotent Double Affine Hecke Algebras
Tomoki NAKANISHI
Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, 464-8604, Japan
E-mail: nakanisi@math.nagoya-u.ac.jp
Received November 15, 2012, in final form December 22, 2012; Published online December 25, 2012
http://dx.doi.org/10.3842/SIGMA.2012.104
Abstract. Recently Cherednik and Feigin [arXiv:1209.1978] obtained several Rogers–Ra-
manujan type identities via the nilpotent double affine Hecke algebras (Nil-DAHA). These
identities further led to a series of dilogarithm identities, some of which are known, while
some are left conjectural. We confirm and explain all of them by showing the connection with
Y -systems associated with (untwisted and twisted) quantum affine Kac–Moody algebras.
Key words: double affine Hecke algebra; dilogarithm; Y -system
2010 Mathematics Subject Classification: 17B37; 13F60
1 Dilogarithm identities from Nil-DAHA
Let Rn be a root system of finite type and of rank n with non-degenerate bilinear form ( , ),
and let αi and ωi be the simple roots and the fundamental weights of Rn. The Cartan matrix
C = (cij)
n
i,j=1 is given by cij = 2(αi, αj)/(αi, αi). Following [3], let A = (aij) and A[ = (a[ij)
be the matrices with aij = 2(ωi, ωj) and a
[
ij = (ωi, ωj), respectively. Set νi = (αi, αi)/2. Then,
ν−1i (αi, ωj) = δij , and we have(
A[
)−1
=
(
cijν
−1
j
)n
i,j=1
. (1)
Below we normalize the bilinear form as (αshort, αshort) = 2 so that νi ∈ {1, 2, 3}.
Let L(x) be the Rogers dilogarithm function
L(x) = −1
2
∫ x
0
{
log(1− y)
y
+
log y
1− y
}
dy.
In [3, equation (3.34)] Cherednik and Feigin presented two (partially conjectural) series of di-
logarithm identities. Let A′ = (a′ij)
n
i,j=1 be either A or A[ as above. Let Qi (i = 1, . . . , n) be the
unique solution of the system of equations
(1−Qi)νi =
n∏
j=1
Q
a′ij
j (2)
in the range 0 < Qi < 1. Then, the following identity was proposed
6
π2
n∑
i=1
νiL(Qi) = LA′ , (3)
where the value LA′ is the rational number given in Table 1.
In addition, there are identities for ‘type Tn’ (tadpole type). We define the ‘Cartan mat-
rix’ C as almost the same as type An except that the last diagonal entry is 1, not 2. Also the
mailto:nakanisi@math.nagoya-u.ac.jp
http://dx.doi.org/10.3842/SIGMA.2012.104
2 T. Nakanishi
Table 1. The value LA′ .
Rn An Bn Cn Dn E6 E7 E8 F4 G2 Tn
LA
n(n+1)
n+3
n(2n−1)
n+1 n n− 1 36
7
63
10
15
2
36
7 3 n(2n+1)
2n+3
LA[
n(n+1)
n+4
2n(2n−1)
2n+3
2n(n+1)
2n+3
2(n−1)n
2n+1
24
5 6 80
11
24
5
8
3
n(2n+1)
2n+4
matrices A and A[ are defined by aij = 2min(i, j) and a[ij = min(i, j), respectively; the latter
is the same matrix A[ for type Cn. Then, (A[)−1 = C holds. We set νi = 1. Again, (3) should
hold for the value LA′ in Table 1, where Tn is formally included as a member of Rn.
In [3] these identities were partially obtained and generally motivated by the Rogers–Rama-
nujan type identities arising from nilpotent affine Hecke algebras (Nil-DAHA), but only some
of them are identified with the known identities.
The authors of [3] expected the connection between (3) and dilogarithm identities from
some Y -systems (and cluster algebras behind them). In this note we answer this question
affirmatively, and, in particular, we confirm all the identities in question. The note has con-
siderable overlap with the paper by Lee [8], but it is written for a different purpose and in
a different perspective.
2 Dilogarithm identities for Y -systems of simply laced type
Let us recall the following dilogarithm identities proved by cluster algebra method [9]. For ` = 2
see also [2].
Let C be any Cartan matrix of simply laced type Rn = An, Dn, E6, E7, E8, and let ` ≥ 2
be any integer (called the level). Let Y
(i)
m (i = 1, . . . , n; m = 1, . . . , ` − 1) be the unique real
positive solution of the system of equations
(
Y (i)
m
)2
=
n∏
j=1
(
1 + Y
(j)
m
)2δij−cij
(
1 + Y
(i)
m−1
−1
)(
1 + Y
(i)
m+1
−1
) , (4)
where Y
(i)
0
−1 = Y
(i)
`
−1 = 0.
Theorem 1 ([9, Corollay 1.9]). The following identity holds
6
π2
n∑
i=1
`−1∑
m=1
L
(
Y
(i)
m
1 + Y
(i)
m
)
=
(`− 1)nh
h+ `
, (5)
where h is the Coxeter number of type Rn, i.e., n + 1, 2n − 2, 12, 18, 30 for An, Dn, E6, E7,
E8, respectively.
The system of equations (4) is called the level ` constant Y -system associated with the
quantum affine Kac–Moody algebra of (untwisted) type R
(1)
n , which is a specialization of the cor-
responding (non-constant) Y -system. It is also known as the (constant) Y -system of Rn ×A`−1.
See [6] for more information.
We explain below that all the identities (3) in question are the ones in (5) for ` = 2, 3, or
their specializations.
Note on Dilogarithm Identities 3
Table 2. The Langlands dual of affine type.
R
(1)
n A
(1)
n B
(1)
n C
(1)
n D
(1)
n E
(1)
6 E
(1)
7 E
(1)
8 F
(1)
4 G
(1)
2
S
(r)
m A
(1)
n A
(2)
2n−1 D
(2)
n+1 D
(1)
n E
(1)
6 E
(1)
7 E
(1)
8 E
(2)
6 D
(3)
4
3 Identification with Y -systems
from quantum affine Kac–Moody algebras
3.1 The non-[ case
We consider the case A′ = A. We use the change of variables
Qi =
Yi
1 + Yi
, (6)
so that the range 0 < Qi < 1 corresponds to the range 0 < Yi. Then, using (1), one can
transform the equations (2) into the form
n∏
j=1
(
1
1 + Yj
)cij
=
(
Yi
1 + Yi
)2
,
which is equivalent to
Y 2
i =
n∏
j=1
(1 + Yj)
2δij−cij . (7)
For simply laced typesAn,Dn, E6, E7, E8, (7) coincides with the level 2 constant Y -system (4)
of untwisted type R
(1)
n by identifying Yi with Y
(i)
1 . Then, the right hand side of (5) with ` = 2
gives the value of LA, agreeing with Table 1.
In contrast, for types Bn, Cn, F4, G2, (7) coincides with the level 2 constant Y -system of
twisted type S
(r)
m (in the sense of [6, Remark 9.22]), where S
(r)
m is the Langlands dual of R
(1)
n .
See Table 2 for the Langlands dual of affine type. Also see [6, Section 9] for the full version
of Y -systems of twisted type. In this case, the direct inspection of the Cartan matrix shows
that the equation (7) can be obtained from the level 2 constant Y -system of the untwisted (and
simply laced) type S
(1)
m by folding, i.e., identifying the variables with the diagram automor-
phism σ of Sm. This is possible, due to the symmetry Y
(i)
m ↔ Y
(σ(i))
m of the Y -system (4).
Furthermore, it is easy to see that νi coincides with the number of elements in the σ-orbit of i.
Thus, we obtain the identity (3) for type Rn with LA(Rn) = LA(Sm). For example, for Rn = Bn,
LA(Bn) = LA(A2n−1). This confirms and explains Table 1.
Finally, for type Tn, (7) coincides with the level 2 constant Y -system of type A
(2)
2n (in the
sense of [6, Remark 9.22]). Note that A
(2)
2n is self-dual under the Langlands duality. Again, this
Y -system is obtained by the folding of level 2 constant Y -system of untwisted A
(1)
2n . Since we set
νi = 1, the multiplicities are discarded in (3). Therefore, LA(Tn) = LA(A2n)/2. Actually, this
connection is known in [3] and other literature.
3.2 The [ case
We consider the case A′ = A[. By the same change of variables (6), one can transform the
equations (2) into the form
n∏
j=1
(
1
1 + Yj
)cij
=
(
Yi
1 + Yi
)
=
(
Yi
1 + Yi
)2 (
1 + Yi
−1),
4 T. Nakanishi
which is equivalent to
Y 2
i =
n∏
j=1
(1 + Yj)
2δij−cij
1 + Yi−1
. (8)
For simply laced types, An, Dn, E6, E7, E8, (8) is obtained from the level 3 constant Y -
system (4) of untwisted type R
(1)
n by the specialization Y
(i)
1 = Y
(i)
2 and identifying it with Yi.
This is possible, due to the symmetry Y
(i)
1 ↔ Y
(i)
2 of level 3 Y -system (4). (One can also view it
as the folding of A2 to T1 in the second component of Rn×A2.) Since we discard the multiplicity
in (3), LA[ is the half of the right hand side of (5) with ` = 3. This agrees with Table 1.
Similarly, for the rest of types, (8) is obtained from the level 3 constant Y -system of type S
(r)
m
or A
(2)
2n (in the sense of [6, Remark 9.22]) by the specialization Y
(i)
1 = Y
(i)
2 , and the latter is
further obtained from the level 3 constant Y -system of type S
(1)
m or A
(1)
2n by the folding. Then,
one can confirm Table 1.
Let us summarize the result.
Theorem 2. The identity (3) holds. Moreover, except for type Tn, the value LA′ in (3) has
a unified expression
LA′ =
mh∗
h∗ + `
,
where ` = 2 for A′ = A and ` = 3 for A′ = A[, and m and h∗ are the rank and Coxeter number
of Sm for the Langlands dual S
(r)
m of R
(1)
n .
We remark that the dilogarithm identities for untwisted and nonsimply laced types B
(1)
n ,
C
(1)
n , F
(1)
4 , G
(1)
2 are also known [4, 5]. It is natural to ask whether they will also appear from
Nil-DAHA.
4 Connection to cluster algebraic method
For the reader’s convenience, we include a brief explanation of the background of the dilogarithm
identity (5), especially in the cluster algebraic method. See [7] and references therein for more
information.
(a) Y -systems and dilogarithm identities. As the name suggests, the constant Y -system (4)
is the constant version of the following (non-constant) Y -system
Y (i)
m (u+ 1)Y (i)
m (u− 1) =
n∏
j=1
(
1 + Y
(j)
m (u)
)2δij−cij
(
1 + Y
(i)
m−1(u)
−1
)(
1 + Y
(i)
m+1(u)
−1
) , (9)
where the variables Y
(i)
m (u) now carry the spectral parameter u ∈ C. The Y -system (9) appears in
the thermodynamic Bethe ansatz (TBA) analysis of the deformation of conformal field theory.
A constant solution Y
(i)
m := Y
(i)
m (u) of (9), which is constant with respect to u, satisfies the
constant Y -system (4), that also appears in the TBA analysis to calculate the effective central
charge of conformal field theory.
The Y -system (9) has the following two remarkable properties.
(i) Periodicity
Y (i)
m (u+ 2(h+ `)) = Y (i)
m (u). (10)
Note on Dilogarithm Identities 5
(ii) Dilogarithm identity. For any positive real solution of (9), the following identity holds
6
π2
2(h+`)−1∑
u=0
n∑
i=1
`−1∑
m=1
L
(
Y
(i)
m (u)
1 + Y
(i)
m (u)
)
= 2(`− 1)nh. (11)
The identity (5) is obtained by specializing the identity (11) to the (unique) positive real
constant solution, and dividing the both hand sides of (5) by the period 2(h+ `).
(b) Cluster algebras and dilogarithm identities. It was once formidable to prove the proper-
ties (10) and (11) in full generality. However, they are now proved and rather well understood by
the cluster algebraic method. In general, to any period of y-variables (coefficients) of a cluster
algebra, the following dilogarithm identity is associated
6
π2
p∑
t=1
L
(
ykt(t)
1 + ykt(t)
)
= N−. (12)
where k1, . . . , kp are the sequence of mutations for which y-variables are periodic, ykt(t) is the
y-variable mutated at t, and N− is the total number of t ∈ {1, . . . , p} such that the tropical sign
of ykt(t) is minus. One can apply this general result to our Y -system (9). First, the Y -system is
embedded into the y-variables of a certain cluster algebra. Then, by proving the periodicity of
y-variables of this cluster algebra, we obtain the periodicity (10) of the Y -system. Finally, from
the general identity (12), we obtain the dilogarithm identity (11) by calculating the constant
term N−.
(c) Quantum cluster algebras and quantum dilogarithm identities. One can lift the result in (b)
to the quantum case. Namely, any period of y-variables of a cluster algebra can be lifted to the
period of quantum y-variables of the corresponding quantum cluster algebra. Then, to any such
period, the quantum dilogarithm identity is associated; furthermore, taking the semiclassical
limit of the quantum dilogarithm identity we recover the classical dilogarithm identity (12). We
expect that the Rogers–Ramanujan type identities of [3] are also deduced from these quantum
dilogarithm identities. Some result in this direction is obtained by [1].
Acknowledgements
I thank Ivan Cherednik for raising this interesting question to me.
References
[1] Cecotti S., Neitzke A., Vafa C., R-twisting and 4d/2d correspondences, arXiv:1006.3435.
[2] Chapoton F., Functional identities for the Rogers dilogarithm associated to cluster Y -systems, Bull. London
Math. Soc. 37 (2005), 755–760.
[3] Cherednik I., Feigin B., Rogers–Ramanujan type identities and Nil-DAHA, arXiv:1209.1978 (especially
version 2).
[4] Inoue R., Iyama O., Keller B., Kuniba A., Nakanishi T., Periodicities of T - and Y -systems, dilogarithm
identities, and cluster algebras I: Type Br, Publ. Res. Inst. Math. Sci., to appear, arXiv:1001.1880.
[5] Inoue R., Iyama O., Keller B., Kuniba A., Nakanishi T., Periodicities of T - and Y -systems, dilogarithm iden-
tities, and cluster algebras II: Types Cr, F4, and G2, Publ. Res. Inst. Math. Sci., to appear, arXiv:1001.1881.
[6] Inoue R., Iyama O., Kuniba A., Nakanishi T., Suzuki J., Periodicities of T -systems and Y -systems, Nagoya
Math. J. 197 (2010), 59–174, arXiv:0812.0667.
[7] Kashaev R.M., Nakanishi T., Classical and quantum dilogarithm identities, SIGMA 7 (2011), 102, 29 pages,
arXiv:1104.4630.
[8] Lee C.H., Nahm’s conjecture and Y -system, arXiv:1109.3667.
[9] Nakanishi T., Dilogarithm identities for conformal field theories and cluster algebras: simply laced case,
Nagoya Math. J. 202 (2011), 23–43, arXiv:0909.5480.
http://arxiv.org/abs/1006.3435
http://dx.doi.org/10.1112/S0024609305004510
http://dx.doi.org/10.1112/S0024609305004510
http://arxiv.org/abs/1209.1978
http://arxiv.org/abs/1001.1880
http://arxiv.org/abs/1001.1881
http://arxiv.org/abs/0812.0667
http://dx.doi.org/10.3842/SIGMA.2011.102
http://arxiv.org/abs/1104.4630
http://arxiv.org/abs/1109.3667
http://arxiv.org/abs/0909.5480
1 Dilogarithm identities from Nil-DAHA
2 Dilogarithm identities for Y-systems of simply laced type
3 Identification with Y-systems from quantum affine Kac-Moody algebras
3.1 The non- case
3.2 The case
4 Connection to cluster algebraic method
References
|
| id | nasplib_isofts_kiev_ua-123456789-149184 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T17:46:39Z |
| publishDate | 2012 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Nakanishi, T. 2019-02-19T18:21:40Z 2019-02-19T18:21:40Z 2012 Note on Dilogarithm Identities from Nilpotent Double Affine Hecke Algebras / T. Nakanishi // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 9 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 17B37; 13F60 DOI: http://dx.doi.org/10.3842/SIGMA.2012.104 https://nasplib.isofts.kiev.ua/handle/123456789/149184 Recently Cherednik and Feigin [arXiv:1209.1978] obtained several Rogers-Ramanujan type identities via the nilpotent double affine Hecke algebras (Nil-DAHA). These identities further led to a series of dilogarithm identities, some of which are known, while some are left conjectural. We confirm and explain all of them by showing the connection with Y-systems associated with (untwisted and twisted) quantum affine Kac-Moody algebras. I thank Ivan Cherednik for raising this interesting question to me. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Note on Dilogarithm Identities from Nilpotent Double Affine Hecke Algebras Article published earlier |
| spellingShingle | Note on Dilogarithm Identities from Nilpotent Double Affine Hecke Algebras Nakanishi, T. |
| title | Note on Dilogarithm Identities from Nilpotent Double Affine Hecke Algebras |
| title_full | Note on Dilogarithm Identities from Nilpotent Double Affine Hecke Algebras |
| title_fullStr | Note on Dilogarithm Identities from Nilpotent Double Affine Hecke Algebras |
| title_full_unstemmed | Note on Dilogarithm Identities from Nilpotent Double Affine Hecke Algebras |
| title_short | Note on Dilogarithm Identities from Nilpotent Double Affine Hecke Algebras |
| title_sort | note on dilogarithm identities from nilpotent double affine hecke algebras |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/149184 |
| work_keys_str_mv | AT nakanishit noteondilogarithmidentitiesfromnilpotentdoubleaffineheckealgebras |