Infinitesimal and tangent to polylogarithmic complexes for higher weight

1.   Introduction

The classical polylogarithms represented by $Li_n$ are one valued functions on a complex plane (see ^[11]). They are called generalization of natural logarithms, which can be represented by an infinite series (power series):

The other versions of polylogarithms are Infinitesimal (see ^[8]) and Tangential (see ^[9]). We will discuss group theoretic form of infinitesimal and tangential polylogarithms in § 2.3, 2.4 and 2.5 below.

Dupont and Sah describe the connection between scissors congruence group and classical dilogarithm (polylogarithm for $n = 2$) (see ^[10]). Suslin (see ^[1]) defines the Bloch group that makes the famous Bloch-Suslin complex which is described in section 2.1 below. Zagier and Goncharov generalize the groups on which polylogarithmic functions are defined. This initiates a new era in the field of polylogarithms, arousing interest of algebraist and geometers. One of the milestones is the proof of Zagier's conjecture for weight $n = 2, 3$ (see ^[2,3]).

On the basis of this study Goncharov introduces a motivic complex (2.1) below, which is called Goncharov's complex (see ^[2]). On the other hand Cathelineau (^[7,8]) uses a differential process to introduce infinitesimal form of motivic (Bloch-Suslin's and Goncharov's) complexes that consists of $\Bbbk$-vector spaces. These $\Bbbk$-vector spaces are algebraic representation of infinitesimal versions of the Bloch-Suslin and Goncharov's complexes for higher weight $n$ (see ^[8]), which satisfies functional equations of infinitesimal polylogarithms. Cathelineau also uses a tangent functor to get the tangential analogue of the Bloch-Suslin complex, that allowing a new approach to view additive dialogalithms (regulator on $T\mathcal{B}_2(F)$) (see ^[9]). The tangent group $T\mathcal{B}_2(F)$ has two parts; first part comes from $\mathcal{B}_2(F)$ and the second part is the derivative of first part. He also suggests a framework for defining the additive trilogarithms.

Our work proposes an improved map (morphism), with the alternate signs between the $\Bbbk$-vector spaces that converts the sequence (2.3) into a complex. Further, we introduce a variant of infinitesimal $\Bbbk$-vector spaces which is structurally infinitesimal but has functional equations similar to classical polylogarithmic groups.

In § 3.1, we are also giving an inductive definition of group $T\mathcal{B}_n(\Bbbk)$ for higher weight $n$ and putting this in a complex with suitable maps that make a tangent complex (3.1) to Goncharov's (motivic) complex.

2.   Materials and method

2.1.   Bloch-Suslin complex

Let $\mathbb{Z}[\Bbbk]$ be a free abelian group generated by $[a]$ for $a\in F$. Suslin defines the following map

where $\wedge^2 \Bbbk^{\times} = \Bbbk^{\times}\otimes \Bbbk^{\times}/\left\langle \theta\otimes \theta, \theta\otimes \phi+\phi\otimes \theta|\quad \theta, \phi\in \Bbbk^{\times}\right\rangle$. The Bloch-Suslin complex is defined as

where $\mathcal{B}_2(\Bbbk)$ is the quotient of $\mathbb{Z}[{\Bbbk}]$ by the subetaoup generated by Abel's five term relation

and $\delta$ is induced by $\delta_2$. When $\Bbbk$ is algebraically closed with characteristic zero, the above complex can be inserted into the algebraic $K$-theory variant of the Bloch-Wigner sequence ^[9]

if this sequence is tensored by $\mathbb{Q}$ then

where $K$-groups $K_n^{(i)}$ are the pieces of the Adams decomposition of $K_n(\Bbbk)\otimes \mathbb{Q}$ (see ^[6]). The homology of Bloch-Suslin complex is the $K_n$-groups for $n = 2, 3$ i.e. $\wedge^2\Bbbk^{\times}/\text{Im}\delta \cong\Bbbk^{\times}\otimes \Bbbk^{\times}/\langle \theta\wedge(1-\theta)|\theta\in \Bbbk^{\times})\rangle$ and $B(F): = \ker \delta$ is called Bloch group, which is isomorphic to $K_3$ group (see ^[11]).

2.2.   Goncharov's (motivic) complexes

The free abelian group $\mathcal{B}_n(\Bbbk)$ is defined by Goncharov (see^[2]) as

with the morphisms for $n = 2$

for $n\geq 3$

where $[x]_n$ is the class of $x$ in $\mathcal{B}_n(\Bbbk)$. The subetaoup $\mathcal{R}_1(\Bbbk)$ of $\mathbb{Z}[\Bbbk]$ is generated by $[x+y-xy]-[x]-[y]$ and $\mathbb{Z}[\Bbbk]$ is a free abelian group generated by the symbol $[x]$ for $0, 1\neq x \in \Bbbk$, where $x, y\in \Bbbk\setminus\{1\}$ then $\mathcal{B}_1(\Bbbk)\cong \Bbbk^\times$. For $n = 2$, $\mathcal{R}_2(\Bbbk)$ is defined

The above relation is the Suslin's form of Abel's relations(^[11]). For $n\geq 2$, $\mathcal{A}_n(\Bbbk)$ is defined as the kernel of $\delta_n$ and $\mathcal{R}_n(\Bbbk)$ is the subetaoup of $\mathbb{Z}[\Bbbk]$ spanned by $[0]$ and the elements $\sum n_i\left([f_i(0)]-[f_i(1)]\right)$, where $f_i$ are rational fractions for indeterminate $T$, such that $\sum n_i[f_i]\in \mathcal{A}_n(\Bbbk(T))$.

Lemma 2.1. (Goncharov ^[2,3]) The following is the (cochain) complex

Proof. Proof requires direct calculation (we work here with modulo 2-torsion means $a\wedge a = 0$ and $a\wedge b = -b\wedge a$).

Example 2.2. For weight $n = 3$ the following is a complex

2.3.   Cathelineau's infinitesimal complexes

Let $\Bbbk$ be a field with a zero characteristic and $\Bbbk^{\bullet\bullet} = K-\{0, 1\}$, subspace $\beta_n(\Bbbk)$ is defined in ^[3,9] as

where $\rho_n(K)$ is the kernel of the following map

where $\langle \theta\rangle_n$ is the coset-class of $\theta$ in $\beta_n(\Bbbk)$ and $\rho_2(\Bbbk)$ generated by Cathelineau's relation,

For $n = 1$ we have $\beta_1(\Bbbk)\cong \Bbbk$.

Vector space $\beta_n(\Bbbk)$ has some non-trivial elements from the functional relations of $Li_n$ for $n\leq 7$ while one can find only inversion and distribution relations in $\beta_n(\Bbbk)$ for $n > 7$(see ^[11]).

The following is the Cathelineau's infinitesimal complex to the Goncharov's complex for weight $n$ (see §2 of ^[4] and ^[9]):

Example 2.3. For weight $n = 3$, the following infinitesimal version satisfying the definition of a complex:

2.4.   Variant of Cathelineau's complex

We put $\left[\kern-0.15em\left[ a \right]\kern-0.15em\right]^D = \frac{D(a)}{a(1-a)}[a]$ where $D(a)\in Der_{\mathbb{Z}}(\Bbbk, \Bbbk)$ and is called general derivation, $\beta^D_n(\Bbbk)$ is defined as

where $\rho_n^D(\Bbbk)$ is a kernel of the following map

and $\left[\kern-0.15em\left[ \theta \right]\kern-0.15em\right]_n^D$ is a class of $\theta$ in $\beta^D_n(\Bbbk)$ which is equal to $\frac{D(\theta)}{\theta(1-\theta)}\langle \theta\rangle_n$. The following is a subspace of $\Bbbk[\Bbbk^{\bullet\bullet}]$:

For $n\geq 4$, one can write only inversion relations in $\beta^D_n(\Bbbk)$ while for $n\leq 3$ we have other non-trivial relations as well. The following sequence is a complex. One can easily prove in a completely analogous way as Lemma 1

Example 2.4. This $D\log$ version of Cathelineau's complex is also satisfying the definition of a complex when the above maps are used for weight $n = 3$.

2.5.   Tangent to Bloch-Suslin complex

We represent a ring of dual numbers by $\Bbbk[\varepsilon]_2 = \Bbbk[\varepsilon]/\langle\varepsilon^2\rangle$ where $\Bbbk$ is algebraically closed field with zero characteristic. There is a $\Bbbk^\star$-action on $\Bbbk[\varepsilon]_2$ for $\lambda\in \Bbbk^\times$

For dual numbers $\Bbbk[\varepsilon]_2$, we define a free abelian group $\mathbb{Z}[\Bbbk[\varepsilon]_2]$ generated by $[\theta+\phi\varepsilon]$ for $\theta+\phi\varepsilon\in \Bbbk[\varepsilon]_2$. Define a morphism

for all $\mu\in \Bbbk[\varepsilon]_2$. Similarly, if we replace $\Bbbk$ by $\Bbbk[\varepsilon]_2$ in the Bloch-Suslin complex, we get

The right hand side of (2.8) is canonically isomorphic to $\bigwedge^2\Bbbk^\times \bigoplus \Bbbk\otimes \Bbbk^{\times}\bigoplus \bigwedge^2\Bbbk$ with

while the left hand side is isomorphic to $\mathcal{B}_2(\Bbbk)\bigoplus \beta_2(\Bbbk)\bigoplus \bigwedge^2\Bbbk\bigoplus \Bbbk$ (see ^[9])

Define a $\mathbb{Z}$-module $\mathbb{Z}'[\Bbbk[\varepsilon]_2]$ generated by $\langle \theta; \phi] = [\theta+\phi\varepsilon]-[\theta]$ for $\theta, \phi\in \Bbbk$ and define $\mathcal{R}_2^\varepsilon(\Bbbk[\varepsilon]_2)$ as a submodule of $\mathbb{Z}'[\Bbbk[\varepsilon]_2]$ generated by the five term relation (see ^[9] and ^[12])

where

and

Define

Remark 2.5. The tangent group $T\mathcal{B}_2(\Bbbk)$ is isomorphic to $\beta_2(\Bbbk)\bigoplus \bigwedge^2\Bbbk\bigoplus \Bbbk$ (Theorem 1.1 of ^[9]) and $\mathbb{Z}'[\Bbbk[\varepsilon]_2]$ is isomorphic to $\mathcal{B}_2(\Bbbk[\varepsilon]_2)$

3.   Main results and discussion

Consider the sequence (2.3) above. Here we suggest a map (morphism) different from the one which is defined in §2 of ^[3] and the relation (2.2) above between the abelian groups of sequence (2.3), since the map without alternate sign does not follow the definition of a complex. Thus, the above sequence becomes a complex if we put alternate signs for $\partial$:

when $n = 2$, we put

and for $n\geq3$, we suggest to use

Theorem 3.1. The sequence (2.3) is a complex for the $\partial$ defined above.

Proof. To prove that the sequence (2.3) is a complex we consider $2\leq k\leq n-2$

Let $\langle u\rangle_{n-k+1}\otimes \bigwedge_{i = 1}^{k-1}v_i+ \theta \otimes [\phi]_{n-k+1}\otimes \bigwedge_{j = 1}^{k-2}\psi_j\in \substack{\beta_{n-k+1}(\Bbbk)\otimes \wedge^{k-1}\Bbbk^\times\\ \oplus\\ K\otimes \mathcal{B}_{n-k+1} (\Bbbk)\otimes \wedge^{k-2}\Bbbk^\times}$

Now compute $\partial\left(\partial\left(\langle u\rangle_{n-k+1}\otimes \bigwedge_{i = 1}^{k-1}v_i + \theta \otimes [\phi]_{n-k+1}\otimes \bigwedge_{j = 1}^{k-2}\psi_j\right)\right)$.

To make calculation simple, first we compute

then find

Now the last case is for $k = 1$ with $\bigwedge_{i = 0}^0v_i = 1\in \mathbb{Z}$ and using $R\otimes_{\mathbb{Z}}\mathbb{Z}\cong R$ for any ring $R$. Similarly, for the variant of Cathelineau's complex (2.5) and tangential version of Goncharv's complex (3.1), we have similar results.

Theorem 3.2. The above sequence (2.5) is a complex.

Proof. There is not much effort required to prove the above sequence is a complex except to use $D\log$ maps. We just follow the steps of Theorem 3.1 and use $D\log$.

3.1.   Tangent to Goncharov's complex

Here, we suggest that how to define a tangent group $T\mathcal{B}_n(\Bbbk)$ for any $n$ in the same spirit as $\beta_n(\Bbbk)$ is defined in ^[3] and give its appropriateness by relating them in a suitable complex.

Inductively, for any $n$, we define a tangent group $T\mathcal{B}_n(\Bbbk)$ by defining the map

thus $T\mathcal{B}_n(\Bbbk)$ is

where $\mathcal{R}_n^\varepsilon(\Bbbk[\varepsilon]_2)$ is a kernel of the following map

where $\langle \theta; \psi] = [\theta+\psi\varepsilon]-[\theta]$ and $\langle \theta; \psi]_n$ is the class of $\langle\theta, \psi]$ in $T\mathcal{B}_n(\Bbbk)$, by using the above definition, the following becomes a complex

where $\partial_\varepsilon$ is induced by $\partial_{\varepsilon, n}$ and when $\partial_\varepsilon$ is applied to the group $\mathcal{B}_n(\Bbbk)$ then it agrees with $\delta_n$ defined above and in ^[11].

Theorem 3.3. For weight $n = 3$, the tangent to Goncharov's complex is also a complex.

where $\partial_{\varepsilon}(\langle \theta; \phi]_3) = \langle \theta; \phi]_2\otimes\theta+\frac{\phi}{\theta}\otimes [\theta]_2$ and

Proof. Here we will prove that how the above sequence is a complex for weight $n = 3$.

Theorem 3.4. The above sequence (3.1) is a complex.

Proof. We can show that (3.1) is a complex, by considering two cases:

Case 1: Consider

Case 2: We consider

Let $\langle \theta; \phi]_{n-k+1}\otimes\wedge_{i = 1}^{k-1}\phi_i+x\otimes [y]\otimes\wedge_{j = 1}^{k-2}z_j\in \substack{T\mathcal{B}_{n-k+1}(\Bbbk) \otimes \wedge^{k-1}\Bbbk^\times\\ \oplus\\ \Bbbk\otimes \mathcal{B}_{n-k+1}(\Bbbk)\otimes \wedge^{k-2}\Bbbk^\times}$

Now applying maps

4.   Conclusion

We have shown that the sequences (2.3), (2.5) and (3.1) are complexes. Complexes (2.3) and (2.5) have only inversion and distribution relations (functional equations) for $n > 3$. However, there are some non-trivial but non-defining relations known for $n\leq 7$ (see ^[5,11]). There is insufficient information for the complex (3.1) (kernels of $\partial_\varepsilon$ and defining relations are unknown) to compute the homologies for $n\geq3$, but it is expected to come out in a similar way as the homology of the complex (2.1).

The original construction of the tangent to Bloch-Suslin complex (see ^[9]) is described by the application of a tangent functor on the Bloch-Suslin, resulting in the first derivative on $\mathcal{B}_2(F)$ and $\wedge^2F^{\times}$. One can find the higher order derivatives (tangent order) on Goncharov's complex or precisely on $T\mathcal{B}_n(F)$ in a similar way as done in ^[13] for Bloch-Suslin complex.

Conflict of interest

The author declares there is no conflicts of interest in this paper.