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Infinitesimal and Tangent to Polylogarithmic Complexes For AIMS Mathematics, 4(4): 1248–1257. DOI:10.3934/math.2019.4.1248 Received: 11 June 2019 Accepted: 11 August 2019 http://www.aimspress.com/journal/Math Published: 02 September 2019 Research article Infinitesimal and tangent to polylogarithmic complexes for higher weight Raziuddin Siddiqui* Department of Mathematical Sciences, Institute of Business Administration, Karachi, Pakistan * Correspondence: Email: [email protected]; Tel: +92-213-810-4700 Abstract: Motivic and polylogarithmic complexes have deep connections with K-theory. This article gives morphisms (different from Goncharov’s generalized maps) between |-vector spaces of Cathelineau’s infinitesimal complex for weight n. Our morphisms guarantee that the sequence of infinitesimal polylogs is a complex. We are also introducing a variant of Cathelineau’s complex with the derivation map for higher weight n and suggesting the definition of tangent group TBn(|). These tangent groups develop the tangent to Goncharov’s complex for weight n. Keywords: polylogarithm; infinitesimal complex; five term relation; tangent complex Mathematics Subject Classification: 11G55, 19D, 18G 1. Introduction The classical polylogarithms represented by Lin 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): X1 zk Li (z) = = − ln(1 − z) 1 k k=1 X1 zk Li (z) = 2 k2 k=1 : : X1 zk Li (z) = for z 2 ¼; jzj < 1 n kn k=1 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 x 2.3, 2.4 and 2.5 below. 1249 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 di fferential process to introduce infinitesimal form of motivic (Bloch-Suslin’s and Goncharov’s) complexes that consists of |-vector spaces. These |-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 TB2(F)) (see [9]). The tangent group TB2(F) has two parts; first part comes from B2(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 |-vector spaces that converts the sequence (2.3) into a complex. Further, we introduce a variant of infinitesimal |-vector spaces which is structurally infinitesimal but has functional equations similar to classical polylogarithmic groups. In x3.1, we are also giving an inductive definition of group TBn(|) 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 [|] be a free abelian group generated by [a] for a 2 F. Suslin defines the following map 2 × δ2 : [|] ! ^ | ; [θ] 7! θ ^ (1 − θ) where ^2|× = |× ⊗ |×= hθ ⊗ θ; θ ⊗ φ + φ ⊗ θj θ; φ 2 |×i. The Bloch-Suslin complex is defined as 2 × δ : B2(|) ! ^ | ;[θ]2 7! θ ^ (1 − θ) where B2(|) is the quotient of [|] by the subgroup generated by Abel’s five term relation " # " # φ 1 − φ 1 − φ−1 [θ] − [φ] + − + θ 1 − θ 1 − θ−1 and δ is induced by δ2. When | is algebraically closed with characteristic zero, the above complex can be inserted into the algebraic K-theory variant of the Bloch-Wigner sequence [9] ind 2 × 0 ! µ(|) ! K3 (|) !B2(|) ! ^ | ! K2(|) ! 0 AIMS Mathematics Volume 4, Issue 4, 1248–1257. 1250 if this sequence is tensored by then (2) 2 × (2) 0 ! K3 (|) !B2(|) ⊗ ! ^ | ! K2 (|) ! 0 (i) where K-groups Kn are the pieces of the Adams decomposition of Kn(|) ⊗ (see [6]). The homology 2 × × × × of Bloch-Suslin complex is the Kn-groups for n = 2; 3 i.e. ^ | =Imδ | ⊗ | =hθ ^ (1 − θ)jθ 2 | )i and B(F):= ker δ is called Bloch group, which is isomorphic to K3 group (see [11]). 2.2. Goncharov’s (motivic) complexes The free abelian group Bn(|) is defined by Goncharov (see[2]) as [|] Bn(|) = Rn(|) with the morphisms for n = 2 V2 |× δ :[|] ! 2 (2 − torsion) 8 >0 where x = 0; 1 [x] 7! < :>x ^ (1 − x) for all other x; for n ≥ 3 × δn :[|] !Bn−1(|) ⊗ | 8 <>0 if x = 0; 1; [x] 7! > :>[x]n−1 ⊗ x for all other x; where [x]n is the class of x in Bn(|). The subgroup R1(|) of [|] is generated by [x + y − xy] − [x] − [y] and [|] is a free abelian group generated by the symbol [x] for 0; 1 , x 2 |, where x; y 2 | n f1g then × B1(|) | . For n = 2, R2(|) is defined * " # " # + φ 1 − φ 1 − φ−1 R2(|) = [θ] − [φ] + − + ; 0; 1 , θ; φ 2 | θ 1 − θ 1 − θ−1 The above relation is the Suslin’s form of Abel’s relations([11]). For n ≥ 2, An(|) is defined as the P kernel of δn and Rn(|) is the subgroup of [|] spanned by [0] and the elements ni ([ fi(0)] − [ fi(1)]), P where fi are rational fractions for indeterminate T, such that ni[ fi] 2 An(|(T)). Lemma 2.1. (Goncharov [2,3]) The following is the (cochain) complex V n × δ δ ^ δ δ ^ δ | B (|) −!B ⊗ |× −!B ⊗ 2|× −!· · · −!B (|) n−2|× −! (2.1) n n−1 n−2 2 2 − torsion Proof. Proof requires direct calculation (we work here with modulo 2-torsion means a ^ a = 0 and a ^ b = −b ^ a). AIMS Mathematics Volume 4, Issue 4, 1248–1257. 1251 Example 2.2. For weight n = 3 the following is a complex δ × δ 3 × B3(|) −!B2(|) ⊗ | −!^ | δδ([θ]3) =δ ([θ]2 ⊗ θ) =(1 − θ) ^ θ ^ θ |{z} 0 =0 2.3. Cathelineau’s infinitesimal complexes •• Let | be a field with a zero characteristic and | = K − f0; 1g, subspace βn(|) is defined in [3,9] as |[|••] βn(|) = ρn(|) where ρn(K) is the kernel of the following map •• × @n : |[| ] ! βn−1 ⊗ | ⊕ (| ⊗ Bn−1(|)) @n :[θ] 7! hθin−1 ⊗ θ + (1 − θ) ⊗ [θ]n−1 (2.2) where hθin is the coset-class of θ in βn(|) and ρ2(|) generated by Cathelineau’s relation, " # φ 1 − φ [θ] − [φ] + θ + (1 − θ) θ 1 − θ For n = 1 we have β1(|) |. Vector space βn(|) has some non-trivial elements from the functional relations of Lin for n ≤ 7 while one can find only inversion and distribution relations in βn(|) for n > 7(see [11]). The following is the Cathelineau’s infinitesimal complex to the Goncharov’s complex for weight n (see x2 of [4] and [9]): × 2 × n−2 × @ βn−1(|)⊗| @ βn−2(|)⊗∧ | @ @ β2(|)⊗∧ | @ n− × β (|) −! ⊕ −! ⊕ −!· · · −! ⊕ −! | ⊗ ^ 1| (2.3) n × n−3 × |⊗Bn−1(|) |⊗Bn−2(|)⊗| |⊗B2(|)⊗∧ | Example 2.3. For weight n = 3, the following infinitesimal version satisfying the definition of a complex: @ × M @ 2 × β3(|) −! β2(|) ⊗ | | ⊗ B2(|) −! | ⊗ ^ | (2.4) @@(hθi3) = @ (hθi2 ⊗ θ + (1 − θ) ⊗ [θ]2) = −θ ⊗ θ ^ θ −(1 − θ) ⊗ (1 − θ) ^ θ + (1 − θ) ⊗ (1 − θ) ^ θ |{z} 0 = 0 AIMS Mathematics Volume 4, Issue 4, 1248–1257. 1252 2.4. Variant of Cathelineau’s complex D D(a) 2 D We put ~a = a(1−a) [a] where D(a) Der(|; |) and is called general derivation, βn (|) is defined as |[|••] D | βn ( ) = D ρn (|) D where ρn (|) is a kernel of the following map D •• D × @n : |[| ] ! βn−1(|) ⊗ | ⊕ (| ⊗ Bn−1) D D D @n :[θ] 7! ~θn−1 ⊗ θ + D log(θ) ⊗ [a]n−1 D D D(θ) h i •• and ~θn is a class of θ in βn (|) which is equal to θ(1−θ) θ n. The following is a subspace of |[| ]: * D D + D 1 − 1 − −1 ρD(|) = ~θD − ~ D + − + ; 0; 1 , θ; φ 2 | 2 θ 1 − θ 1 − θ−1 D For n ≥ 4, one can write only inversion relations in βn (|) while for n ≤ 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 3.1 D βD | ⊗|× D D βD(|)⊗∧n−2|× D D @ n−1( ) @ @ 2 @ n−1 × β (|) −−! ⊕ −−!· · · −−! ⊕ −−! | ⊗ ^ | (2.5) n n−3 × |⊗Bn−1(|) |⊗B2(|)⊗∧ | 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. D @ D × M @ 2 × β3 (|) −! β2 (|) ⊗ | | ⊗ B2(|) −! | ⊗ ^ | (2.6) D D @@(hθi3 ) = @ hθi2 ⊗ θ + D log θ ⊗ [θ]2 = −D log(1 − θ) ⊗ θ ^ θ +D log θ ⊗ (1 − θ) ^ θ + D log θ ⊗ θ ^ (1 − θ) |{z} 0 = D log θ ⊗ (1 − θ) ^ θ − D log θ ⊗ (1 − θ) ^ θ = 0 2.5. Tangent to Bloch-Suslin complex 2 We represent a ring of dual numbers by |["]2 = |["]=h" i where | is algebraically closed field with ? × zero characteristic.
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