Geometry of configurations in tangent groups

Raziuddin Siddiqui; Raziuddin Siddiqui

doi:10.3934/math.2020035

AIMS Mathematics

2020, Volume 5, Issue 1: 522-545. doi: 10.3934/math.2020035

Previous Article Next Article

Research article

Geometry of configurations in tangent groups

Raziuddin Siddiqui ^,

Department of Mathematical Sciences, Institute of Business Administration, Karachi, Pakistan

Received: 05 October 2019 Accepted: 27 November 2019 Published: 09 December 2019
MSC : 11G55, 18G, 19D

This article relates the Grassmannian complexes of geometric configurations to the tangent to the Bloch-Suslin complex and to the tangent to Goncharov's motivic complex. By means of morphisms, we bring the geometry of configurations in tangent groups, $T\mathcal{B}_2(F)$ and $T\mathcal{B}_3(F)$ that produce commutative diagrams. To show the commutativity of diagrams, we use combinatorial techniques that include permutations in symmetric group S₆. We also create analogues of the Siegel's cross-ratio identity for the truncated polynomial ring F[ε]_ν.V
- affine spaces,
- cross-ratio,
- tangent complex,
- odd permutation,
- symmetric group
Citation: Raziuddin Siddiqui. Geometry of configurations in tangent groups[J]. AIMS Mathematics, 2020, 5(1): 522-545. doi: 10.3934/math.2020035

Related Papers:

Abstract

This article relates the Grassmannian complexes of geometric configurations to the tangent to the Bloch-Suslin complex and to the tangent to Goncharov's motivic complex. By means of morphisms, we bring the geometry of configurations in tangent groups, $T\mathcal{B}_2(F)$ and $T\mathcal{B}_3(F)$ that produce commutative diagrams. To show the commutativity of diagrams, we use combinatorial techniques that include permutations in symmetric group S₆. We also create analogues of the Siegel's cross-ratio identity for the truncated polynomial ring F[ε]_ν.V

References

[1]	J. L. Cathelineau, The tangent complex to the Bloch-Suslin complex, B. Soc. Math. Fr., 135 (2007), 565-597. doi: 10.24033/bsmf.2546
[2]	P. Elbaz-Vincent, H. Gangl, On Poly (ana) logs I, Compos. Math., 130 (2002), 161-214. doi: 10.1023/A:1013757217319
[3]	A. B. Goncharov, Geometry of configurations, polylogarithms and motivic cohomology, Adv. Math., 114 (1995), 197-318. doi: 10.1006/aima.1995.1045
[4]	A. B. Goncharov, Polylogarithms and motivic Galois groups, In: Proceedings of Symposia in Pure Mathematics, Providence: American Mathematical Society, 55 (1994), 43-96.
[5]	S. Hussain, R. Siddiqui, Projective configurations and the variant of Cathelineau's complex, J. Prime Res. Math., 12 (2016), 24-34.
[6]	C. Siegel, Approximation algebraischer zahlen, Math. Z., 10 (1921), 173-213. doi: 10.1007/BF01211608
[7]	S. Ünver, Additive polylogarithms and their functional equations, Math. Ann., 348 (2010), 833-858. doi: 10.1007/s00208-010-0493-7
[8]	S. Ünver, On the additive dilogarithm, Algebr. Number Theory, 3 (2009), 1-34. doi: 10.2140/ant.2009.3.1

Reader Comments

Your name:*

Email:*
© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)