Elliptic multizetas and the elliptic double shuffle relations
We define an elliptic generating series whose coefficients, the elliptic multizetas, are related to the elliptic analogues of multiple zeta values introduced by Enriquez as the coefficients of his elliptic associator; both sets of coefficients lie in O(H), the ring of functions on the Poincaré uppe...
| Main Authors: | , , |
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| Format: | Journal article |
| Language: | English |
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Oxford University Press
2020
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| _version_ | 1826282309515476992 |
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| author | Lochak, P Matthes, N Schneps, L |
| author_facet | Lochak, P Matthes, N Schneps, L |
| author_sort | Lochak, P |
| collection | OXFORD |
| description | We define an elliptic generating series whose coefficients, the elliptic multizetas, are related to the elliptic analogues of multiple zeta values introduced by Enriquez as the coefficients of his elliptic associator; both sets of coefficients lie in O(H), the ring of functions on the Poincaré upper half-plane H. The elliptic multizetas generate a Q-algebra E, which is an elliptic analogue of the algebra of multiple zeta values. Working modulo 2πi, we show that the algebra E decomposes into a geometric and an arithmetic part and study the precise relationship between the elliptic generating series and the elliptic associator defined by Enriquez. We show that the elliptic multizetas satisfy a double shuffle type family of algebraic relations similar to the double shuffle relations satisfied by multiple zeta values. We prove that these elliptic double shuffle relations give all algebraic relations among elliptic multizetas if (1) the classical double shuffle relations give all algebraic relations among multiple zeta values and (2) the elliptic double shuffle Lie algebra has a certain natural semi-direct product structure analogous to that established by Enriquez for the elliptic Grothendieck–Teichmüller Lie algebra. |
| first_indexed | 2024-03-07T00:41:54Z |
| format | Journal article |
| id | oxford-uuid:835610ac-9fc7-4ed3-af67-98ef8bdd0114 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T00:41:54Z |
| publishDate | 2020 |
| publisher | Oxford University Press |
| record_format | dspace |
| spelling | oxford-uuid:835610ac-9fc7-4ed3-af67-98ef8bdd01142022-03-26T21:43:28ZElliptic multizetas and the elliptic double shuffle relationsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:835610ac-9fc7-4ed3-af67-98ef8bdd0114EnglishSymplectic ElementsOxford University Press 2020Lochak, PMatthes, NSchneps, LWe define an elliptic generating series whose coefficients, the elliptic multizetas, are related to the elliptic analogues of multiple zeta values introduced by Enriquez as the coefficients of his elliptic associator; both sets of coefficients lie in O(H), the ring of functions on the Poincaré upper half-plane H. The elliptic multizetas generate a Q-algebra E, which is an elliptic analogue of the algebra of multiple zeta values. Working modulo 2πi, we show that the algebra E decomposes into a geometric and an arithmetic part and study the precise relationship between the elliptic generating series and the elliptic associator defined by Enriquez. We show that the elliptic multizetas satisfy a double shuffle type family of algebraic relations similar to the double shuffle relations satisfied by multiple zeta values. We prove that these elliptic double shuffle relations give all algebraic relations among elliptic multizetas if (1) the classical double shuffle relations give all algebraic relations among multiple zeta values and (2) the elliptic double shuffle Lie algebra has a certain natural semi-direct product structure analogous to that established by Enriquez for the elliptic Grothendieck–Teichmüller Lie algebra. |
| spellingShingle | Lochak, P Matthes, N Schneps, L Elliptic multizetas and the elliptic double shuffle relations |
| title | Elliptic multizetas and the elliptic double shuffle relations |
| title_full | Elliptic multizetas and the elliptic double shuffle relations |
| title_fullStr | Elliptic multizetas and the elliptic double shuffle relations |
| title_full_unstemmed | Elliptic multizetas and the elliptic double shuffle relations |
| title_short | Elliptic multizetas and the elliptic double shuffle relations |
| title_sort | elliptic multizetas and the elliptic double shuffle relations |
| work_keys_str_mv | AT lochakp ellipticmultizetasandtheellipticdoubleshufflerelations AT matthesn ellipticmultizetasandtheellipticdoubleshufflerelations AT schnepsl ellipticmultizetasandtheellipticdoubleshufflerelations |