论文标题
椭圆相关者满足的Fay关系
The Fay relations satisfied by the elliptic associator
论文作者
论文摘要
令$a_τ$表示由Enriquez构建的椭圆相关器，这是两个非交换变量$ a，b $定义为Kronecker函数的迭代积分$f_τ$的功率系列。我们研究了一个由$a_τ$满足的{\ it fay关系}的家庭，该家庭源自$f_τ$满足的原始fay关系。 $a_τ$的Fay关系由Broedel，Matthes和Schlotterer研究，并确定为非正规化非代表积分的必要性而产生的非明确校正项。在这里，我们研究还原版本$ \ bar {a}_τ$ mod $2πi$。 We recall a different construction of $\bar{A}_τ$ in three steps, due to Matthes, Lochak and the author: first one defines the reduced {\it elliptic generating series} $\bar{E}_τ$ which comes from the reduced Drinfeld associator $\overlineΦ_{KZ}$ and whose coefficients generate the same ring $\bar{R}$ as those $ \ bar {a}_τ$;然后，一个人将$ψ$定义为免费的关联环$ \ bar {r} \ langle \ langle a，b \ rangle \ rangle $ $由$ψ（a）= \ bar {e} _ _} _ _°}_τ$和$ψ（a，b] = [a，b] = [a，a，b] $;最后，一个人表明，还原的椭圆形相关器$ \ bar {a}_τ$等于$ψ\ bigl（{{ad（b）} \ over {e^{ad（b）} - 1}}}}}（a）\ bigr）$。使用这种结构和模具理论，并与椭圆形生成系列和关联的类似谎言版本一起工作，我们证明了以下结果：（1）当且仅当一个紧密相关的模具满足“交换循环中的关系”的关系时，霉菌才能满足Fay的关系，定义了椭圆形的Kashiwara-vergne-vergne lie elgebra $ krv_ krv_ krv _ eell ofer eell eell fim e el eell fime e el e e el eell} $（2）只要非常简单的校正项直接来自德林菲尔德关联公司的校正术语，并且（3）可以从中明确推导减少的椭圆相关者满足的FAY关系的校正项。
Let $A_τ$ denote the elliptic associator constructed by Enriquez, a power series in two non-commutative variables $a,b$ defined as an iterated integral of the Kronecker function $F_τ$. We study a family of {\it Fay relations} satisfied by $A_τ$, derived from the original Fay relation satisfied by the $F_τ$. The Fay relations of $A_τ$ were studied by Broedel, Matthes and Schlotterer, and determined up to non-explicit correction terms that arise from the necessity of regularizing the non-convergent integral. Here we study a reduced version $\bar{A}_τ$ mod $2πi$. We recall a different construction of $\bar{A}_τ$ in three steps, due to Matthes, Lochak and the author: first one defines the reduced {\it elliptic generating series} $\bar{E}_τ$ which comes from the reduced Drinfeld associator $\overlineΦ_{KZ}$ and whose coefficients generate the same ring $\bar{R}$ as those of $\bar{A}_τ$; then one defines $Ψ$ to be the automorphism of the free associative ring $\bar{R}\langle\langle a,b\rangle\rangle$ defined by $Ψ(a)=\bar{E}_τ$ and $Ψ([a,b])=[a,b]$; finally one shows that the reduced elliptic associator $\bar{A}_τ$ is equal to $Ψ\bigl({{ad(b)}\over{e^{ad(b)}-1}}(a)\bigr)$. Using this construction and mould theory and working with Lie-like versions of the elliptic generating series and associator, we prove the following results: (1) a mould satisfies the Fay relations if and only if a closely related mould satisfies the "swap circ-neutrality" relations defining the elliptic Kashiwara-Vergne Lie algebra $krv_{ell}$, (2) the reduced elliptic generating series satisfies a family of Fay relations with extremely simple correction terms coming directly from those of the Drinfeld associator, and (3) the correction terms for the Fay relations satisfied by the reduced elliptic associator can be deduced explicitly from these.
