arXiv subjects

Benjamin Enriquez

Publications and source records attributed to Benjamin Enriquez.

33 records · Page 2Linked to original sources

arXiv2017

A Tannakian interpretation of the elliptic infinitesimal braid Lie algebras

Let $n\geq 1$. The pro-unipotent completion of the pure braid group of $n$ points on a genus 1 surface has been shown to be isomorphic to an explicit pro-unipotent group with graded Lie algebra using two types of tools: (a) minimal models (Bezrukavnikov), (b) the choice of a complex structure on the genus 1 surface, making it into an elliptic curve $E$, and an appropriate flat connection on the configuration space of $n$ points in $E$ (joint work of the authors with D. Calaque). Following a suggestion by P. Deligne, we give an interpretation of this isomorphism in the framework of the Riemann-Hilbert correspondence, using the total space $E^\#$ of an affine line bundle over $E$, which identifies with the moduli space of line bundles over $E$ equipped with a flat connection.

math.AG↗

arXiv2017

A stabilizer interpretation of double shuffle Lie algebras

According to Racinet's work, the scheme of double shuffle and regularization relations between cyclotomic analogues of multiple zeta values has the structure of a torsor over a pro-unipotent $\mathbb Q$-algebraic group $\sf{DMR}_0$, which is an algebraic subgroup of a pro-unipotent $\mathbb Q$-algebraic group of outer automorphisms of a free Lie algebra. We show that the harmonic (stuffle) coproduct of double shuffle theory may be viewed as an element of a module over the above group, and that $\sf{DMR}_0$ identifies with the stabilizer of this element. We identify the tangent space at origin of $\sf{DMR}_0$ with the stabilizer Lie algebra of the harmonic coproduct, thereby obtaining an alternative proof of Racinet's result stating that this space is a Lie algebra (the double shuffle Lie algebra).

math.QA↗

arXiv2015

Analogues elliptiques des nombres multizétas

We study functions of an elliptic parameter, which are defined as iterated integrals of elliptic functions. We establish their relation with the "elliptic associators" of our previous work, by means of a functional realization of Lie algebras appearing in that work.

math.NT↗

arXiv2014

Homology of depth-graded motivic Lie algebras and koszulity

The Broadhurst-Kreimer (BK) conjecture describes the Hilbert series of a bigraded Lie algebra A related to the multizeta values. Brown proposed a conjectural description of the homology of this Lie algebra (homological conjecture (HC)), and showed it implies the BK conjecture. We show that a part of HC is equivalent to a presentation of A, and that the remaining part of HC is equivalent to a weaker statement. Finally, we prove that granted the first part of HC, the remaining part of HC is equivalent to either of the following equivalent statements: (a) the vanishing of the third homology group of a Lie algebra with quadratic presentation, constructed out of the period polynomials of modular forms; (b) the koszulity of the enveloping algebra of this Lie algebra.

math.RT↗

arXiv2014

On a lower central series filtration of the Grothendieck-Teichmüller Lie algebra grt_1

The Grothendieck-Teichmüller Lie algebra is a Lie subalgebra of a Lie algebra of derivations of the free Lie algebra in two generators. We show that the lower central series of the latter Lie algebra induces a decreasing filtration of the Grothendieck-Teichmüller Lie algebra and we study the corresponding graded Lie algebra. Its degree zero part had been previously computed by the second author. We show that the degree one part is a module over a symmetric algebra, which are both equipped with compatible decreasing filtrations, and we exhibit an explicit lower bound for the associated graded module. We derive from there some information on explicit expression of the depth 3 part of the depth-graded of the Grothendieck-Teichmüller Lie algebra.

math.AG↗

arXiv2011

Groups and Lie algebras corresponding to the Yang-Baxter equations

For a positive integer n we introduce quadratic Lie algebras tr_n qtr_n and discrete groups Tr_n, QTr_n naturally associated with the classical and quantum Yang-Baxter equation, respectively. We prove that the universal enveloping algebras of the Lie algebras tr_n, qtr_n are Koszul, and find their Hilbert series. We also compute the cohomology rings of these Lie algebras (which by Koszulity are the quadratic duals of the enveloping algebras). We construct cell complexes which are classifying spaces of the groups Tr_n and QTr_n, and show that the boundary maps in them are zero, which allows us to compute the integral cohomology of these groups. We show that the Lie algebras tr_n, qtr_n map onto the associated graded algebras of the Malcev Lie algebras of the groups Tr_n, QTr_n, respectively. We conjecture that this map is actually an isomorphism (this is now a theorem due to P. Lee). At the same time, we show that the groups Tr_n and QTr_n are not formal for n>3.

math.RA↗

arXiv2011

Mixed Pentagon, octagon and Broadhurst duality equation

This paper is on elimination of defining equations of the cyclotomic analogues, introduced by the first author, of Drinfeld's scheme of associators. We show that the mixed pentagon equation implies the octagon equation for N=2 and the particular distribution relation. We also explain that Broadhurst duality is compatible with the torsor structure. We develop a formalism of infinitesimal module categories and use it for deriving a proof left implicit in the first named author's earlier work.

math.QA↗

arXiv2010

Half-balanced braided monoidal categories and Teichmueller groupoids in genus zero

We introduce the notions of a half-balanced braided monoidal category and of its contraction. These notions give rise to an explicit description of the action of the Galois group of QQ on Teichmueller groupoids in genus 0, equivalent to that of L. Schneps. We also show that a prounipotent version of this action is equivalent to a graded action.

math.QA↗

arXiv2009

A quasi-Lie bialgebra formulation of the Pohlmeyer-Rehren Poisson algebra

We present a quasi-Lie bialgebra (QLBA) quantization problem which comes from an algebraic reformulation of the Nambu-Goto string theory and invariant charges by Pohlmeyer and Rehren. This QLBA structure depends on a symmetric bivector (coming from a Minkowski metric) and is built on the free Lie algebra on a finite dimensional vector space. We solve this problem when the bivector has rank 1 or 2.

math.QA↗

arXiv2007

Compatibility of quantization functors of Lie bialgebras with duality and doubling operations

We study the behavior of the Etingof-Kazhdan quantization functors under the natural duality operations of Lie bialgebras and Hopf algebras. In particular, we prove that these functors are "compatible with duality", i.e., they commute with the operation of duality followed by replacing the coproduct by its opposite. We then show that any quantization functor with this property also commutes with the operation of taking doubles. As an application, we show that the Etingof-Kazhdan quantization of some affine Lie superalgebras coincide with their Drinfeld-Jimbo-type quantizations.

math.QA↗

arXiv2006

A Formula for the Logarithm of the KZ Associator

We prove that the logarithm of a group-like element in a free algebra coincides with its image by a certain linear map. We use this result and the formula of Le and Murakami for the Knizhnik-Zamolodchikov (KZ) associator $Φ$ to derive a formula for $\log(Φ)$ in terms of MZV's (multiple zeta values).

math.QA↗

arXiv2006

Weight functions and Drinfeld currents

A universal weight function for a quantum affine algebra is a family of functions with values in a quotient of its Borel subalgebra, satisfying certain coalgebraic properties. In representations of the quantum affine algebra it gives off-shell Bethe vectors and is used in the construction of solutions of the qKZ equations. We construct a universal weight function for each untwisted quantum affine algebra, using projections onto the intersection of Borel subalgebras of different types, and study its functional properties.

math.QA↗

arXiv2003

Quantization of Alekseev-Meinrenken dynamical r-matrices

We quantize the Alekseev-Meinrenken solution r to the classical dynamical Yang-Baxter equation, associated to a Lie algebra g with an element t in S^2(g)^g. Namely, we construct a dynamical twist J with nonabelian base in the sense of P. Xu, whose quasiclassical limit is r-t/2. This twist gives rise to a dynamical quantum R-matrix, and also provides a quantization of the quasi-Poisson manifold and Poisson groupoid associated to r. The twist J is obtained by an appropriate renormalization of the Knizhnik-Zamolodchikov associator for g, introduced by Drinfeld.

math.QA↗

arXiv1996

Geometric interpretation of the Poisson structure in affine Toda field theories

We express the Poisson brackets of local fields of the affine Toda field theories in terms of the Drinfeld-Sokolov dressing operator. For this, we introduce a larger space of fields, containing ``half screening charges'' and ``half integrals of motions''. In addition to local terms, the Poisson brackets contain nonlocal terms related to trigonometric r-matrices.

q-alg↗

arXiv1996

Equivalence of Two Approaches to the mKdV Hierarchies

The equivalence between the approaches of Drinfeld-Sokolov and Feigin-Frenkel to the mKdV hierarchies is established. A new derivation of the mKdV equations in the zero curvature form is given. Connection with the Baker-Akhiezer function and the tau-function is also discussed.

q-alg↗