Crystals and coboundary categories

Henriques, André; Kamnitzer, Joel

doi:10.1215/s0012-7094-06-13221-0

Cited by 76 publications

(166 citation statements)

References 14 publications

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“…The next result is a development of an idea introduced in [Henriques and Kamnitzer 2006] to study the crystal commutor; the proof is a straightforward exercise.…”

Section: Bmentioning

confidence: 98%

The half-twist forU_q(g) representations

Snyder

Tingley

2009

ANT

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We introduce the notion of a half-ribbon Hopf algebra, which is a Hopf algebra H along with a distinguished element t ∈ H such that (H, R, C) is a ribbon Hopf algebra, where R = (t −1 ⊗ t −1 ) (t) and C = t −2 . The element t is closely related to the topological "half-twist", which twists a ribbon by 180 degrees. We construct a functor from a topological category of ribbons with half-twists to the category of representations of any half-ribbon Hopf algebra. We show that U q (g) is a (topological) half-ribbon Hopf algebra, but that t −2 is not the standard ribbon element. For U q (sl 2 ), we show that there is no half-ribbon element t such that t −2 is the standard ribbon element. We then discuss how ribbon elements can be modified, and some consequences of these modifications.

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“…The next result is a development of an idea introduced in [Henriques and Kamnitzer 2006] to study the crystal commutor; the proof is a straightforward exercise.…”

Section: Bmentioning

confidence: 98%

The half-twist forU_q(g) representations

Snyder

Tingley

2009

ANT

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“…Furthermore, it is explained in [Dev99], [DJS03], [HK06] that the group J n has the following presentation: it is generated by elements s p;q , 1 Ä p < q Ä n, with defining relations The above map J n ! S n is defined by sending s p;q to the involution that reverses the interval OEp; q and keeps the indices outside of this interval fixed.…”

Section: Indeed the Equivalence Follows By Applying The Lemma Formentioning

confidence: 99%

“…Recall ( [Dri89], see also [HK06]) that a coboundary monoidal category is a monoidal category Ꮿ together with a commutor morphism c X;Y W X˝Y ! Y˝X, functorial in X; Y , such that c X;Y c Y;X D 1, and…”

Section: Indeed the Equivalence Follows By Applying The Lemma Formentioning

confidence: 99%

“…Also its homology operad is well known to be the operad of Gerstenhaber algebras, which has two binary generators. The analogy between M n and C n 1 and between their fundamental groups (the pure cactus group n and the pure braid group PB n 1 ), discussed already in [Dev99], [Mor], and [HK06], is very useful and has been a source of inspiration for us while writing this paper.…”

Section: Introductionmentioning

confidence: 99%

“…The representation category of such a quasiHopf algebra is a coboundary category. The group n acts on a tensor product Y 1˝ ˝Y n 1 in any coboundary monoidal category (see [HK06] and 3.4). From this, we get an action of ᏸ n on X 1˝ ˝X n 1 .…”

Section: Introductionmentioning

confidence: 99%

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The cohomology ring of the real locus of the moduli space of stable curves of genus 0 with marked points

et al. 2010

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We compute the Poincaré polynomial and the cohomology algebra with rational coefficients of the manifold M n of real points of the moduli space of algebraic curves of genus 0 with n labeled points. This cohomology is a quadratic algebra, and we conjecture that it is Koszul. We also compute the 2-local torsion in the cohomology of M n . As was shown by the fourth author, the cohomology of M n does not have odd torsion, so that the above determines the additive structure of the integral homology and cohomology. Further, we prove that the rational homology operad of M n is the operad of 2-Gerstenhaber algebras, which is closely related to the Hanlon-Wachs operad of 2-Lie algebras (generated by a ternary bracket). Finally, using Drinfeld's theory of quantization of coboundary Lie quasibialgebras, we show that a large series of representations of the quadratic dual Lie algebra L n of H .M n ; Q/ (associated to such quasibialgebras) factors through the the natural projection of L n to the associated graded Lie algebra of the prounipotent completion of the fundamental group of M n . This leads us to conjecture that the said projection is an isomorphism, which would imply a formula for lower central series ranks of the fundamental group. On the other hand, we show that the spaces M n are not formal starting from n D 6.

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Towards Quantum Cohomology of Real Varieties

Ceyhan

2010

Arithmetic and Geometry Around Quantization

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Summary. This paper is devoted to a discussion of Gromov-Witten-Welschinger (GWW) classes and their applications. In particular, Horava's definition of quantum cohomology of real algebraic varieties is revisited by using GWW-classes and it is introduced as a DG-operad. In light of this definition, we speculate about mirror symmetry for real varieties.The strangeness and absurdity of these replies arise from the fact that modern history, like a deaf man, answers questions no one has asked.

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Crystals and coboundary categories

Cited by 76 publications

References 14 publications

The half-twist forU_q(g) representations

The half-twist forU_q(g) representations

The cohomology ring of the real locus of the moduli space of stable curves of genus 0 with marked points

Towards Quantum Cohomology of Real Varieties

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Crystals and coboundary categories

Cited by 76 publications

References 14 publications

The half-twist forUq(g) representations

The half-twist forUq(g) representations

The cohomology ring of the real locus of the moduli space of stable curves of genus 0 with marked points

Towards Quantum Cohomology of Real Varieties

Contact Info

Product

Resources

About

The half-twist forU_q(g) representations

The half-twist forU_q(g) representations