Pentagon and hexagon equations

Furusho, Hidekazu

doi:10.4007/annals.2010.171.545

Cited by 44 publications

(45 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…ψ(x, y) + ψ(y, z) + ψ(z, x) = 0 (12) for x + y + z = 0 (that is, one can put z = −x − y), and (13) Note that defining equations of grt 1 have no solutions in degrees one and two. The Lie bracket induced on solutions of (11), (12), and (13) is called Ihara bracket:…”

Section: The Grothendieck-teichmüller Lie Algebramentioning

confidence: 99%

See 1 more Smart Citation

The Kashiwara-Vergne conjecture and Drinfeld's associators

2012

View full text Add to dashboard Cite

The Kashiwara-Vergne (KV) conjecture is a property of the CampbellHausdorff series put forward in 1978. It has been settled in the positive by E. Meinrenken and the first author in 2006. In this paper, we study the uniqueness issue for the KV problem. To this end, we introduce a family of infinite-dimensional groups KRV 0 n , and a group KRV2 which contains KRV 0 2 as a normal subgroup. We show that KRV2 also contains the Grothendieck-Teichmüller group GRT1 as a subgroup, and that it acts freely and transitively on the set of solutions of the KV problem SolKV. Furthermore, we prove that SolKV is isomorphic to a direct product of affine line A 1 and the set of solutions of the pentagon equation with values in the group KRV 0 3 . The latter contains the set of Drinfeld's associators as a subset. As a by-product of our construction, we obtain a new proof of the Kashiwara-Vergne conjecture based on the Drinfeld's theorem on existence of associators.

show abstract

Section: The Grothendieck-teichmüller Lie Algebramentioning

confidence: 99%

“…To establish the second equality, we make use of (dg l s /ds)(g l s ) −1 = ν(ψ) 1,2 + ν(ψ) 12,3 and (dg r s /ds)(g r s ) −1 = ν(ψ) 2,3 + ν(ψ) 1,23 . The third equality follows from Proposition 4.7.…”

Section: Associatorsmentioning

confidence: 99%

The Kashiwara-Vergne conjecture and Drinfeld's associators

2012

View full text Add to dashboard Cite

show abstract

“…The indices of the F -matrix in expressions (28) and (29) belong to the four faces of a tetrahedron. In general, the F -matrix also depends on the spectral parameter x [36], [37], and the series expansion in this parameter for small values leads to the Hamiltonian.…”

Section: Resultsmentioning

confidence: 99%

“…We do not here present the explicit expressions for the exchange matrix for two anyons R c ab and the hexagon equation containing this matrix. The hexagon equation is a consequence of the pentagon equation [29].…”

Section: Chain Of Su (2) 4 Anyonsmentioning

confidence: 99%

Chain of interacting SU(2)4 anyons and quantum SU(2) k × $\overline {SU(2)_k } $ doubles

2011

View full text Add to dashboard Cite

We consider a chain of SU(2)4 anyons with transitions to a topologically ordered phase state. For half-integer and integer indices of the type of strongly correlated excitations, we find an effective lowenergy Hamiltonian that is an analogue of the standard Heisenberg Hamiltonian for quantum magnets. We describe the properties of the Hilbert spaces of the system eigenstates. For the Drinfeld quantum SU(2) k ×SU(2) k doubles, we use numerical computations to show that the largest eigenvalues of the adjacency matrix for graphs that are extended Dynkin diagrams coincide with the total quantum dimensions for the levels k = 2, 3, 4, 5. We also formulate a hypothesis about the reason for the universal behavior of the system in the long-wave limit.

show abstract

“…The connection between the various associator relations has been elucidated in [12]. It can be explained as follows.…”

Section: Introductionmentioning

confidence: 99%

Homology of depth-graded motivic Lie algebras and koszulity

Enriquez¹,

Lochak²

2016

J. Théor. Nombres Bordeaux

View full text Add to dashboard Cite

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.

show abstract

Pentagon and hexagon equations

Abstract: The author will prove that Drinfel'd's pentagon equation implies his two hexagon equations in the Lie algebra, pro-unipotent, pro-l and pro-nilpotent contexts.

Cited by 44 publications

References 12 publications

The Kashiwara-Vergne conjecture and Drinfeld's associators

The Kashiwara-Vergne conjecture and Drinfeld's associators

Chain of interacting SU(2)4 anyons and quantum SU(2) k × $\overline {SU(2)_k } $ doubles

Homology of depth-graded motivic Lie algebras and koszulity

Contact Info

Product

Resources

About