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infinitesimal braid relation

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Knot theory

Lie theory

∞-Lie theory (higher geometry)

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Smooth structure

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\infty-Lie groupoids

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\infty-Lie algebroids

\infty-Lie algebras

Contents

Definition

Definition

(infinitesimal braid Lie algebra)

For D,nD,n \in \mathbb{N} natural numbers, let F({t ij} ij{1,,n})F(\{t_{i j}\}_{i\neq j \in \{1,\cdots, n\}}) be the free Lie algebra (over a given ground field) on generators t ijt_{i j} in degree D2D-2 with two indices ranging from 1 to nn and distinct.

The infinitesimal braid relations are the following relations on the underlying vector space of the free Lie algebra on generators {t ij} ij{1,,n}\{t_{i j}\}_{i\neq j \in \{1,\cdots, n\}}

(1)R0: t ij(1) Dt ji =0 R1: [t ij,t kl] =0 R2: [t ik+t jk,t ij] =0}AAAfor all pairwise distincttuples of indices \left. \array{ R0: \;\; & t_{i j} - (-1)^D t_{j i} & = 0 \\ R1: \;\; & [t_{i j}, t_{k l}] & = 0 \\ R2: \;\; & [t_{i k} + t_{j k}, t_{i j}] & = 0 } \;\; \right\} \phantom{AAA} { \text{for all pairwise distinct} \atop \text{tuples of indices} }

This defines a quotient Lie algebra often denoted

n(D)F({t ij} ij{1,,n})/(R0,R1,R2). \mathcal{L}_n(D) \;\coloneqq\; F(\{t_{i j}\}_{i\neq j \in \{1,\cdots, n\}}) /(R0, R1, R2) \,.

This is originally due to Kohno 87 (1.1.4), it appears also in Kohno 88, Bar-Natan 96, Fact 3, Fadell-Husseini 01, Cohen-Gitler 01, Section 3, Cohen-Gitler 02, p. 2.

Properties

Relation to horizontal chord diagrams

If F({t ij} ij{1,,n})/R0F(\{t_{i j}\}_{i\neq j \in \{1,\cdots, n\}})/R0 is identified with the commutator Lie algebra of horizontal chord diagrams on nn strands, generated by

and forming an associative algebra under concatenation of diagrams along strands, as in

then the remaining infinitesimal braid relations (1) are equivalently the following:

R1 is the the 2T relation:

R2 is the 4T relation:

graphics from Sati-Schreiber 19c

Hence:

Proposition

(universal enveloping algebra of infinitesimal braid Lie algebra is horizontal chord diagrams modulo 2T&4T)

The associative algebra

(𝒜 n pbSpan(𝒟 n pb)/(2T,4T),) \Big( \mathcal{A}_n^{pb} \;\coloneqq\; Span \big( \mathcal{D}_n^{pb} \big)/(2T, 4T) , \circ \Big)

of horizontal chord diagrams on nn strands with product given by concatenation of strands (this Def.), modulo the 2T relations and 4T relations (this Def.) is isomorphic to the universal enveloping algebra of the infinitesimal braid Lie algebra (Def. ):

(𝒜 n pb,)𝒰( n(D)). \big(\mathcal{A}_n^{pb}, \circ\big) \;\simeq\; \mathcal{U}(\mathcal{L}_n(D)) \,.
chord diagramsweight systems
linear chord diagrams,
round chord diagrams
Jacobi diagrams,
Sullivan chord diagrams
Lie algebra weight systems,
stringy weight system,
Rozansky-Witten weight systems


knotsbraids
chord diagram,
Jacobi diagram
horizontal chord diagram
1T&4T relation2T&4T relation/
infinitesimal braid relations
weight systemhorizontal weight system
Vassiliev knot invariantVassiliev braid invariant
weight systems are associated graded of Vassiliev invariantshorizontal weight systems are cohomology of loop space of configuration space

References

Last revised on January 12, 2020 at 14:00:10. See the history of this page for a list of all contributions to it.