nLab infinitesimal braid relation

Contents

Context

Knot theory

knot theory

Examples/classes:

Types

knot invariants

Related concepts:

category: knot theory

Contents

Definition

Definition

(infinitesimal braid Lie algebra)

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

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

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

This defines a quotient Lie algebra often denoted

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

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 $F(\{t_{i j}\}_{i\neq j \in \{1,\cdots, n\}})/R0$ is identified with the commutator Lie algebra of horizontal chord diagrams on $n$ 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)

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

of horizontal chord diagrams on $n$ 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. ):

$\big(\mathcal{A}_n^{pb}, \circ\big) \;\simeq\; \mathcal{U}(\mathcal{L}_n(D)) \,.$

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.