nLab
stringy weight systems span classical Lie algebra weight systems

Contents

Contents

Idea

The linear span of the image of stringy weight systems inside weight systems on Jacobi diagrams (under passage to the point particle limit) coincides with that of Lie algebra weight systems coming from metric Lie representations of the general linear Lie algebras 𝔤𝔩(n)\mathfrak{gl}(n) and special orthogonal Lie algebras 𝔰𝔬(n)\mathfrak{so}(n), which in turn contains also the Lie algebra weight systems coming from abelian and symplectic Lie algebras 𝔰𝔭(n)\mathfrak{sp}(n).

Bar-Natan 95, Theorem 11

Ingredients

Details

Given any ground field 𝔽\mathbb{F}, consider

(1)Span(MarkedSurfaces /)Vect Span\big( MarkedSurfaces_{/\sim} \big) \;\in\; Vect

the linear span of the set of diffeomorphism classes of surfaces with boundary equipped with a finite number of chosen oriented points on their boundary, at least one on each connected component (Bar-Natan 95, Def. 1.12).

A stringy weight system is a linear map from this space to the ground field, hence the space of stringy weight systems is the dual vector space

(2)𝒮(Span(MarkedSurfaces /)) *. \mathcal{S} \;\coloneqq\; \Big( Span\big( MarkedSurfaces_{/\sim} \big) \Big)^\ast \,.

Of course a linear function Span(MarkedSurfaces /)𝔽Span\big( MarkedSurfaces_{/\sim}\big) \to \mathbb{F} is equivalently a plain function MarkedSurfaces𝔽MarkedSurfaces \to \mathbb{F}, but for the following construction it is necessary to consider formal linear combinations of marked surfaces:

The 't Hooft double line construction constitutes a linear map from the linear span of Jacobi diagrams modulo STU-relations to the linear span of marked surfaces (1)

(Bar-Natan 95, Theorem 10 with Theorem 8)

graphics from Sati-Schreiber 19c

Under passage to dual vector spaces this yields a linear map

(3)𝒮(tHperm) *n(𝒲 n) \mathcal{S} \overset{ (tH \,\circ\, perm)^\ast }{\longrightarrow} \underset{n \in \mathbb{N}}{\prod} \big( \mathcal{W}^n \big)

from stringy weight systems (2) to ordinary weight systems. This may be regarded as the point particle limit of stringy weight systems.

Statement

Remark

(stringy weight systems span classical Lie algebra weight systems)

The image of the vector space (2) of stringy weight systems under the point particle limit map (3) inside the space of ordinary weight systems coincides with the direct sum of the linear span of Lie algebra weight systems for the general linear Lie algebras 𝔤𝔩(N)\mathfrak{gl}(N) and the special orthogonal Lie algebras 𝔰𝔬(N)\mathfrak{so}(N), and contains that of symplectic Lie algebras 𝔰𝔭(N)\mathfrak{sp}(N):

(Bar-Natan 95, Theorem 11)

graphics from Sati-Schreiber 19c

Facts about chord diagrams and their weight systems:

References

See also

Last revised on April 25, 2021 at 03:19:45. See the history of this page for a list of all contributions to it.