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Adams operation on Jacobi diagrams

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Idea

On the vector space 𝒜\mathcal{A} of Jacobi diagrams modulo STU-relations (equivalently chord diagrams modulo 4T relations) there is a system of linear maps (for qq \in \mathbb{Z}, q0q \neq 0)

ψ q:𝒜𝒜 \psi^q \;\colon\; \mathcal{A} \longrightarrow \mathcal{A}

which respect the coalgebra structure and satisfy

ψ q 2ψ q 1=ψ q 1q 2 \psi^{q_2} \circ \psi^{q_1} \;=\; \psi^{q_1 \cdot q_2}

and as such are (dually) analogous to the Adams operations on topological K-theory.

(Bar-Natan 95, Def. 3.11 & Theorem 7)

In fact, when evaluated in Lie algebra weight systems w Nw_{\mathbf{N}} and under the identification (see here) of the representation ring of a compact Lie group GG with the GG-equivariant K-theory of the point, these Adams operations on Jacobi diagrams correspond to the Adams operations on equivariant K-theory:

w N(ψ qD)=w ψ qN(D). w_{\mathbf{N}}(\psi^q D) \;=\; w_{\psi^q \mathbf{N}}(D) \,.

(Bar-Natan 95, Exc. 6.24 )

References

Last revised on January 3, 2020 at 06:21:51. See the history of this page for a list of all contributions to it.