Odd-Even Decomposition

Non-Planar

This module provides instances of kauri.Map related to the odd-even decomposition applied to the BCK Hopf algebra [ABS06, SEFTS25].

id_sqrt = <kauri.maps.Map object>

The square root of the identity map, \(\mathrm{Id}^{1/2}\). The unique multiplicative map such that \(\mathrm{Id}^{1/2} \cdot \mathrm{Id}^{1/2} = \mathrm{Id}\) [SEFTS25].

Example usage:

minus = <kauri.maps.Map object>

The minus operation, defined by [SEFTS25]

\[\tau^- = \mu \circ (\overline{S} \otimes \mathrm{Id}) \circ \Delta \circ \mathrm{Id}^{1/2}(\tau)\]

where \(\overline{S}(\tau) := (-1)^{|\tau|}S(\tau)\).

Example usage:

plus = <kauri.maps.Map object>

The plus operation, defined by [SEFTS25]

\[\tau^+ = \mu \circ (\mathrm{Id} \otimes (\cdot)^- \circ S) \circ \Delta(\tau)\]

Example usage:

Planar

This module provides instances of kauri.Map related to the odd-even decomposition applied to the NCK Hopf algebra [ABS06].

The minus map is computed via the convolution formula

\[\tau^- = \mu \circ (\overline{S} \otimes \mathrm{Id}) \circ \Delta(\mathrm{Id}^{1/2}(\tau))\]

where \(\overline{S}(\tau) := (-1)^{|\tau|}S(\tau)\) and the coproduct is extended to the ForestSum returned by \(\mathrm{Id}^{1/2}\). The plus map is then derived recursively from the factorisation \(\mathrm{Id} = \mathrm{Id}^+ \cdot \mathrm{Id}^-\) in the NCK convolution algebra:

\[\tau^+ = \tau - \tau^- - \sum_{(\tau)}' (\tau'_{(1)})^+ \cdot (\tau'_{(2)})^-\]

id_sqrt = <kauri.maps.Map object>

The square root of the identity map in the NCK Hopf algebra, \(\mathrm{Id}^{1/2}\). The unique multiplicative map such that \(\mathrm{Id}^{1/2} \cdot \mathrm{Id}^{1/2} = \mathrm{Id}\) where the product is the convolution in the NCK Hopf algebra [ABS06].

Example usage:

minus = <kauri.maps.Map object>

The minus (odd) part of the identity in the NCK Hopf algebra. Satisfies \(\mathrm{Id}^+ \cdot \mathrm{Id}^- = \mathrm{Id}\) where \(\cdot\) is the NCK convolution product [ABS06].

Example usage:

plus = <kauri.maps.Map object>

The plus (even) part of the identity in the NCK Hopf algebra. Satisfies \(\mathrm{Id}^+ \cdot \mathrm{Id}^- = \mathrm{Id}\) where \(\cdot\) is the NCK convolution product [ABS06].

Example usage: