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Uniqueness of universal dimensions and configurations of points and lines

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Abstract

The problem of uniqueness of universal formulae for (quantum) dimensions of simple Lie algebras is investigated and connection of some of these functions with geometrical configurations, such as the famous Pappus–Brianchon–Pascal \((9_3)_1\) configuration of points and lines, is established by proposing some generic functions, which multiplied by a universal (quantum) dimension formula, preserve both its structure and its values at the points from Vogel’s table. We deduce, that the appropriate realizable configuration \((144_3 36_{12})\) (yet to be found) will provide a symmetric non-uniqueness factor for any universal dimension formula.

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Notes

  1. In [7] Vogel’s plane is defined as the factor of projective plane by permutations of the homogeneous parameters. Here we refer as Vogel’s the projective plane itself.

  2. The following notation is used

    $$\begin{aligned} \sinh \left[ x: \right. \,\, \frac{A\cdot B...}{M\cdot N...}\equiv \frac{\sinh (xA)\sinh (xB)...}{\sinh (xM)\sinh (xN)...} \end{aligned}$$

  3. The labeling of the lines in the following figures is meant to identify the corresponding colors they are given. For example, in Fig. 1, \(r_2\) identifies the line, associated to the second factor in the numerator of (3), and \(g_3\) - to the third factor in the corresponding denominator.

  4. One has to take into account that as the equations of the three distinguished lines are \(\alpha =0, \beta =0\) and \(\gamma =0\), one of them unavoidably will be the ideal line of the projective plane, i.e. the line in the infinity (we choose \(\alpha =0\)).

References

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Acknowledgements

We are indebted to the referee of our paper [11] for a question which is partially answered by the present investigation. We are grateful to M. Mkrtchyan for his invaluable support provided to our research. MA is grateful to the organizers of RDP Online Workshop on Mathematical Physics (December 5-6, 2020) for invitation.

The work of MA was fulfilled within the Regional Doctoral Program on Theoretical and Experimental Particle Physics Program sponsored by VolkswagenStiftung. The work of MA and RM is partially supported by the Science Committee of the Ministry of Science and Education of the Republic of Armenia under contracts 20AA-1C008 and 21AG-1C060.

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  1. Yerevan Physics Institute, Yerevan, Armenia

    M. Y. Avetisyan & R. L. Mkrtchyan

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  1. M. Y. Avetisyan
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Correspondence to M. Y. Avetisyan.

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Avetisyan, M.Y., Mkrtchyan, R.L. Uniqueness of universal dimensions and configurations of points and lines. Geom Dedicata 216, 41 (2022). https://doi.org/10.1007/s10711-022-00699-2

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