Abstract
The affine space of traceless complex matrices in which the sum of all elements in every row and every column is equal to one is presented as an example of an affine space with a Lie bracket or a Lie affgebra.
In memory of Anatol Odzijewicz
The research is partially supported by the National Science Centre, Poland, Grant no. 2019/35/B/ST1/01115
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Notes
- 1.
The definition (18) is equivalent to say that \([x,y]= \zeta \triangleright _x y\), for all \(x,y\in {\mathbb A}\). The latter Lie bracket can be defined on any affine space, not necessarily one-dimensional one.
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Acknowledgements
I am grateful to Janusz Grabowski for suggesting to consider the Lie algebra hull of \(\mathrm {sna}{(n , {\mathbb C})}\). The interpretation of \(\mathcal {L}(\mathrm {sna}{(n , {\mathbb C})} ;\mathbf {o})\) as the Lie ideal (33) in the non-standard Lie algebra structure (32) on \(\mathrm {gl}{(n , {\mathbb C})}\) presented in Remark 7 is also due to him.
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Brzeziński, T. (2024). Special Normalised Affine Matrices: An Example of a Lie Affgebra. In: Kielanowski, P., Beltita, D., Dobrogowska, A., Goliński, T. (eds) Geometric Methods in Physics XL. WGMP 2022. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-62407-0_9
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