If $A$ is a G-graded algebra then one can define on it a color involution, i.e. a bijective linear map preserving the grading such that the image of a product of two homogeneous elements is defined through a bicharacter of the group G. Color algebras are strictly related with color Lie and Jordan algebras.
Does exist a classification of simple (associative) algebras with color involution? Is there at least a classification of color involutions on matrix algebras?
Yes, see e.g., Article Classification with Noisy Labels by Importance Reweighting
https://openaccess.thecvf.com/content_ICCV_2017/papers/Li_Learning_From_Noisy_ICCV_2017_paper.pdf and attached book:
Take the example of the basic algebraic structure like group. it can be partitioned into cyclic subgroups and each can be given a colour that readily shows the behaviour of the group.
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