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Finite and symmetrized colored multiple zeta values

Colored multiple zeta values are special values of multiple polylogarithms evaluated at Nth roots of unity. In this paper, we define both the finite and the symmetrized versions of these values and show that they both satisfy the double shuffle relations. Further, we provide strong evidence for an isomorphism connecting the two spaces generated by these two kinds of values. This is a generalization of a recent work of Kaneko and Zagier on finite and symmetrized multiple zeta values and of the second author on finite and symmetrized Euler sums.