Is there any etymological relationship between “arche” and “arete”?

Zoef De Haas answered a version of the question involving argē and aretē. OP actually intended archē and aretē; they are still not related.

  • ἀρχή: Verbal noun from ἄρχω ‎(árkhō, “I begin”), from Proto-Indo-European *h₂ergʰ- ‎(“to begin, rule, command”), according to Wiktionary.
  • ἀρετή: From Proto-Indo-European *h₂erh₁- ‘plough’ according to Wiktionary; from areskō ‘to like’ or areiōn ‘better’, according to Frisk.

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