v ∈ U. Here aγ ∈ Hom(Vγ , Vαγ ). Denote aγk by ak, k = 1, . . . , p. Note that if γ ∈ Γk, αγ ∈ Γm then emaek ∈ Aα, so P aγ ∈ Aα for every k = 1, . . . , p. γ∈Γk If αγk ∈/ Γ0 then Wα,γ = {b ∈ Hom(Vγ , Vαγ ) | ∃a ∈ Aα : aγ = b} 6= 0 for every γ ∈ Γk. Since Wα,γ is a (End Vαγ )-(End Vγ )-bimodule embedded into Hom(Vγ , Vαγ ), we have Wα,γ = Hom(Vγ , Vαγ ). Given γ ∈ Γk, α ∈ Γ, αΓk = Γm, the map σα,γ : ak 7→ aγ, a ∈ Aα, is a linear bijection between Hom(Vγk , Vαγk ) and Hom(Vγ , Vαγ ). Assume σα,γk = id and σe,γ (x) = ιγ xιγ−1, since we have already established the structure of Ae. Suppose a = P akσα,γ ∈ Aα, b = P ιβbmιβ−1Tβ ∈ Ae, and consider γ∈Γk β∈Γm

[7] Frenkel, I.B., Lepowsky, J., Meurman, A.: Vertex operator algebras and the Monster. Pure and Applied Math. 134, Academic Press, Boston (1998).

[8] Roitman, M.: On free conformal and vertex algebras. J. Algebra 217, no. 2, 496-527 (1999).

[9] Ginzburg, V., Kapranov, M.: Kozul duality for operads. Duke Math. J. 76, no. 1, 203-272 (1994). [OpenAIRE]

[10] Kolesnikov, P.: Identities of conformal algebras and pseudoalgebras. Comm. Algebra 34, no. 6, 1965-1979 (2006).

[11] Golenishcheva-Kutuzova, M.I., Kac, V.G.: Γ-conformal algebras. J. Math. Phys. 39, no. 4, 2290-2305 (1998).

[12] Brown, K.S.: Cohomology of groups. Graduate Texts in Mathematics, 87. Springer-Verlag, New York-Berlin (1982).

[13] Berezin, F.A., Introduction to superanalysis. D. Reidel Publishing Company, Dordrecht (1987).

[14] Bahturin, Yu.A., Sehgal, S.K., Zaicev, M.V.: Group gradings on associative algebras. Journal of Algebra 241, no. 2, 677-698 (2001). [OpenAIRE]

[15] Nastasescu, C., van Oystaeyen, F.: Graded ring theory. North-Holland Mathematical Library, 28. North-Holland Publishing Co., Amsterdam-New York (1982).

[16] Boyallian, C., Kac, V.G., Liberati, J.-I.: On the classification of subalgebras of CendN and gcN . J. Algebra 260, no. 1, 32-63 (2003). [OpenAIRE]

[17] Kolesnikov, P.: Associative conformal algebras with finite faithful representation. Adv. Math. 202, no. 2, 602-637 (2006).