Second cohomology groups of the Hopf * -algebras associated to universal unitary quantum groups

Abstract

We compute the second (and the first) cohomology groups of *algebras associated with the universal quantum unitary groups of not necessarily Kac type, extending our earlier results for the free unitary group U + d . The extended setup forces us to use infinite-dimensional representations to construct the cocycles.Résumé. -Nous calculons les deuxièmes (et premiers) groupes de cohomologi des algèbres involutives associées aux groupes quantiques unitaires universels pas nécessairement du type Kac, étendant ainsi nos résultats précédents pour le groupe unitaire libre U + d . Dans ce cadre nous sommes obligés d'utiliser des représentations de dimension infinie pour la construction des cocycles.

Free resolutions for free unitary quantum groups and universal cosovereign Hopf algebras

“…[11,19] Example 6.3 (Cohomology of Pol(𝑈 + 𝐹 ) with 𝐹 = diag(𝑞 −1 , 1, 𝑞) and with trivial coefficients). In [20], it was shown that for 𝐹 ∈ 𝑀 3 (ℂ) positive, H 2 (Pol(𝑈 + 𝐹 ), 𝜀 ℂ 𝜀 ) ≅ 𝑠𝑙 𝐹 (𝑛) = {𝐴 ∈ 𝑀 𝑛 (ℂ)∶ 𝐴𝐹 = 𝐹𝐴, Tr(𝐴𝐹) = Tr(𝐴𝐹 −1 ) = 0}, provided that the eigenvalues of 𝐹 do not form a geometric progression. This agrees with Theorem 6.1 as the only case in which 𝐹 is a positive matrix and satisfies 𝐹 2 = ±𝐼 is for 𝐹 = 𝐼.…”

“…Example In [20], it was shown that for $$F\in M_3(\mathbb {C})$$ positive, $$\mathrm{H}^2(\operatorname{Pol} (U_F^+), {_\varepsilon \mathbb {C}_\varepsilon }) \cong sl_F(n) =\lbrace A\in M_n(\mathbb {C}) \colon AF=FA, \operatorname{Tr}(AF) = \operatorname{Tr}(AF^{-1}) = 0\rbrace$$, provided that the eigenvalues of $$F$$ do not form a geometric progression. This agrees with Theorem 6.1 as the only case in which $$F$$ is a positive matrix and satisfies $$F^2=\pm I$$ is for $$F=I$$.…”

“…‡ as the free unitary and free orthogonal quantum group of Kac type. In [19,20], the first and the second Hochschild cohomologies for the Hopf algebras Pol(𝑈 + 𝐾 ) and Pol(𝑂 + 𝐿 ) are determined with concrete formulas for cocycles and coboundaries for most parameter matrices 𝐾, 𝐿; however, the case of 𝑈 + 𝐾 where 𝐾 * 𝐾 has three distinct Eigenvalues in "geometric progression" (𝑞 −1 , 1, 𝑞, with 𝑞 ∈ ℝ + ) remained open.…”

“…In particular, for $$F = K^*K = \operatorname{diag}(q^{-1},1,q)$$, $$q\in \mathbb {R}^+$$, it turns out that the second Hochschild cohomology of $$\operatorname{Pol}(U_{K}^+)\cong \mathcal {H}(F)$$ with trivial coefficients is one‐dimensional; this was the missing piece in [20], where the second Hochschild cohomology of all other free unitary quantum groups was determined. Theorem B also yields explicitly for every generic matrix $$F$$ a (one‐dimensional) bimodule $$M$$ with $$\mathrm{H}^3(\mathcal {H}(F),M)\ne 0$$, which confirms that the cohomological dimension of $$\mathcal {H}(F)$$ is three; a fact Bichon conjectured in [11, Remark 5.15] and proved by more abstract considerations in [12, Theorem 8.1].…”

“…We denote the associated Hopf algebras $$\operatorname{Pol}(U_K^+)$$ and $$\operatorname{Pol}(O_L^+)$$, respectively; for $$n\in \mathbb {N}$$, we refer to $$U_n^+:=U_{I_n}^+$$ and $$O_n^+:=O_{I_n}^+$$ as the free unitary and free orthogonal quantum group of Kac type . In [19, 20], the first and the second Hochschild cohomologies for the Hopf algebras $$\operatorname{Pol}(U_K^+)$$ and $$\operatorname{Pol}(O_L^+)$$ are determined with concrete formulas for cocycles and coboundaries for most parameter matrices $$K,L$$; however, the case of $$U_K^+$$ where $$K^*K$$ has three distinct Eigenvalues in “geometric progression” ($$q^{-1},1,q$$, with $$q\in \mathbb {R}^+$$) remained open.…”