Do you know of any self-injective basic algebra $A$ over a field $k$ such that its enveloping algebra $A^{\mathrm{op}}\otimes\_k A$ is **not** self-injective?


The algebra $A$ cannot be finite-dimensional, since then $A$ is Frobenius and so is $A^{\mathrm{op}}\otimes\_k A$.



Self-injective basic algebras
[Nagata, Masayoshi. A conjecture of O'Carroll and Qureshi on tensor products of fields. Japan. J. Math. (N.S.) 10 (1984), no. 2, 375--377. MR0884425] proved that the Krull dimension of the tensor product $\mathbb C(x,y)\otimes\_{\mathbb C}\mathbb C(x,y)$ is $2$. If this tensor product is reduced (I think it is, but it is late...) then it cannot be self-injective. In that case, $\mathbb C(x,y)$ is an example.


