# Is there a notion of tensor product of perfect bases of representations of Lie algebras?

Berenstein and Kazhdan define perfect bases as an "unquantized" version of crystal bases. A perfect basis is roughly a basis with a crystal structure such that $E_i\cdot v=\mathbb{C}\cdot \tilde{e}_iv+\cdots$ where the $\cdots$ indicates terms in basis vectors killed by $\tilde{e}_i^{\epsilon_i(v)-2}$ (here $E_i$ is an element of the Lie algebra, and $\tilde{e}_i$ is a Kashiwara operator).

The cool theorem is that any given finite-dimensional representation only has one possible crystal attached to it.

Note that many of the "nicest" crystal bases (in particular, the global crystal basis) are perfect bases when specialized at q=1, this is far from universally true. In particular, taking the tensor product of perfect bases in the naive sense doesn't result in a new perfect basis.

Does anyone know of a way of fixing this, and getting in a canonical perfect basis on the tensor product from perfect bases on the factors?

What I particularly want is a natural bijection from the basis in the tensor product to the product of the original bases, sending the induced crystal structure to the crystal tensor product.

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