These contrasting notions appear in several places in mathematics. Real or imaginary numbers, real or imaginary quadratic fields, real or imaginary roots in a root system ... In this talk I will consider new items in this list: real or imaginary elements of a crystal basis, real or imaginary objects of a monoidal category.
The story starts from a paper on "string bases for quantum groups" published by Berenstein and Zelevinsky in 1993. The paper contained their "main conjecture on the multiplicative property of string bases". After several attempts at proving this conjecture, I finally succeeded in disproving it in 2001. The rather mysterious basis elements I found which did not satisfy the conjecture were called "imaginary", because their square is not what was expected. I also formulated a modified conjecture, involving the additional assumption that one of the two factors of the product is "real". The proof of this modified conjecture was eventually given by Kang, Kashiwara, Kim and Oh (JAMS 2018).
In the meantime, string bases (that is, Lusztig's dual canonical bases or Kashiwara's upper global crystal bases) have been recognized as classes of simple objects in certain monoidal categories (representations of quiver Hecke algebras or Khovanov-Lauda-Rouquier algebras, representations of quantum affine algebras). This leads to the notion of "real or imaginary simple object" in these categories. Actually, the proof of KKKO relies heavily on categorification techniques.
It is an interesting open question to give an explicit parametrization of the real and imaginary simple representations of KLR-algebras or quantum affine algebras. Important progress in this direction were done recently by Lapid and Minguez (2018, 2020) and Brito and Chari (2023).
In my talk I will give an introduction to these ideas. I will explain from scratch the BZ-conjecture and its modified version. Then I will sketch the proof given by KKKO. Finally I will discuss the above recent results.