Double canonical bases

29.11.2022 15:15 – 17:00

The goal of my talk (based on joint work with Jacob Greenstein) is to introduce a new basis B_g for the quantized universal enveloping algebra U_q(g) of any semisimple or Kac-Moody Lie algebra g. As part of the construction (and a free byproduct), a ``canonically looking" basis B'_g emerges in the quantum Heisenberg double H_q(g) and other relevant doubles.

These bases contain dual canonical bases of the upper and lower halves of U_q(g) and H_q(g) and are invariant under all known symmetries including (yet conjecturally) Lusztig’s braid group action on U_q(g) when g is semisimple.

We expect that the bases B_g and B'_g carry cluster-like structures extending those on the upper and lower halves. We also expect that the center of U_q(g) (resp. H_q(g)) is spanned by a part of B_g (resp. B'_g) and this part of B_g can be identified with "Schur functions», i.e., the characters of simple g-modules.

No prior knowledge is required for the first half of my talk: I will review the quantized enveloping algebras, Heisenberg doubles, and the Lusztig-Kashiwara theory of (dual) canonical bases of their upper/lower halves.

In the second half, I will focus on the construction of double canonical bases of H_q(g) and U_q(g) and their (mostly conjectural) properties.

Lieu

Bâtiment: Conseil Général 7-9

Room 1-07, Séminaire "Groupes de Lie et espaces de modules"

entrée libre