A Frobenius homomorphism for Lusztig’s quantum groups for arbitrary roots of unity

Abstract

For a finite dimensional semisimple Lie algebra and a root of unity, Lusztig defined an infinite dimensional quantum group of divided powers. Under certain restrictions on the order of the root of unity, he constructed a Frobenius homomorphism with finite dimensional Hopf kernel and with image the universal enveloping algebra.In this article we define and completely describe the Frobenius homomorphism for arbitrary roots of unity by systematically using the theory of Nichols algebras. In several new exceptional cases the Frobenius-Lusztig kernel is associated to a different Lie algebra than the initial Lie algebra. Moreover, the Frobenius homomorphism often switches short and long roots and/or maps to a braided category.

The unrolled quantum group inside Lusztig’s quantum group of divided powers

“…The original result [Lus90] requires prime to all 2d α and has no g ∨ . The extended result [Len14] for arbitrary modifies the Lie algebras on the left and right hand side accordingly. Here we have written out the result under the assumptions we have in place for in this article (divisibility by 2d α and excluded small degenerate values).…”

“…This is an infinite-dimensional Hopf algebras constructed by specialization. It fits into a Hopf algebra extension, see [Lus90][A96] for q odd and for G 2 not divisible by 3, which was generalized in [Len14] to divisible cases (as in this article) where the dual Lie algebra appears:…”

Logarithmic conformal field theories of type B n, ℓ = 4 and symplectic fermions

“…The case of arbitrary is more involved and treated in the second author work [Lent15]: For B n , = 4 it turns out that the Lusztig quantum group decomposes into the small quantum group u q (A n 1 ), associated to only the short root vectors, and U (g ∨ ) = U (C n ) acting on it by adjoint action.…”

“…Moreover there are additional degeneracies for small values of . The second author has studied these non-relatively-prime cases in [Lent15], and the result in particular with respect to B n , = 4 are as follows:…”

“…In fact from a theoretic view U L q (g) is the more natural object, which is defined for every datum by specializing an integral form of the quantum group over the field C(q). In [Lent15] the second author has worked out the cases where the divisibility condition is dropped. As turns out, the infinite quantum group again decomposes into a short exact sequence of Hopf algebras (at least on the level of Borel parts and up to a symmetric braiding, which we omit here for simplicity) u q (g (0) ) + −→ U L q (g) + −→ U (g ( ) ) + with possibly different Lie algebras g (0) , g ( ) depending on the divisibilities as follows: g = ord(q) g (0) g ( ) Trivial cases: all = 1 0 g all = 2 0 g Generic cases: ADE = 1, 2 g g B n 4 = 1, 2 B n B n C n 4 = 1, 2 C n C n F 4 4 = 1, 2 F 4 F 4 G 2 3 = 1, 2, 4 G 2 G 2…”