Local finiteness of the adjoint action for quantized enveloping algebras

The Harish-Chandra homomorphism for a quantized classical hermitian symmetric pair

“…If A € P" 1 " then we denote by V{\) the irreducible finite dimensional U-module of highest weight A as defined in [9]. We denote by P^~ the set of weights in P that are dominant for n\{a^}.…”

The quantum analog of a symmetric pair: a construction in type (𝐶_{𝑛},𝐴₁×𝐶_{𝑛-1})

“…If λ ∈ P + we let V (λ) denote the finite dimensional irreducible U-module of highest weight λ as defined in [7] or [4].…”

“…Proof of Lemma 5.9. For the proof of the lemma we will need to use the following formulas that can be derived easily from [2], Theorem 9.3: if i < j then X −e1+ei X −e1+ej = qX −e1+ej X −e1+ei , X −e1+ei X −e1−ej = qX −e1−ej X −e1+ei , X −e1−ei X −e1+ej = q −1 X −e1+ej X −e1−ei , X −e1−ei X −e1−ej = q −1 X −e1−ej X −e1−ei , 4) and, furthermore,…”

Singular localization for quantum groups at generic q

“…Another important feature is that, contrary to U q , U int q is free over its Harish-Chandra center, except possibly for a finite set of roots of unity, see [JL92,B00,BK11]. This freeness property holds only for the simply connected version of U q which is a main reason we work with that version.…”

“…U int q was first systematically studied in [JL92]. (They called it the "ad-finite" subalgebra, but since this is misleading at a root of unity we prefer the name "ad-integrable".…”