Positive Casimir and Central Characters of Split Real Quantum Groups

2015/03/02 by Ivan Chi-Ho Ip, Ip, Ivan Chi-Ho
Mathematics · #55R80 #81R50 #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1503.00543

openalex publication_date 2015/03/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We describe the generalized Casimir operators and their actions on the positive representations P_λ of the modular double of split real quantum groups U_qq(g_R). We introduce the notion of virtual highest and lowest weights, and show that the central characters admit positive values for all parameters λ. We show that their image defines a semi-algebraic region bounded by real points of the discriminant variety independent of q, and we discuss explicit examples in the lower rank cases.

Citations

• On tensor products of positive representations of split real quantum Borel subalgebra U_qq(b_R)

Related

• Positive Representations with Zero Casimirs
• Positive representations of split real quantum groups and future perspectives
• Positive Representations of Split Real Quantum Groups: The Universal R\n Operator
• Positive representations of non-simply-laced split real quantum groups
• On tensor products of positive representations of split real quantum Borel subalgebra U_qq(b_R)