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PreprintCC BY 4.0arXiv

Haar measure for totally positive split real semisimple quantum groups of classical type

  • Kenny De Commer
  • Geoffrey Janssens
  • Wannes Malfait
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Abstract

Let ℏ be a non-zero real number and put q= e−πi ℏ. Let G be a semisimple split real Lie group of classical type, with Lie algebra g and associated Borel subalgebra b ⊆ g. We show that there exists a locally compact quantum group (Lℏ(B),Δ) which integrates the Borel part Uq(b) of the Drinfeld--Jimbo quantized enveloping algebra Uq(g), viewed as a Hopf *-algebra. We further compute the associated Haar weights and modular data for (Lℏ(B),Δ). Up to an excess Cartan part, the Drinfeld double of (Lℏ(B),Δ) can be seen as an analytic quantization of the function algebra of the totally positive semigroup G+⊆ G.

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Cite

BibTeXRIS
BibTeX
@misc{commer2026haar,
  title = {{Haar measure for totally positive split real semisimple quantum groups of classical type}},
  author = {Kenny De Commer and Geoffrey Janssens and Wannes Malfait},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.13213},
  eprint = {2610.13213},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.13213},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Haar measure for totally positive split real semisimple quantum groups of classical type
AU  - Kenny De Commer
AU  - Geoffrey Janssens
AU  - Wannes Malfait
PY  - 2026
DA  - 2026/10/09
PB  - arXiv
DO  - 10.48550/arxiv.2610.13213
UR  - https://arxiv.org/abs/2610.13213
AB  - Let ℏ be a non-zero real number and put q= e^−πi ℏ. Let G be a semisimple split real Lie group of classical type, with Lie algebra g and associated Borel subalgebra b ⊆ g. We show that there exists a locally compact quantum group (L_ℏ(B),Δ) which integrates the Borel part U_q(b) of the Drinfeld--Jimbo quantized enveloping algebra U_q(g), viewed as a Hopf *-algebra. We further compute the associated Haar weights and modular data for (L_ℏ(B),Δ). Up to an excess Cartan part, the Drinfeld double of (L_ℏ(B),Δ) can be seen as an analytic quantization of the function algebra of the totally positive semigroup G^+⊆ G.
ER  -