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PreprintCC BY 4.0arXiv

On Some Lie Group Invariant noncommutative functions

  • Jiachen Lu
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Abstract

We study noncommutative polynomials and functions invariant under linear actions of infinite (compact Lie) groups, with a focus on SU(2) acting on two noncommuting variables. For SU(2), we give a complete description of the invariant noncommutative polynomials: there are no invariants of odd degree, and in degree 2n the invariants are exactly the span of the S2n-orbit of (XY−YX)n. We also construct an explicit basis recursively from the operation (A,B)↦ XAYB−YAXB, and show that the space of degree-2n invariants has dimension equal to the n-th Catalan number. All proofs proceed by direct computation with the Haar integral.

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Cite

BibTeXRIS
BibTeX
@misc{lu2026some,
  title = {{On Some Lie Group Invariant noncommutative functions}},
  author = {Jiachen Lu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.37185},
  eprint = {2610.37185},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.37185},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On Some Lie Group Invariant noncommutative functions
AU  - Jiachen Lu
PY  - 2026
DA  - 2026/10/27
PB  - arXiv
DO  - 10.48550/arxiv.2610.37185
UR  - https://arxiv.org/abs/2610.37185
AB  - We study noncommutative polynomials and functions invariant under linear actions of infinite (compact Lie) groups, with a focus on SU(2) acting on two noncommuting variables. For SU(2), we give a complete description of the invariant noncommutative polynomials: there are no invariants of odd degree, and in degree 2n the invariants are exactly the span of the S_2n-orbit of (XY−YX)^n. We also construct an explicit basis recursively from the operation (A,B)↦ XAYB−YAXB, and show that the space of degree-2n invariants has dimension equal to the n-th Catalan number. All proofs proceed by direct computation with the Haar integral.
ER  -