Smith Algebra from sl2 Tensor Product Branching via the Mickelsson-Zhelobenko Algebra
Abstract
Let Ln be the finite-dimensional irreducible sl2-module of highest weight n. We study the space of highest-weight vectors in Ln⊗ Lm for the diagonal sl2-action. The extremal projector constructs these singular vectors and gives operators connecting the highest-weight lines in the Clebsch-Gordan decomposition. We interpret these operators in the Mickelsson-Zhelobenko algebra of the diagonal pair (sl2⊕sl2, sl2diag). A rescaling of their quadratic relation yields the cubic Smith relation [E,F]=2(H3−cH). This cubic algebra is also known as a Higgs algebra in the physics literature. We express both the central parameter c and the Smith Casimir in terms of the quadratic sl2 Casimir elements of the two tensor factors. The singular subspace becomes an irreducible Smith module of dimension min{n,m}+1, with explicit formulas for the actions of the raising and lowering operators E and F. Finally, comparison with finite-dimensional simple Smith modules gives a necessary and sufficient condition for a module to arise from this tensor-product construction.
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BibTeX
@misc{hasegawa2026smith,
title = {{Smith Algebra from $\mathfraksl_2$ Tensor Product Branching via the Mickelsson-Zhelobenko Algebra}},
author = {Kohei Hasegawa and Takuya Matsumoto and Akito Takashima},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.41150},
eprint = {2610.41150},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.41150},
}
RIS
TY - UNPB
TI - Smith Algebra from sl_2 Tensor Product Branching via the Mickelsson-Zhelobenko Algebra
AU - Kohei Hasegawa
AU - Takuya Matsumoto
AU - Akito Takashima
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.41150
UR - https://arxiv.org/abs/2610.41150
AB - Let L_n be the finite-dimensional irreducible sl_2-module of highest weight n. We study the space of highest-weight vectors in L_n⊗ L_m for the diagonal sl_2-action. The extremal projector constructs these singular vectors and gives operators connecting the highest-weight lines in the Clebsch-Gordan decomposition. We interpret these operators in the Mickelsson-Zhelobenko algebra of the diagonal pair (sl_2⊕sl_2, sl_2^diag). A rescaling of their quadratic relation yields the cubic Smith relation [E,F]=2(H^3−cH). This cubic algebra is also known as a Higgs algebra in the physics literature. We express both the central parameter c and the Smith Casimir in terms of the quadratic sl_2 Casimir elements of the two tensor factors. The singular subspace becomes an irreducible Smith module of dimension min{n,m}+1, with explicit formulas for the actions of the raising and lowering operators E and F. Finally, comparison with finite-dimensional simple Smith modules gives a necessary and sufficient condition for a module to arise from this tensor-product construction.
ER -