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Lie Ideals and Lie Derivations of the Algebraic Toeplitz Algebra
Abstract
Let K be a field and let T=LK(ET) be the algebraic Toeplitz algebra. We classify the Lie ideals of T-. If w is the sink of the Toeplitz graph and F=I(w), then F≅ M∞(K), T/F≅ K[t,t−1], and [T,T]=F. We prove that every noncentral Lie ideal contains s=[F,F], the finitary trace-zero Lie algebra. Thus the classification reduces to the Heisenberg-type quotient T/s. The Lie ideals are precisely $0$, K1, the spaces s+U, where U≤ PK⊕ Kw and w∉ U, and the inverse images ρ−1(W), where W≤ K[t,t−1]. Here PK=K1 in characteristic $0$, and PK=spanK{1,emℓ,(e*)mℓ:m≥1} if charK=ℓ>0. We also describe the Lie derivations of T- as sums of associative derivations of T and central Lie derivations factoring through T/F.
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BibTeX
@misc{huynh2026ideals,
title = {{Lie Ideals and Lie Derivations of the Algebraic Toeplitz Algebra}},
author = {Thao Nhi Nguyen Huynh and Viet Khanh Huynh},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.04745},
eprint = {2609.04745},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.04745},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Lie Ideals and Lie Derivations of the Algebraic Toeplitz Algebra
AU - Thao Nhi Nguyen Huynh
AU - Viet Khanh Huynh
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2609.04745
UR - https://arxiv.org/abs/2609.04745
AB - Let K be a field and let T=L_K(E_T) be the algebraic Toeplitz algebra. We classify the Lie ideals of T^-. If w is the sink of the Toeplitz graph and F=I(w), then F≅ M_∞(K), T/F≅ K[t,t^−1], and [T,T]=F. We prove that every noncentral Lie ideal contains s=[F,F], the finitary trace-zero Lie algebra. Thus the classification reduces to the Heisenberg-type quotient T/s. The Lie ideals are precisely $0$, K1, the spaces s+U, where U≤ P_K⊕ Kw and w∉ U, and the inverse images ρ^−1(W), where W≤ K[t,t^−1]. Here P_K=K1 in characteristic $0$, and P_K=span_K{1,e^mℓ,(e^*)^mℓ:m≥1} if charK=ℓ>0. We also describe the Lie derivations of T^- as sums of associative derivations of T and central Lie derivations factoring through T/F.
ER -