Ideals in Enveloping Algebras of Loop Algebras

Abstract

Let g be a finite-dimensional simple complex Lie algebra, and consider the loop algebra Lg = g[s, s⁻¹] and the affine Lie algebra ĝ, which is an extension of Lg by a central element c. It is known that the rings U(ĝ)/(c−λ) are simple for any nonzero scalar λ. On the other hand, the universal enveloping algebra U(Lg) ≅ U(ĝ)/(c) has many two-sided ideals, and in this paper we investigate their structure. We show that the two-sided structure of U(Lg) is both intricate and well-behaved. Specifically, U(Lg) does not satisfy the ascending chain condition on two-sided ideals, but the two-sided ideals still have a nice structure: there is a canonical nested collection of ideals I_n, parameterised by positive integers, so that any nonzero two-sided ideal of U(Lg) contains all I_n for n ≫ 0. The ideals I_n can be thought of as universal annihilators of natural classes of finite-dimensional representations of Lg. We also study the Poisson structure of the symmetric algebra S(Lg) and give applications to the representation theory of Lg.

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BibTeX RIS

BibTeX

@misc{petukhov2026ideals,
  title = {{Ideals in Enveloping Algebras of Loop Algebras}},
  author = {Alexey V. Petukhov and Susan J. Sierra},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.22040},
  eprint = {2610.22040},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.22040},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Ideals in Enveloping Algebras of Loop Algebras
AU  - Alexey V. Petukhov
AU  - Susan J. Sierra
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2610.22040
UR  - https://arxiv.org/abs/2610.22040
AB  - Let g be a finite-dimensional simple complex Lie algebra, and consider the loop algebra Lg = g[s, s^−1] and the affine Lie algebra ĝ, which is an extension of Lg by a central element c. It is known that the rings U(ĝ)/(c−λ) are simple for any nonzero scalar λ. On the other hand, the universal enveloping algebra U(Lg) ≅ U(ĝ)/(c) has many two-sided ideals, and in this paper we investigate their structure. We show that the two-sided structure of U(Lg) is both intricate and well-behaved. Specifically, U(Lg) does not satisfy the ascending chain condition on two-sided ideals, but the two-sided ideals still have a nice structure: there is a canonical nested collection of ideals I_n, parameterised by positive integers, so that any nonzero two-sided ideal of U(Lg) contains all I_n for n ≫ 0. The ideals I_n can be thought of as universal annihilators of natural classes of finite-dimensional representations of Lg. We also study the Poisson structure of the symmetric algebra S(Lg) and give applications to the representation theory of Lg.
ER  -

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