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This paper presents a classification of right coideal subalgebras for the finite-dimensional quantum groups Uq(sl2) and Oq(SL2) at a root of unity q of odd order, providing explicit generators and relations for each subalgebra.

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arxiv.org

Shimizu, K., & Sugitani, R. (2024). Coideal subalgebras of quantum SL2 at roots of unity. arXiv preprint arXiv:2410.10064.

This paper aims to classify all right coideal subalgebras of the finite-dimensional quotient of the quantized enveloping algebra Uq(sl2) and the quantized coordinate algebra Oq(SL2) at a root of unity q of odd order.

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by Kenichi Shim... at **arxiv.org** 10-15-2024

Deeper Inquiries

Classifying coideal subalgebras for even order roots of unity is significantly more intricate than the odd order case. Here's why:
Complexity of the Small Quantum Group: When q is an even order root of unity, the structure of Uq(sl2) and its dual, Oq(SL2), becomes more complex. New classes of indecomposable representations emerge, leading to a richer, more challenging classification problem for coideal subalgebras.
Restricted Methods: The techniques employed in the paper, such as the partial derivative operators and the reliance on the specific form of the minimal polynomial for certain elements, might not directly generalize to the even order case. The relations defining Uq(sl2) at even roots of unity exhibit different behavior.
Potential New Coideal Subalgebras: It's highly plausible that entirely new families of coideal subalgebras, not present in the odd order case, arise when q is an even order root of unity. These new subalgebras would reflect the increased complexity of the underlying quantum group.
In essence, while the classification for odd order roots of unity provides a valuable starting point, the even order case demands a more sophisticated approach and is likely to unveil a more intricate landscape of coideal subalgebras.

Yes, exploring alternative approaches to classifying coideal subalgebras of quantum groups is a promising avenue. Here are a few potential directions:
Geometric Techniques: Quantum groups are intimately connected to quantum homogeneous spaces. Leveraging this connection, one could attempt to classify coideal subalgebras by studying the geometry of these quantum spaces. This approach might offer a more intuitive understanding of the classification.
Representation-Theoretic Methods: Coideal subalgebras are closely tied to the representation theory of quantum groups. Investigating the structure of certain classes of representations, such as those related to quantum symmetric pairs, could lead to a more streamlined classification of the corresponding coideal subalgebras.
Computational Methods: For specific quantum groups and roots of unity, computational algebra systems could be employed to systematically explore and classify coideal subalgebras. This approach might be particularly fruitful for small rank quantum groups or for identifying patterns and conjectures that could guide a more general classification.
By pursuing these alternative approaches, we can hope to gain deeper insights into the structure of coideal subalgebras and potentially uncover connections to other areas of mathematics and physics.

The classification of coideal subalgebras has profound implications for the representation theory of quantum groups at roots of unity, with connections rippling out to other areas:
Fusion Rules and Modular Invariants: Coideal subalgebras play a crucial role in understanding the fusion rules of Rep(Uq(sl2)) at roots of unity. These rules govern how tensor products of representations decompose, and their structure is deeply intertwined with the coideal subalgebra lattice. This has implications for constructing modular invariants, which are central objects in conformal field theory.
Quantum Symmetric Pairs: Certain coideal subalgebras correspond to quantum symmetric pairs, which are quantum analogs of symmetric spaces. This connection provides a rich interplay between the representation theory of quantum groups and the geometry of quantum symmetric spaces.
Integrable Systems: Quantum groups at roots of unity appear in the study of integrable systems, such as certain spin chains. Coideal subalgebras could provide insights into the structure of these systems and their solutions.
Topological Quantum Computation: The representation theory of quantum groups at roots of unity, particularly those related to coideal subalgebras, has found applications in topological quantum computation. These subalgebras could be relevant for constructing new topological quantum codes.
In summary, the classification of coideal subalgebras is not merely an algebraic exercise but a gateway to a deeper understanding of the representation theory of quantum groups at roots of unity and their connections to diverse areas of mathematics and physics.

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