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On the structure and representation theory of q-deformed Clifford algebras

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Abstract

We provide a generalized definition for the quantized Clifford algebra introduced by Hayashi using another parameter k that we call the twist. For a field of characteristic not equal to 2, we provide a basis for our quantized Clifford algebra, show that it can be decomposed into rank 1 components, and compute its center to show it is a classical Clifford algebra over the group algebra of a product of cyclic groups of order 2k. In addition, we characterize the semisimplicity of our quantum Clifford algebra in terms of the semisimplicity of a cyclic group of order 2k and give a complete set of irreducible representations. We construct morphisms from quantum groups and explain various relationships between the classical and quantum Clifford algebras. By changing our generators, we provide a further generalization to allow k to be a half integer, where we recover certain quantum Clifford algebras introduced by Fadeev, Reshetikhin, and Takhtajan as a special case.

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Notes

  1. Modifications can be made for the characteristic 2 case, but the situation is drastically different.

  2. While \({{\,\textrm{Cl}\,}}_r(R \oplus R^*)\) can be considered simply as notation for an algebra given by a presentation, if we consider \(R^*\) as the (rank 1) dual free R-module of R, then \({{\,\textrm{Cl}\,}}_r(R \oplus R^*)\) is defined analogously to the usual Clifford algebras.

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Acknowledgements

The authors thank Daniel Bump and Jae-Hoon Kwon for useful discussions. The first author also thanks Daniel Bump for his patient guidance and caring support throughout the years. The second author thanks Stanford University for its hospitality during his visit in May, 2022. The authors thank the referee for useful comments. This work benefited from computations using SageMath [37]. Some of the results in this work were discovered by the first author as part of his dissertation research. This work was partly supported by Osaka City University Advanced Mathematical Institute (MEXT Joint Usage/Research Center on Mathematics and Theoretical Physics JPMXP0619217849).

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Authors and Affiliations

  1. The Institute for Computational and Mathematical Engineering (ICME), Stanford University, 450 Serra Mall, Stanford, CA, 94305, USA

    Willie Aboumrad

  2. Faculty of Science, Hokkaido University, 5 Chōme Kita 8 Jōnishi, Kita Ward, Sapporo, Hokkaidō, 060-0808, Japan

    Travis Scrimshaw

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T.S. was partially supported by Grant-in-Aid for JSPS Fellows 21F51028.

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Aboumrad, W., Scrimshaw, T. On the structure and representation theory of q-deformed Clifford algebras. Math. Z. 306, 10 (2024). https://doi.org/10.1007/s00209-023-03402-7

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