### Abstract

This paper considers an invariant of modules over a finite-dimensional Hopf algebra, called the critical group. This generalises the critical groups of complex finite group representations studied in [1, 11]. A formula is given for the cardinality of the critical group generally, and the critical group for the regular representation is described completely. A key role in the formulas is played by the greatest common divisor of the dimensions of the indecomposable projective representations.

Original language | English (US) |
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Journal | Mathematical Proceedings of the Cambridge Philosophical Society |

DOIs | |

State | Accepted/In press - Jan 1 2018 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Mathematical Proceedings of the Cambridge Philosophical Society*. https://doi.org/10.1017/S0305004118000786

**Critical groups for Hopf algebra modules.** / Grinberg, D. A.R.I.J.; Huang, J. I.A.; Reiner, Victor.

Research output: Contribution to journal › Article

*Mathematical Proceedings of the Cambridge Philosophical Society*. https://doi.org/10.1017/S0305004118000786

}

TY - JOUR

T1 - Critical groups for Hopf algebra modules

AU - Grinberg, D. A.R.I.J.

AU - Huang, J. I.A.

AU - Reiner, Victor

PY - 2018/1/1

Y1 - 2018/1/1

N2 - This paper considers an invariant of modules over a finite-dimensional Hopf algebra, called the critical group. This generalises the critical groups of complex finite group representations studied in [1, 11]. A formula is given for the cardinality of the critical group generally, and the critical group for the regular representation is described completely. A key role in the formulas is played by the greatest common divisor of the dimensions of the indecomposable projective representations.

AB - This paper considers an invariant of modules over a finite-dimensional Hopf algebra, called the critical group. This generalises the critical groups of complex finite group representations studied in [1, 11]. A formula is given for the cardinality of the critical group generally, and the critical group for the regular representation is described completely. A key role in the formulas is played by the greatest common divisor of the dimensions of the indecomposable projective representations.

UR - http://www.scopus.com/inward/record.url?scp=85056298535&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85056298535&partnerID=8YFLogxK

U2 - 10.1017/S0305004118000786

DO - 10.1017/S0305004118000786

M3 - Article

AN - SCOPUS:85056298535

JO - Mathematical Proceedings of the Cambridge Philosophical Society

JF - Mathematical Proceedings of the Cambridge Philosophical Society

SN - 0305-0041

ER -