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## Abstract

Assume that F is an algebraically closed field with characteristic zero. The universal Racah algebra ℜ is a unital associative F-algebra defined by generators and relations. The generators are A,B,C,D and the relations state that [A,B]=[B,C]=[C,A]=2D and each of [A,D]+AC−BA,[B,D]+BA−CB,[C,D]+CB−AC is central in ℜ. The universal additive DAHA (double affine Hecke algebra) H of type (C_{1}^{∨},C_{1}) is a unital associative F-algebra generated by t_{0},t_{1},t_{0}^{∨},t_{1}^{∨} and the relations state that t_{0}+t_{1}+t_{0}^{∨}+t_{1}^{∨}=−1 and each of t_{0}^{2},t_{1}^{2},t_{0}^{∨2},t_{1}^{∨2} is central in H. Each H-module is an ℜ-module by pulling back via the algebra homomorphism ℜ→H given by [Formula presented] Let V denote any finite-dimensional irreducible H-module. The set of ℜ-submodules of V forms a lattice under the inclusion partial order. We classify the lattices that arise by this construction. As a consequence, the ℜ-module V is completely reducible if and only if t_{0} is diagonalizable on V.

Original language | English |
---|---|

Article number | 106653 |

Journal | Journal of Pure and Applied Algebra |

Volume | 225 |

Issue number | 8 |

DOIs | |

State | Published - Aug 2021 |

## Keywords

- Additive DAHA
- Irreducible modules
- Racah algebras

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## Projects

- 1 Finished