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We characterize mutation-finite cluster algebras of rank at least 3 using positive semi-definite quadratic forms. In particular, we associate with every unpunctured bordered surface a positive semi-definite quadratic space V , and with every triangulation a basis in V , such that any mutation of a cluster (i.e., a flip of a triangulation) transforms the corresponding bases into each other by partial reflections. Furthermore, every triangulation gives rise to an extended affine Weyl group of type A, which is invariant under flips. The construction is also extended to exceptional skew-symmetric mutation-finite cluster algebras of types E.
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Conformal Invariance of Clifford Monogenic Functions in the Indefinite Signature Case
We extend constructions of classical Clifford analysis to the case of indefinite non-degenerate quadratic forms. Clifford analogues of complex holomorphic functions—called monogenic functions—are defined by means of the Dirac operators that factor a certain wave operator. One of the fundamental features of quaternionic analysis is the invariance of quaternionic analogues of holomorphic function under conformal (or Möbius) transformations. A similar invariance property is known to hold in the context of Clifford algebras associated to positive definite quadratic forms. We generalize these results to the case of Clifford algebras associated to all non-degenerate quadratic forms. This result puts the indefinite signature case on the same footing as the classical positive definite case.
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- Award ID(s):
- 2051032
- PAR ID:
- 10658759
- Publisher / Repository:
- Springer Nature
- Date Published:
- Journal Name:
- Complex Analysis and Operator Theory
- Volume:
- 18
- Issue:
- 4
- ISSN:
- 1661-8254
- Subject(s) / Keyword(s):
- Conformal transformations on R^{p,q} Möbius transformations Vahlen matrices Monogenic functions Clifford analysis Clifford algebras Conformal compactification
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/s11785-024-01528-y
