The centre of the Dunkl total angular momentum algebra. (English) Zbl 07905965
Let \(V\) be a Euclidean space of dimension \(d\), \(W\) be a reflection group, \(R\) be its root system, \(t\neq 0\) and \(c: R\to \mathbb C\) a \(W\)-invariant parameter function. From this data, the rational Cherednik algebra \(H_{t,c}(V,W)\) can be defined. It is an associative algebra that admits a representation, called the Dunkl representation, defined by the multiplication operators, the Dunkl operators and the group algebra \(\mathbb C W\). Let \(\mathrm{Cl}(V)\) be the Clifford algebra on \(V\). There is a realisation of the Lie superalgebra \(\mathfrak{osp}(1|2)\) inside \(H_{t,c}(V,W)\otimes \mathrm{Cl}(V)\) [B. Ørsted, P. Somberg, V. Souček , Adv. Appl. Clifford Algebr. 19, No. 2, 403–415 (2009; Zbl 1404.17018)]. The Dunkl total angular momentum algebra, \(O_{t,c}(V,W)\) is the supercentraliser of a realisation of this Lie superalgebra \(\mathfrak{osp}(1|2)\) present in \(H_{t,c}(V,W)\otimes Cl(V)\) in the context of the Howe dual pair (\(\mathsf{Pin}(d),\mathfrak{osp}(1|2,\mathbb C)\)).
The generators of the Dunkl total angular momentum algebra were given, in a slightly more general context, in [H. De Bie, R. Oste, J. Van der Jeugt, Lett. Math. Phys. 108, No. 8, 1905–1953 (2018; Zbl 1397.81085)]. The total ideal of relations of this algebra is still not completely known.
The main result of the work under review is the characterisation of the (graded) centre of the algebra \(O_{t,c}(V,W)\) for \(d\geq 3\). It is a univariate polynomial ring whose generator has its expression depends on whether the longest element of the (real) reflection group is \(-1\) (in the realisation \(V\)) or not.
Representation of \(O_{t,c}(V,W)\) are related to the spin representation of \(W\), that is the representations of the double covering \(\widetilde{W}\) that are not equivalent to those of the representation of \(W\). Building from the results on the centre, the authors relate the centres of \(O_{t,c}(V,W)\) and \(\mathbb C \widetilde{W}_-\). This allows them to define a cohomology and to build results à la Vogan, in an analogous fashion to [D. Ciubotaru, Sel. Math., New Ser. 22, No. 1, 111–144 (2016; Zbl 1383.20007)]).
The generators of the Dunkl total angular momentum algebra were given, in a slightly more general context, in [H. De Bie, R. Oste, J. Van der Jeugt, Lett. Math. Phys. 108, No. 8, 1905–1953 (2018; Zbl 1397.81085)]. The total ideal of relations of this algebra is still not completely known.
The main result of the work under review is the characterisation of the (graded) centre of the algebra \(O_{t,c}(V,W)\) for \(d\geq 3\). It is a univariate polynomial ring whose generator has its expression depends on whether the longest element of the (real) reflection group is \(-1\) (in the realisation \(V\)) or not.
Representation of \(O_{t,c}(V,W)\) are related to the spin representation of \(W\), that is the representations of the double covering \(\widetilde{W}\) that are not equivalent to those of the representation of \(W\). Building from the results on the centre, the authors relate the centres of \(O_{t,c}(V,W)\) and \(\mathbb C \widetilde{W}_-\). This allows them to define a cohomology and to build results à la Vogan, in an analogous fashion to [D. Ciubotaru, Sel. Math., New Ser. 22, No. 1, 111–144 (2016; Zbl 1383.20007)]).
Reviewer: Alexis Langlois-Rémillard (Bonn)
MSC:
| 16S80 | Deformations of associative rings |
| 17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |
| 20F55 | Reflection and Coxeter groups (group-theoretic aspects) |
| 81R12 | Groups and algebras in quantum theory and relations with integrable systems |
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