Etingof's nondegeneracy conjecture for the q-deformed inner product

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Let Hc,k(Γn)H_{c,k}(\Gamma_n) be a symplectic reflection algebra, and let M,NM,N be finite-dimensional Hc,k(Γn)H_{c,k}(\Gamma_n)-modules. Define the q-deformed form by

(M,N)q=∑(−q)jdim⁡Ext⁡j(M,N).(M,N)_q=\sum (-q)^j\dim \operatorname{Ext}^j(M,N).

Etingof's q-nondegeneracy conjecture. If qq is not a root of unity, then the form (,)q(,)_q is nondegenerate.

The conjecture is a q-analogue of nondegeneracy of the finite-dimensional inner product. The paper notes that it is proved for cyclic groups under some restrictions on the parameters by Gordon and Losev, while the unrestricted general case remains open.

References

Primary source

Pavel Etingof, “Symplectic reflection algebras and affine Lie algebras”, arXiv:1011.4584 (2012).

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