[Paper Review] Twisted symplectic reflection algebras

Tatyana Chmutova|ArXiv.org|May 30, 2005

Algebraic structures and combinatorial models10 references11 citations

TL;DR

This paper introduces twisted symplectic reflection algebras (SRAs) for non-faithful group actions, generalizing Etingof and Ginzburg's original construction. It establishes a categorical equivalence between representations of a twisted SRA for a non-faithful action and those of a twisted SRA for a faithful action on a quotient group, showing that nontrivial cocycles can emerge even from trivial ones during reduction, thus extending the representation theory of SRAs to non-injective group actions.

ABSTRACT

In this paper we introduce the notion of twisted symplectic reflection algebras and describe the category of representations of such an algebra associated to a non-faithful G-action in terms of those for faithful actions of G.

Motivation & Objective

• To generalize symplectic reflection algebras to cases where the group action on a symplectic vector space is not necessarily injective.
• To define and study twisted symplectic reflection algebras using projective group representations and 2-cocycles.
• To reduce the representation theory of non-faithful actions to that of faithful actions via categorical equivalences.
• To show that nontrivial cocycles may arise in the reduction process even when starting from a trivial cocycle.
• To establish a correspondence between representations of twisted SRAs for non-faithful actions and those for faithful actions on quotient groups.

Proposed method

• Introduces twisted group algebras ℂψG via 2-cocycles ψ: G×G → ℂ*, defining multiplication g1∘ψg2 = ψ(g1,g2)g1g2.
• Defines ψ-regular elements g ∈ G as those for which ψ(g,h) = ψ(h,g) for all h in the centralizer of g.
• Uses the category of representations of twisted group algebras and constructs a functor F from representations of e_iℋ_c(G,U,ψ) to representations of ℋ_c'(H,U,ζ) via a module category equivalence.
• Applies the theory of fusion categories and character theory for projective representations to relate twisted characters and conjugacy classes.
• Constructs an equivalence of categories between ℋ_c(G,U,ψ)-mod and ℋ_c'(H,U,ζ)-mod by lifting U-action and using the centralizer structure of ψ-regular elements.
• Defines the twisted parameter c' via averaging over preimages: c'(C_i^j) = (α(C_i^j)/(α(e)|C_i^j|)) × ∑_{π(𝓒)=C_i} c(𝓒)|𝓒|, ensuring compatibility of actions.

Experimental results

Research questions

• RQ1 How can symplectic reflection algebras be generalized to non-faithful group actions?
• RQ2 What happens to the cocycle structure when reducing a non-faithful action to a faithful one?
• RQ3 Can the representation category of a twisted SRA for a non-faithful action be described in terms of a faithful action?
• RQ4 Under what conditions does a trivial cocycle give rise to a nontrivial cocycle in the reduced setting?
• RQ5 How do twisted characters and ψ-regular conjugacy classes relate in the context of projective representations?

Key findings

• The representation category of a twisted symplectic reflection algebra for a non-faithful G-action is categorically equivalent to that of a twisted SRA for a faithful H-action, where H is a quotient of G.
• Even if the original cocycle ψ is trivial, the reduced algebra may carry a nontrivial cocycle ζ, demonstrating that nontriviality can emerge in the reduction process.
• The parameter c' in the reduced algebra is defined by averaging c(𝓒) over ψ-regular conjugacy classes 𝒞 mapping to a class C_i in the quotient, ensuring compatibility of the SRA relations.
• The functor F: ℋ_c(G,U,ψ)-mod → ℋ_c'(H,U,ζ)-mod is an equivalence, preserving the U-action and inducing a correspondence between modules.
• The number of irreducible representations of ℂψG equals the number of ψ-regular conjugacy classes, and these classes are preserved under the reduction map when restricted to reflections.
• For a generalized symmetric group G = S_n ⋉ (ℤ/mℤ)^n, the Schur multiplier H^2(G,ℂ*) is nontrivial when m is even and n ≥ 3, leading to nontrivial cocycles that affect the structure of the twisted SRA.