Degree-one rational Cherednik algebras for the symmetric group

Drinfeld orbifold algebras deform skew group algebras in polynomial degree at most one and hence encompass graded Hecke algebras, and in particular symplectic reflection algebras and rational Cherednik algebras. We introduce parametrized families of Drinfeld orbifold algebras for symmetric groups acting on doubled representations that generalize rational Cherednik algebras by deforming in degree one. We characterize rich families of maps recording commutator relations with their linear parts supported only on and only off the identity when the symmetric group acts on the natural permutation representation plus its dual. This produces degree-one versions of gl_n-type rational Cherednik algebras. When the symmetric group acts on the standard irreducible reflection representation plus its dual there are no degree-one Lie orbifold algebra maps, but there is a three-parameter family of Drinfeld orbifold algebras arising from maps supported only off the identity. These provide degree-one generalizations of the sl_n-type rational Cherednik algebras H_{0,c}.


Introduction
Skew group or smash-product algebras S(V )#G twist the symmetric algebra S(V ) of a finite-dimensional vector space V together with the action of the group algebra CG of a finite group G acting linearly on V . The center is the invariant polynomial ring S(V ) G and there is a natural grading by polynomial degree, with elements in V of degree one and elements in CG of degree zero.
Utilizing parameter maps that originate as Hochschild 2-cocycles to explore formal deformations of S(V )#G has proven useful because although the resulting algebras are noncommutative they give rise to deformations of S(V ) G (by examining centers), yet are easily described as quotient algebras. Both the polynomial degree and the support, i.e., which group elements appear in the nonzero image, are helpful descriptors for the parameter maps and hence the relations for the quotient algebras.
Degree-zero deformations of skew group algebras involve parameter maps that identify commutators of elements in V with certain elements of the group algebra. Several important families of these are of broad interest in noncommutative geometry, combinatorics, and representation theory and are the subject of an already extensive literature (see [Gor08] and [Gor10] and further references therein). By comparison, finding elements of degree one with which to identify commutators of elements in V requires a more intricate analysis of which cocycles pass obstructions in cohomology in order to determine if the resulting deformations satisfy PBW properties [SW12,FGK17]. As a result, degree-one deformations are not as well understood or as often studied, yet could also be significant in giving insight into deformations of the invariant algebra S(V ) G and in connection with singularities of orbifolds.
Degree-zero deformations of skew group algebras are called Drinfeld graded Hecke algebras in recognition of their origins in [Dri86] (see also [Lus88]). These include the important special cases when G acts on a symplectic vector space, and more particularly when G is a complex reflection group acting by the sum of a reflection representation and its dual (a doubled representation). The latter leads to the rational Cherednik algebras, first introduced in [Che91] as rational degenerations of double affine Hecke algebras and later highlighted as an important subfamily of the more general symplectic reflection algebras introduced in [EG02]. When built from an action of the symmetric group, rational Cherednik algebras model Hamiltonian reduction in quantum mechanics and are used to show integrability of Calogero-Moser systems [Eti07].
Degree-one deformations of skew group algebras were termed Drinfeld orbifold algebras and characterized in [SW12] via explicit PBW conditions on parameter maps that are also interpreted in Hochschild cohomology. In [FGK17] we describe the Drinfeld orbifold algebras for S n acting on its natural permutation representation, W , by starting with candidate 2-cocycles and imposing the PBW conditions from [SW12]. Here we expand on that class of examples by considering S n acting on both its doubled permutation representation W * ⊕ W and the doubled representation h * ⊕ h, where W = h ⊕ ι is the sum of the (n − 1)-dimensional irreducible standard and the trivial representations. This not only results in much richer families of algebras, but also yields degree-one generalizations of rational Cherednik algebras for the same doubled representations.
More specifically, in [FGK17] we describe all Drinfeld orbifold algebras where the linear parts of the maps recording commutator relations are supported only on or only off the identity in S n , and show that there are no such maps with linear part supported both on and off the identity. For the two doubled representations of S n considered here we describe all degree-one families of Drinfeld orbifold algebras whose maps have linear part supported only on or only off the identity (Theorems 3.1-3.2). For the case of maps with linear part supported both on and off the identity we provide a family of examples involving the doubled permutation representation (Theorem 5.4) and observe there are no corresponding such maps for the doubled standard representation (Remark 8.5).
We summarize our main results.
Theorem (A). For the symmetric group S n (n ≥ 4) acting on V ∼ = C 2n by the doubled permutation representation, there is (1) a 17-parameter family of Lie orbifold algebras described by 20 homogeneous quadratic equations, and (2) a seven-parameter family of Drinfeld orbifold algebras described in terms of parameter maps with linear part supported only off the identity that are controlled by four homogeneous quadratic equations in six of the parameters.
These are the only degree-one deformations of the skew group algebra for S n acting by the doubled permutation representation whose parameter maps have linear part supported only on or only off the identity.
See Theorems 5.1 and 5.3 for more details about the maps, Theorems 3.1-3.2 for the resulting quotient algebras, and Table 2 in Section 3 for a summary.
Theorem (B). For the symmetric group S n (n ≥ 4) acting on V ∼ = C 2n−2 by the doubled standard representation, there are no degree-one Lie orbifold algebras, but there is a three-parameter family of Drinfeld orbifold algebras described by parameter maps with linear part supported only off the identity.
These are the only degree-one deformations of the skew group algebra for S n acting by the doubled standard representation whose parameter maps have linear part supported only on or only off the identity.
See Theorem 8.2 and 8.3 for details, Theorem 3.3 for the resulting algebras, and Table 3 in Section 3 for a summary.
The algebras in Theorems (A) and (B) specialize to the well-known rational Cherednik algebras for the symmetric group, described as of gl n -and sl n -type respectively in [GG06], and hence should be of substantial interest. In particular, Theorem (B) provides a degree one version of the sl n -type rational Cherednik algebras H 0,c . We refer to the algebras as degree-one rational Cherednik algebras. Investigating their structure, properties, combinatorics, representation theory, geometric significance, and potential importance in physics should provide fertile ground for future research. It would also be natural to explore whether similar algebras exist for other complex reflection groups.
We summarize the organization of the paper. Following a brief summary of general preliminaries in Section 2 we restrict to the setting of the symmetric group and the two doubled representations of interest. In Section 3 we collect together the definitions of the relevant maps developed in Sections 5 and 8 and present the resulting degreeone rational Cherednik algebras as quotient algebras. All pre-Drinfeld orbifold algebra maps are constructed in Section 4. We analyze when these lift in Section 5, proving Theorems 5.1, 5.3, and 5.4 using details delayed to Sections 6 and 7. Section 8 begins with Proposition 8.1 relating certain Drinfeld orbifold algebra maps for G acting on a vector space V to such maps when G acts on a subrepresentation of V . This is used with the results from Section 5 to describe in Theorems 8.2 and 8.3 all Drinfeld orbifold algebra maps for S n acting by the doubled standard representation on the subspace h * ⊕ h when the linear part is supported only on or only off the identity.

Preliminaries
Throughout, we let G be a finite group acting linearly on a vector space V ∼ = C n . All tensors will be over C.
Skew group algebras. Let G be a finite group that acts on a C-algebra R by algebra automorphisms, and write g s for the result of acting by g ∈ G on s ∈ R. The skew group algebra R#G is the semi-direct product algebra R CG with underlying vector space R ⊗ CG and multiplication of simple tensors defined by (r ⊗ g)(s ⊗ h) = r( g s) ⊗ gh for all r, s ∈ R and g, h ∈ G. The skew group algebra becomes a G-module by letting G act diagonally on R ⊗ CG, with conjugation on the group algebra factor: g (s ⊗ h) = ( g s) ⊗ ( g h) = ( g s) ⊗ ghg −1 .
In working with elements of skew group algebras, we commonly omit tensor symbols unless the tensor factors are lengthy expressions. If G acts linearly on a vector space V ∼ = C n , then G also acts on the tensor algebra T (V ) and symmetric algebra S(V ) by algebra automorphisms. Assign elements of V degree one and elements of G degree zero to make the skew group algebras T (V )#G and S(V )#G graded algebras.

Cochains.
A k-cochain is a G-graded linear map α = g∈G α g g with components α g : k V → S(V ). If each α g maps into V , then α is called a linear cochain, and if each α g maps into C, then α is called a constant cochain.
We regard a map α on k V as a multilinear alternating map on V k and write α(v 1 , . . . , v k ) in place of α(v 1 ∧ · · · ∧ v k ). Of course, if α(v 1 , . . . , v k ) = 0, then α is zero on any permutation of v 1 , . . . , v k . Also, if α is zero on all k-tuples of basis vectors, then α is zero on any k-tuple of vectors. We exploit these facts in the computations in Sections 6 and 7.
The support of a cochain α is the set of group elements for which the component α g is not the zero map. For X a subset of G, we say a cochain α is supported only on X if α g = 0 for all g not in X. Similarly, we say α is supported only off X if α g = 0 for all g in X. At times, it is convenient to talk about support in a weaker sense, so we say α is supported on X if α g = 0 for some g in X and that α is supported off X if α g = 0 for some g not in X. (Hence, it is possible for a cochain to be simultaneously supported on and off of a set.) The kernel of a cochain α is the set of vectors v 0 such that α(v 0 , v 1 , . . . , v k−1 ) = 0 for all v 1 , . . . , v k−1 ∈ V .
The group G acts on the components of a cochain. Specifically, for a group element h and component α g , the map h α g is defined by In turn, the group acts on the space of cochains by letting h α = g∈G h α g ⊗ hgh −1 .
Thus α is a G-invariant cochain if and only if h α g = α hgh −1 for all g, h ∈ G.
Drinfeld orbifold algebras. For a parameter map κ = κ L + κ C , where κ L is a linear 2-cochain and κ C is a constant 2-cochain, the quotient algebra is called a Drinfeld orbifold algebra if the associated graded algebra gr H κ is isomorphic to the skew group algebra S(V )#G. The condition gr H κ ∼ = S(V )#G is called a Poincaré-Birkhoff-Witt (PBW) condition, in analogy with the PBW Theorem for universal enveloping algebras. Further, if H κ is a Drinfeld orbifold algebra and t is a complex parameter, then Lie orbifold algebras. The parameter maps of Drinfeld orbifold algebras decompose as κ = g κ g g. When κ is a parameter map for a Drinfeld orbifold algebra and the linear part κ L = κ L 1 is supported only on the identity then the map gives rise to a Lie orbifold algebra (see [SW12, Section 4] and Definition 2.1). Lie orbifold algebras deform universal enveloping algebras twisted by a group action just as certain symplectic reflection algebras deform Weyl algebras twisted by a group action.
Drinfeld orbifold algebra maps. Though the defining PBW condition for a Drinfeld orbifold algebra H κ involves an isomorphism of algebras, Shepler and Witherspoon proved an equivalent characterization [SW12, Theorem 3.1] in terms of properties of the parameter map κ.
Definition 2.1. Let κ = κ L + κ C where κ L is a linear 2-cochain and κ C is a constant 2-cochain, and let Alt 3 denote the alternating group on three elements. We say κ is a Drinfeld orbifold algebra map if the following conditions are satisfied for all g ∈ G and v 1 , v 2 , v 3 ∈ V : In the special case when the linear component κ L of a Drinfeld orbifold algebra map is supported only on the identity, we call κ a Lie orbifold algebra map.
To simplify reference to the expressions appearing in the last three Drinfeld orbifold algebra map properties, we define operators ψ and φ that convert 2-cochains (such as κ L and κ C ) into the 3-cochains we see evaluated within the properties.
For the interested reader, we indicate in [FGK17] how the maps ψ and φ relate to coboundary and bracket operations in Hochschild cohomology of a skew group algebra.
Drinfeld orbifold algebra maps (condensed definition). Equipped with the definitions of ψ and φ, the properties of a Drinfeld orbifold map κ = κ L +κ C (Definition 2.1) may be expressed succinctly: Note that any G-invariant 2-cochain whose linear part is supported only on the identity trivially satisfies Properties (2.0) and (2.2), so in this case it is enough to analyze conditions under which Properties (2.3) and (2.4) hold (see Theorem 5.1 and Section 6).
Remark 2.3. If H κ is a Drinfeld orbifold algebra, then κ must satisfy conditions (2.1)-(2.4), but not necessarily the image constraint (2.0). However, [SW12, Theorem 7.2 (ii)] guarantees there will exist a Drinfeld orbifold algebra H κ such that H κ ∼ = H κ as filtered algebras and κ satisfies the image constraint im κ L g ⊆ V g for each g in G. Thus, in classifying Drinfeld orbifold algebras, it suffices to only consider Drinfeld orbifold algebra maps.
Strategy. As described and utilized in [FGK17], the process of determining the set of all Drinfeld orbifold algebra maps consists of two phases, and language from cohomology and deformation theory can be used to describe each phase. First, one finds all pre-Drinfeld orbifold algebra maps, i.e., all G-invariant linear 2-cochains satisfying the image condition (2.0) and the mixed Jacobi identity (2.2). To find such maps supported on reflections in Section 4 we utilize a bijection between pre-Drinfeld orbifold algebra maps and a particular set of representatives of Hochschild cohomology classes (see Lemma 2.5). But to find such maps supported only on the identity in Section 4 we present a simpler argument based on Lemma 4.1 analyzing the eigenvector structure of images dependent on the group action on input vectors. In the second phase, outlined in Section 5 with computational details given in Sections 6 and 7, we determine for which pre-Drinfeld orbifold algebra maps κ L there exists a compatible G-invariant constant 2-cochain κ C such that Properties (2.3) and (2.4) hold. We say κ C clears the first obstruction if Property (2.3) holds and clears the second obstruction if Property (2.4) holds. If a G-invariant constant 2-cochain κ C clears both obstructions, then we say κ L lifts to the Drinfeld orbifold algebra map κ = κ L + κ C .
Hochschild cohomology to pre-DOA maps. We briefly recall how Hochschild cohomology can be used in general to find linear and constant 2-cochains κ that are both G-invariant and satisfy Property 2.2. For more detailed background discussion about the connections to deformations and further references see [FGK17].
For an algebra A over C with bimodule M , the Hochschild cohomology of A with coefficients in M is HH For any finite group G acting linearly on a vector space V ∼ = C n , and for A = S(V )#G, using results of Ştefan [Şte95] yields the following, where R G denotes the set of elements in R fixed by every g in G, Here first described independently by Farinati [Far05] and Ginzburg-Kaledin [GK04]. Note that H • is tri-graded by cohomological degree p, homogeneous polynomial degree d, and group element g.
Since the exterior factors of H p,d g can be identified with a subspace of p V * , and since , the space H • may be identified with a subspace of the cochains introduced earlier in this section. The next lemma records the relationship between Properties (2.1) and (2.2) of a Drinfeld orbifold algebra map and Hochschild cohomology. When d = 1, the lemma is a restatement of [SW12, Theorem 7.2 (i) and (ii)]. When d = 0, the lemma is a restatement of [SW08, Corollary 8.17(ii)]. It is also possible to give a linear algebraic proof in the spirit of [RS03, Lemma 1.8].
Lemma 2.5. For a 2-cochain κ = g∈G κ g g with im κ g ⊆ S d (V g ) for each g ∈ G, the following are equivalent: (a) The map κ is G-invariant and satisfies the mixed Jacobi identity, i.e., for all where [·, ·] denotes the commutator in S(V )#G.
(c) The map κ is an element of Remark 2.6. Part (b)(ii) of Lemma 2.5 is 2ψ(κ) = 0. Part (c) of Lemma 2.5 shows that κ can only be supported on elements g with codim V g ∈ {0, 2} since negative exterior powers are zero and an element g with codimension one acts nontrivially on H 2,d g .

Descriptions of degree-one rational Cherednik algebras
Doubled Representations. For the rest of the paper, let G = S n be the symmetric group, let W ∼ = C n denote its natural permutation representation and consider the doubled permutation representation of S n on V = W * ⊕ W . Let {y 1 , . . . , y n } be an orthonormal basis for W and {x 1 , . . . , x n } be the corresponding dual basis for W * . Then the action of σ ∈ S n is given by σ y i = y σ(i) and where S n acts trivially on the 1-dimensional subspace ι * of W * spanned by x i and by the standard reflection representation on its (n − 1)-dimensional orthogonal complement h * , and similarly W ∼ = h ⊕ ι. We also consider the doubled standard representation of S n on the subspace h * ⊕ h spanned by In this section we present, via generators and relations, degree-one PBW deformations of the skew group algebras S(W * ⊕ W )#S n and S(h * ⊕ h)#S n that result from Theorems 5.1, 5.3, and 8.3. This facilitates comparison with degree-zero deformations (i.e., rational Cherednik algebras) and with the PBW deformations of S(W )#S n in [FGK17]. The classifications are summarized in Tables 2 and 3.
Definitions of linear and constant cochains. For convenience we collect here the definitions of the components of the maps given in Proposition 4.2, and Definitions 4.3 and 7.5. The parameters and elements of support for each map are recorded in Table 1.
Some parts of the descriptions below involve sums of basis vectors over subsets of For 1 ≤ k ≤ 7, let a k , b k , α, β ∈ C be complex parameters. The S n -invariant linear and constant 2-cochains κ L 1 : 2 V → V and κ C 1 : 2 V → C are the alternating bilinear maps defined by Let a, a ⊥ , b, b ⊥ , c be complex parameters. The component κ C g + κ L g of the cochain κ ref corresponding to a transposition g ∈ S n is G-invariant, has V g ⊆ ker κ * g , and has values Algebras for the doubled permutation representation. First, the Lie orbifold algebra maps involving 17 parameters classified in Theorem 5.1 yield a projective variety controlling the Lie orbifold algebras that deform S(W * ⊕ W )#S n in degree one. Based on representative calculations in Macaulay2 [GS] we conjecture that this variety is of dimension eight. Some subvarieties are indicated in Figure 1 in Section 7. When κ L 1 ≡ 0 these Lie orbifold algebras specialize to rational Cherednik algebras corresponding to the parameter c and the general G-invariant skew-symmetric bilinear form κ C 1 involving α and β (because W is decomposable -see [EG02, Proof of Theorem 1.3]).
Second, for κ L supported only off the identity, Theorem 5.3 shows that by comparison there is only a seven-parameter family of Drinfeld orbifold algebra maps and these are controlled by a projective variety which, according to a few representative calculations in Macaulay2 [GS], appears to be five-dimensional. The resulting algebras also specialize to rational Cherednik algebras parametrized by α, β, and c when κ L ref = κ C 3-cyc ≡ 0. Theorem 3.2 (Drinfeld orbifold algebras for doubled permutation representation over C[t]). Let S n (n ≥ 4) act on V = W * ⊕ W with basis B = {x 1 , . . . , x n , y 1 , . . . , y n } by the doubled permutation representation. Suppose a, a ⊥ , b, b ⊥ , c, α, β ∈ C such that and define κ L = κ L 1 and κ C = κ C 1 + κ C ref + κ C 3-cyc to be the linear and constant cochains such that for 1 ≤ i = j ≤ n, Then the quotient H κ,t of T (V )#S n [t] by the ideal generated by is a Drinfeld orbifold algebra over C[t]. Further, the algebras H κ,1 are precisely the Drinfeld orbifold algebras such that im κ L g ⊆ W g for each g ∈ S n and κ L is supported only off the identity.
An analogous statement may be made for algebras constructed from the family of lifts of κ L 1 + κ L ref described in Theorem 5.4 but is omitted here.
When the indicated parameter relations are satisfied, the map κ = κ L + κ C is a Drinfeld orbifold algebra map. The question marks indicate there could be further Algebras for the doubled standard representation. By Theorem 8.2 the only Lie orbifold algebras for S n acting on h * ⊕ h by the doubled standard representation are the known rational Cherednik algebras H nβ,c . However, by Theorem 8.3 there is in this case a three-parameter family of Drinfeld orbifold algebras which are not graded Hecke algebras, but which specialize when a ⊥ = b ⊥ = 0 to the rational Cherednik algebras H 0,c for S n .

Theorem 3.3 (Drinfeld orbifold algebras for doubled standard representation over C[t]).
Let S n (n ≥ 4) act on h * ⊕ h by the doubled standard representation. For a ⊥ , b ⊥ , c ∈ C andx i andȳ i as in (3.1) define κ L and κ C to be the linear and constant cochains such that for Then the quotient H κ,t of T (h * ⊕ h)#S n [t] by the ideal generated by . Further, the algebras H κ,1 are precisely the Drinfeld orbifold algebras such that im κ L g ⊆ h g for each g ∈ S n and κ L is supported only off the identity. Specializing a ⊥ = b ⊥ = 0 yields the rational Cherednik algebra H 0,c . Table 3. Classification of Drinfeld Orbifold Algebra Maps for S n Acting by Doubled Standard Representation.
When the indicated parameter relations are satisfied, the map κ = κ L + κ C is a Drinfeld orbifold algebra map.

Pre-Drinfeld Orbifold Algebra Maps
In this section we identify all pre-Drinfeld orbifold algebra maps for S n acting on W * ⊕ W by the doubled permutation representation. That is, we identify all linear 2cochains κ L satisfying the image condition (2.0), the G-invariance condition (2.1), and the mixed Jacobi identity ψ(κ L ) = 0 (2.2). To organize computations we make use of Lemma 2.5 relating Hochschild cohomology and pre-Drinfeld orbifold algebra maps.
By Remark 2.6, we need only consider elements of codimensions zero and two. Thus for S n acting by the doubled permutation representation we consider two cases: κ L supported only on the identity and κ L supported only on the set of transpositions (which act as reflections on W and bireflections on W * ⊕ W ).
Pre-Drinfeld orbifold algebra maps supported only on the identity. We first prove a general lemma describing all G-invariant maps κ 1 : 2 V → V ⊕ C, where V is a permutation representation of G. Since Properties (2.0) and (2.2) are trivially satisfied when g = 1, this will produce pre-Drinfeld orbifold algebra maps supported only on the identity. Recall that κ 1 is G-invariant if and only if for all g in G and u, v ∈ V . The following lemma shows that how G acts on a set of representative basis vector pairs determines a G-invariant linear cochain.
Lemma 4.1. Suppose G is a finite group acting on a complex vector space V by a permutation representation. If κ L 1 is G-invariant, then the following two conditions hold for all g in G and all basis vector pairs v i and v j .
These imply that the unique way to extend κ L 1 to be G-invariant is well-defined.
Linear cochains. If V ∼ = C 2n is the doubled permutation representation of S n and κ L 1 : 2 V → V is an S n -invariant map, then the value on any pair of basis vectors in {x 1 , . . . , x n , y 1 , . . . , y n } can be obtained by acting by an appropriate permutation on one of the representative values κ L 1 (x 1 , x 2 ), κ L 1 (y 1 , y 2 ), κ L 1 (x 1 , y 1 ), or κ L 1 (x 1 , y 2 ). Consider κ L 1 (x 1 , x 2 ). The permutation σ = (12) swaps x 1 and x 2 , so κ L 1 (x 1 , x 2 ) must be a (−1)-eigenvector of (12), i.e., a linear combination of the vectors x 1 −x 2 and y 1 −y 2 . Both of these vectors are fixed by the group S {3,...,n} of permutations that fix both x 1 and x 2 . Thus, for a choice of complex parameters a 1 and b 1 , we let Consider κ L 1 (x 1 , y 1 ). There are no permutations that will swap x 1 and y 1 . The group of permutations that fix both x 1 and y 1 is S {2,...,n} , so κ L 1 (x 1 , y 1 ) must be an element of the subspace We define κ L 1 (x 1 , y 1 ) to be a linear combination of the basis elements, using complex parameters a 3 , a 4 , b 3 , and b 4 as weights. Orbiting yields the definition Consider κ L 1 (x 1 , y 2 ). There are no permutations that will swap x 1 and y 2 . The group of permutations that fix both x 1 and y 2 is S {3,...,n} , so κ L 1 (x 1 , y 2 ) must be an element of the subspace We define κ L 1 (x 1 , y 2 ) to be a linear combination of the basis elements using complex parameters a 5 , a 6 , a 7 , b 5 , b 6 , and b 7 as weights. Orbiting yields the definition Constant cochains. By comparison there is only a two-parameter family of G-invariant constant cochains. To see this we combine ad hoc and cohomological approaches. First, Next, we appeal to the discussion of Hochschild cohomology in Section 2. The Hochschild 2-cohomology for g = 1 in polynomial degree zero is H 2,0 1 = ( 2 V * ) Sn and has dimension two by the following calculation, involving characters of 2 V and various subrepresentations, where V = W * ⊕W , with W ∼ = C n the natural permutation representation of S n and W = h⊕ι with ι the trivial representation and h the irreducible standard representation.
By Lemma 2.5 it follows that κ C 1 can be defined in terms of two parameters. A similar calculation yields dim H 2,1 1 G = 14, consistent with the number of parameters for κ L 1 .
Definitions. The following proposition summarizes the definitions of all G-invariant skew-symmetric bilinear maps, i.e., describes H 2,0 . , x n , y 1 , . . . , y n } by the doubled permutation representation. For 1 ≤ k ≤ 7, let a k , b k , α, β ∈ C be complex parameters. The S n -invariant linear and constant 2-cochains κ L 1 : 2 V → V and κ C 1 : 2 V → C are the alternating bilinear maps defined by Pre-Drinfeld orbifold algebra maps supported only off the identity. Now we when g is a transposition. Let g = (12) and first note that Z(g) = (12) × S {3,...,n} , . , x n , y 1 + y 2 , y 3 , . . . , y n }, and 12) and After orbiting the centralizer invariants to obtain G-invariants (see the end of Section 4.1 in [FGK17] for more detail), these yield the following definition of a cochain Definition 4.3. Let S n (n ≥ 4) act on V = W * ⊕ W ∼ = C 2n , equipped with the basis {x 1 , . . . , x n , y 1 , . . . , y n }, by the doubled permutation representation. Let a, a ⊥ , b, b ⊥ , c be complex parameters. The component κ C g + κ L g of the cochain κ ref corresponding to a transposition g ∈ S n is defined by being G-invariant, by V g ⊆ ker κ * g , and by Remark 4.4. Note that im κ L g ⊆ V g and since κ * g is G-invariant then in particular (4.6) Pre-Drinfeld orbifold algebra maps. By Lemma 2.5 and Remark 2.6, the polynomial degree one elements of Hochschild 2-cohomology we found in Proposition 4.2 and Definition 4.3 provide a description of all pre-Drinfeld orbifold algebra maps.
Corollary 4.5. The pre-Drinfeld orbifold algebra maps for S n (n ≥ 4) acting on V = W * ⊕ W ∼ = C 2n by the doubled permutation representation are the linear 2-cochains κ L = κ L 1 + κ L ref for κ L 1 described in terms of the parameters a 1 , . . . , a 7 , b 1 , . . . , b 7 as in Proposition 4.2 and κ L ref controlled by the parameters a, a ⊥ , b, b ⊥ as in Definition 4.3. In Theorem 5.1 and Theorem 5.3 we will characterize when the maps κ L 1 and κ L ref lift separately to Drinfeld orbifold algebra maps and in Theorem 5.4 we will show it is also possible to lift κ L 1 + κ L ref .
Any two lifts of a particular pre-Drinfeld orbifold algebra map must differ by a constant 2-cochain that satisfies the mixed Jacobi identity. Lemma 2.5 and the analysis in Section 4 and this section yield the following corollary describing these maps.
Corollary 4.6. For S n (n ≥ 4) acting on V = W * ⊕ W ∼ = C 2n by the doubled permutation representation, the S n -invariant constant 2-cochains satisfying the mixed Jacobi identity are the maps κ C = κ C 1 + κ C ref with κ C 1 given in terms of parameters α and β in Proposition 4.2 and κ C ref described using parameter c in Definition 4.3.

5.
Lifting to deformations of S(W * ⊕ W )#S n We now consider when or whether it is possible to lift each of κ = κ L 1 , κ = κ L ref , and κ = κ L 1 + κ L ref to a Drinfeld orbifold algebra map. In other words, we consider for which pre-Drinfeld orbifold algebra maps κ L there exists a constant 2-cochain κ C such that κ = κ L + κ C also satisfies the remaining Properties (2.3) and (2.4).
We explain in Theorem 5.1 that κ L 1 can be lifted to Characterizing in general when κ L 1 +κ L ref lifts is straightforward but rather more involved. Instead, in Theorem 5.4 we provide a nontrivial choice of parameters and verify in that case that it is possible to lift simultaneously to Most computational details needed for the three proofs are postponed to Sections 6 and 7. The degree-one rational Cherednik algebras corresponding to the maps characterized in Theorems 5.1 and 5.3 are described in Section 3.
Lie orbifold algebra maps. First we describe all Drinfeld orbifold algebra maps with linear part supported only on the identity, i.e., all Lie orbifold algebra maps κ L 1 + κ C . The corresponding Lie orbifold algebras are described in Theorem 3.1.
Theorem 5.1. Let S n (n ≥ 4) act on V = W * ⊕ W ∼ = C 2n by the doubled permutation representation, and let κ L 1 and κ C 1 be as described in Proposition 4.2 and κ C ref be as in Definition 4.3 with complex parameters a 1 , . . . , a 7 , b 1 , . . . , b 7 , α, β, and c. The Lie orbifold algebra maps are precisely the maps of the form κ = κ L 1 + κ C 1 + κ C ref satisfying conditions (6.1)-(6.20). This includes the following natural special cases: (1) κ = κ L 1 + κ C 1 satisfying conditions (6.1)-(6.18), and Proof. Let κ L be a pre-Drinfeld orbifold algebra map supported only on the identity. By Corollary 4.5 we know κ L = κ L 1 given in terms of a i and b i for 1 ≤ i ≤ 7 is as in Proposition 4.2.
Remark 5.2. Notice that in case (2) of Theorem 5.3, by (6.31) and (6.32), either α = −(n − 1)β in κ C 1 or we also have a 4 = a 7 and b 4 = b 7 in κ L 1 . In the latter case, a 3 = a 5 + a 6 , a 4 = a 7 , and describe a compatibility between the definition of κ L 1 (x i , y j ) in (4.5) but with j allowed to be i, and the definition of κ L 1 (x i , y i ) in (4.4). Also note that a 4 = a 7 and b 4 = b 7 together with (6.21)-(6.22) allow further simplification and collapsing in conditions (6.1)-(6.16) from (6.23)-(6.30) to just the three equations Drinfeld orbifold algebra maps. Next we describe all Drinfeld orbifold algebra maps with linear part supported only off the identity. The proof is outlined here but the details of clearing the obstructions appear in Section 7. The corresponding Drinfeld orbifold algebras are described in Theorem 3.2.
Theorem 5.3. For S n (n ≥ 4) acting on V = W * ⊕W ∼ = C 2n by the doubled permutation representation, the Drinfeld orbifold algebra maps supported only off the identity are precisely the maps of the form κ 3-cyc as in Definition 7.5, κ C 1 as in Proposition 4.2, and with the parameters a, a ⊥ , b, b ⊥ , c, α, and β satisfying the conditions αa + β(a + (n − 2)a ⊥ ) = 0, αa ⊥ + β(2a + (n − 3)a ⊥ ) = 0, (7.3) Proof. Suppose κ L is a pre-Drinfeld orbifold algebra map supported only off the identity. By Corollary 4.5 we must have κ L = κ L ref for some parameters a, a ⊥ , b, b ⊥ ∈ C as in Definition 4.3. It remains to find all G-invariant maps κ C such that Properties (2.3) and (2.4) of a Drinfeld orbifold algebra map also hold.
First we find a particular lift.
, with κ C ref as in Definition 4.3 for some parameter c ∈ C and κ C 1 as in Proposition 4.2 for some parameters α, β ∈ C. • Second obstruction. By Corollary 7.9, φ(κ C  Lastly, by specifying parameters we obtain some Drinfeld orbifold algebra maps that are supported both on and off the identity. The proof uses results related to clearing obstructions that appear in Section 6 and Section 7.  Proof. As in the proof of Theorem 5.3, even without the given parameter choices we have . Setting a i = b i = 0 for i = 1, 2, 3, 5, 6, a 4 = a 7 , and b 4 = b 7 in (6.1)-(6.16) and in the values of φ L g,1 (u, v, w) in Sections 6.1-6.3 shows φ(κ L 1 , κ L 1 ) = 0 and φ(κ L ref , κ L 1 ) = 0 respectively. Using the forms of φ * 1,g given in Lemma 7.8 when * = L and the assumptions that a 4 = a 7 and b 4 = b 7 yield that and similarly for φ L 1,g (y i , y j , x k ) and φ L 1,g (y i , y j , x j ), but replacing b with a and b ⊥ with a ⊥ . By the hypothesis on a, a ⊥ , b, and b ⊥ , all of these are zero, so φ(κ L 1 , κ L ref ) = 0 and hence Thus with the given parameter choices κ C 1 + κ C ref + κ C 3-cyc clears the first obstruction for κ L 1 + κ L ref .
Thus with the given choices of parameters, κ C 1 + κ C ref + κ C 3-cyc clears the second obstruction as well and lifts κ L 1 + κ L ref to a Drinfeld orbifold algebra map.

Invariant Lie Brackets
In Section 4, as summarized in Proposition 4.2, we determined the pre-Drinfeld orbifold algebra maps κ L 1 supported only on the identity. In this section, we determine conditions under which these maps also endow V with a Lie algebra structure -i.e., under which they lift to Lie orbifold algebra maps. This provides details used in the proof of Theorem 5.1.
Our main goal is to write down conditions on the parameters involved in the definitions of κ L 1 , κ C 1 , and κ C ref such that Properties (2.3) and (2.4) hold, i.e., such that φ(κ L 1 , κ We use this to arrive at characterizing conditions on parameters as summarized in Section 6.4. It will be convenient along the way to also consider φ(κ L ref , κ L 1 ), for use in Theorem 5.4, by using * to denote either C or L and x to denote either a transposition or the identity and computing, for v 1 , v 2 , v 3 ∈ V , ) as uniformly as possible. This notation omits a factor of two (and hence differs from that in [FGK17]) because ψ(κ C 1 + κ C ref ) = 0 means the factor of 2 is irrelevant to clearing the first obstruction and it is also irrelevant to clearing the second obstruction.
First note that due to bilinearity and skew-symmetry it suffices to compute φ * x,1 , with x equal to the identity or a transposition, on basis triples of six main types for 1 ≤ i, j, k ≤ n.
1. All basis vectors in W or in W * and i, j, k distinct: (x i , x j , x k ), (y i , y j , y k ).
2. Two basis vectors in W or in W * and i, j, k distinct: (x i , x j , y k ), (y i , y j , x k ).
3. Two basis vectors in W or W * and i, j distinct: This is done in corresponding subsections below.
6.1. All basis vectors in W or in W * and three distinct indices. For any distinct indices i, j, k with 1 ≤ i, j, k ≤ n, we have ). Using bilinearity, skew-symmetry, and the definitions of κ 1 and κ ref in Proposition 4.2 and Definition 4.3 yield for x either the identity or any transposition, Combining these shows that φ * x,1 (x i , x j , x k ) = 0 and similarly φ * x,1 (y i , y j , y k ) = 0, for any (distinct i, j, k with) 1 ≤ i, j, k ≤ n, for x either the identity or a transposition, and with * = C or * = L. Thus this case imposes no restrictions on any parameters. 6.2. Two basis vectors in W or W * and three distinct indices. For any distinct indices i, j, k with 1 ≤ i, j, k ≤ n, using the definition of κ L 1 , bilinearity, and skewsymmetry yields ). In particular, for x = 1 and * = C, since κ C 1 (v, w) = 0 when v, w ∈ W or v, w ∈ W * , we have φ C 1,1 (x i , x j , y k ) = b 5 (β − β) + (b 6 − a 1 )(β − β) + b 7 (α − α + (n − 1)(β − β)) = 0, and for x = 1 and * = L using the definition of κ L 1 yields while when x = g is a transposition, by Definition 4.3 we have Interchanging the roles of x and y and recomputing yields that for any distinct i, j, k with 1 ≤ i, j, k ≤ n, , y i − y j ).
6.4. Summary of PBW conditions. The expressions for φ * x,1 (u, v, w) in Sections 6.1-6.3 yield the following PBW conditions for Lie orbifold algebras obtained by deforming the skew group algebra for S n acting on V = W * ⊕ W by the doubled permutation representation.
It contains the lattice of subvarieties shown in Figure 1 arising from subrepresentations of the doubled permutation representation, where varieties corresponding to subrepresentations that differ only by a trivial summand are equal (see Proposition 8.1). The subvarieties are the zero sets of the quadratic polynomials from the PBW conditions (6.1)-(6.20), together with additional quadratic or linear polynomials that arise from image constraints for κ L 1 . Finding these is facilitated by Remark 6.1. Remark 6.1. Observe that x i =x i + 1 n x [n] and y i =ȳ i + 1 n y [n] can be used in (4.2)-(4.5) to decompose the results according to V ∼ = h * ⊕ ι * ⊕ h ⊕ ι: . In particular, if the image of κ L 1 is contained within h * ⊕ h then the coefficients of x [n] and y [n] are zero, leading to conditions (6.36) and (6.37), and if the image of κ L 1 contains a copy of the trivial representation, i.e., some linear combination of x [n] and y [n] , then the coefficients of x [n] and y [n] satisfy the weaker condition shown in condition (6.38). Similarly, if the image of κ L 1 contains a copy of the standard representation then the coefficients ofx i andȳ i satisfy conditions (6.39). Lastly, if the image of κ L 1 is contained within ι * ⊕ ι then the coefficients ofx i andȳ i are zero, as in conditions (6.40). Note that each condition involves a quadratic or linear polynomial. For each subvariety the list of polynomials and the dimension and degree (computed for some specific values of n) are indicated in Figure 1. Conditions when im κ L 1 is contained in the doubled standard representation a 3 + na 4 = 0 a 5 + a 6 + na 7 = 0 (6.36) b 3 + nb 4 = 0 b 5 + b 6 + nb 7 = 0 (6.37) Condition when im κ L 1 contains the trivial representation (a 3 + na 4 )(b 5 + b 6 + nb 7 ) = (b 3 + nb 4 )(a 5 + a 6 + na 7 ) (6.38) Conditions when im κ L 1 contains the standard representation a i b j = b i a j for 1 ≤ i < j ≤ 6 and i, j = 4 (6.39)  on the parameters α and β for κ C 1 and a, a -cyc is always a Drinfeld orbifold algebra map and κ L ref 3-cyc is a Drinfeld orbifold algebra map when conditions (7.3) and (7.4) hold. First we recall a lemma from [FGK17] that allows for a reduction in the computations necessary to remove obstructions and lift κ maps.
Invariance relations. Recall that a cochain α = g∈G α g g with components α g : for all g, h ∈ G and v 1 , . . . , v k ∈ V . Thus a G-invariant cochain is determined by its components for a set of conjugacy class representatives.
In the following lemma, one can let α = κ L or α = κ C and let β = κ L to see that if κ L and κ C are G-invariant, then φ(κ * , κ L ) and ψ(κ * ) are also G-invariant. This is helpful because, for instance, if φ g = 2ψ g for some g ∈ G, then acting by h ∈ G on both sides shows φ hgh −1 = 2ψ hgh −1 also. Thus if φ g = 2ψ g for all g in a set of conjugacy class representatives, then φ(κ L , κ L ) = 2ψ(κ C ). Similar reasoning applies to Properties (2.2) and (2.4) of a Drinfeld orbifold algebra map.
Clearing the First Obstruction. We begin by recording simplifications of a sum- , where * stands for L or C. Simplification of φ * σ,τ (u, v, w) depends on the location of the basis vectors relative to W and W * and relative to the fixed spaces V σ and V τ , so recall that v * is the vector dual to v and define the following indicator function. For g ∈ S n and v ∈ V , let Note that for g ∈ G and v ∈ V , δ g (v * ) = δ g (v). Remark. Let φ * σ,τ be as in Lemma 7.2. Then for all u, v, w ∈ V we have . This follows from the definition of φ * σ,τ and that κ L τ is τ -invariant. (1) If u, v, w ∈ W , u, v, w ∈ W * , u, v ∈ V τ , or u ∈ V τ ∩ V σ , then φ * σ,τ (u, v, w) = 0.
(2) If u ∈ V τ \ V σ and v / ∈ V τ , then the basis vectors moved by τ are of the form v, τ v, v * , τ v * . We have φ * σ,τ (v, τ v, u * ) = 0, and by (7.1), and by (7.1), )). and that κ L τ is zero whenever both input vectors are in W * or both are in W shows φ * σ,τ (u, v, w) = 0 whenever u, v, w ∈ W or u, v, w ∈ W * . As in [FGK17], φ * σ,τ (u, v, w) = 0 when u, v ∈ V τ follows from (7.2) and that V τ ⊆ ker κ * τ , while φ * σ,τ (u, v, w) = 0 when u ∈ V τ ∩ V σ uses also V σ ⊆ ker κ * σ . Case (2). Assume u ∈ V τ \ V σ and v / ∈ V τ . First note that φ * σ,τ (v, τ v, u * ) = 0 by using (7.1) and the alternating property to see that φ ) by using u ∈ V τ ⊆ ker κ * τ in (7.2). Using bilinearity and V σ ⊆ ker κ * σ , the right hand side is a linear combination of expressions κ * σ (u, h u) and κ * σ (u, h u * ) for h ∈ σ . The appropriate coefficients in terms of a, a ⊥ , b, b ⊥ can be described in terms of the indicator function for the fixed space of τ and depend on whether u ∈ W * or u ∈ W . Also note that h∈ σ h u ∈ V σ ⊆ ker κ * σ . Thus, for u ∈ W * we have . When u ∈ W , the calculation of κ * σ (u, κ L τ (v, v * )) involves a sign difference, and the coefficients on the two sums are reversed, so the one with coefficient a ⊥ − a survives and As mentioned in the outline of the proof of Theorem 5.3, the next two propositions are used to evaluate both φ(κ Proof. Since κ * ref is supported only on transpositions and the only cycle types that arise as a product of two transpositions are the identity, double transpositions, and 3-cycles, it suffices to consider only the components κ 1 and κ g where g is a double transposition. Since h σ h τ = h (στ ), it suffices to use only representatives of orbits of factor pairs under the action of S n by diagonal conjugation.
For each of these three basis triples, u, v, w, the result φ * g (u, v, w) will be the same by Lemma 7.2 (although the reason varies for a given term on different triples), namely A similar reduction and computation applies to the nine basis triples with two elements in W and one element in W * , and that combined with the orbiting properties in Lemma 7.1 lead to the conclusion in the statement.
Definition 7.5. Let S n (n ≥ 4) act by the doubled permutation representation on V = W * ⊕ W ∼ = C 2n equipped with basis B = B x ∪ B y where B x = {x 1 , . . . , x n } and B y = {y 1 , . . . , y n }. For parameters a, b, a ⊥ , b ⊥ ∈ C as in Definition 4.3, define an S ninvariant map κ C 3-cyc = g∈Sn κ C g g with component maps κ C g : 2 V → C. If g is not a 3-cycle, let κ C g ≡ 0. If g is a 3-cycle, define κ C g by V g ⊆ ker κ C g , and if v ∈ B is not in V g then . Alternatively, when g is a 3-cycle, we could define κ C g by the skew-symmetric matrix where [g] denotes the matrix of g with respect to the basis x 1 , . . . , x n , or equivalently y 1 , . . . , y n , and the (i, j)-entry of [κ C g ] records κ C g (v i , v j ) for the ordered basis {v 1 , . . . , v 2n } = {x 1 , . . . , x n , y 1 , . . . , y n } of V .
Proposition 7.6. Let κ L ref and κ C 3-cyc be as in Definitions 4.3 and 7.5, with common parameters a, a . If g is not a 3-cycle, then φ g ≡ 0 by Proposition 7.3; and κ C 3-cyc is not supported on g, so 2ψ g ≡ 0 as well. Now assume g is a 3-cycle, and note that it suffices to compare components φ g and ψ g on basis triples of the form in the statement of Proposition 7.4: Case (1). If u ∈ V g , then because φ g (u, v, w) = 0 by Proposition 7.4, and because g u − u = 0 and V g ⊆ ker κ C because the left hand side is zero by Proposition 7.4, and because g-invariance of κ C g yields Case (3). Also by Proposition 7.4, for v / ∈ V g we have and by definition of ψ g we have Comparing terms and applying the consequences of Definition 7.5 yield for all v ∈ V , Clearing the Second Obstruction. As mentioned in the outline of the proof of Theorem 5.3, the final step in determining when , which is stated as Corollary 7.9 and follows immediately from Propositions 7.3 and 7.4 and Lemmas 7.7 and 7.8. This clears the second obstruction and completes the proof of Theorem 5.3.
Proof. The proof proceeds by considering the same exhaustive cases as in Lemma 7.2, but using the definition of κ C 3-cyc to show in fact φ σ,τ ≡ 0 in cases (2) and (3). Showing that φ * σ,τ (u, v, w) = 0 when u, v, w ∈ W , u, v, w ∈ W * , u, v ∈ V τ , or u ∈ V τ ∩ V σ proceeds exactly as in the proof of Lemma 7.2 since the methods did not depend on anything about κ * σ other than V σ ⊆ ker κ * σ . As in the proof of case (2) in Lemma 7.2, assume u ∈ V τ \ V σ and v / ∈ V τ and note ). Using bilinearity and V σ ⊆ ker κ * σ , the right hand side is a linear combination of expressions κ * σ (u, h u) for h ∈ σ . The appropriate coefficients in terms of a, a ⊥ , b, b ⊥ can be described in terms of the indicator function for the fixed space of τ and depend on whether u ∈ W * or u ∈ W . Also, h∈ σ h u ∈ V σ ⊆ ker κ * σ . Thus for u ∈ W * , and hence φ σ,τ (u, v, v * ) = 0. A similar calculation shows that and applying Remark 4.4 leads to the same conclusion when u ∈ W . Then (7.1) also implies For case (3) assume v / ∈ V τ and note that as in the proof of Lemma 7.2, = 0 by the same calculation as in case (2) except with τ v and v in place of u, it follows that Lastly, (7.1) yields The case where * = L is included in the preliminary calculations of the following lemma because it will be useful as a starting point in the proof of Theorem 5.4.
Lemma 7.8. Let κ C 1 be as in Proposition 4.2 and κ L ref be as in Definition 4.3, with parameters α, β ∈ C and a, a ⊥ , b, b ⊥ ∈ C respectively. Denote a term of the component φ g of φ(κ * 1 , κ L ref ) by φ * 1,g , where * = C or * = L and g is a transposition. Then φ C 1,g = 0 if and only if the following conditions hold Proof. As in Section 6, it suffices to compute φ * 1,g on basis triples of the following forms for 1 ≤ i, j, k ≤ n.
1. All basis vectors in W or in W * and i, j, k distinct: (x i , x j , x k ), (y i , y j , y k ).
2. Two basis vectors in W or in W * and i, j, k distinct: (x i , x j , y k ), (y i , y j , x k ).
3. Two basis vectors in W or W * and i, j distinct: (x i , x j , y j ), (y i , y j , x j ).
Case 1. For any distinct indices i, j, k with 1 ≤ i, j, k ≤ n, we have By Definition 4.3 of κ L ref it is immediate that φ * 1,g (x i , x j , x k ) = 0, and in similar fashion φ * 1,g (y i , y j , y k ) = 0, for any (distinct i, j, k with) 1 ≤ i, j, k ≤ n. Thus this case imposes no conditions on any parameters. Case 2. For any distinct indices i, j, k with 1 ≤ i, j, k ≤ n, using the definitions of κ L ref and κ C 1 , bilinearity, and skew-symmetry yields if g = (ik) and * = C, −[αb ⊥ + β(2b + (n − 3)b ⊥ )] if g = (jk) and * = C, 0 otherwise.
Interchanging the roles of x and y and recomputing yields that for any distinct indices i, j, k with 1 ≤ i, j, k ≤ n, Case 3. For any distinct indices i, j with 1 ≤ i, j ≤ n, using the definitions of κ L ref and κ C 1 , bilinearity, and skew-symmetry yields Interchanging the roles of x and y and recomputing yields that for any distinct indices i, j with 1 ≤ i, j ≤ n, Setting the results in cases 2 and 3 equal to zero yields conditions (7.3) and (7.4).
Corollary 7.9. Let κ C 3-cyc and κ L ref be as in Definitions 7.5 and 4.3, with common parameters a, a ⊥ , b, b ⊥ ∈ C. For every g ∈ S n , the component Conditions (7.3) and (7.4) given in Lemma 7.8 give rise to a projective variety that controls the parameter space for the family of maps in Theorem 5.3 and Drinfeld orbifold algebras in Theorem 3.2. We conjecture that this variety has dimension five based on computations done for a few specific values of n in Macaulay2 [GS] with the graded reverse lexicographic monomial ordering and the parameter order a, a ⊥ , b, b ⊥ , α, β, c.

Deformations of S(h * ⊕ h)#S n
We use the results in Section 5 on Lie and Drinfeld orbifold algebra maps that produce deformations of the skew group algebra S(W * ⊕ W )#S n in order to understand which maps produce deformations of S(h * ⊕ h)#S n .
In contrast to the complicated families of Lie orbifold algebras and maps in Theorems 3.1 and 5.1, when S n instead acts on its doubled standard subrepresentation h * ⊕ h there are no Lie orbifold algebra maps with nonzero linear part (Theorem 8.2). However, Theorem 8.3 describes a three-parameter family of Drinfeld orbifold algebra maps that do provide polynomial degree one deformations generalizing the sl n -type rational Cherednik algebras H 0,c (see also Theorem 3.3). We begin with a result that relates some Drinfeld orbifold algebra maps for G acting on a vector space V to such maps when G acts on a subrepresentation of V .
Proposition 8.1. Let G be a finite group acting linearly on V ∼ = C n and suppose U and U 0 are subrepresentations such that V = U ⊕ U 0 and G acts trivially on U 0 . There is a bijection between Drinfeld orbifold algebra maps for G acting on U and Drinfeld orbifold algebra maps κ : 2 V → (V ⊕ C) ⊗ CG for G acting on V such that U 0 ⊆ ker κ and im κ L g ⊆ U for all g ∈ G.
Proof. Each Drinfeld orbifold algebra map κ : 2 V → (V ⊕C)⊗CG such that U 0 ⊆ ker κ and im κ L g ⊆ U for all g ∈ G restricts to 2 U to produce a Drinfeld orbifold algebra map for G acting on U . Conversely, given a Drinfeld orbifold algebra map κ for G acting on U , extend κ to 2 V by letting κ(u 0 , v) = 0 for all u 0 ∈ U 0 and v ∈ V . We show this extension is a Drinfeld orbifold algebra map for G acting on V = U ⊕ U 0 .
Note that U g ⊆ V g so the extension still satisfies the image containment condition (2.0) and it is straightforward to verify that the extended maps are G-invariant (2.1).
By hypothesis the Drinfeld orbifold algebra map properties (2.2)-(2.4) hold for triples of vectors in U . Using multilinearity, all that remains is to consider triples with at least one vector in U 0 . We claim that in this case ψ * g (v 1 , v 2 , v 3 ) = 0 and φ * x,y (v 1 , v 2 , v 3 ) = 0 for all g ∈ G, and so the properties (2.2)-(2.4) easily follow. Indeed, recalling that , we see that if, for example, v 3 is in U 0 , then the first term is zero by the assumption that G acts trivially on U 0 and the last two terms are zero because v 3 ∈ U 0 ⊆ ker κ * g . Recalling that , we see that if, for example, v 3 is in U 0 , then the first two terms are zero because v 3 ∈ U 0 ⊆ ker κ L y and the last term is zero because v 3 + y v 3 = 2v 3 ∈ U 0 ⊆ ker κ * x . This completes the details showing the extension of κ is a Drinfeld orbifold algebra map for G acting on U ⊕ U 0 . Since the restriction and extension maps are inverses, we obtain the desired bijection.
We now use this result to try to extend κ from 2 (h * ⊕ h) to 2 (W * ⊕ W ). However, the form of the Lie orbifold algebra maps κ L 1 + κ C 1 + κ C ref for S n acting on its doubled permutation representation will imply that in fact κ L 1 ≡ 0.
Theorem 8.2. For S n (n ≥ 4) acting on h * ⊕ h ∼ = C 2n−2 by the doubled standard representation there are no degree-one Lie orbifold algebra maps.
Proof. If there were a Lie orbifold algebra map κ L + κ C for the doubled standard representation with κ C : 2 (h * ⊕ h) → C and nonzero κ L : 2 (h * ⊕ h) → h * ⊕ h, then it could be extended as described in Proposition 8.1 to yield a Lie orbifold algebra map for S n acting on V = W * ⊕ W ∼ = C 2n via the doubled permutation representation. The possible forms of such extensions κ are controlled by Theorem 5.1. By imposing first the image constraint im κ L ⊆ h * ⊕ h, then the extension conditions in Proposition 8.1, and then PBW conditions (6.1)-(6.4), we find that in fact κ L ≡ 0. First, use x i =x i + 1 n x [n] and y i =ȳ i + 1 n y [n] in (4.2)-(4.5) to write the values of κ L 1 according to the decomposition V ∼ = h * ⊕ ι * ⊕ h ⊕ ι: 1 (x i , y j ) = a 5xi + a 6xj + 1 n (a 5 + a 6 + na 7 )x [n] + b 5ȳi + b 6ȳj + 1 n (b 5 + b 6 + nb 7 )y [n] .
To analyze the PBW conditions (6.1)-(6.4) it will help to first observe that the above constraints a 5 = a 6 and a 5 + a 6 + na 7 = 0 yield that a 5 = a 6 = − n 2 a 7 , and hence that a 4 = a 5 + a 7 = − n−2 2 a 7 and a 3 = −na 4 = n(n−2) 2 a 7 , with corresponding expressions in terms of b 7 for b 3 , b 4 , b 5 , and b 6 . These allow the simplification Similarly, φ 1,1 (y i , y j , x k ) = n 2 4 a 2 7 (x i − x j ) + n 2 4 a 7 b 7 (y i − y j ). Requiring each of these to be zero forces a 7 = b 7 = 0, and thus a i = b i = 0 for 3 ≤ i ≤ 6 as well. Since we already have a 1 = b 1 = a 2 = b 2 = 0, this proves there are no Lie orbifold algebra maps for S n acting on the doubled standard subrepresentation h * ⊕ h with κ L ≡ 0.
For maps with linear part supported only off the identity there is instead a threeparameter family of Drinfeld orbifold algebra maps that generalize the commutator relations for the rational Cherednik algebra H 0,c . Theorem 8.3. For S n (n ≥ 4) acting on h * ⊕ h ∼ = C 2n−2 by the doubled standard representation, all Drinfeld orbifold algebra maps with nonzero linear part supported only off the identity have the form κ L + κ C defined by Proof. Suppose κ = κ L + κ C is a Drinfeld orbifold algebra map for the doubled standard representation with κ C : 2 (h * ⊕h) → CS n and nonzero κ L : 2 (h * ⊕h) → (h * ⊕h)⊗CS n supported only off the identity. Extend κ as described in Proposition 8.1 to yield a Drinfeld orbifold algebra map for S n acting on V = W * ⊕ W ∼ = C 2n via the doubled permutation representation. By Theorem 5.3 the possible forms of such extensions are κ L ref + κ C 1 + κ C ref + κ C 3-cyc satisfying the PBW conditions (7.3) and (7.4).
For g a three-cycle, by the orbit property in (8.2) and by (8.1) we see that and κ C g ( gv ,v * ) = −κ C g (v, gv * ) = n 2 4 a ⊥ b ⊥ . These components produce the given definition of κ L + κ C .
In the theory of rational Cherednik algebras, H t,c , for the symmetric group, a natural isomorphism between H t,c and H λt,λc when λ ∈ C × means that only two distinct cases need be considered, t = 0 and t = 0. Theorems 8.2 and 8.3 show that in the first case there are no further deformations in polynomial degree one with the linear part of the parameter map supported only on the identity while there is a three-parameter family of such deformations in the second case with the linear part of the parameter map supported only off the identity. , and κ L (ij) (x i , y i ) = −κ L (ij) (x i , y j ) = − n 2 a ⊥ (x i +x j ) − n 2 b ⊥ (ȳ i +ȳ j ). But then im(κ L 1 + κ L ref ) ⊆ h * ⊕ h would require a 4 = b 4 = 0 so κ L 1 ≡ 0. This combined with part (2) of Theorem 5.4 shows there is no Drinfeld orbifold algebra map for S n on h * ⊕ h with linear part supported both on and off the identity which extends to a map κ of the form in Theorem 5.4. But since Theorem 5.4 is not exhaustive, it is not clear whether there exist such maps in general and this is one possible area for further work.