Primitive ideals for W-algebras in type A

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<ul><li><p>Journal of Algebra 359 (2012) 8088</p><p>Contents lists available at SciVerse ScienceDirect</p><p>P</p><p>Iv</p><p>No</p><p>a</p><p>ArReAvCo</p><p>M1617</p><p>KeWPr</p><p>1.</p><p>beWofre</p><p>unanan80</p><p>PrJva</p><p>00htJournal of Algebra</p><p>www.elsevier.com/locate/jalgebra</p><p>rimitive ideals for W-algebras in type A</p><p>an Losev</p><p>rtheastern University, Department of Mathematics, 360 Huntington Avenue, Boston, MA 02115, United States</p><p>r t i c l e i n f o a b s t r a c t</p><p>ticle history:ceived 10 November 2011ailable online 28 March 2012mmunicated by J.T. Stafford</p><p>SC:S99B35</p><p>ywords:-algebrasimitive ideals</p><p>In this note we classify the primitive ideals in nite W-algebras oftype A.</p><p> 2012 Elsevier Inc. All rights reserved.</p><p>Introduction</p><p>Let g be a semisimple Lie algebra over an algebraically closed eld K of characteristic 0 and e ga nilpotent element. Then to the pair (g, e) one can assign an associative algebra W called the</p><p>-algebra. This algebra was dened in full generality by Premet in [P1]. Two equivalent denitionsW-algebras are provided in Section 2. For other details on W-algebras the reader is referred to theview [L5].One of the reasons to be interested in W are numerous connections between this algebra and theiversal enveloping algebra U of g. For example, the sets of primitive ideals Pr(W) and Pr(U) of Wd U are closely related. Recall that an ideal in an associative algebra is called primitive if it is thenihilator of some irreducible module. The structure of Pr(U) was studied extensively in 70s ands.One of manifestations of a relationship between Pr(W) and Pr(U) is a map I I : Pr(W) </p><p>(U) constructed in [L1]. One can describe the image of this map. Namely, to each primitive ideal Pr(U) one assigns its associated variety V(U/J ). According to a theorem of Joseph, the associatedriety is the closure of a single nilpotent orbit in g = g. Thanks to [L1, Theorem 1.2.2(vii)], an</p><p>Supported by the NSF grant DMS-0900907.E-mail address: i.loseu@neu.edu.</p><p>21-8693/$ see front matter 2012 Elsevier Inc. All rights reserved.</p><p>tp://dx.doi.org/10.1016/j.jalgebra.2012.03.010</p></li><li><p>I. Losev / Journal of Algebra 359 (2012) 8088 81</p><p>elfo</p><p>Th</p><p>J</p><p>2.</p><p>2.</p><p>de[Ath</p><p>AonLe</p><p>Fi[isth(uin</p><p>wthof</p><p>1</p><p>Ato</p><p>eqth1h2</p><p>1</p><p>acaneq</p><p>1</p><p>tioement J Pr(U) is of the form I for some I Pr(W) if and only if O V(U/J ), where O standsr the adjoint orbit of e.In general, the map is not injective. However, the following result holds.</p><p>eorem 1.1. The map : Pr(W) Pr(U) is an injection provided g= sln.</p><p>This theorem provides a classication of primitive ideals in W because the set of primitive ideals U with V(U/J ) =O is known thanks to the work of Joseph, [J].</p><p>W-algebras and the map between ideals</p><p>1. Quantum slice</p><p>Let Y be an ane Poisson scheme equipped with a K-action such that the Poisson bracket hasgree 2. Let Ah be an associative at graded K[h]-algebra (where h has degree 1) such thath,Ah] h2Ah and K[Y ] = Ah/(h) as a graded Poisson algebra. Pick a point Y . Let I bee maximal ideal of in K[Y ] and let I be the inverse image of I in Ah . Consider the completionh := limnAh /In . This is a complete topological Kh-algebra with A</p><p>h /(h) =K[Y ] , where</p><p>the right-hand side we have the usual commutative completion. Moreover, as we have seen in [L4,mma A2],1 the algebra Ah is at over Kh.The cotangent space T Y = I/I2 comes equipped with a natural skew-symmetric form, say .</p><p>x a maximal symplectic subspace V T Y . One can choose an embedding V I such that(u), (v)] = h2(u, v) and whose composition with the projection I T Y is the identity. Thisproved similarly to Proposition 3.3 in [Ka] (or can be deduced from that proposition, compare withe argument of Subsection 7.2 in [L3]). Consider the homogenized Weyl algebra Ah(V ) = T (V )[h]/ v v u h2(u, v)) and its completion A0h (V ) at zero. Similarly to the proof of Lemma 3.2[Ka], one can show that</p><p>Ah = A0h (V ) Kh Ah,</p><p>here Ah is the centralizer of V in Ah . We remark that the algebra Ah is complete with respect to</p><p>e topology induced by its maximal ideal. The symbol stands for the completed tensor producttopological vector spaces.The argument in the proof of [L6, Proposition 6.6.1, Step 2], shows that any two embeddings</p><p>, 2 : V I satisfying the conditions in the previous paragraph differ by an automorphism ofh of the form exp(</p><p>1h2</p><p>ad(z)) with z (I )3. In particular, the algebra Ah is dened uniquely upa Kh-linear isomorphism.Now let us consider a compatibility of our construction with certain derivations. Suppose Ah isuipped with a derivation D such that Dh = h. The derivation extends to Ah . According to [L4],ere is a derivation D of Ah with the following properties. First, D h = h. Second, we have D D =ad(a) for some element a Ah , where D means the derivation that equals D on Ah and acts byon the symplectic space V generating A0h (V ).An easy special case of the previous construction is when the derivation D comes from a K-tion on Ah preserving . Here we can choose a K-stable V and a K-equivariant : V Id so we get a K-action on Ah . Moreover, the algebra Ah is now dened uniquely up to a K-uivariant Kh-linear isomorphism.</p><p>In [L4] we had an assumption that Y has only nitely many leaves but this assumption is not necessary for many construc-</p><p>ns and results.</p></li><li><p>82 I. Losev / Journal of Algebra 359 (2012) 8088</p><p>su</p><p>ThA</p><p>thar</p><p>toQ</p><p>A</p><p>eaphS2</p><p>So</p><p>secoqu</p><p>to</p><p>Qwto</p><p>2.</p><p>Yorassure</p><p>T[h</p><p>byNow let Z(Ah) denote the center of Ah . Consider a K-equivariant Kh-linear homomorphism: Z(Ah) K[h] and the corresponding central reduction A,h :=Ah/Ah ker. Until the end of thebsection we assume that lies in the spectrum of A,h/(h).Consider the induced homomorphism Z(Ah) Ah = A0h (V ) Kh Ah . The image is central.e center of A0h (V ) coincides with Kh, so the image of Z(Ah) is contained in Ah . Set A,h =h/Ah ker.Then we have the completion A</p><p>,h of A,h , and A,h = A</p><p>h /A</p><p>h ker. Furthermore, we have</p><p>e following commutative diagram, where the horizontal arrows are isomorphisms and the verticalrows are the natural quotients.</p><p>Ah A0h (V ) Kh Ah</p><p>A,h A</p><p>0h (V ) Kh A,h</p><p>Now suppose that we have a reductive group Q that acts on Ah rationally by K[h]-algebra au-morphisms xing . Further, suppose that there is quantum comoment map A : q Ah , i.e., a-equivariant linear map with the property that [A(),a] = h2.a, where on the right-hand sideis the derivation of Ah coming from the Q -action. Composing A with a natural homomorphismh Ah we get a quantum comoment map qA</p><p>h again denoted by </p><p>A .We remark that we can choose V to be Q -stable. This gives rise to a Q -action on V by lin-r symplectomorphisms and hence to an action of Q on Ah(V ) by K[h]-linear algebra automor-isms. There is a quantum comoment map A : q Ah(V ) that is the composition q sp(V ) =V Ah(V ).Further, since Q is reductive, we can assume that the embedding : V Ah is Q -equivariant.we get a Q -action on Ah . Let us produce a quantum comoment map for this action. For q</p><p>t () = A() A(). The quantum comoment map conditions for A,A imply that ()mmutes with Ah(V )0 and hence the image of is in Ah . Now it is clear that : q Ah is aantum comoment map for the action of Q on Ah .We remark that the Q -action and the map are dened by the previous construction uniquely upan isomorphism of the form exp( 1</p><p>h2ad(z)), where, in addition to conditions mentioned above, z is</p><p>-invariant. Also we remark that if we have a K-action on Ah as above but additionally commutingith Q and such that A() has degree 2 for each , then the Q -action on Ah also may be assumedcommute with K and () may be assumed to have degree 2.</p><p>2. W-algebras via quantum slices</p><p>We are going to consider a special case of the construction explained in the previous subsection.Let G be a simply connected semisimple algebraic group and let g be the Lie algebra of G . Set</p><p>:= g . We remark that we can identify g with g by means of the Killing form. Pick a nilpotentbit O g and an element e O. We take e for . Let us equip Y with a Kazhdan K-action denedfollows. Pick an sl2-triple (e,h, f ), where h is semisimple. Let :K G be the one-parameterbgroup corresponding to h. We dene a K-action on g by t. = t2 (t), g , t K . Wemark that t. = .For Ah we take the homogenized version Uh of the universal enveloping algebra dened by Uh :=</p><p>(g)[h]/( y y h2[, y]). We can extend the Kazhdan action to Uh . Explicitly, for g with, ] = i we have t. = ti+2 and we set t.h := th.Apply the construction of the previous subsection to Ah, . We get the Kh-algebra Ah acted on</p><p> 0 </p><p>K together with a K -equivariant isomorphism Ah = Ah (V ) Kh Ah . Here V has the same</p></li><li><p>I. Losev / Journal of Algebra 359 (2012) 8088 83</p><p>mT</p><p>oraidthtrman</p><p>coth</p><p>Th</p><p>th</p><p>ishatoofau</p><p>wimasde</p><p>2.</p><p>wt.</p><p>Sisonac</p><p>thvi</p><p>pe</p><p>(1(2</p><p>(3(4(5eaning as before but we can describe it explicitly: namely, for V we take the subspace [g, f ] g= Y .Let Ah denote the subalgebra of all K-nite vectors in Ah . It turns out that the algebra Ah/hAh ,</p><p>, more precisely, the corresponding variety is well known in the theory of nilpotent orbits this isso-called Slodowy slice. In more detail, set S := e + zg( f ) and view S as a subvariety in g via theentication g = g . Then S is K-stable and is transverse to Ge at e: g = Te S TeGe. Moreover,e Kazhdan action contracts S to e, meaning that limt t.s = e for all s S . This implies that S isansversal to any G-orbit it intersects g = Ts S + TsGs for all s S . Also the contraction propertyeans that the grading on K[S] induced by the K-action is positive: there are no negative degrees,d the only elements in degree 0 are constants.The contraction property for the K-action on S implies that the subalgebra of K[S] =Ah/hAhnsisting of the K-nite elements coincides with K[S]. Moreover, since the degree of h is positive,is implies that Ah/hAh =K[S].We set A :=Ah/(h 1)Ah . This is a ltered associative algebra whose associated graded is K[S].e algebra Ah can be recovered as the Rees algebra of A, while Ah is the completion A</p><p>h .</p><p>Let us remark that the algebra A comes equipped with a homomorphism Z A, where Z ise center of the universal enveloping algebra U(= Uh/(h 1)Uh) of g. Indeed, set Zh := UGh . Thisthe center of Uh . Consider Zh as a subalgebra of Ah . According to the previous subsection, weve Zh Ah . It is easy to see that Zh consists of K-nite vectors so Zh Ah . This gives risean embedding Z A. We remark that this embedding does not depend on the choice (thatthe embedding V I ) we have made. This is because any two choices are conjugate by antomorphism that commutes with all elements of the center.Consider the subgroup Q := ZG(e,h, f ) G . This group acts on Uh and stabilizes (and S as</p><p>ell). The Q -action commutes with K and there is a quantum comoment map q Ah whoseage consists of functions of degree 2 with respect to the K-action. So we get a Q -action on Ahwell as a quantum comoment map q Ah . Both the action and the quantum comoment mapscend to A.</p><p>3. Equivalence with a previous denition</p><p>Recall that the cotangent bundle T G carries a natural symplectic form . This form is invariantith respect to the natural G G-action. Moreover, for the K-action by berwise dilations we have</p><p>= t1, t K .We remark that we can trivialize T G by using left-invariant 1-forms hence T G = Gg . Consideras a subvariety in g and set X := G S . It turns out that the subvariety X T G is symplectic. Iteasy to see that X is stable with respect to the left G-action, as well as to the Kazhdan K-actionT G given by t.(g,) = (g (t)1, t2 (t)). Clearly, has degree 2 with respect to the Kazhdan</p><p>tion. Also Q acts on T G by q.(g,) = (gq1,q) and X is Q -stable.We remark that G : T G g , G(g,) = g is a moment map, i.e., a G-equivariant map such</p><p>at for any g the derivation {G(), } of K[T G] coincides with the derivation produced by a the G-action. Similarly, Q : T G q , Q (x,) = |q is a moment map.In [L1] the author proved that there is an associative product on K[X][h], where h is an inde-ndent variable, satisfying the following properties:</p><p>) is G K-equivariant, where G K acts on K[X] as usual, g.h = h, t.h = th.) For f , g K[X] we have f g =i=0 Di( f , g)h2i , where Di is a bi-differential operator of orderat most i.</p><p>) f g f g mod h2.) f g g f h2{ f , g} mod h4.</p><p> ) The maps G : gK[X][h], Q : qK[X][h] are quantum comoment maps.</p></li><li><p>84 I. Losev / Journal of Algebra 359 (2012) 8088</p><p>Thinth</p><p>Prbe</p><p>Pr(1refrK</p><p>an(tThAisininsi</p><p>2.</p><p>ex</p><p>thAA</p><p>fo</p><p>saThse</p><p>is</p><p>biA</p><p>ofw</p><p>th</p><p>stFogride last property was established in [L2]. By denition, the W-algebra W is the quotient of thevariant subalgebra K[X][h]G by h 1. The quantum comoment map : g K[X][h] gives rise toe homomorphism Z = U (g)G W .</p><p>oposition 2.1. We have a ltration preserving Q -equivariant isomorphism W A intertwining the em-ddings of Z and the quantum comoment maps from q.</p><p>oof. Similarly to the above we have a star-product on T G having the properties analogous to)(5). Set x := (1,) X T G . Consider the completions K[T G]hGx ,K[X]hGx of the cor-sponding algebras (with respect to star-products) at the ideals of Gx. According to Theorem 2.3.1om [L2], we have a G K-equivariant (where we consider the Kazhdan K-actions) topologicalh-algebra isomorphism</p><p>K[T G</p><p>]hGx A0h (V ) Kh K[X]hGx ,</p><p>d this isomorphism intertwines the quantum comoment maps for the G-action and Q -actionhe Weyl algebra component of the quantum comoment map for G on the right-hand side is 0).e algebra of G-invariants of the left-hand side is Uh , while on the right-hand side we get0</p><p>h (V ) Kh Wh , see [L2]. So we can take W</p><p>h for Ah . It follows that we have Q -equivariant</p><p>omorphisms Wh =Ah of graded K[h]-algebras and W =A of ltered algebras. Both isomorphismstertwine the quantum comoment maps from q. Moreover, the embedding UGh Uh =K[T G]hGduced by the quantum comoment map is just the inclusion. This completes the proof of the propo-tion. 4. Map between the set of ideals</p><p>Let us construct the map mentioned in Theorem 1.1. We will start with the general settingplained in Section 2.1.Consider the set Idh(Ah) of all K-stable h-saturated ideals Jh Ah , where h-saturated means</p><p>at Ah/Jh is at over K[h]. Similarly, consider the set Idh(Ah) of all D -stable h-saturated ideals inh . The discussion of D</p><p> in Section 2.1 implies that an h-saturated I h Ah is D -stable if and only if0h Kh I h A</p><p>h is D-stable. In particular, the set Idh(Ah) does not depend on the choice of D .</p><p>We have maps between Idh(Ah) and Idh(Ah) constructed as follows. Take an ideal Jh Ah andrm its closure J h A</p><p>h . This ideal is D-stable but also one can check that it is actually h-</p><p>turated. As such, the ideal J h has the form A0h Kh I h for a unique two-sided ideal I h in Ah .e ideal I h is automatically D -stable and h-saturated. We consider the map : Idh(Ah) Idh(Ah)nding Jh to I h .Let us produce a map in the opposite direction. Take I h Idh(Ah). Then Jh :=Ah A0h Kh I ha K-stable h-saturated ideal in Ah . Consider the map : Idh(Ah) Idh(Ah) sending I h to Jh .Now suppose that the grading on K[Y ] induced by the K-action is positive. Then Idh(Ah) is in</p><p>jection with the set Id(A) of two-sided ideals in A :=Ah/(h 1). Under this...</p></li></ul>