FORMALITY QUANTIZATION OF LIE BIALGEBRAS FROM E 2

Transcription

1 QUANTIZATION OF LIE BIALGEBRAS FROM E 2 FORMALITY PAVEL SAFRONOV 1. Outlie 1.1. Statemet. Let k be a field of characteristic zero. The goal of this talk is to prove the followig statemet: Theorem. Let C be a k-liear symmetric mooidal category. The there is a equivalece betwee coilpotet Lie bialgebras ad coilpotet Hopf algebras i C. Recall that a Lie bialgebra g is coilpotet if for every x g we have δ (x) = 0 for large eough. Similarly, a uital ad couital bialgebra A is coilpotet if for every x A such that ɛ(a) = 0 we have ( ) (x) = 0 for large eough, where (x) = (x) x 1 1 x. Let us explai how we get the Etigof-Kazhda quatizatio theorem. Start with a Lie bialgebra g over k. The g := g[ ]/ is a coilpotet Lie bialgebra, where the bracket is liearly exteded from the oe o g ad the cobracket is δ g = δ g. Q(g ) is a coilipotet Hopf algebra i k[ ]/ -modules ad the iverse limit of the system gives a QUE algebra with classical limit g.... Q(g ) Q(g 1 )... Remark. It is easy to see that coilpotet Lie bialgebras are classical versios of coilpotet bialgebras: by the Cartier-Milor-Moore theorem a cocommutative coilpotet bialgebra is a uiversal evelopig algebra of some Lie algebra g. The co-poisso structure o the classical limit gives the Lie bialgebra structure o g. Remark. A coilpotet bialgebra has a uique atipode, so we will just talk about bialgebras Uiversal reformulatio. We will use the laguage of PROPs to give a uiversal statemet. Thus, we have PROPs BA ad LBA of o-uital o-couital bialgebras ad Lie bialgebras. Remark. Uless otherwise oted, from ow o we will oly cosider o-uital algebras ad o-couital coalgebras. Oe would expect a uiversal quatizatio statemet to be a fuctor of PROPs BA LBA. The, give a Lie algebra i C, i.e. a fuctor LBA C, we get a bialgebra BA LBA C i C. However, this does t happe for the followig reaso: a fuctor BA LBA is a bialgebra i LBA. As a algebra we simply expect the uiversal evelopig algebra which is costructed usig direct sums. But we do t have ay direct sums i LBA. 1

2 2 PAVEL SAFRONOV Thus, let s freely complete our category: for a liear category C we defie C to be the category of liear fuctors C op Vect fd. The category C has fiite direct sums ad fuctors C D commutig with fiite direct sums are the same as fuctors C D. It is hard to get a hadle o C, so let C pro be the idempotet completio of the subcategory of C spaed by direct sums of represetable objects. As we have oly added fiite direct sums, we also eed to restrict the (Lie) bialgebras we cosider; this is doe as follows. Let BA (LBA ) be the PROP of (Lie) bialgebras where we declare that is a zero object. The a algebra over BA is a bialgebra A with the coditio that A = 0. I particular, it is automatically coilpotet (the caoical filtratio has fiite legth). Let BA be the iverse limit of the categories BA ad let BA pro be the iverse limit of BA pro. Similar otatios will be used for Lie bialgebras. The we have the followig propositio: Propositio. Assume C is a liear abelia symmetric mooidal category. The direct sum preservig fuctors BA pro C are the same as coilpotet bialgebras. Similarly, direct sum preservig fuctors LBA pro C are the same as coilpotet Lie bialgebras. Thus we ca reformulate our origial theorem i the followig uiversal way: Theorem. There is a equivalece Q : BA LBA of k-liear categories. It restricts to a equivalece BA pro LBA pro, which, as we have explaied, gives the Etigof-Kazhda quatizatio theorem Strategy of the proof. The mai idea of the proof is to use Koszul duality betwee bialgebras ad brace algebras ad duality betwee Lie bialgebras ad Gerstehaber algebras to reduce the theorem to a equivalece betwee brace ad Gerstehaber algebras, which is provided by the formality of the E 2 operad. Let BRACES be the PROP of brace algebras, GER the PROP of Gerstehaber algebras ad HOGER the PROP of homotopy Gerstehaber algebras. The operad of homotopy Gerstehaber algebras is a cofibrat replacemet of the operad of Gerstehaber algebras; explicitly this meas that we replace the commutative product by a E product, the Poisso bracket by a L bracket ad add homotopies expressig compatibilities betwee these structures. As explaied i the previous talk, formality of the E 2 operad gives a equivalece betwee Gerstehaber ad brace algebras, i.e. we have a zig-zag of equivaleces of PROPs BRACES HOGER GER. For a dg category C let C proj (ote the differece from C pro which was defied for a o-dg category!) be the idempotet completio of the subcategory of fuctors C op Ch b k spaed by fiite sums ad shifts of represetable objects. The the quatizatio fuctor is obtaied

3 from the followig diagram: QUANTIZATION OF LIE BIALGEBRAS FROM E 2 FORMALITY 3 BA proj Q LBA proj cobar bar BRACES proj formality Harr HOGER proj W GER proj Every arrow i this diagram is a weak equivalece of symmetric mooidal dg categories. Vertical arrows are Koszul dualities that will be explaied i the ext sectio. This diagram is compatible with projectio fuctors BA +1 BA, so we get a correspodig diagram for the iverse limits. Hece, we get a weak equivalece Q : BA proj LBA proj. This is ot yet the statemet that we wat sice it ivolves dg categories ad ot BA pro or BA. The fial step is to put a t-structure o each dg category, show that the hearts of BA proj ad LBA proj are BA ad LBA ad show that each fuctor i the diagram is t-exact. This is doe i the last sectio. Note that this is ot a tautological step as Koszul duality ivolves shifts. 2. Koszul duality 2.1. Geeral picture. Recall that a bialgebra A is the same as a mooidal category A mod together with the forgetful fuctor F : A mod Vect. The bialgebra A ca be recovered as the space of atural trasformatios Ed(F ). O the other had, we also have the fuctor T : Vect A mod give by sedig a vector space to the correspodig trivial A-module. What is Ed(T )? First of all, we ca idetify Ed(T ) = Ed A (k) =:! A. O! A we have two commutig multiplicatios: oe comig from the compositio of edomorphisms ad the other oe comig from the mooidal structure. Therefore,! A is a E 1 E 1 -algebra, i.e. a brace algebra (as will be explaied later). This is a example of Koszul duality: the brace algebra! A is Koszul dual to the bialgebra A Basic istaces. There are basic Koszul duality costructios betwee the followig objects: associative algebras ad coassociative coalgebras, commutative algebras ad Lie coalgebras Associative algebras. Let A be a vector space ad cosider the reduced tesor coalgebra A! := T (A[1]) (A[1]). A differetial o A! is uiquely determied by the maps m : (A[1]) A[2]. Apart from m 2 : A A A, all these maps have the wrog degree, so they must be zero. The fact that d 2 = 0 the traslates ito the associativity coditio o m 2. I other words, differetials o A! are the same as (o-uital) associative algebra structures o A. Remark. If oe istead starts with a complex A, the a differetial o A! would be the same as a (o-uital) A structure o A. =1 CE

4 4 PAVEL SAFRONOV Give a associative algebra A, the complex A! is called the reduced bar complex. We ca do a dual versio of this costructio. Let A agai be a vector space ad cosider the reduced tesor algebra! A := T (A[ 1]) (A[ 1]). As before, a differetial o! A is equivalet to a (o-couital) coassociative coalgebra structure o A. The complex! A for a give coalgebra is called the reduced cobar complex Lie ad commutative algebras. Now let s repeat these costructios for commutative ad Lie algebras. Let g be a vector space ad cosider the free commutative algebra CE(g) := Sym (g[ 1]) Sym (g[ 1]). A differetial o CE(g) is uiquely determied by the map δ : g[ 1] Sym(g[ 1])[1] whose oly otrivial compoet is δ 2 : g 2 g. The square-zero coditio is equivalet to the Jacobi idetity for δ 2, i.e. g is a Lie coalgebra. I other words, differetials o CE(g) are equivalet to Lie coalgebra structures o g. Give a Lie coalgebra g, the complex CE(g) is called the Chevalley-Eileberg cochai complex. Fially, give a vector space A, we ca form a cofree Lie coalgebra =1 =1 Harr(A) := cofr(a[1]). A differetial o Harr(A) is the same as a commutative product o A. The complex Harr(A) for a give commutative algebra is called the Harriso chai complex (aka the cotaget complex) Duality for bialgebras. So far we have described Koszul duality for basic structures cotrolled by (co)operads. We will ow use these basic dualities to costruct dualities betwee more complicated structures cotrolled by PROPs, such as: brace algebras ad bialgebras, Gerstehaber algebras ad Lie bialgebras Brace algebras. Cosider a E 1 E 1 -algebra A. That is, we have a complex A together with two commutig operatios: oe of them is a A product ad the other oe we assume to be strictly associative. Let us try to uderstad what kid of operatios A has more explicitly. Let A! be the bar complex of A with respect to the A product. We have a lax mooidal structure o the tesor coalgebra fuctor give by the shuffle product. T (V ) T (W ) T (V W ) Remark. Note that the ordiary cocateatio map v 1... v w1... w m v 1... v w 1... w m

5 QUANTIZATION OF LIE BIALGEBRAS FROM E 2 FORMALITY 5 does ot respect the coalgebra structure. To get a bialgebra structure o T (V ), oe has to either cosider the shuffle product ad decocateatio coproduct or vice versa, the cocateatio product ad shuffle coproduct. Usig the lax mooidal structure we obtai that A! is a algebra i dg coalgebras. I other words, it is a dg bialgebra. A product T (A[1]) T (A[1]) T (A[1]) is uiquely determied by the maps m k,l : T (A[1]) T (A[1]) A[1]. Let us assume that m k,l = 0 for k 1. The we recover the structure of a brace algebra o A: it is a A algebra together with a collectio of brace operatios A A A of degree satisfyig certai compatibilities [GV94]. We will deote x{x 1,..., x } := m 1, (x, x 1,..., x ). Now let s go back. Let A be a bialgebra ad cosider! A, the cobar complex of A with respect to the coproduct. We already have a dg algebra structure o! A. The brace operatios are uiquely characterized by the followig: x{y 1,..., y m } = 0, m > 1 x{y 1... y k } = k (x) (y 1... y k ). Therefore, for a bialgebra A the complex! A is a brace algebra (with a strictly associative multiplicatio) Gerstehaber algebras. Let A be a P 2 -algebra (aka a Gerstehaber algebra). This is a commutative dg algebra with a Poisso bracket {, } of degree 1. We ca cosider the Harriso complex Harr(A) of A regarded as a commutative dg algebra. Let us defie a Lie bracket o the Harriso complex i the followig way. A Lie bracket is uiquely determied by the map which we defie to be Harr(A) Harr(A) A[1] Harr(A) Harr(A) A[1] A[1] {,} A[1]. It is clearly a Lie bracket ad it uiquely exteds to the Harriso complex sice it is cofree. Therefore, the Harriso complex of a P 2 algebra is a dg Lie bialgebra. Coversely, suppose g is a Lie bialgebra ad cosider the Chevalley-Eileberg complex C (g) of g with respect to the cobracket. We get a Poisso bracket uiquely determied by the map Sym(g[ 1]) Sym(g[ 1]) Sym(g[ 1])[ 1] g[ 1] g[ 1] Sym(g[ 1])[ 1], which we defie to be give by the Lie bracket o g. Remark. If g is a (dg) Lie algebra, the space Sym(g[ 1]) is the free P 2 algebra o g.

6 6 PAVEL SAFRONOV 2.4. Uiversal Koszul duality. Let 1 BA be the uiversal bialgebra. We deote by h 1 BA proj the fuctor represeted by 1. The cobar complex! h 1 does ot make sese i BA proj as it ivolves ifiite direct sums ad ifiite shifts. This is why we cosider istead the cobar complex! h 1 i BA proj. As we have explaied, the cobar complex of a bialgebra is a brace algebra, so we get a symmetric mooidal fuctor BRACES BA proj. This iduces a symmetric mooidal fuctor BRACES proj BA proj. Similarly, we have h 1 BRACES proj represetig the uiversal brace algebra. The bar complex of h 1 is a well-defied elemet h! 1 BRACESproj, so we get a fuctor which iduces a fuctor BA BRACES proj, BA proj Theorem. The bar ad cobar fuctors BA proj BRACES proj. BRACES proj are weak equivaleces. Oe similarly has a pair of fuctors betwee Lie bialgebras ad Gerstehaber algebras, which gives a uiversal pair of fuctors LBA proj GER proj. Theorem. The Harriso ad Chevalley-Eileberg fuctors LBA proj equivaleces. GER proj are weak The equivaleces i the theorems are compatible with the projectios BA proj +1 BA proj, so we also have a statemet o the level of iverse limits BA proj. 3. t-structures 3.1. t-structures o (Lie) bialgebras. We have obvious t-structures o the categories BA proj ad LBA proj defied as follows. We defie the subcategory D 0 BA proj BA proj of objects of degree 0 to cosists of fuctors havig cohomology i oegative degrees. Similarly, D 0 BA proj BA proj cosists of fuctors havig cohomology i opositive degrees. The heart of this t-structure is BA. Oe similarly defies a t-structure o LBA proj with heart LBA t-structures o Gerstehaber ad brace algebras. We will defie the t-structure o GER proj motivated by Koszul duality. That is, we wat the Harriso fuctor LBA GER proj 1 Harr(h 1 ) to be exact. I particular, X = Harr(h 1 ) should be i degree 0 with respect to the t- structure. We have the followig property: the cohomology H j (X( i )) is zero uless j i. This uses the fact that the operad of Gerstehaber algebras is geerated by a degree 1

7 QUANTIZATION OF LIE BIALGEBRAS FROM E 2 FORMALITY 7 operatio of arity 2. Let s defie D 0 GER proj to cosist of those fuctors whose cohomology is cocetrated i degrees i whe evaluated o i. We ca defie D 0 GER proj as a complemet: it cosists of all objects X, such that Hom(X, Y ) is cocetrated i oegative degrees for alll Y D 0 GER proj. A similar t-structure exists o BRACES proj. Theorem. The Koszul duality fuctors are exact fuctors with respect to these t-structures. Therefore, we obtai that Q is exact, hece seds BA to LBA. Refereces [EE03] B. Eriquez, P. Etigof, O the ivertibility of quatizatio fuctors, arxiv:math/ [GV94] M. Gerstehaber, A. Voroov, Homotopy G-algebras ad moduli space operad, arxiv:hepth/ [Tam07] D. Tamarki, Quatizatio of Lie bialgebras via the formality of the operad of little disks.