Frobenius manifolds from subregular classical $W$-algebras

FR OBENIUS MANIF OLDS FR OM SUBREGULAR CLASSICAL W -ALGEBRAS Y ASSIR IBRAHIM DINAR Abstract. W e obtain alg e br aic F rob enius manifolds from class ical W -algebra s asso ci- ated to subregular nilp otent elements in simple Lie algebra s of type D r where r is even and E r . The resulting F rob enius ma nifolds ar e certain hypersurface s in the total spa ces of semiuniv ersal deformation of simple hypersurface singular ities of the same types. Contents 1. In tro duction 1 2. Preliminaries 5 2.1. Flat p encil of metrics and F rob enius manifolds 5 2.2. Lo cal Poiss on brack e ts 7 3. Subregular nilp o ten t elemen ts 8 3.1. Opp osite Cart a n subalgebra 10 3.2. Slo dow y co ordinates 11 4. Drinfeld-Sok olov reduction 13 4.1. The nondegeneracy condition 15 5. Dirac reduction 17 5.1. The quasihomogeneit y condition 18 5.2. Subregular classical W - a lgebra 19 6. Algebraic F rob enius manifold 21 6.1. The algebraic F rob enius manifo ld of D 4 ( a 1 ) 22 7. Conclusions and remarks 24 References 25 1. Intr oduction A F robenius manifold M is a manifold with the structure of F rob enius algebra on the tangen t space T z M at an y p oin t z ∈ M with a flat in v ar ian t bilinear for m ( ., . ) and an iden tit y e plus some compatibilit y conditions [13 ]. W e say M is semisim ple or massiv e if T z M is semisimple for generic z . Lo cally , in the fla t co ordinates ( t 1 , ..., t r ), the structure of F rob enius manifold is enco ded in a p oten tial F ( t 1 , ..., t r ) satisfying a system of partial differen tial equations kno wn in to p ological field theory as the Witten-Dijkgraaf- V erlinde- V erlinde (WDVV) equations. W e consider F rob enius manifolds where the quasihomogene- it y condition take s the from r X i =1 d i t i ∂ t i F ( t ) = (3 − d ) F ( t ) (1.1) 2000 Mathematics Subje ct Classific atio n. Primary 37 K10; Secondar y 35 D45. Key wor ds and phr ases. Int egra ble s y stems, F r ob enius manifolds, Classical W -algebr as, Drinfeld- Sokolo v reduction, Dirac r eduction, Slodowy s lice, Simple hypersur face singular it y . 1 2 Y A SSIR DINAR where e = ∂ t r − 1 and d r − 1 = 1. This condition defines t he degrees d i and t he c harge d of M . If F ( t ) is an algebraic function w e call M an algeb raic F rob enius mani- fold . Dubro vin conjecture on classification o f algebraic F rob enius manifolds is stated as fo llo ws: semisimple irreducible algebraic F rob enius manifolds with p ositiv e degrees d i corresp ond to quasi-Co xeter (primitiv e) conjugacy classes in irreducible Co xeter groups. A quasi-Co xeter conjugacy class in an irreducible Co xeter group is a Conjugacy class whic h has no represen tativ e in a prop er Co xeter subgroup [3]. There are t w o ma j o r results supp ort the conjecture. First, the conjecture arises fro m studying the algebraic solutions to a sso ciated equations of isomono dro mic defor ma t ion of algebraic F rob enius manifolds [13],[16]. It leads to quasi-Coxete r conjugacy classes in Co x- eter groups b y considering the classification of finite orbits of the braid g roup a ctio n on tuple of reflections obta ined in [28]. T herefore, it remains the problem of constructing all these algebraic F rob enius manifolds. Second, Dubrov in constructed p o lynomial F rob enius structures on t he orbit spaces of Cox eter groups [12 ] using the w ork o f [24]. Then Hertling [17] pr ov ed that these ar e a ll p ossible p olynomial F rob enius manifolds . The isomon- o dromic deformation of Poly nomial F rob enius manifo lds lead to Co xeter conjugacy classes [13]. The classification of p olynomial F rob enius manifolds rev eals a relation b etw een the order and eigenv alues of the conjug a cy class, and the c harge and degrees of the correspo nding F rob enius manifold. More precisely , If the order of the conjugacy class is κ + 1 and the eigen v alues are exp 2 η i π i κ +1 then the c harge of the F rob enius manifold is κ − 1 κ +1 and the degrees are η i +1 κ +1 . W e dep end o n this w eak relation in considering a new examples of algebraic F rob enius manifolds. In [8] w e con tin ue the work of [23] and we b egan to dev elop a construction of algebraic F rob enius manifolds using D rinfeld-Sok olo v reduction. This means w e r estrict ourself to conjugacy classes in W eyl groups. The examples o btained corresp ond, in the notatio ns o f [3], to the conjugacy classes D 4 ( a 1 ) and F 4 ( a 2 ). In [10] w e succeeded to uniform the con- struction of all p olynomial F rob enius manifolds. In this work we unifor m t he construction of algebraic F rob enius manifolds whic h corresp ond to the conjuga cy classes D r ( a 1 ) where r is ev en and E r ( a 1 ). In order t o form ulate the main results of this work, let us recall the relation b et w een subregular nilp oten t elemen ts and deformatio n o f simple hypersurface singularities [2 6]. Let g b e a simple Lie algebra of t yp e D r where r is ev en or E r . Fix a subregular nilpot en t elemen t e in g . By definition a nilp ot en t elemen t is called subregular if its cen tralizer in g is of dimension r + 2. W e fix, by using t he Jacobson-Morozov theorem, a semisimple elemen t h and a nilp ot en t elemen t f suc h that A = { e, h, f } is an sl 2 -triple, i.e [ h, e ] = 2 e ; [ h, f ] = − 2 f ; [ e, f ] = h. (1.2) The action of A decomp ose g t o r + 2 irreducible A -submo dules g = r +2 M i =1 V i (1.3) Let dim V i = 2 η i + 1. W e call the set W t ( e ) = { η i : i = 1 , ..., r + 2 } (1.4) the w eigh ts of e . Let η 0 + 1 b e the Coxete r n um b er of g . The set W t ( e ), under o ur c hoice of a total order, together with η 0 are given in the followin g table. FROBENIUS MANIFOLDS AND W -ALGEBRAS 3 g W t ( e ) η 0 η 1 , . . . , η r − 1 η r η r +1 , η r +2 D r 2r-3 1 , 3 , . . . , r − 1; r − 1 , r + 1 , . . . , 2 r − 5 r-3 1 ,r - 2 E 6 11 1 , 4 , 5 , 7 , 8 2 3 , 5 E 7 17 1 , 5 , 7 , 9 , 11 , 13 3 5 , 8 E 8 29 1 , 7 , 11 , 13 , 17 , 19 , 23 5 9 , 14 E t ( g ) W e observ e tha t the set E t ( g ) of the exp onents of g is E t ( g ) = { η i : i = 0 , ..., r − 1 } . (1.5) Let G b e the adjoint group of g . By Chev alley theorem, the algebra S ( g ∗ ) G of in v a rian t p olynomials under the adjoin t a ctio n of G is g enerated by r homogenous p olynomials of degrees η i + 1 , i = 0 , ..., r − 1. W e fix a homogenous generator χ 0 , ..., χ r − 1 of the algebra S ( g ∗ ) G with degree χ i equals η i + 1. Then let us consider the adjoin t quotient map χ = ( χ 0 , . . . , χ r − 1 ) : g → C r (1.6) and define the Slo dowy slice Q = e + g f ; g f := k er ad f . (1.7) Then Briesk orn pro v ed that the restriction of χ to Q is semiuniv e rsal deformation of the simple hypersurface singularit y N ∩ Q [2 ] whic h is of the same type as g . Let us in tro duce the follow ing co ordinates on Q r +2 X i =1 z i X i − η i + e ∈ Q where X i − η i is a normalized minimal w eigh t vec tor of V i . W e assign the degree 2 η i + 2 to z i . Then a simple mo dificatio n of the work in [25] we get the follow ing Prop osition 1.1. The m ap χ | Q has r an k r − 1 at e . We c an normaliz e the mo dules V i and cho ose a homo genous gener ators χ 0 , ..., χ r − 1 for S ( g ∗ ) G such that the r estriction t i of χ i to Q with i > 0 take the fo rm t i = z i + non line ar terms . (1.8) In p articular, setting t r + i = z r + i , i = 0 , 1 , 2 (1.9) we get a quasiho m o genous c o or dinates ( t 1 , ..., t r +2 ) on Q with de gr e e t i e quals de gr e e z i . W e will call the co ordinates ( t 1 , . . . , t r +2 ) on Q o btained in this prop osition Slo dowy co ordinates . In this co ordinates the restriction of the quotien t map to Q take the form χ | Q : ( t 1 , ...., t r +2 ) 7→ ( t 0 , t 1 , ..., t r − 1 ) (1.10) where t 0 is the restriction to Q of the inv arian t p olynomial χ 0 . Let us consider the Lie-Poiss on brack e t P on g . The Dirac reduction of P to Q give a non trivial Pois son brac k et { ., . } Q . It is kno w in the literature as the adjoint transve rse P oisson structure to the nilp oten t or bit of e . In [5] they pr ov e the follow ing Theorem 1.2. T h e matrix F ij ( t ) = { t i , t j } Q is c onstant multiple of the matrix  0 0 0 Ω  (1.11) 4 Y A SSIR DI NAR wher e Ω is a 3 × 3 matrix of the fo rm   0 ∂ t 0 ∂ t r +2 − ∂ t 0 ∂ t r +1 − ∂ t 0 ∂ t r +2 0 ∂ t 0 ∂ t r ∂ t 0 ∂ t r +1 − ∂ t 0 ∂ t r 0   . (1.12) Let N ⊂ Q b e the hypersurface of dimension r defined as follow s N = n t ∈ Q : ∂ t 0 ∂ t r +2 = ∂ t 0 ∂ t r +1 = 0 o (1.13) It will follo w t hat ∂ t 0 ∂ t r +2 dep ends linearly on t r +2 and ∂ t 0 ∂ t r +1 is a p olynomial in t r +1 of degree r − 2 (resp. 2 ) if g is a Lie algebra of t yp e D r (resp. E r ). In particular, ( t 1 , ...., t r ) are w ell defined co ordinates o n N . Let κ = max W t ( e ). Then we pro v ed the fo llo wing Theorem 1.3. The sp ac e N has a natur al structur e of algebr aic F r ob e n ius manif o ld with char ge κ − 1 κ +1 and de gr e es η i +1 κ +1 , i = 1 , ..., r . By natural w e mean that it can b e form ulated entirely in terms of the represen tation theory of A . The p otential F of this F rob enius structure dep ends on the solution of the equation ∂ t 0 ∂ t r +1 = 0. The set E t ( e ) = { η i : i = 1 , ..., r } (1.14) pla ys the same role as the set E t ( g ) for p olynomial F rob enius manifolds and we call it the exp onen t s of e . Here is some brief details ab out what we did in order to prov e the theorem ab ov e. (1) W e review the relation bet w een the nilp otent elemen t and the conjugacy class. Using the w ork of [27] and [7], w e fix a n um b er ρ suc h that y 1 = e + ρX κ − η κ is regular semisimple. Then h ′ = k er y 1 is Cartan subalgebra and it is kno wn as the opp osite Cartan subalgebra . T he elemen t w := exp 2 π i κ +1 ad h , acts on h ′ as a represen tative of the conjugacy class D r ( a 1 ) (resp. E r ( a 1 )) if g is of t yp e D r (resp. E r ). (2) W e consider the standard lo cal P oisson brack et o n the lo op algebra L ( g ). Then we use the subalgebra A to p erform D rinfeld-Sok o lo v reduction ([8],[19],[1]) or equiv a- len tly the D irac reduction [19],[9] to get a lo cal P oisson structure { ., . } e Q on Slo do wy slice e Q = e + L ( g f ); . This P oisson brac k et is kno wn in the litera t ur e as the classical W -algebras asso- ciated to the nilp oten t o rbit of e [19]. The leading term of this P o isson brac ket is the adjoint transv erse P oisson structure { ., . } Q . (3) w e preform the Dirac reduction on { ., . } e Q in order to obtain a lo cal P o isson structure whic h admits a disp ersionless limit. This is p ossible on a lo op space e N := L ( N ) of the h yp ersurface N defined ab ov e. The new lo cal P oisson bra c ke t { ., . } e N is also a classical W -alg ebra. W e call it subregular classical W -algebra . Then from { ., . } e N w e get a lo cal P o isson brack et of h ydro dynamic t yp e { ., . } [0] . In the co ordinates ( t 1 , ..., t r ) of N w e hav e b y definition { t i ( x ) , t j ( y ) } [0] = g ij ( t ( x )) δ ′ ( x − y ) + Γ ij k ( t ( x )) t k x δ ( x − y ) . (1.15) where g ij ( t ( x )) and Γ ij k ( t ( x )) ar e p olynomials in t i ( x ). FROBENIUS MANIFOLDS AND W -ALGEBRAS 5 (4) W e transfer to finite dimensional geometry . W e ch ec k that the matrix g ij ( t ( x )) is nondegenerate. It f o llo ws that the nondegeneracy condition could b e traced bac k to the existence of opp osite Cartan subalgebra h ′ . Then from Dubrovin- No vik ov theorem, g ij ( t ) define a flat contra v ariant metric on N . Moreo v er, form the structure of the matrix g ij ( t ), it follows that the matrix ∂ t r − 1 g ij define another flat contra v ariant metric on N . F urthermore, the tw o metrics form a flat p encil of metrics on N . (5) W e pro ve that the flat p encil of metrics on N satisfies the quasihomogeneity a nd the regularit y conditions. The pro of depends on the definition of classical W -algebra and the structure of the set E t ( e ). T hen w e o btain a F rob enius structure on N using a theorem and construction due to D ubro vin [14 ]. The resulting F rob enius manifold satisfy the w eak relat io n. 2. Preliminaries 2.1. Flat p encil of metrics and F rob enius manifolds. In this section w e review the re- lation b et wee n the geometry of flat p encil of metrics and the theory of F rob enius manifolds outlined in [14]. Let M b e a smo oth manifold of dimension r . A symmetric bilinear f orm ( ., . ) on T ∗ M is called con tra v arian t metric if it is inv ertible on an op en dense subset M 0 ⊂ M . In a lo cal co or dina t es ( u 1 , ..., u r ), if w e set g ij ( u ) = ( du i , du j ); i, j = 1 , ..., r , (2.1) Then the in ve rse matrix g ij ( u ) of g ij ( u ) determines a metric < ., . > on T M 0 . W e define the contra v ariant Levi-Civita connection Γ ij k for ( ., . ) b y Γ ij k := − g is Γ j sk (2.2) where Γ j sk is the Levi-Civita connection of < ., . > . W e sa y the metric ( ., . ) is flat if < ., . > is flat. Let g ij 1 ( u ) and g ij 2 ( u ) b e tw o con trav ariant flat metrics on M and denote the corresp ond- ing Levi-Civita connections by Γ ij 1; k ( u ) and Γ ij 2; k ( u ), resp ectiv ely . W e say g ij 1 ( u ) and g ij 2 ( u ) form a flat p encil of metrics if (1) g ij λ ( u ) := g ij 2 ( u ) + λg ij 1 ( u ) defines a flat metric on T ∗ M for a generic λ and, (2) The Levi-Civita connection of g ij λ ( u ) is giv en b y Γ ij λ ; k ( u ) = Γ ij 2; k ( u ) + λ Γ ij 1; k ( u ) . The flat p encil of metrics in this work is obta ined b y using the following lemma Lemma 2.1. [1 2] If for a c ontr avariant flat metric g ij 2 in some c o or dinate ( u 1 , ..., u r ) the entries of g ij 2 ( u ) and its L evi-C i v ita c onne ction Γ ij 2; k dep end lin e arly on u r then the metric g ij 1 = ∂ u r g ij 2 (2.3) with g ij 2 form a flat p encil of metrics. The L evi-Ci v ita c onne ction of the m e tric g ij 1 has the form Γ ij 1; k = ∂ u r Γ ij 2; k . (2.4) W e are concern with the following particular class of flat p encil of metrics. 6 Y A SSIR DI NAR Definition 2.2. A con trav ariant flat p encil of metrics on a manifold M defined b y the matrices g ij 1 and g ij 2 is called quasihomogenou s of degree d if there exists a function τ on M suc h that the ve ctor fields E := ∇ 2 τ , E i = g is 2 ∂ s τ (2.5) e := ∇ 1 τ , e i = g is 1 ∂ s τ satisfy the follow ing prop erties (1) [ e, E ] = e . (2) L E ( , ) 2 = ( d − 1)( , ) 2 . (3) L e ( , ) 2 = ( , ) 1 . (4) L e ( , ) 1 = 0. Here, for example L E denote the Lie deriv ative along the v ector field E and ( , ) 1 denote the metric defined b y the matrix g ij 1 . In a ddition, the quasihomogenous flat p encil of metrics is called regular if the (1,1)-tensor R j i = d − 1 2 δ j i + ∇ 1 i E j (2.6) is nondegenerate on M . 2.1.1. F r ob enius manifolds . A F rob enius algebra is a comm utativ e asso ciativ e alg ebra with unit y e and an in v a rian t nondegenerate bilinear form ( ., . ). A F r ob enius manifold is a manifo ld M with a smo oth structure of F rob enius algebra on the tangent space T t M at an y p oint t ∈ M with certain compatibility conditions [13]. Globally , w e require the metric ( ., . ) to b e flat a nd the unit y v ector field e is constant with resp ect to it. In the flat co ordinates ( t 1 , ..., t r ) where e = ∂ ∂ t r − 1 the compatibility conditions implies that there exist a function F ( t 1 , ..., t r ) suc h that η ij = ( ∂ t i , ∂ t j ) = ∂ t r − 1 ∂ t i ∂ t j F ( t ) and the structure constants of the F rob enius algebra is given b y C k ij = η k p ∂ t p ∂ t i ∂ t j F ( t ) where η ij denote the in v erse of the matrix η ij . In this w ork, w e consider F rob enius manifolds where the quasihomogeneit y condition tak es the f o rm r X i =1 d i t i ∂ t i F ( t ) = (3 − d ) F ( t ); d r − 1 = 1 . (2.7) This condition defines the degrees d i and the charge d of the F rob enius structure. If F ( t ) is a n a lgebraic function w e call M an algebraic F rob enius manifold . The asso ciativity of F rob enius algebra implies the p o ten tial F ( t ) satisfy a system of partial differen tial equations whic h app ears in top olog ical field theory and called WDVV equations: ∂ t i ∂ t j ∂ t k F ( t ) η k p ∂ t p ∂ t q ∂ t n F ( t ) = ∂ t n ∂ t j ∂ t k F ( t ) η k p ∂ t p ∂ t q ∂ t i F ( t ) . (2.8) The following theorem g iv es a connection b et w een t he geometry of F rob enius manifolds and flat p encil o f metrics. Theorem 2.3. [14] A c ontr avaria n t quasihom o genous r e g ular flat p encil of m etrics of de gr e e d on a manifo ld M defines a F r ob enius structur e on M of de gr e e d . FROBENIUS MANIFOLDS AND W -ALGEBRAS 7 It is w ell kno wn that from a F rob enius manifold w e a lw ays hav e a flat p encil of metrics but it do es not necessary satisfy the regularit y condition ( 2 .6) [14]. Lo cally , in the co ordinates defining equation (2.8), the flat p encil of metrics is found fr o m the equations η ij = g ij 1 (2.9) g ij 2 = ( d − 1 + d i + d j ) η iα η j β ∂ α ∂ β F This fla t p encil o f metric is quasihomogenous of degree d with τ = t 1 . F ur t hermore, w e ha v e E = X i d i t i ∂ t i ; e = ∂ t r − 1 . (2.1 0 ) 2.2. Lo cal P o isson brac kets. In this section w e fix notations a nd w e review the Dir a c reduction for lo cal P oisson brac k ets o n lo o p spaces. Let M be a ma nif o ld. The lo op spaces L ( M ) of M is the space of smo o t h functions from the circle to M . A lo cal P oisson brac k et { ., . } is a P oisson brack et o n the space of lo cal functional on L ( M ). If w e c ho ose a lo cal co ordinates ( u 1 , ..., u r ) then { ., . } is a finite summation of the fo rm { u i ( x ) , u j ( y ) } = ∞ X k = − 1 ǫ k { u i ( x ) , u j ( y ) } [ k ] (2.11) { u i ( x ) , u j ( y ) } [ k ] = k +1 X s =0 A i,j k ,s ( u ( x )) δ ( k − s +1) ( x − y ) , where ǫ is just a parameter, A i,j k ,s ( u ( x )) are homogenous p olynomials in ∂ j x u i ( x ) of degree s when w e assign ∂ j x u i ( x ) degree j , and δ ( x − y ) is the Dirac delta function defined b y Z S 1 f ( y ) δ ( x − y ) dy = f ( x ) . In particular, the first t erms can b e written as follow s { u i ( x ) , u j ( y ) } [ − 1] = F ij ( u ( x )) δ ( x − y ) (2.12) { u i ( x ) , u j ( y ) } [0] = g ij ( u ( x )) δ ′ ( x − y ) + Γ ij k ( u ( x )) u k x δ ( x − y ) . where g ij ( u ), F ij ( u ) and Γ ij k ( u ) are smo oth functions on the finite dimensional space M . It follo ws from the definition that the matrix F ij ( u ) defines a Poiss on structure on M . W e sa y the P oisson brac k et admits a disp ersionless limit if F ij ( u ) = 0 and { u i ( x ) , u j ( y ) } [0] 6 = 0. In this case { u i ( x ) , u j ( y ) } [0] defines a P oisson brack et on L ( M ) know n a s Poisson brac k et of h ydro dynamic type . W e call it nondegenerate if det( g ij ( u )) 6 = 0 in an op en dense subset of M . Theorem 2.4. [15 ] In the notations given ab ove. If { u i ( x ) , u j ( y ) } [0] defines a nonde gen- er ate Poisson br ac kets of hydr o dynamic typ e then the matrix g ij ( u ) defines a c ontr avariant flat metric on M an d Γ ij k ( u ) is its c ontr avaria n t L evi-Civi ta c onne ction. 2.2.1. Dir ac r e duction . Assume we hav e a lo cal P o isson brac k et on the lo o p space L ( M ) of a manifo ld M . Let N ⊂ M b e a submanifold of dimension m . Then under some assumptions the P oisson bra ck et can b e reduced to N using Dirac reduction. F or this end w e assume N is defined b y t he equations u α = 0 for α = m + 1 , ..., r . W e in tro duce three t yp es o f indexes; capital letters I , J, K , ... = 1 , .., r , small letters i, j, k , ... = 1 , ...., m whic h parameterize t he submanifold N a nd Greek letters α, β , δ, ... = m + 1 , ..., r . 8 Y A SSIR DI NAR Prop osition 2.5. [8] I n the notations of e quations (2.12) . Assume the min or matrix F αβ is nonde ge ner ate. T h en Dir ac r e duction is wel l define d on L ( N ) and gives a lo c al Pois son br acket. If we write the le ading terms of the r e duc e d Poisson br a cket in the form { u i ( x ) , u j ( y ) } [ − 1] N = e F ij ( u ) δ ( x − y ) , (2.13) { u i ( x ) , u j ( y ) } [0] N = e g ij 0 ( u ) δ ′ ( x − y ) + e Γ ij k u k x δ ( x − y ) . (2.14) Then e F ij = ( F ij − F iβ F β α F αj , ) (2.15) e g ij = g ij 0 − g iβ F β α F αj + F iβ F β α g αϕ F ϕγ F γ j − F iβ F β α g αj , (2.16) and e Γ ij k u k x =  Γ ij k − Γ iβ k F β α F αj + F iλ F λα Γ αβ k F β ϕ F ϕj − F iβ F β α Γ αj k  u k x −  g iβ − F iλ F λα g αβ  ∂ x ( F β ϕ F ϕj ) (2.17) and the other terms c ould b e found b y solving c e rtain r e cursive e quations. Corollary 2.6. If the entries F iα = 0 on N , then the r e d uc e d Poisson br acket on L ( N ) wil l have the same le ading terms, i.e e F ij = F ij . (2.18) e g ij = g ij . e Γ ij k = Γ ij k . 3. Subregular nilpote nt elements W e review some facts ab o ut the theory of subregular nilp otent elemen ts in simple Lie algebras and the related structure of o pp osite Cartan subalgebra. Let g b e simple Lie algebra o f rank r . W e assume the Lie a lgebra g is of type D r where r is even or E r . This assumption is due to t he fact that for simply laced Lie a lg ebra, the opp osite Carta n subalgebra for a subregular nilp oten t elemen t exists only for these t yp es of Lie algebra. Let us fix a subregular nilp otent elemen t e ∈ g . By definition, a nilp oten t elemen t is called subregular if g e := k er a d e has dimension equal to r + 2 [4]. Using t he Jacobson- Morozo v theorem, w e fix a semisimple elemen t h and a nilp otent elemen t f in g suc h that { e, h, f } generate a sl 2 -subalgebra A ⊂ g satisfying [ h, e ] = 2 e, [ h, f ] = − 2 f , [ e, f ] = h. (3.1) Let us consider the adjoin t represen tation of A on g . Then g decomp oses to ir r educible A -submo dules g = r +2 M i =1 V i . (3.2) Let dim V i = 2 η i + 1 and assume V 1 is isomorphic to A as a v ector space. W e call t he set W t ( e ) = { η i : i = 1 , ..., r + 2 } (3.3) the w eights of e . Let η 0 + 1 b e the Coxete r n um b er of g . The set W t ( e ), under o ur c hoice of a total order, together with η 0 are given in the followin g table. FROBENIUS MANIFOLDS AND W -ALGEBRAS 9 g W t ( e ) η 0 η 1 , . . . , η r − 1 η r η r +1 , η r +2 D r 2r-3 1 , 3 , . . . , r − 1; r − 1 , r + 1 , . . . , 2 r − 5 r-3 1 ,r - 2 E 6 11 1 , 4 , 5 , 7 , 8 2 3 , 5 E 7 17 1 , 5 , 7 , 9 , 11 , 13 3 5 , 8 E 8 29 1 , 7 , 11 , 13 , 17 , 19 , 23 5 9 , 14 E t ( g ) W e observ e that the set E t ( g ) of exp onen ts o f g is give n by E t ( g ) = { η i : i = 0 , ..., r − 1 } . (3.4) W e emphasis that the statemen ts and pro ofs in this w ork dep end explicitly on the total ordering o f the set W t ( e ) and E t ( g ) given in this table. W e fix on g the inv ariant bilinear form h . | . i suc h tha t h e | f i = 1. W e no rmalize the decomp osition (3.2 ) and w e fix a ba sis fo r eac h V i b y using the fo llo wing prop o sition. Prop osition 3.1. The r e exists a de c omp osition of g into a sum of irr e ducib l e A - s ubmo dules g = ⊕ r +2 i =1 V i in such a way that ther e is a b asis X i I , I = − η i , − η i + 1 , ..., η i for e ach V i , i = 1 , . . . , r + 2 satisfying the fol lowing r elations X i I = 1 ( η i + I )! ad e η i + I X i − η i , I = − η i , − η i + 1 , . . . , η i . (3.5) and ad h X i I = 2 I X i I . (3.6) ad e X i I = ( η i + I + 1 ) X i I +1 . ad f X i I = ( η i − I + 1) X i I − 1 . F urthermor e < X i I , X j J > = δ i,j δ I , − J ( − 1) η i − I +1  2 η i η i − I  . (3.7) Pr o of. The pro of is similar to the pro of of prop osition 2.3 in [10], since for a ll simple Lie algebras, except D 4 , there a r e at most t w o w eigh ts of the same v alue. The Lie algebra D 4 has three w eigh t s equal one. W e giv e such a normalizat io n for D 4 in section 6.1.  W e observ e that the norma lized basis for V 1 are X 1 1 = − e, X 1 0 = h, X 1 − 1 = f . W e recall that the semisimple elemen t h define the follo wing Z -grading on g and it is called the Dynkin grading g = M i ∈ Z g i ; g i = { q ∈ g : ad h ( q ) = iq } . (3.8) W e observ e that g i = 0 if i is o dd. 10 Y ASS IR DINA R 3.1. Opp osite Cartan subalgebra. The f ollo wing theorem summarize the relation b e- t w een the subregular nilp ot ent elemen t e and a quasi-Co xeter conjugacy class in the W eyl group of g . T o simplify the notations let κ denote the maxim um w eigh t η r − 1 . Theorem 3.2. T h er e exists a nonzer o el e m ent X ′ ∈ g − 2 κ such that the ele m ent y 1 = e + X ′ is r e gular semisimple. L et h ′ b e the Cartan sub a lgebr a c ontainin g y 1 , i.e h ′ = ke r ad y 1 and c onsi d er the ad j o int gr oup e lement w d efine d by w = exp 2 π i κ + 1 ad h. Then w ac ts on h ′ as a r epr es e ntative of a r e gular quasi -Coxeter c onjugacy class in the Weyl gr oup acting on h ′ . The c on jugacy class is of typ e D r ( a 1 ) (r esp. E r ( a 1 ) ) if g is of typ e D r (r esp. E r ). F urthermor e, the e lement y 1 c an b e c om p lete d to a b asis y i , i = 1 , . . . , r for h ′ having the form y i = v i + u i , u i ∈ g 2 η i , v i ∈ g 2 η i − 2( κ +1) and such that y i is an eigenve ctor of w with eigenvalue exp 2 π i η i κ +1 . Pr o of. The pro of for a Lie a lg ebra of t yp e E i , i = 6 , 7 , 8 is obtained by Springer [27]. The case of a Lie algebra of type D r , the pro of is g iven in the app endix of [7 ].  W e fix an elemen t X ′ satisfy the h yp othesis of the t heorem ab ov e. In the literature, the elemen t y 1 = e + X ′ is called a cyclic elemen t and h ′ = k er ad y 1 is called the opp osite Cartan subalgebra . W e will call the set E t ( e ) := { η i , i = 1 , ...r } ⊂ W t ( e ) (3.9) the exp onents of e as it plays the same role of the exp onen ts of ( g ) to the regular nilp oten t elemen ts [10]. Let us consider a basis y i = u i + v i for h ′ satisfying the hy p othesis of the theorem ab o v e. W e normalize them b y using the following theorem Prop osition 3.3. The b asis y i = u i + v i c an b e chosen in such a way that u i = − X i η i , i = 1 , . . . , r (3.10) and v 1 = X ′ = ρX r − 1 − κ (3.11) for some nonzer o numb er ρ Pr o of. F rom the construction w e kno w that u 1 = e = − X 1 1 . Then, it is easy to see that u i , i = 1 , ..., r generate a comm utativ e subalgebra of g e . But X i η i are homogenous basis for g e . Hence for a Lie algebra of ty p e E 8 and D r , r > 4 the norma lizat io n of u i , i = 2 , ..., r and v 1 follo ws from the structure of the set W t ( e ). F or a Lie algebra of t yp e E 6 , E 7 and D 4 w e obtained suc h normalization by direct computations.  Let us consider the matrix A i,j of the in v a r ia n t bilinear fo rm on h ′ under the basis y i = − X i η i + v i . A ij = h y i | y j i = −h X i η i | v j i − h v i | X i η j i ; i, j = 1 , . . . , r . (3.12) W e kno w from the theory of Cartan subalgebras that the matrix A ij is nondegenerate. Some useful prop erties w e gain from h ′ are summarized in the fo llo wing prop osition. FROBENIUS MANIFOLDS AND W -ALGEBRAS 11 Prop osition 3.4. The matrix A ij is an tidia gonal with r esp e ct to the set E t ( e ) in the se n se that A ij = 0 , if η i + η j 6 = κ + 1 . Ther efor e, after total ly r e or dering the set E t ( e ) in the form µ 1 ≤ µ 2 . . . ≤ µ r , we have the pr op erty µ j + µ r − j +1 = κ + 1 = η 1 + η r − 1 , j = 1 , ..., r . (3.13) Pr o of. W e will use the fact that the matrix h . | . i is a nondegenerate inv ariant bilinear for m on h ′ . Hence for any elemen t y i there exists an elemen t y j suc h that h y i | y j i 6 = 0. But if w is the quasi-Co xeter elemen t w e defined in theorem 3 .2 then t he equalit y h y i | y j i = h w y i | w y j i = exp 2( η i + η j ) π i κ + 1 h y i | y j i implies tha t in case h y i | y j i 6 = 0 we m ust ha v e η i + η j = κ + 1. Hence, the matrix A ij is an tidiagonal with resp ect to the set E t ( e ).  In the remainder of this pap er let a denote the elemen t X r − 1 − κ . Prop osition 3.5. The c ommutators of a and X i η i satisfy the r elation h [ a, X i η i ] | X j η j − 1 i 2 η j + h [ a, X j η j ] | X i η i − 1 i 2 η i = 1 ρ A ij (3.14) for al l i, j = 1 , . . . , r . Her e the nonzer o numb er ρ is the sam e as in pr op osition 3.3. Pr o of. W e note t ha t the commu tator of y 1 = e + ρX r − 1 − κ and y i = v i − X i η i giv es the r elat io n [ e, v i ] = ρ [ a, X i η i ] , i = 1 , ..., r . (3.15) This in turn g ive the follow ing equality for ev ery i, j = 1 , ..., r h ρ [ a, X i η i ] | X j η j − 1 i = h [ e, v i ] | X j η j − 1 i = −h v i | [ e, X j η j − 1 ] i (3.16) = − 2 η j h v i | X j η j i but then h [ a, X i η i ] | X j η j − 1 i 2 η j + h [ a, X j η j ] | X i η i − 1 i 2 η i = − 1 ρ ( h v i | X j η j i + h v j | X i η i i ) = 1 ρ A ij . (3.17)  3.2. Slo dowy co ordinates. W e review the relatio n b et w een subregular nilp oten t ele- men ts and deformation of simple hy p ersurface singularities [26 ]. Let G b e the adjoint group of g . By Chev alley theorem, t he algebra S ( g ∗ ) G of in v a rian t p olynomials under the adjoin t action of G is generated by r homogenous p olynomials of degrees η i + 1 , i = 0 , ..., r − 1. The inclusion homomorphism S ( g ∗ ) G ֒ → S ( g ∗ ) , is dual to a morphism χ : g → g /G called t he adjoin t quotien t . W e fix a homogenous generator χ 0 , ..., χ r − 1 of t he algebra S ( g ∗ ) G with degree χ i equals η i + 1. Then the a dj oin t quotient map is given b y χ = ( χ 0 , . . . , χ r − 1 ) : g → C r . (3.18) 12 Y ASS IR DINA R The fib er N := χ − 1 ( χ (0)) is called the nilp ot en t v ariety of g , it consists of all nilp oten t elemen ts o f g . W e define the Slo dowy slice to b e the a ffine subspace Q = e + g f (3.19) where g f = ke r ad f . Then Briesk orn pro v ed that the restriction of χ to Q is semiuniv ersal deformation of the simple h yp ersurface singularity N ∩ Q whic h is of the same type as g . Let us in tro duce the following co ordinates on Q r +2 X i =1 z i X i − η i + e ∈ Q and assign degree 2 η i + 2 to z i . Prop osition 3.6. ( [25] ,se ction 7.2) L et χ i , i = 0 , ..., r − 1 b e a homo genous gener ators of the ring S ( g ∗ ) G . Th e n the r estriction of χ i to Q w i l l b e quasihomo genous of de g r e e 2 η i + 2 . Prop osition 3.7. The m ap χ | Q has r an k r − 1 at e . We c an normalize the mo dules V i and cho ose a homo gen ous gener a tors χ 0 , ..., χ r − 1 for S ( g ∗ ) G such that the r estriction t i of χ i to Q with i > 0 take the fo rm t i = z i + non line ar terms . (3.20) In p articular, setting t r + i = z r + i , i = 0 , 1 , 2 (3.21) we get a quasiho m o genous c o or dinates ( t 1 , ..., t r +2 ) on Q with de gr e e t i e quals de gr e e z i . Pr o of. Let χ 0 , ..., χ r − 1 b e a homogenous generators f o r S ( g ∗ ) G and denote t 1 , ..., t r − 1 their restriction to Q . Using the structure of the set W t ( e ) and E t ( g ) together with the isolat- edness of the singularit y of N ∩ Q , Slo do wy prov ed the following [2 5] (section 8.3 ). W e can c ho o se a co or dinates v 1 , v 2 , v 3 from z 1 , ...., z r +2 of degrees 2 η r + 2 , 2 η r +1 + 2 , 2 η r +2 + 2, resp ectiv ely , suc h that ( t 1 , . . . , t r − 1 , v 1 , v 2 , v 3 ) are homogenous co ordinat es o n Q . There fore, the statemen t follo ws up on proving that v i = z r + i for i = 0 , 1 , 2 when considering the normalization of prop osition 3 .1 . This is ob vious for the Lie algebra E 8 since the n umbers in the set W t ( e ) are all differen t. It is also true fo r the Lie alg ebra D r , r is ev en, since t he restriction of the first in v ariant χ 1 to S is, up to constan t, equal to z 1 . F or the Lie algebra E 6 (resp ectiv ely E 7 ) we v erify b y direct computation t ha t the inv ariant χ 3 (resp. χ 2 ) dep ends explicitly on z 3 (resp. z 2 ).  W e will call the co ordinates ( z 1 , . . . , z r +2 ) on Q obtained in this pro p osition Slo dowy co ordinates . W e observ e that in this co ordinates the quotien t map tak e the fo r m χ | Q : ( t 1 , ...., t r +2 ) 7→ ( t 0 , t 1 , ..., t r − 1 ) (3.22) where t 0 is the restriction to Q of the in v arian t p olynomial χ 0 . Setting t 1 , ..., t r equal zero in t 0 w e get, from the quasihomogeneit y , a p olynomial function f ( t r , t r +1 , t r +2 ) of the f orm sho wn in the table b elow (w e lo w er the index for conv enience). Here c 1 , c 2 , c 3 , c 4 are some constan ts. Note that the h yp ersurface N ∩ Q will b e giv en b y setting f ( t r , t r +1 , t r +2 ) = 0. Moreo v er, fro m t he isolatedness of the singularit y the num b ers c 1 , c 2 , c 3 are nonzero con- stan t s. The constant c 4 could b e eliminated b y c ha nge of v ariables to obta in the standard equation defining the simple h yp ersurface singularity . FROBENIUS MANIFOLDS AND W -ALGEBRAS 13 g f ( t r , t r +1 , t r +2 ) D r c 1 t r − 1 r +1 + c 2 t r +1 t 2 r + c 3 t 2 r +2 + c 4 t r 2 r +1 t r E 6 c 1 t 4 6 + c 2 t 3 7 + c 3 t 2 8 + c 4 t 2 6 t 8 E 7 c 1 t 3 7 t 8 + c 2 t 3 8 + c 3 t 2 9 E 8 c 1 t 5 8 + c 2 t 3 9 + c 3 t 2 10 4. Drinfeld-Sokolo v re duction In this section w e review the construction of the classical W - algebra asso ciated t o the nilp oten t elemen t e using Drinfeld-Sok olov reduction. W e use the D ynkin grading of e to define the follow ing subalgebras b := M i ≤ 0 g i , (4.1) n := M i ≤− 2 g i = [ b , b ] . Then w e consider the action of the adjo in t group N of L ( n ) on L ( g ) defined b y q ( x ) → exp ad s ( x )( ∂ x + q ( x )) − ∂ x (4.2) where s ( x ) ∈ L ( n ) , q ( x ) ∈ L ( g ). Let us extend t he in v ariant bilinear form fro m g t o L ( g ) b y setting ( u | v ) = Z S 1 h u ( x ) | v ( x ) i dx, u, v ∈ L ( M ) . (4.3) Then we iden tif y L ( g ) with L ( g ) ∗ b y means of this bilinear form. W e define the gradien t δ F ( q ) for a functional F on L ( g ) to b e the unique elemen t in L ( g ) satisfying d dθ F ( q + θ ˙ s ) | θ = 0 = Z S 1 h δ F | ˙ s i dx fo r all ˙ s ∈ L ( g ) . (4.4) W e fix on L ( g ) the fo llowing P oisson brac k et {F [ q ( x )] , I [ q ( y )] } = 1 ǫ  δ F ( x ) | [ ǫ∂ x + q ( x ) , δ I ( x )]  (4.5) for ev ery functional F and I on L ( g ). Prop osition 4.1. ( [8] ) The action of N on L ( g ) with Poiss on br ac k et { ., . } is Hamiltonian. It adm its a m omentum map J to b e the pr o j e ction J : L ( g ) → L ( n + ) wher e n + is the image of n under the K i l ling map. Mor e over, J is A d ∗ -e quivariant. W e tak e e a s a regular v alue of J . Since b is the or t ho gonal complemen t t o n under h . | . i , w e get the affine space S = J − 1 ( e ) = L ( b ) + e . Moreo v er, it follows from the D ynkin grading that the isotro p y group of e is N . Let R b e the r ing of in v arian t diff erential p olynomials o f S under the a ction of N . Then, from Marsden-Ratiu reduction theorem, the set R of functiona ls on S whic h hav e densities in the ring R is closed under the Pois son brac k ets { ., . } . Let us define the space e Q to b e t he Slo dowy slice e Q := e + L ( g f ) . (4.6) The following prop osition iden t ifies the space S/ N with e Q . 14 Y ASS IR DINA R Prop osition 4.2. [8] The sp ac e e Q is a c r o s s se ction for the action of N on S , i.e for any element q ( x ) + e ∈ S ther e is a unique e l e m ent s ( x ) ∈ L ( n ) such that z ( x ) + e = exp a d s ( x )( ∂ x + q ( x )) − ∂ x ∈ e Q. (4.7) Ther efor e, the entries o f z ( x ) ar e gener ators of the ring R . Hence, the space e Q has a P oisson structure { ., . } e Q from { ., . } . This P oisson brac ke t is kno wn in the literature as classical W -algebra asso ciated t o e . F or a formal definition of classical W -algebras see [20]. Let us obtain the linear terms of the inv arian ts z i ( x ). W e in tro duce a para meter τ and write q ( x ) + e = τ r +2 X i =1 η i X I =0 q I i X i − I + e ∈ S z ( x ) + e = τ r +2 X i =1 z i ( x ) X i − η i + e ∈ e Q s ( x ) = τ r +2 X i =1 η i X I =1 s I i ( x ) X i − I ∈ L ( n ) . Then equation (4.7) expands to r +2 X i =1 z i ( x ) X i − η i + r +2 X i =1 η i X I =1 ( η i − I + 1) s I i X i − I +1 = r +2 X i =1 η i X I =0 q I i ( x ) X i − I − r +2 X i =1 η i X I =1 ∂ x s I i ( x ) X i − I + O ( τ ) . (4.8) Hence, any in v ar ia n t z i ( x ) will tak e the from z i ( x ) = q η i i − ∂ x s η i i + O ( τ ) (4.9) = q η i i ( x ) − ∂ x q η i − 1 i + O ( τ ) . F urthermore, using h e | f i = 1 w e get z 1 ( x ) = q 1 1 ( x ) − ∂ x s 1 1 + τ h e | [ s 1 i ( x ) X i − 1 , q 0 i X i 0 ] i (4.10) + 1 2 τ h e | [ s 1 i ( x ) X i − 1 , [ s I i ( x ) X i − 1 , e ]] i = q 1 1 ( x ) − ∂ x q 0 1 ( x ) + 1 2 τ h e | [ s 1 i ( x ) X i − 1 , q 0 i X i 0 ] i = q 1 1 ( x ) − ∂ x q 0 1 ( x ) + 1 2 τ X i ( q 0 i ( x )) 2 h X i 0 | X i 0 i . The in v ariant z 1 ( x ) is kno wn in the literature a s the Virasoro densit y . W e observ e that the reduced P oisson structure could b e obta ined as follo ws. W e write the co ordinates of e Q as a differen tial p olynomials in the co ordinates of S using equation (4.7) and then w e apply the Leibnitz rule. The Leibnitz rule for u, v ∈ R hav e t he follo wing form { u ( x ) , v ( y ) } = ∂ u ( x ) ∂ ( q I i ) ( m ) ∂ m x  ∂ v ( y ) ∂ ( q J j ) ( n ) ∂ n y  { q I i ( x ) , q J j ( y ) }   . (4.11) FROBENIUS MANIFOLDS AND W -ALGEBRAS 15 Our analysis f o r the P oisson brac k ets will rela y on the quasihomogeneit y o f the in v aria nts z i ( x ) in the co or dina t es of q ( x ) ∈ L ( b ) and their deriv a t ives . Lemma 4.3. If we assign de gr e e 2 J + 2 l + 2 to ∂ l x ( q J i ( x )) then z i ( x ) wil l b e quasihom o genous of de gr e e 2 η i + 2 . F urthermor e, e ach invariant z i ( x ) d e p ends line arly only on q η i i ( x ) and ∂ x q η i − 1 i ( x ) , i.e z i ( x ) = q η i i ( x ) − ∂ x q η i − 1 i + non l i n e ar terms . (4.12) F urthermor e z 1 ( x ) = q 1 1 ( x ) − ∂ x q 0 1 ( x ) + 1 2 X i ( q 0 i ( x )) 2 h X i 0 | X i 0 i . (4.13) In p articular, z i ( x ) with i 6 = r − 1 do es not dep en d on q κ r − 1 ( x ) or its de rivatives. W e fix the follow ing nota t io ns for the leading terms of the P oisson brac k et { z i ( x ) , z j ( y ) } e Q = ∞ X k = − 1 ǫ k { z i ( x ) , z j ( y ) } [ k ] 1 (4.14) where { z i ( x ) , z j ( y ) } [ − 1] = F ij ( z ( x )) δ ( x − y ) (4.15) { z i ( x ) , z j ( y ) } [0] = g ij ( z ( x )) δ ′ ( x − y ) + Γ ij k ( z ( x )) z k x δ ( x − y ) 4.1. The nondegeneracy condition. W e w an t to prov e that the minor matrix g mn ( z ) , m, n = 1 , . . . , r is nondegenerate generically on e Q . F or t his end w e define the matrix g ij 1 ( z ) = ∂ z r − 1 g ij ( z ) (4.16) and w e will prov e tha t the minor matrix g mn 1 ( z ) , m, n = 1 , . . . , r is low er a n tidia gonal with resp ect to the set E t ( e ), i.e g mn 1 =  0 if η m + η n < κ + 1 1 ρ A mn if η m + η n = κ + 1 where the ma t r ix 1 ρ A mn is defined b y equation (3.12) and its pro p erties were obtained in prop osition 3.4. W e recall tha t z r − 1 is the co ordinate of the low est weigh t ro ot v ector a = X r − 1 κ . W e denote Ξ i I the v alue h X i I | X i I i and set [ a, X i I ] = X j ∆ ij I X j I − κ . The fact that z r − 1 ( x ) is the only inv ariant whic h dep ends on q κ r − 1 ( x ) implies tha t the in v ar ian t z r − 1 ( x ) will app ear in the express ion o f { z i ( x ) , z j ( y ) } e Q only if, when using the Leibnitz rule, w e encoun ter terms of t he original P o isson brac k et { ., . } dep end explicitly on q κ r − 1 ( x ). The later a pp ear as a r esult of the f o llo wing “brac k ets” [ q κ − I j ( x ) , q I i ( y ) ] := q κ r − 1 ( x ) ∆ ij I Ξ i I δ ( x − y ) . (4.17) 16 Y ASS IR DINA R Hence the dep endence of { z i ( x ) , z j ( y ) } e Q on z r − 1 ( x ) can b e ev aluated b y imp osing the Leibnitz rule on the “brack ets” ab ov e. W e get [ z m ( x ) , z n ( y )] = X i,I ; j X l,h ∆ ij I Ξ i I ∂ z m ( x ) ∂ ( q κ − I j ) ( l ) ∂ l x  ∂ z n ( y ) ∂ ( q I i ) ( h ) ∂ h y ( q κ r − 1 ( x ) δ ( x − y ))  = X i,I ; j X l,h, α, β ( − 1) h  h α  l β  ∆ ij I Ξ i I ∂ z m ( x ) ∂ ( q κ − I j ) ( l )  q κ r − 1 ( x )  ∂ z n ( x ) ∂ ( q I i ) ( h )  ( α )  ( β ) δ ( h + l − α − β ) ( x − y ) . Here w e omitted the ranges of the indices since no confusion can arise. W e o bserv e that the co efficien t of δ ′ ( x − y ) of this expression which con tributes to the v alue of g mn ( z ) is giv en by B ( z m , z n ) = X i,I ,J X h,l ( − 1) h ( l + h ) ∆ ij I Ξ i I q κ r − 1 ( x ) ∂ z m ( x ) ∂ ( q κ − I j ) ( l )  ∂ z n ( x ) ∂ ( q I i ) ( h )  h + l − 1 (4.18) Ob viously , W e g et g ij 1 ( z ) from the expression A ( z m , z n ) = ∂ q κ r − 1 B = X i,I ,J X h,l ( − 1) h ( l + h ) ∆ ij I Ξ i I ∂ z m ( x ) ∂ ( q κ − I j ) ( l )  ∂ z n ( x ) ∂ ( q I i ) ( h )  h + l − 1 . (4.19) Lemma 4.4. The matrix A ( z m , z n ) is l o wer a ntidiagonal w ith r es p e ct to W t ( e ) and the antidiagonal entries ar e c onstants. In o ther wor d A ( z m , z n ) is c on s tant i f η m + η n ≤ κ + 1 and e quals zer o if η m + η n < κ + 1 . Pr o of. W e note that if z m ( x ) and z n ( x ) are quasihomogenous of degree 2 η m + 2 and 2 η n + 2, resp ectiv ely , then A ( z m , z n ) will b e quasihomogenous of degree 2 η m + 2 + 2 η n + 2 − (2 κ + 2 ) − 4 = 2 η m + 2 η n − 2 κ − 2 . The pro of is complete.  Prop osition 4.5. Th e mi n or matrix g mn 1 , m, n = 1 , . . . , r is nonde gener ate and its deter- minant is e qual to the determin ant of the matrix 1 ρ A mn . Pr o of. W e observ e t ha t, from our c hoice of co o r dina t es, the minor matrix g mn 1 will b e low er an tidiagonal with resp ect to the set E t ( e ). Hence, form the second part of prop osition 3.4 w e need only to prov e that A ( z n , z m ) with η n + η m = κ + 1 is nonzero constant. In this case z m and z n are quasihomogenous of degree 2 η m + 2 and 2 κ − 2 η m + 4, respectiv ely . The expression A ( z n , z m ) = X i,I ,J X h,l ( − 1) h ( l + h ) ∆ ij I Ξ i I ∂ z n ( x ) ∂ ( q κ − I j ) ( l )  ∂ z m ( x ) ∂ ( q I i ) ( h )  h + l − 1 (4.20) giv es the constrains 2 I + 2 ≤ 2 η m + 2 (4.21) 2 κ − 2 I + 2 ≤ 2 κ − 2 η m + 4 whic h implies η m − 1 ≤ I ≤ η m Therefore the o nly p ossible v alues for the index I in the expression of A ( z n , z m ) that make sense are η m and η m − 1. Consider the partial summation of A ( z n , z m ) when I = η m . The degree of z m ( x ) yields h = 0 and that z m ( x ) dep ends linearly on q η m i ( x ). But then equation (4.9) implies that i is fixed a nd equals to m . A similar argumen t on z n ( x ) we find that the FROBENIUS MANIFOLDS AND W -ALGEBRAS 17 indices l and j are fixed and equal to 1 and n , resp ectiv ely . Hence, the partial summation when I = η m giv es the v alue ∆ mn η m Ξ m η m ∂ z n ( x ) ∂ ( q κ − η m n ) (1) ∂ z m ( x ) ∂ ( q η m m ) (0) = − ∆ mn η m Ξ m η m . W e now turn to the partia l summation of A ( z n , z m ) when I = η m − 1. The p o ssible v alues for h are 1 and 0. When h = 0 we get zero since l and h can only b e zero. When h = 1 w e get, similar to the ab ov e calculation, the v a lue ( − 1) ∆ mn η m − 1 Ξ i I ∂ z n ( x ) ∂ ( q κ − η m n ) (0) ∂ z m ( x ) ∂ ( q η m − 1 m ) (1) = ∆ mn η m − 1 Ξ m η m − 1 . Hence we end with the expression A ( z n , z m ) = ∆ mn η m − 1 Ξ m η m − 1 − ∆ mn η m Ξ m η m = h [ a, X n η n ] | X m η m − 1 i 2 η m + h [ a, X m η m ] | X n η n − 1 i 2 η n = 1 ρ A mn where the last equalit y was obtained in prop osition 3.5. Hence, the determinant of minor matrix g mn 1 , m, n = 1 , . . . , r equals to the determinan t of 1 ρ A mn whic h is nondegenerate.  5. Dirac re duction In this section we extract information ab out { ., . } e Q using t he fa ct that the P oisson brack et { ., . } e Q can b e obtained b y preforming the Dirac reduction on { ., . } . Let n denote the dimension of g . W e use prop osition 3.1 to fix a total order ξ I , I = 1 , . . . , n of the basis X i I suc h that : (1) The first r + 2 are the low est w eigh t v ectors in the following order X 1 − η 1 < X 2 − η 2 < . . . < X r +2 − η r +2 . (5.1) (2) The mat r ix h ξ I | ξ J i , I , J = 1 , . . . , n (5.2) is antidiagonal. Let ξ I b e the dual basis of ξ I under h . | . i . W e observ e that if ξ I ∈ g µ then ξ I ∈ g − µ . Let c I J K denotes the structure constan t of g under the dual basis a nd w e set e g I J = h ξ I | ξ J i . W e extend the co ordinates z i ( x ) on e Q to all L ( g ) by setting for q ( x ) ∈ L ( g ) z I ( q ( x )) = h q ( x ) − e | ξ I i , I = 1 , . . . , n . (5.3) Then w e consider the following matrix differen tia l op erator F I J = ǫ e g I J ∂ x + e F I J , (5.4) where, e F I J = X K  c I J K z K ( x )  . In this notations the Poiss on brack et { ., . } is giv en by { z I ( x ) , z J ( y ) } = F I J 1 ǫ δ ( x − y ) . (5.5) F or the rest of this section w e consider three t yp es of indices whic h ha v e differen t ranges; capital letters I , J, K , ... = 1 , .., n , small letters i, j, k , ... = 1 , ...., r + 2 and Greek letters α, β , δ, ... = r + 3 , ..., n . 18 Y ASS IR DINA R W e observ e that the matrix e F I J define the finite L ie- P oisson structure on g . It is w ell kno wn that the symplectic subspaces of this structure are the orbit spaces of g under the adjoin t action and there ar e r global Casimirs [22]. Since t he Slo dow y slice Q = e + g f is transv ersal to the orbit of e , the minor matrix e F αβ is nondegenerate. Let e F αβ denote its in v erse. Prop osition 5.1. ( [9 ] , [1 9] ) T h e Poisso n br acket { ., . } e Q c an b e obtaine d by p erforming the Dir ac r e d uction on { ., . } o n e Q . By prop o sition 2.5, the leading terms of { ., . } e Q are giv en b y F ij ( z ( x )) = ( e F ij − e F iβ e F β α e F αj ) (5.6) g ij ( z ( x )) = e g ij − e g iβ e F β α e F αj + e F iβ e F β α e g αϕ e F ϕγ e F γ j − e F iβ e F β α e g αj . (5.7) and Γ ij k ( z ( x )) z k x = −  e g iβ − e F iλ e F λα e g αβ  ∂ x ( e F β ϕ e F ϕj ) (5.8) A consequence of this prop osition is the follow ing Prop osition 5.2. [19] The Poisson br a cket { ., . } e Q has the fol lowing form { z 1 ( x ) , z 1 ( y ) } e Q = ǫδ ′′′ ( x − y ) + 2 z 1 ( x ) δ ′ ( x − y ) + z 1 x δ ( x − y ) , (5.9) { z 1 ( x ) , z i ( y ) } e Q = ( η i + 1) z i ( x ) δ ′ ( x − y ) + η i z i x δ ( x − y ) , i = 1 , ..., r + 2 . Indeed, equations (5.9) are exactly t he iden tities whic h define Virasoro densit y and classical W - a lgebras [20]. 5.1. The quasihomogeneit y condition. w e w an t to study the quasihomogeneit y of the en tries g ij 2 ( z ). W e use the definition of t he co ordinates z I ( x ) and we a ssign degree µ I + 2 to z I ( x ) if ξ I ∈ g µ I . Thes e degrees agree with t hose given in corollary 4.3. F rom the tot a l order of the basis, it follo ws that if z I ( x ) has degree µ I + 2 then degree z n − I +1 ( x ) equals − µ I + 2. F urther, since [ g µ I , g µ J ] ⊂ g µ I + µ J , an en try e F I J ( x ) is quasihomogenous of degree µ I + µ J + 2. Definition 5.3. in considering the degrees of the co or dinat es z I , W e sa y a matrix B I J ( z ) with p o lynomial entries is quasihomogenous of degree n if eac h en tr y B I J ( z ) is quasiho- mogenous of degree µ I + µ J + n . Prop osition 5.4. [5 ] The matrix e F β α ( z ) r estricte d to e Q is p olynomial and quasihomo ge- nous of de gr e e − 2 . Prop osition 5.5. The matrix g ij ( z ) is quasihomo genous of de gr e e − 4 while the ma trix F ij ( z ) i s quasih o mo genous of de gr e e − 2 a nd the m a trix Γ ij k ( z ) i s quasih o mo genous of de gr e e − (2 η k + 2) − 4 . Pr o of. The statemen t ab out the quasihomogeneity of the matr ix F ij ( z ) w as prov ed in [5]. W e will deriv e the quasihomogeneity of g ij ( z ). W e kno w that the matrix e g I J is constant an tidiagonal. Hence w e can write e g I J = C I δ I n − J +1 where C I are nonzero constan t. In particular, w e hav e e g ij ( z ) = 0. Now consider the expression (5.7 ) . Then for a fixed i we ha v e e g iβ e F β α e F αj = C i e F n − i +1 ,α e F αj . (5.10) FROBENIUS MANIFOLDS AND W -ALGEBRAS 19 Hence, the left hand sigh t is quasihomogenous of degree µ j + µ α + 2 − µ α − ( − µ i ) − 2 = µ j + µ i = 2 η i + 2 η j . A similar arg umen t sho ws that e F iβ e F β α e g αj is quasihomogenous of degree 2 η i + 2 η j . Finally , the summation e F iβ e F β α e g αϕ e F ϕγ e F γ j = X α C α e F iβ e F β α e F n − α +1 ,γ e F γ j . (5.11) Then, it has the degree µ i + µ β + 2 − µ β − µ α − 2 − µ n − α +1 − µ γ − 2 + µ γ + µ j + 2 = 2 η i + 2 η j . This pro v es that g ij ( z ) is quasihomogenous o f degree − 4. F or the last statemen t in the prop osition we observ e tha t the formula fo r Γ ij 2; k ( z ( x )) is giv en b y Γ ij k ( z ( x )) = −  e g iβ − e F iλ e F λα e g αβ  ∂ z k ( e F β ϕ e F ϕj ) . (5.12) Hence the calculation of quasihomogeneit y will b e same a s equations (5.10) and (5.11) with subtracting 2 η k + 2. This complete the pro of.  5.2. Subregular classical W -algebra. Let us consider the P oisson brac k et { ., . } e Q in Slo do wy co o rdinates ( t 1 , ..., t n ) and write { t i ( x ) , t j ( y ) } [ − 1] = F ij ( t ( x )) δ ( x − y ) (5.13) { t i ( x ) , t j ( y ) } [0] = g ij ( t ( x )) δ ′ ( x − y ) + Γ ij k ( t ( x )) t k x δ ( x − y ) Prop osition 5.6. Th e minor matrix ∂ t r − 1 g mn 1 ( t ) , m, n = 1 , . . . , r is nonde gener ate and its determinant is e qual to the determinant of the ma trix 1 ρ A ij . In p articular, the minor matrix g mn , m, n = 1 , . . . , r is generic al ly nonde ge n er ate. Mor e over, we have the same identities which defines class i c al W -algebr as, i . e { t 1 ( x ) , t 1 ( y ) } e Q = ǫδ ′′′ ( x − y ) + 2 t 1 ( x ) δ ′ ( x − y ) + t 1 x δ ( x − y ) (5.14) { t 1 ( x ) , t i ( y ) } e Q = ( η i + 1) t i ( x ) δ ′ ( x − y ) + η i t i x δ ( x − y ) . Pr o of. The nondegeneracy statements follo ws from the fact that t he pro of of prop osition 4.5 dep ends only on the linear terms of the in v arian t differential p olynomials z i (see prop osition 3.7). F or the second pa rt of the stat ement, w e need only to sho w that g 1 ,n ( t ) = ( η i + 1) t i , Γ 1 j k ( t ) = η j δ j k . (5.15) Note that, from prop osition 5 .2 , w e ha v e g 1 ,n ( z ) = ( η i + 1) z i , Γ 1 j k ( z ) = η j δ j k . (5.16) If we in tro duce the Euler v ector field E ′ := X i ( η i + 1) z i ∂ z i . (5.17) Then the f orm ula fo r c hange of co ordinates gives g 1 j ( t ) = ∂ z a t 1 ∂ z b t j g ab 2 ( z ) = E ′ ( t j ) = ( η j + 1) t j . (5.18) Where the last equalit y comes from quasihomogeneit y o f the co ordinates t i . F or Γ 1 j k ( z ), the change of co ordinates has t he following form ula Γ ij k ( t ) dt k =  ∂ z a t i ∂ z c ∂ z b t j g ab 2 ( z ) + ∂ z a t i ∂ z b t j Γ ab c ( z )  dz c . (5.19) 20 Y ASS IR DINA R But then w e get Γ 1 j k dt k =  E ′ ( ∂ z c t j ) + ∂ z b t j Γ 1 b c  dz c (5.20) =  ( η j − η c ) ∂ z c t j + η c ∂ z c t j  dz c = η j ∂ z c t j dz c = η j dt j  The fo llo wing theorem w as pro v ed in [5] using Slo do wy co ordinates Theorem 5.7. T h e matrix F ij ( t ) is a c on stant multiple of the matrix  0 0 0 Ω  (5.21) wher e Ω is a 3 × 3 matrix of the fo rm   0 ∂ t 0 ∂ t r +2 − ∂ t 0 ∂ t r +1 − ∂ t 0 ∂ t r +2 0 ∂ t 0 ∂ t r ∂ t 0 ∂ t r +1 − ∂ t 0 ∂ t r 0   (5.22) wher e t 0 is the r estriction to Q o f the invariant p ol yno m ial χ 0 define d af ter pr op osition 3.7. Let N ⊂ Q b e the hypersurface of dimension r defined as follow s N = n t ∈ Q : ∂ t 0 ∂ t r +2 = ∂ t 0 ∂ t r +1 = 0 o (5.23) F rom the quasihomogeneit y o f t 0 and the table in page 1 3 , w e observ e that ∂ t 0 ∂ t r +2 dep ends linearly on t r +2 and ∂ t 0 ∂ t r +1 is a p olynomial in t r +1 of degree ι = r − 2 (resp. ι = 2 ) if g is a Lie algebra of t yp e D r (resp. E r ). In particular, ( t 1 , ...., t r ) is well defined co ordinates on N . Theorem 5.8. The Dir ac r e duction of { ., . } e Q to e N = L ( N ) is wel l defin e d and gives a lo c al Poisson br ackets { ., . } e N . The Poisson br acket { ., . } e N is a class i c al W -algebr a. It admits a di s p ersionless lim i t and the le ading term is a nonde gen er a te Poisson br acket of hydr o dynamic typ e. Pr o of. W e observ e that the minor matrix  0 ∂ t 0 ∂ t r − ∂ t 0 ∂ t r 0  (5.24) of the matrix F ij ( t ) is nondegenerate. Hence, fro m theorem 2.5, it follows that the Dira c reduction of { ., . } e Q to e N = L ( N ) is we ll defined and gives a lo cal P oisson brack ets { ., . } e N . Let us write the reduced P oisson brac k et on e N in the form { t m ( x ) , t n ( y ) } e N = ∞ X k = − 1 ǫ k { t m ( x ) , t n ( y ) } [ k ] e N where { t m ( x ) , t n ( y ) } [ − 1] e N = b F mn ( t ( x )) δ ( x − y ) (5 .25) { t m ( x ) , t n ( y ) } [0] e N = b g mn ( t ( x )) δ ′ ( x − y ) + b Γ mn k ( t ( x )) t k x δ ( x − y ) . Then, it follo ws fro m corollary 2.6 that the en tries b g mn ( t ) and b F ij ( t ) equal g mn ( t ) and F mn ( t ), resp ectiv ely , where t r +1 and t r +2 are solutio ns of equations (5.2 3). F rom pro p o- sition 5.6, t his implies that { ., . } e N is a classical W -a lgebra. Moreo v er, f r o m prop osition FROBENIUS MANIFOLDS AND W -ALGEBRAS 21 5.7, w e ha v e b F ij = 0. Hence { ., . } e N admits a disp ersionless limit. F urthermore, propo - sition 5.6 implies that b g mn ( t ( x )) is generically nondegenerate. Hence, { t m ( x ) , t n ( y ) } [0] e N is nondegenerate Poisson brac k ets of hydrodynamics type.  In addition to the fact t ha t { ., . } e N giv es a F rob enius structure. T he construction b y considering the t heory of opp osite Cart a n subalgebra implies that it is v ery a sso ciat ed to the Drinfeld-Sokolo v hierarc hy obtained in [6] that { ., . } e Q . Therefore, w e call it subregular classical W -algebra . 6. Algebraic Frobenius manifold In this section w e obtain the pr o mised algebraic F rob enius structure. Let us consider, using the Dubrov in-Novik ov t heorem 2.4, the contra v ariant metric b g mn ( t ) on N and its Levi-Civita connection b Γ mn k ( t ). F rom prop osition 5.5, these ma t r ices ar e linear in t r − 1 . Hence, lemma 2.1 implies that the mat r ices b g mn 2 ( t ) = b g mn ( t ) , b g mn 1 ( t ) = ∂ t r − 1 b g mn 2 ( t ) (6.1) form a fla t p encil of metrics on N . Prop osition 6.1. Ther e ex ist quasihomo gen o us p olynomi a ls c o or dinates of de gr e e s 2 η i + 2 in the form s i = t i + T i ( t 1 , ..., t i − 1 ) such that the ma trix b g ij 1 ( s ) is c onstant antidiagonal. F urthermor e, in this c o o r dinates the metric g ij 2 ( s ) an d its L e v i -Civita c onne c tion have the fol lowing entries g 1 ,n 2 ( s ) = ( η i + 1) s i , Γ 1 j 2 k ( s ) = η j δ j k (6.2) Pr o of. The pro o f of the first part of the prop osition is giv en in [12] using the quasihomo- geneit y pro p ert y of the matrix b g mn . The second part is obtained in the same manor as in prop osition 5.2.  W e assume without lost of generalit y that the co ordinates t i are the flat co ordinates for b g ij 1 . Theorem 6.2. The flat p encil of metrics given by b g mn 1 ( t ) and b g mn 2 ( t ) on the sp ac e N is r e gular quasihomo genous of de gr e e d = κ − 1 κ +1 . Pr o of. In the notatio ns of definition 2.2 w e tak e τ = 1 κ +1 t 1 then E = g ij 2 ∂ t j τ ∂ t i = 1 κ + 1 X i ( η i + 1) t i ∂ t i , (6.3) e = g ij 1 ∂ t j τ ∂ t i = ∂ t r − 1 . W e see immediately tha t [ e, E ] = e The iden tit y L e ( , ) 2 = ( , ) 1 (6.4) follo ws fro m the fact that ∂ t r − 1 = ∂ z r − 1 . Then L e ( , ) 1 = 0 . (6.5) 22 Y ASS IR DINA R is a consequence from the quasihomogeneity of the mat r ix g ij 1 (see lemma 4.4). W e also obtain from prop o sition 5.5 that L E ( , ) 2 ( dt i , dt j ) = E ( g ij 2 ) − η i + 1 κ + 1 g ij 2 − η j + 1 κ + 1 g ij 2 = − 2 κ + 1 g ij 2 . (6.6) Hence, L E ( , ) 2 = ( d − 1)( , ) 2 (6.7) It remains to pro v e the regularity condition. But the (1,1)- tensor is nondegenerate since it has the en tries R j i = d − 1 2 δ j i + ∇ 1 i E j = η i κ + 1 δ j i . (6.8) This complete the pro of.  No w w e hav e all the to ols to prov e the follo wing Theorem 6.3. The sp ac e N has a natur al structur e of algebr aic F r ob enius ma nifold with char ge κ − 1 κ +1 and de gr e es η i +1 κ +1 , i = 1 , ..., r . Pr o of. It f ollo ws from theorem 6.2 and 2.3 that N has a F rob enius structure of degree κ − 1 κ +1 . T his F rob enius structure is algebraic since in the co ordinates t i the p oten tia l F is constructed using equations (2.9). Besides we hav e from theorem 5.8 the matrix b g mn 2 dep ends on the nontrivial solutions of equations (5 .2 3).  6.1. The algebraic F rob enius manifold of D 4 ( a 1 ) . W e v erify the pro cedure, outlined in this w ork, of constructing algebraic F rob enius manifold when g is of type D 4 . F or this end w e choo se the realization of D 4 as a subalgebra of g l 8 ( C ) giv en in the a pp endix of [11]. In this case it is easy to obtain a represen tation of e whic h b elongs to strictly lo w er diagonal matrices. In what fo llo ws we will denote b y σ i,j the standard matrix defined by ( σ i,j ) k ,l = δ i,k δ j,l ∈ g l 8 ( C ). W e fix the subregular nilp oten t elemen t e and the s l 2 -triple { e, h, f } as follows e = σ 2 , 1 − σ 3 , 1 + σ 4 , 3 − σ 5 , 2 2 + σ 6 , 5 + σ 7 , 4 2 + σ 8 , 6 + σ 8 , 7 (6.9) h = − 4 σ 1 , 1 − 2 σ 2 , 2 − 2 σ 3 , 3 + 2 σ 6 , 6 + 2 σ 7 , 7 + 4 σ 8 , 8 (6.10) f = 2 σ 1 , 2 − 2 σ 1 , 3 − 2 σ 2 , 4 − 8 σ 2 , 5 + 4 σ 3 , 4 + 4 σ 3 , 5 + 4 σ 4 , 6 (6.11) +8 σ 4 , 7 + 4 σ 5 , 6 + 2 σ 5 , 7 + 2 σ 6 , 8 + 2 σ 7 , 8 W e observ e tha t W t ( e ) = { 1 , 3 , 3 , 1 , 1 , 3 } and E t ( e ) = { 1 , 3 , 3 , 1 } . W e construct a ba sis for g satisfy the hypotheses of prop osition 3.1 f rom the f o rm ula X i I = 1 ( η i + I ) ad η i + I e X i − η i , i = 1 , ..., 6 . (6.12) FROBENIUS MANIFOLDS AND W -ALGEBRAS 23 where the low est ro ot v ectors X i − η i are X 2 − 3 = 24 √ 3 σ 3 , 8 − 2 4 √ 3 σ 1 , 6 (6.13) X 3 − 3 = − 24 σ 1 , 6 − 4 8 σ 1 , 7 − 4 8 σ 2 , 8 + 24 σ 3 , 8 X 4 − 1 = − 4 r 3 5 σ 1 , 2 − 2 r 3 5 σ 1 , 3 + 2 r 3 5 σ 2 , 4 + 2 r 3 5 σ 3 , 4 − 1 2 r 3 5 σ 3 , 5 − 12 r 3 5 σ 4 , 6 + 2 r 3 5 σ 5 , 6 − 2 r 3 5 σ 5 , 7 + 2 r 3 5 σ 6 , 8 − 4 r 3 5 σ 7 , 8 X 5 − 1 = − 8 σ 1 , 2 √ 5 − 2 √ 5 σ 1 , 3 − 2 √ 5 σ 2 , 4 + 8 σ 2 , 5 √ 5 + 2 σ 3 , 4 √ 5 − 4 σ 3 , 5 √ 5 − 4 σ 4 , 6 √ 5 − 8 σ 4 , 7 √ 5 + 2 σ 5 , 6 √ 5 + 2 √ 5 σ 5 , 7 + 2 √ 5 σ 6 , 8 − 8 σ 7 , 8 √ 5 . X 6 − 2 = − 4 √ 3 σ 1 , 4 + 8 √ 3 σ 1 , 5 + 8 √ 3 σ 2 , 6 + 8 √ 3 σ 3 , 7 + 8 √ 3 σ 4 , 8 − 4 √ 3 σ 5 , 8 . The opp osite Cartan subalgebra h ′ ha v e the following normalized basis y 1 = e + X 4 − 3 (6.14) y 2 = − X 2 3 − 3 √ 5 X 4 − 1 − 1 5 √ 3 X 3 − 1 + 1 2 X 6 − 1 y 3 = − X 3 3 + 3 X 1 − 1 − 1 5 √ 3 X 2 − 1 + 3 √ 5 X 5 − 1 y 4 = − X 4 1 − 1 √ 5 X 2 − 3 . The matrix of the restriction of h . | . i to h ′ under the order { y 1 , y 4 , y 2 , y 3 } equals     0 0 0 4 0 0 − 4 √ 5 0 0 − 4 √ 5 0 0 4 0 0 0     (6.15) W e write an elemen t z in Slo dowy slice Q in t he form z = z 1 X 1 − 1 + z 2 X 3 − 3 + z 3 X 4 − 3 + z 4 X 2 − 1 + z 5 X 5 − 1 + z 6 X 6 − 2 + e. (6.16) Here w e low er the index fo r con ve nience. Then the restriction of the in v ar ia n t p olynomials to Q can b e found form the co efficien ts of the indeterminan t P in the equation det ( z − P ) . After normalization w e get the followin g Slo dowy co ordinates on Q t 1 = z 1 (6.17) t 2 = z 2 − 1 2 √ 3 z 2 1 + 4 z 5 z 1 √ 15 + 7 z 2 4 5 √ 3 − 7 z 2 5 5 √ 3 t 3 = z 3 − 3 z 2 1 2 − 4 z 4 z 1 √ 15 − 14 z 4 z 5 5 √ 3 t i = z i , i = 4 , 5 , 6 . W e take the following as the restriction to Q of a highest degree inv ariant p olynomial t 0 =20 t 3 1 + 18 √ 15 t 4 t 2 1 − 1 8 √ 5 t 5 t 2 1 + 60 t 2 4 t 1 + 60 t 2 5 t 1 + 6 √ 3 t 2 t 1 + 18 t 3 t 1 − 20 √ 5 t 3 5 − 2 7 t 2 6 + 12 √ 15 t 3 t 4 + 60 √ 5 t 2 4 t 5 − 12 √ 15 t 2 t 5 (6.18) 24 Y ASS IR DINA R Note that in the case t i = 0 , i = 1 , 2 , 3 we get the equation f ( t 4 , t 5 , t 6 ) = − 20 √ 5 t 3 5 + 60 √ 5 t 2 4 t 5 − 2 7 t 2 6 (6.19) whic h define a simple hypersurface singularity of t yp e D 4 . It fo llows that the leading term of the classical W -algebra on e Q is giv en by F ij ( t ) = 75  0 0 0 Ω  (6.20) where Ω is a 3 × 3 matrix o f theorem 5.7. The h yp ersurface N ⊂ Q is defined by the equations ∂ t 0 ∂ t 6 = 270 t 6 = 0 (6.21) ∂ t 0 ∂ t 5 = − 5  − 18 √ 5 t 2 1 + 12 0 t 5 t 1 + 60 √ 5 t 2 4 − 6 0 √ 5 t 2 5 − 12 √ 15 t 2  = 0 W e c ho ose the flat co o rdinates s 1 = t 1 (6.22) s 2 = t 2 + √ 3 4 t 2 1 − 5 √ 34 t 2 4 s 3 = t 3 + 3 2 t 2 1 + √ 15 2 t 4 t 1 s 4 = t 4 Then the p oten tial F o f the F rob enius structure reads F = Z 180  − √ 5 s 4 1 − 1 0 √ 5 s 2 4 s 2 1 + 8 √ 15 s 2 s 2 1 − 2 5 √ 5 s 4 4 − 48 √ 5 s 2 2 + 40 √ 15 s 2 s 2 4  (6.23) + 1 2880  35 s 5 1 + 51 0 s 2 4 s 3 1 − 48 √ 3 s 2 s 3 1 + 77 5 s 4 4 s 1 + 36 0 s 2 3 s 1 − 720 √ 5 s 2 s 3 s 4 + 1128 s 2 2 s 1 − 1 840 √ 3 s 2 s 2 4 s 1  where Z is a solution of the quadratic equation Z 2 − 2 √ 5 s 1 Z + 3 20 s 2 1 − 1 4 s 2 4 + √ 3 5 s 2 = 0 . (6.24) It is straigh t f orw a r d to c hec k v alidity of the WD VV equations for this p oten tial. The iden tity v ector field is ∂ ∂ s 3 and the quasihomogeneit y reads 1 2 s 1 ∂ F ∂ s 1 + s 2 ∂ F ∂ s 2 + s 3 ∂ F ∂ s 3 + 1 2 s 4 ∂ F ∂ s 4 = 5 2 F . (6.25) 7. Conclusions and remarks In this work w e obta ined infinite nu m b er of examples of algebraic F rob enius manifolds. These examples corr esp ond to the regular quasi-Co xeter conjugacy classes D r ( a 1 ) where r is even and E r ( a 1 ). One of the to ols w e use is t he structure of opp osite Cartan subalgebra whic h relate the subregular nilp otent orbit to t he conjugacy class. The structure of opp o- site Carta n subalgebra exists only fo r regular conjugacy class. But ta king the subregular nilp oten t orbit in t he Lie algebra D 5 w e o btain a lgebraic F rob enius manifold related to the nonregular quasi-Cox eter conjugacy class D 5 ( a 1 ). This implies that the existence of FROBENIUS MANIFOLDS AND W -ALGEBRAS 25 algebraic F rob enius manifold is a far deep er than the not io n of opp osite Cartan subalge- bra. This fact will b e the frame w ork of o ur future researc h. Our next step is to dev elop a metho d to uniform the construction of all algebraic F rob enius manifolds that could b e obtained fr o m quasi-Co xeter conjugacy classes in W eyl groups. In this work we g iv e, fo r the first time, a geometric realization o f algebraic F rob enius manifolds. The examples obtained are certain hy p ersurfaces in the total spaces of semi- univ ersal deformations of simple h yp ersurface singularities. W e hop e this will ric h the relation b et wee n F rob enius manifolds and singularity theory . Whic h one of its main con- tributions is the existence of p olynomial F rob enius structures on the univ ersal unfolding of simple h yp ersurface singularities. F or the definition a nd deference b et wee n semiuniv ersal and unfolding see c hapt er 2 section 1 of [18]. Ac kno wledgmen ts. The author tha nks B. Dubrovin f o r useful discussions. This w ork w as done primarily during the author p o stdo ctoral fellows hip at the Ab dus Salam In ternationa l Cen tre for Theoretical Phys ics (ICTP). Italy . Reference s [1] Burroug hs, N., de Gro ot, M., Hollowoo d, T. and Miramo ntes, J., Generalized Drinfeld-Sokolov hier- archies I I: the Hamiltonian structur es, Comm. Math. Phys.153, 187 (1993). [2] Brieskorn, E. Singular elements of semi-s imple algebraic groups. Actes du Congres International des Mathematiciens, Nice, (1970). [3] Carter, R., Conjugacy clas ses in the W eyl group, Compos itio Math.2 5 , 1 (1972). [4] Collingwoo d, David H.; Mc Govern, William M., Nilpotent or bits in semisimple Lie algebr a s. V an Nostrand Reinhold Mathematics Ser ies. ISBN: 0-5 34-188 34-6 (199 3). [5] Damianou, P . A., Sab ourin, H., V anha ec ke, P ., T ransverse Poisson structur es to adjoin t or bits in semisimple Lie algebras. Pacific J . Math., no. 1, 1 1 1–138 2 32 (200 7). [6] De Gro ot, M., Hollowoo d, T. and Mira montes, J., Gener alized Drinfeld-Sokolo v hierarchies, Comm. Math. Phys. 145 1 57 (199 2). [7] Delduc, F.; F eher, L., Reg ular conjuga cy classes in the W eyl group and integrable hierar chies. J. Phys. A 2 8, no . 20, 5843 –5882 (1995). [8] Dinar, Y assir, On classificatio n and constructio n of algebr aic F r ob enius manifolds. Jo urnal o f Ge o m- etry and P h ysics, V olume 5 8, Is s ue 9, September (2008). [9] Dinar, Y assir, Remarks on Bihamiltonian Geometry and Classical W -algebras. ht tp://ar xiv.org/ abs/091 1.2116v1 (20 09). [10] Dina r , Y assir, F rob enius manifolds from regular class ical W-alg ebras. Adv a nces in Mathematics, V ol- ume 226, Issue 6, Pages 5018-5 040 (2011). [11] Dr infeld, V. G.; Sokolo v, V. V., Lie algebras and eq uations of Korteweg-de V ries t yp e. (Russia n) Cur- rent problems in mathematics, V o l. 24, 81–180 , Itogi Nauki i T ekhniki, Ak a d. Nauk SSSR, Vsesoyuz. Inst. Nauc hn. i T ek hn. Infor m., Moscow, (1 984). [12] Dubrovin, Bor is, Differential geometr y of the space of o rbits of a Coxeter gro up. Surveys in differential geometry IV: in tegrable systems, 1 8 1–211 (1998 ). [13] Dubrovin, Boris, Geometr y of 2D top o logical field theories. Integrable systems a nd quantum groups (Montecatini T erme, 1993 ), 120–3 48, Lecture Notes in Math., 1620 , Springer, Ber lin, (199 6). [14] Dubrovin, Boris, Flat pencils of metrics a nd F rob enius ma nifolds. Integrable systems and alg ebraic geometry (Kob e/Kyoto, 1997), 4 7–72, W or ld Sci. Publ. (1998 ). [15] Dubrovin, B . A.; Novik ov, S. P ., Poisson bra ckets of hydro dynamic type. (Russian) Do k l. Ak ad. Nauk SSSR 279, no . 2, 294– 2 97 (1984). [16] Dubrovin, B ., Painlev ´ e tra nscendent s in tw o-dimensio na l topolo gical field theory . The Painlev´ e prop- erty , 287, ISBN 0-387- 98888 -2 (199 9 ). [17] Her tling, Claus , F rob enius manifolds and mo duli s paces for singularities. Cambridge T racts in Math- ematics, 151. Ca m bridge Univ ersity Pres s, ISBN: 0-521- 81296 -8 (2 002). [18] Gr euel, G.-M.; Lo ssen, C.; Sh ustin, E ., Introductio n to singular ities and defo rmations, Spring er Mono- graphs in Mathematics , B erlin ISBN-1 0 3-54 0-283 80-3, (200 7 ). 26 Y ASS IR DINA R [19] F eher, L.; O’Raifeartaig h, L.; Ruelle, P .; Tsutsui, I.; Wipf, A. On Hamiltonian reductions o f the W ess-Zumino-Noviko v-Witten theories. Ph ys. Rep. 2 22, no. 1 (1992). [20] F eher, L.; O’Raifear ta igh, L.; Ruelle, P .; Tsutsui, I., O n the completeness o f the set of classic a l W -algebra s obtained from DS reductions. Comm. Math. Phys. 1 62 , no. 2, 399–4 3 1 (19 94). [21] K ostant, B., The pr incipal thr ee-dimensional subgr o up and the Betti num b ers of a complex simple Lie group, Amer. J. Math. 81, 9 73(195 9). [22] Ma rsden, J errold E.; Ratiu, T udor S., In tro duction to mechanics and symmetry . Springer -V erlag, ISBN: 0-387-9 7275- 7; 0 - 387-9 4347-1 (19 94). [23] Pavlyk, O ., Solutions to WD VV from genera lized Drinfeld-So kolo v hiera rchies, www.arx iv.org math-ph/000 3020 (2003). [24] Sa ito, K.; Y ano, T.; Sekig uch i, J. , On a certain gener ator sys tem o f the ring of in v aria n ts of a finite reflection group. Comm. Algebra 8 , no. 4, 373 –408 (1980). [25] Slo dowy P ., Simple singularities and simple algebr aic gro ups, Lect.Notes in Math. 815 Springer V erlag , Berlin.(1980 ). [26] Slo dowy P ., F our lectures on simple groups a nd singularities , Co mm unications of the Math.inst.Rijksun.Utrech t, 1 1 (1980 ). [27] Spr inger, T., Reg ular elements o f finite reflection groups, In ven t. Math. 25,159 (1974). [28] Stefanov, A., Finite orbits of the braid group ac tio n on sets of r eflections, www.a r xiv.org math-ph/040 9026 (2004). F acul ty of Ma thema tical Sciences, University of Kh ar toum, Sud an. Email: dinar@ictp.it. E-mail addr ess : dinar@ ictp.i t