Functional Analysis and Its Applications, Vol. 36, No. 4, pp. 267–280, 2002 Translated from Funktsional nyi Analiz i Ego Prilozheniya, Vol. 36, No. 4, pp. 18–34, 2002 c by V. M. Buchstaber and D. V. Leykin Original Russian Text Copyright
Polynomial Lie Algebras V. M. Buchstaber and D. V. Leykin Received May 5, 2002
Abstract. We introduce and study a special class of infinite-dimensional Lie algebras that are finite-dimensional modules over a ring of polynomials. The Lie algebras of this class are said to be polynomial. Some classification results are obtained. An associative co-algebra structure on the rings k[x1 , . . . , xn ]/(f1 , . . . , fn ) is introduced and, on its basis, an explicit expression for convolution matrices of invariants for isolated singularities of functions is found. The structure polynomials of moving frames defined by convolution matrices are constructed for simple singularities of the types A, B , C , D, and E6 . Key words: Lie algebra, moving frame, convolution of invariants, co-algebra.
Introduction The present paper is devoted to the theory and applications of a special class of infinitedimensional Lie algebras that are finite-dimensional modules over a ring of polynomials. We call such Lie algebras polynomial Lie algebras. They have canonical representations (which are exact in many important cases) in the ring of linear operators on a ring of polynomials. As a toy-model, consider a 3-dimensional module L over C[x, y] with generators (1, 1 , 2 ), where 1 = x2 ∂x + y 2 ∂y and 2 = x3 ∂x + y 3 ∂y . The module L is closed with respect to commutation (namely, [2 , 1 ] = xy1 − (x + y)2 , [1 , x] = x2 , . . . ). In the ring of all linear operators on C[x, y], the associative algebra generated by the operators {1, x, y, 1 , 2 } corresponds to it. Polynomial Lie algebras naturally appear in the theory of integrable systems [10, 14], singularity theory [1], and the theory of Hopf algebra doubles, which includes quantum group theory, [2, 15, 17]. In our investigations of Abelian functions [4, 5], we arrived at polynomial Lie algebras while solving the problem on the structure of the annihilator ideal of σ-functions [6, 7]. It turned out that the classical result by Weierstrass [18], the description of the elliptic sigma function σ(u, g2 , g3 ) as a power series in u whose coefficients are polynomials in the parameters g2 and g3 of the elliptic curve w2 = 4z 3 − g2 z − g3 , reduces to constructing a representation of the generators of a certain polynomial algebra in the form of vector fields tangent to the hypersurface Σ = {(u, g2 , g3 ) ∈ C3 | σ(u, g2 , g3 ) = 0}. Besides its immediate significance in the theory of Abelian functions, the problem on annihilator ideals of σ-functions associated with plane algebraic curves is of importance to the current investigations in quantum theory [8, 11, 13]. The reason for this is that the restrictions of the τ -functions of integrable (e.g., KdV) hierarchies of evolution equations to the finite-gap solutions can be expressed in terms of σ-functions associated with the related (e.g., hyperelliptic) curves. In this paper, some classification results on the structure of polynomial Lie algebras are obtained. The following assertion is a clue for them (see Theorem 2.4): the polynomial vector fields Lk = q vk,q ∂q defining the canonical representation of a given polynomial Lie algebra are tangent to the hypersurface det{vk,q } = 0. This result reveals a profound interrelation between polynomial Lie algebras and singularity theory [1, 12, 16, 19, 20]. (In the above toy-model we have ∆ = det{vk,q } = −x2 y 2 (x − y), 1 (∆) = 3(x + y)∆, and 2 (∆) = (3(x + y)2 − 5xy)∆.) Section 1 contains the definition of a polynomial Lie algebra a(C, V ) with polynomial structure data C = {cki,j (λ)} and V = {vi,q (λ)}. Applying the scaling transformation (C(λ), V (λ)) → (βC(αλ), αβ V (αλ)) to the data, we find the limits with C = const of polynomial Lie algebras in the 0016–2663/02/3604–0267 $27.00
c 2002 Plenum Publishing Corporation
267
class of polynomial Lie algebras. In Sec. 2, we obtain a necessary condition on a nonsingular polynomial matrix V (λ) for the existence of the polynomial Lie algebra a(C(V ), V ). Here cki,j = Γkj,i −Γki,j , where Γki,j are the Christoffel symbols of the torsion-free connection of the nonholonomic moving 0 2 and frame determined by the matrix V (λ). (In the toy-model, we have {Γki,1 } = −3xy 3(x+y) −2xy 2(x+y) k {Γi,2 } = −3xy(x+y) 3(x2 +xy+y2 ) .) In Sec. 3, we give the main constructions for the theory of transformations of polynomial Lie algebras. It is shown that if a nonsingular matrix V (λ) satisfies the above necessary condition, then the matrix det(V (λ)) V (λ) determines a polynomial Lie algebra. In Sec. 4, we construct graded polynomial Lie algebras using generating functions. We elaborate a complete description of polynomial Lie algebras that play an important rˆ ole in the theory of hyperelliptic Abelian functions and in singularity theory (for Aµ -type singularities). In Sec. 5, we construct a symmetrization operator determining an associative co-algebra structure on the rings k[x1 , . . . , xn ]/(f1 , . . . , fn ). In Sec. 6, using the structure polynomials of the co-multiplication in the co-algebra, we obtain an explicit expression for convolution matrices of invariants for isolated singularities of functions. The authors are grateful to V. M. Zakalyukin, S. K. Lando, S. P. Novikov, and A. N. Tyurin for useful and encouraging discussions in the preparation of the present paper for publication. 1. Definition and Examples Let P be the polynomial ring in the variables λ0 , . . . , λn over the ground field C. Denote by LP the free left P -module with generators {1, 0 , . . . , n }. For brevity, the elements p(λ) · 1, where p(λ) ∈ P , will be written as p(λ). Let us define a skew-symmetric operation [ · , · ] on the generators 0 , . . . , n and λ0 , . . . , λn by means of structure polynomials vi,q (λ) and cki,j (λ) in P , [i , j ] =
n
cki,j (λ)k ,
[i , λq ] = vi,q (λ),
[λq , λr ] = 0.
k=0
The skew-symmetry of the operation [ · , · ] leads only to the condition cki,j (λ) = −ckj,i (λ). Using the fact that [i , λq ] ∈ P , we extend the operation [ · , · ] to the entire module LP by the Leibnitz rule. In particular, for p(λ) ∈ P , we have [i , p(λ)] = nq=0 vi,q ∂λ∂ q p(λ). We require that the operation [ · , · ] satisfy the Jacobi identity. This results in n q=0
vi,q
∂cm j,k ∂λq
+ vk,q n q=0
∂cm ∂cm k,i i,j q m q m + vj,q + cqj,k cm i,q + ci,j ck,q + ck,i cj,q = 0, ∂λq ∂λq cqi,j vq,k + vj,q
∂vi,k ∂vj,k − vi,q = 0. ∂λq ∂λq
(1.1)
(1.2)
Under all the above conditions, we obtain a Lie algebra structure on LP , and we denote this Lie algebra by a(C, V ) with C = {cki,j (λ)} and V = {vi,q (λ)}. Definition 1.1. The Lie algebras a(C, V ) are called polynomial Lie algebras. In the subsequent text, we use the abbreviation “p. Lie algebra.” A solution (C(λ), V (λ)) of system (1.1)–(1.2) can be subjected to the scaling transformation (C(λ), V (λ)) → (βC(αλ), αβ V (αλ)). Let us use this fact to clarify what kinds of p. Lie algebras correspond to the constant parts C(0) and V (0) of the structure polynomials. We have to consider two cases. (1) Let V (0) = 0. The limit limα→0 α1 V (αλ) exists and is equal either to the zero matrix or to a matrix V (1) (λ) linear in λ. Setting β = 1, we arrive either at the p. Lie algebra a(C(0), 0) or at a(C(0), V (1) (λ)). 268
(2) Let V (0) = 0. Set β = α and pass to the limit as α → 0. This gives the p. Lie algebra a(0, V (0)). Example 1.2. The polynomials vi,q (λ) are zero and cki,j (λ) are constants. The Lie algebra a(C, 0n+1 ) includes the (n + 1)-dimensional Lie algebra over C with generators {j }, i = 0, . . . , n. Lemma 1.3. If {vi,q (0)} = 0, then {cki,j (0)} are the structure constants of a Lie algebra over C, say, with generators 00 , . . . , 0n . The mapping i → 0i , λ → 0 determines a Lie algebra homomorphism. Example 1.4. The polynomials vi,q (λ) are constants and cki,j (λ) are zero. An arbitrary matrix V ∈ Mat(n + 1, C) satisfies system (1.2). The most important example of such a p. Lie algebra is a(0, 1n+1 ), which contains the (2n + 2)-dimensional Heisenberg algebra given by the commutation relations [i , j ] = 0, [i , λq ] = δi,q , [λq , λr ] = 0, where δi,q is the Kronecker delta. The p. Lie algebra a(0, 1n+1 ) is isomorphic to the canonical Lie algebra L on the free left P -module with generators {1, ∂λ1 , . . . , ∂λn }. Here and henceforth, ∂λi = ∂/∂λi . Denote this isomorphism by ρ, i.e., ρ(i ) = ∂λi and ρ(p) = p, p ∈ P . Example 1.5. If cki,j (λ) are constants satisfying system (1.1), then the matrix V (C) = {vi,k = r r=0 λr ci,q } linear in λ gives a solution of system (1.2). The p. Lie algebra a(C, V (C)) includes the 2(n + 1)-dimensional Lie algebra given by the following commutation relations: n
[i , j ] = cki,j k ,
[i , λq ] =
n r=0
There is a representation ρ : a(C, V (C)) → L, ρ(i ) =
λr cri,q ,
[λq , λr ] = 0.
q vi,q ∂λq
and ρ(p) = p, p ∈ P .
Lemma 1.6. The homomorphism of P -modules ρV : LP → L that takes the generators i to the vector fields Li according to the formulas n vj,q (λ)∂λq , j = 0, 1, . . . , n, ρV (j ) = Lj = q=0
is a Lie algebra homomorphism. The representation ρV of a p. Lie algebra a(C, V ) is exact if and only if det V = 0. Denote by Der(P ) the Lie algebra of differentiations for the polynomial ring P . Corollary 1.7. The data (C, V ) with constant C = {cki,j } determine a p. Lie algebra a(C, V ) if and only if (1) the structure constants C define a Lie algebra A structure in the (n + 1)-dimensional space with basis 0 , . . . , n , (2) the matrix V defines a representation γ : A → Der(P ) by the formulas γ(j )λq = vj,q , j, q = 0, . . . , n. Note that the universal enveloping algebras of the p. Lie algebras a(C, V ) described in Corollary 1.7 are operator doubles [2, 15, 17] (defined by the Milnor action [15] on the polynomial ring P ) of Hopf algebras multiplicatively generated by the primitive elements. Let us give an example of a p. Lie algebra a(C, V ) with constant cki,j and nonlinear vi,q (λ). Example 1.8. The data (C, V ), where C = {cki,j } with nontrivial ci+j i,j = (i−j) for i+j n and n i+2 q+1 and [f (t)]α stands for the coefficient before V = {vi,q = [Λ(t) ]q−i }, where Λ(t) = 1+ q=0 λq t α t in a polynomial f (t), determine a p. Lie algebra a(C, V ). The universal enveloping algebras of such p. Lie algebras are algebras of differential operators on the topological monoid determined by the right regular representation of the group of jets J n+1 (1, 1) on the polynomial ring P (see [3]). The left regular representation of the group J n+1 (1, 1) on the ring P leads to p. Lie algebras of the type described in Example 1.5. 269
2. Nondegenerate Polynomial Lie Algebras Let us set ∆ = det V . Definition 2.1. The polynomial Lie algebras a(C, V ) with ∆ = 0 are called nondegenerate polynomial Lie algebras, or, briefly, “n.p. Lie algebras.” According to Lemma 1.6, each n.p. Lie algebra a(C, V ) is isomorphic to the Lie algebra of operators on P that is generated by the operators of multiplication by λi , i = 0, . . . , n, and the polynomial vector fields Lj , j = 0, . . . , n. In this case, according to relations (1.2), the structure polynomials C = {cki,j (λ)} are uniquely defined by the polynomial matrix V , i.e., C = C(V ). Thus, it is natural to denote n.p. Lie algebras a(C(V ), V ) by a(V ). Since in the operator algebras a solution C(V ) of system (1.2) is automatically a solution of system (1.1), we arrive at the following problem: to classify the matrices V (λ) for which system (1.2) admits a polynomial solution C(V ). Note that if ∆ = const = 0, then a polynomial solution C(V ) is sure to exists. This is implied by the following property of system (1.2): for any matrix V (λ) with ∆ = 0, system (1.2) admits the solution C(V ) in the ring P [t]/(t∆ − 1). Example 2.2. Let V = diag(p0 , . . . , pn ), pi ∈ P , and let p0 · · · pn = const. We find cki,j =
1 (δi,k pj ∂λj pi − δj,k pi ∂λi pj ) pk
from (1.2), whence easily follows that, first, only the polynomials cji,j = −cjj,i are nonzero and, second, cji,j are in P if and only if pi = αi q(λ) for some q ∈ P and (α0 , . . . , αn ) ∈ Cn+1 , α0 · · · αn = 0, and cji,j = −αi ∂λi q.
Set ci = {cki,j (V )}, j, k = 0, . . . , n, vi = (vi,0 , . . . , vi,n ), and ∂λ = (∂λ0 , . . . , ∂λn )t . Then ci = (V (∂λ vi ) − Li (V ))V −1 .
(2.1)
Lemma 2.3. Let C(V ) = {cki,j } be the solution of system (1.2) for some matrix V with ∆ = 0. Then n n cki,k = −Li (log ∆) + ∂λq vi,q . (2.2) q=0
k=0
Proof. According to (2.1), the trace of ci is given by formula (2.2). Theorem 2.4. Let a matrix V determine a nondegenerate polynomial Lie algebra a(V ). Then (1) Li (∆) = φi ∆, where φi ∈ P , i = 0, . . . , n, (2) the vector fields Li are tangent to the hypersurface Σ = {(λ0 , . . . , λn ) ∈ Cn+1 | ∆ = 0}. Proof. (1) Let us rewrite relations (2.2) in the following form: Li (∆) = φi ∆,
where φi =
n
∂λk vi,k − cki,k ,
i = 0, . . . , n.
(2.3)
k=0
Under the assumptions of the theorem, the functions φi belong to P . (2) The operators Li satisfying (2.2) define differentiations of the local ring P/(∆). In the case n = 1, we obtain the corollary below. v0,0 v0,1 over the ring P = C[λ0 , λ1 ] determines Corollary 2.5. A nonsingular matrix V = v1,0 v1,1 a nondegenerate polynomial Lie algebra a(V ) if and only if the vector fields L0 = v0,0 ∂λ0 + v0,1 ∂λ1 and L1 = v1,0 ∂λ0 +v1,1 ∂λ1 are tangent to the plane curve Σ = {(λ0 , λ1 ) ∈ C2 | v0,0 v1,1 −v0,1 v1,0 = 0}. 270
Proof. For n = 1, the p. Lie algebra structure is defined by only two structure functions, c00,1 and c10,1 . By (2.2), we have: c00,1 =
∂v1,0 ∂v1,1 + − L1 (log ∆), ∂λ0 ∂λ1
c10,1 = −
∂v0,0 ∂v0,1 − + L0 (log ∆). ∂λ0 ∂λ1
(2.4)
The tangency condition Li (log ∆) ∈ P proves the desired assertion. λ0 λ 1 Example 2.6. Let V = . Then Σ is the set of zeros of the linear function ∆ = −β α αλ0 + βλ1 . The vector fields L0 = λ0 ∂λ0 + λ1 ∂λ1 and L1 = −β∂λ0 + α∂λ1 are tangent to Σ since L0 (∆) = ∆ and L1 (∆) = 0. Hence, V determines an n.p. Lie algebra. From (2.4), we find c00,1 = 0 and c10,1 = 1. The next example shows that if n > 1, then the necessary condition Li (log ∆) ∈ P is no longer sufficient for system (1.2) to have the polynomial solution C(V ). Example 2.7. Consider the family of matrices wλ1 + xλ0 yλ2 vλ0 zλ2 0 , ∆ = −y 2 zλ32 , V = wλ1 + xλ0 0 0 yλ2
(v, w, x, y, z) ∈ C5 .
Impose the conditions y = 0 and z = 0. We have L0 (∆) = 3y∆,
L1 (∆) = 0,
L2 (∆) = 0.
(2.5)
Solving system (1.2), we find c10,1 = y − w −
x(xλ0 + wλ1 ) x(xλ0 + wλ1 )2 (2w − y)(xλ0 + wλ1 ) − vwλ1 , c20,1 = + , zλ2 yλ2 yzλ22 xy x(xλ0 + wλ1 ) c10,2 = − , c20,2 = y − v + , c21,2 = −x, z zλ2
and the values of the other cki,j , i < j , are zero. With regard to (2.5), it follows from Lemma 2.3 that k k ci,k belongs to P although individual terms in the sum are not polynomials. For the parameter values x = 0, v = 2w − y and x = 0, w = 0, the matrices V determine n.p. Lie algebras a(V ). The following relations, which are well known in classical differential geometry (e.g., see [9]), are useful: 1 cki,j = Γkj,i − Γki,j and Γki,j = − (cki,j + g k,r gi,q cqj,r + g k,r gj,q cqi,r ), (2.6) 2 where Γki,j are the Christoffel symbols of the connection that is torsion-free and compatible with the metric {gi,j } = { nq=0 vi,q vq,j } of the nonholonomic moving frame {L0 , . . . Ln }. In formula (2.6), {g k,r } is the inverse matrix of {gi,j }, and the summation extends over the repeated indices. Let us set Γi = {Γkj,i }, j, k = 0, . . . , n. By the definition of Christoffel symbols, we have Li (V ) = Γi V.
(2.7)
Corollary 2.8. Let the Christoffel symbols Γki,j = Γki,j (V ) corresponding to the moving frame {L0 , . . . Ln } be polynomial. Then the matrix V determines an n.p. Lie algebra. Definition 2.9. An n.p. Lie algebra a(V ) is said to be strictly polynomial if the Christoffel symbols Γki,j (V ) are polynomial. Example 2.10. Let V = {vi,q = λi+1 i, q = 0, . . . , n. We define the polynomials e1 , . . . , en+1 q }, n n+1 n+1−k using the relation t − k=0 ek t = nq=0 (t − λq ) and set Q = {qi,j = δi+1,j + en+1−j δi,n }, 271
i, j = 0, . . . , n. Then Γk = diag(1, . . . , n + 1) Qk , k = 0, . . . , n. Thus the Lie algebras a(V ) are strictly polynomial for all positive n. In particular, for n = 2, we have: 0 0 1 1 0 0 0 1 0 2e2 2e1 . 0 2 , Γ0 = 0 2 0 , Γ2 = 2e3 Γ1 = 0 2 0 0 3 3e3 3e2 3e1 3e3 e1 3(e3 + e1 e2 ) 3(e1 + e2 ) 3. Transformations of Polynomial Lie Algebras Let X = {xi,j (λ)} be a nonsingular matrix. Consider the transformation V → XV . Below, for the elements of the n.p. Lie algebra structure such as vector fields, structure polynomials, etc., we will indicate explicitly the dependence on their defining matrix, e.g., Lj (XV )( · ) and cki,j (XV ) for a(XV ). Lemma 3.1. (1) The structure polynomials of n.p. Lie algebras a(V ) and a(XV ) are related by the formula cm (XV )x − L (XV )(x ) − L (XV )(x ) = xi,q xj,s ckq,s (V ). (3.1) j i m,k i,k j,k i,j m
q,s
(2) Let X be a polynomial matrix and let det X = const = 0. Then, for any n.p. Lie algebra a(V ), the n.p. Lie algebra a(XV ) is defined. Definition 3.2. We say that a matrix X determines a left transformation of an n.p. Lie algebra a(V ) if the matrix XV is polynomial and system (3.1) admits the polynomial solution {cm i,j (XV )}. Theorem 3.3. Let a nonsingular matrix V satisfy the condition Li (log ∆) = ϕi ∈ P
for all i = 0, . . . , n.
Then the matrix ∆V determines an n.p. Lie algebra. Proof. Set X = {xi,k = ∆δi,k } and let {ai,j } = ∆V −1 be the adjoint of the matrix V . Consider formula (3.1) with {cki,j (V )} equal to the solution of system (1.2). As is seen from (2.1), the functions ∆cki,j (V ) are polynomials. Since Li (XV )( · ) = ∆Li (V )( · ), the assumptions of the theorem lead to the polynomials cki,j (∆V
) = ∆ϕj δi,k − ∆ϕi δj,k +
n
ar,k (vi,q ∂λq vj,r − vj,q ∂λq vi,r ).
q,r=0
Thus, for a given nonsingular polynomial matrix V , the polynomiality condition for all the functions Li (log ∆) is necessary for the existence of the n.p. Lie algebra a(V ) and sufficient for the existence of the n.p. Lie algebra a(∆V ). Definition 3.4. We say that n.p. Lie algebras a(V1 ) and a(V2 ) are left-equivalent if there exists a polynomial matrix X such that det X = const and V2 = XV1 . The group of nonsingular polynomial matrices X with det X = const will be called the left equivalence group of n.p. Lie algebras. All n.p. Lie algebras determined by matrices V with det V = const are left-equivalent to the n.p. Lie algebra a(1n+1 ) (see Example 1.4). The Christoffel symbols of the frames defined by matrices V and XV are related by the formula xi,k XΓk (V )X −1 , (3.2) Γi (XV ) = Li (XV )(X)X −1 + k
which is a direct consequence of (2.7). Thus, the left equivalence group acts on the set of strictly polynomial Lie algebras. Corollary 3.5. Let an n.p. Lie algebra a(V ) be not strictly polynomial. Then the Lie algebra a(∆V ) is strictly polynomial. 272
We now consider the transformations of p. Lie algebras that are induced by the polynomial mappings f : Cn+1 → Cn+1 . Denote by Jf the Jacobi matrix Jf (z) = ∂(f0 , . . . , fn )/∂(z0 , . . . , zn ) of a mapping f . Definition 3.6. A polynomial mapping f is said to be compatible with a polynomial matrix V if the matrix V (f (z))Jf (z)−1 is polynomial. Lemma 3.7. Let a(V (λ)) be an n.p. Lie algebra with structure polynomials cki,j (λ) and let f be a mapping compatible with V . Then the matrix W (z) = V (f (z))Jf (z)−1 determines an n.p. Lie algebra a(W (z)) with structure polynomials cki,j (f (z)). Definition 3.8. We say that n.p. Lie algebras a(V1 (λ)) and a(V2 (z)) are right-equivalent if there exists a polynomial mapping z → f (z) such that det Jf (z) = const and V1 (f (z)) = V2 (z)Jf (z), where Jf (z) is the Jacobi matrix of the mapping. The group of nondegenerate polynomial mappings f with det Jf (z) = const is called the right equivalence group of n.p. Lie algebras. All n.p. Lie algebras defined by the matrices V = Jf (z)t with det V = const are right-equivalent to the Lie algebra a(1n+1 ). We present the transformation rule for the Christoffel symbols induced by the mappings f : Cn+1 → Cn+1 , Γki,j (W (z)) = Γki,j (V (f (z))) + vi,r vj,q j q,p (∂zp j r,m )jm,s v s,k ,
where W (z) = V (f (z))Jf (z)−1 , vi,q = vi,q (f (z)), V (f (z))−1 = {v i,q }, Jf = {jq,r }, and Jf−1 = {j q,r }, and the summation extends over the repeated indices. Thus, the right equivalence group acts on the set of strictly polynomial Lie algebras. Example 3.9. The matrix 2λ1 3λ2 λ0 1 4 2 2 3λ2 V (λ) = 2λ1 6 (λ0 − 12λ0 λ1 + 12λ1 + 24λ0 λ2 ) 1 4 1 5 2 2 3 2 3λ2 6 (λ0 − 12λ0 λ1 + 12λ1 + 24λ0 λ2 ) 6 (λ0 − 10λ0 λ1 + 15λ0 λ2 + 30λ1 λ2 ) determines an n.p. Lie algebra a(V ) with structure polynomials (we present only the nontrivial ones) c10,1 = 1, c20,2 = 2, c01,2 = λ2 − λ1 λ0 + 16 λ30 , c11,2 = λ1 − 12 λ20 , c21,2 = λ0 . It is easy to verify that the Christoffel symbols Γki,j (V ) are not polynomial. Consider a mapping that gives λ in the form of symmetric polynomials, λ = f (z0 , z1 , z2 ) = (z0 + z1 + z2 , 12 (z02 + z12 + z22 ), 13 (z03 + z13 + z23 )). The mapping f is compatible with the matrix V since the matrix W (z) = V (f (z))Jf−1 is polynomial. Its exact form is W = {zqi+1 }, i, q = 0, 1, 2. It follows from Example 2.10 that the Lie algebra a(W ) is strictly polynomial. In Example 3.9, we have encountered a set of potential vector fields. Let us state the following problem: to describe the polynomial mappings f determining n.p. Lie algebras a(Jf (z)t ). In this connection, the notion below is important Definition 3.10. We say that a polynomial mapping ϕ : Cn+1 → Cn+1 is compatible with a mapping f if there exists a polynomial matrix V such that Jf (z)t Jϕ (z) = V (ϕ(z)). Lemma 3.11. Let a polynomial mapping f determine an n.p. Lie algebra a(Jf (z)t ) with structure polynomials cki,j (z) and let ϕ be a mapping compatible with f . Then (1) there exist polynomials χki,j such that cki,j (z) = χki,j (ϕ(z)), (2) the polynomials χki,j are the structure polynomials of the n.p. Lie algebra a(V ) with V (ϕ(z)) = Jf (z)t Jϕ (z). Example 3.12. Denote by Λ(z)the ring of symmetric polynomials in the variables z = (z0 , z1 , . . . , zn ). As usual, we set pi = nk=0 zki , and write p = (p1 , p2 , . . . , pn+1 ). As is well known, Λ(z) = C[p]. Denote by A the mapping given by the formula A(z) = (pi /i), i = 1, . . . , i + 1. 273
The Jacobi matrix JA (z) is the Vandermonde matrix with rows (1, zi , . . . , zin ), i = 0, 1, . . . , n. The following important fact can be verified directly. Lemma 3.13. The matrix JA (z)t determines an n.p. Lie algebra structure with constant structure polynomials cki,j (JA (z)t ) = (j − i)δi+j,k . Let S : Cn+1 → Cn+1 be a mapping acting by the formula S(z) = (s0 (z), . . . , sn (z)), where {si (z)} is a set of algebraically independent symmetric polynomials in z. According to the classical theory of symmetric polynomials, the mapping S can be represented as a composition of the mapping A and a uniquely determined polynomial mapping BS . Theorem 3.14. (1) The mapping A is compatible with S . (2) The functions cki,j (z) satisfying system (1.2) with V (z) = JS (z)t are symmetric functions in z. Proof. (1) We have S = BS ◦ A. Therefore V (z) = X t (p(z))JA (z)t , where X(p) = {xi,k } is the Jacobi matrix of the mapping BS and the entries of X(p) are polynomials in p. Set Q = JA (z)t JA (z). We have Q = {qi,j = pi+j }, i, j = 0, . . . , n, and, since pj with j > n + 1 can be expressed polynomially in terms of p, it finally follows that V (z)JA (z) = X(p)Q(p), which is what we had to prove. (2) To calculate cki,j (z), we apply formula (3.1). We have cm (r − q)xi,q xj,r = 0. i,j (z)xm,k − Lj (xi,k ) − Li (xi,k ) − m
q+r=k
To complete the proof, note that since sk (z) ∈ Λ(z), the differential operators Lk = are differentiations of the ring Λ(z).
Set (x) =
q (∂zq sk (z))∂zq
4. Graded Polynomial Lie Algebras k q k0 x k and λ(x) = 1 + q1 λq x . Then
[(x), (y)] =
xi y j cki,j (λ)k
and
[λ(x), (y)] = −
vp,q (λ)xq y p .
q,p0
i,j,k0
To investigate p. Lie algebras whose structure polynomials cki,j are at most linear in {λq } and for which vp,q are at most of the second degree, we set (4.1) xi y j cki,j (λ)k = 2Rx,y λ(y)(x) and vp,q (λ)xq y p = −Sx,y λ(y)λ(x) , i,j,k0
q,p0
where Rx,y is a linear skew-symmetric operator on C[x, y], i.e., Rx,y (f (x, y)) = −Ry,x (f (y, x)), and Sx,y is a linear operator on C[x, y] annihilating the skew-symmetric polynomials in x and y. The operators Rx,y and Sx,y possess natural extensions to LP [x,y] such that the elements {k } and the polynomials in {λq } play the role of constants for them. The Jacobi identities (1.1)–(1.2) have the following form in terms of (4.1): 2 } Rx,y (2Rx,z + Sy,z )(λ(z)λ(y)(x)) = 0, {1 + Tx,y,z + Tx,y,z (4.2) {2Rx,y Sz,x − Sz,x (Sx,y + Sz,y ) + Sz,y (Sy,x + Sz,x )}(λ(z)λ(y)λ(x)) = 0, where Tx,y,z is the cyclic permutation operator, Tx,y,z (F (x, y, z)) = F (z, x, y). (1) (1) (2) (2) Let operator pairs (Rx,y , Sx,y ) and (Rx,y , Sx,y ) each of which satisfies (4.2) be given. The pairs (1) (1) (2) (2) (1) (2) (1) (2) (Rx,y , Sx,y ) and (Rx,y , Sx,y ) are said to be compatible if their sum (Rx,y + Rx,y , Sx,y + Sx,y ) also satisfies (4.2). 274
(1)
(1)
(2)
(2)
Lemma 4.1. The pairs (Rx,y , Sx,y ) and (Rx,y , Sx,y ) are compatible if and only if
(1) 2 (2) (2) (2) (1) (1) (λ(z)λ(y)(x)) = 0, } Rx,y (2Rx,z + Sy,z ) + Rx,y (2Rx,z + Sy,z {1 + Tx,y,z + Tx,y,z (1) (2) (2) (1) (2) (1) (1) (2) (2) (2) Sz,x + 2Rx,y Sz,x − Sz,x (Sx,y + Sz,y ) − Sz,x (Sx,y + Sz,y ) {2Rx,y
(4.3)
(2) (1) (1) (1) (2) (2) (Sy,x + Sz,x ) + Sz,y (Sy,x + Sz,x )}(λ(z)λ(y)λ(x)) = 0. + Sz,y
The above identities are an analog of the vanishing condition of the Schouten bracket for a pair of skew-symmetric rank 2 tensors defining compatible Poisson brackets [10, 14]. Theorem 4.2. The pairs x2 ( · ), x−y
(2) − 2 ◦ ∂x,y ◦ x ( · ), Rx,y ( · ) = τx,y
x2 + ◦ (∂x − ∂y ) ◦ τx,y ( · ), x−y
(2) + Sx,y ( · ) = x2 ◦ ∂x,y ◦ τx,y ( · ),
(1) − ◦ (∂x − ∂y ) ◦ ( · ) = τx,y Rx,y (1) Sx,y (·) =
± is the (skew-)symmetrization operator, τ ± (F (x, y)) = (F (x, y) ± F (y, x))/2, satisfy where τx,y x,y identities (4.3), and each of them satisfies identities (4.2).
Proof. One should carry out the calculations according to (4.2) and (4.3) assuming that λ(x), (x), etc. are symbols of functions, that is, without expanding them into series. We obtain a family aα,β (c(x, y, z), v(x, y)) of p. Lie algebras with the structure polynomials of the family given by the generating functions, ψ(x, y, z; λ) cki,j (λ)xi y j z k = α (∂x − ∂y ) + β ∂x,y ψ(x, y, z; λ) , c(x, y, z) = (x − y) i,j,k0
where x2 (1 − yz)λ(y) − y 2 (1 − xz)λ(x) , (1 − xz)(1 − yz) λ(x)λ (y) − λ (x)λ(y) v(x, y) = − βx2 λ (x)λ (y). vp,q (λ)xq y p = αx2 x−y ψ(x, y, z; λ) =
(4.4)
p,q0
Note that vp,0 = 0 and vp,1 = 0 for all p 0, i.e., [(x), 1] = 0, which is natural, and, moreover, [(x), λ1 ] = 0. Thus, C[λ1 ] belongs to the center of the p. Lie algebra aα,β (c, v). Let us set deg x = deg y = −2 and deg(xz) = 0. Then, under the conditions deg (x) = 0 and deg λ(x) = 0, we obtain deg i = deg λi = 2i. By (4.4), we have deg c(x, y, z) = deg v(x, y) = 0, and hence, deg cki,j = 2(i + j − k) and deg vp,q = 2(p + q). The p. Lie algebra aα,β (c, v) is graded with respect to this grading. Consider the quotient Lie algebra an+1 (c, v) of the graded p. Lie algebra aα,β (c, v) modulo the relations {j = 0 | j > n}. Theorem 4.3. The graded p. Lie algebra an+1 (c, v) exists if and only if β(n + 2) − α = 0 and λq = 0 for q > n + 2. Proof. We have to verify that the commutation relations following from (4.4) are compatible with the truncation condition j = 0 for j > n. Taking the fact that Rx,y preserves the grading into account, it is necessary and sufficient to verify that Rx,y (xi y j ) = 0 if i + j > n. 5. Associative Co-Algebra Structure on the Rings k[x1 , . . . , xn]/(f1 , . . . , fn) Let x = (x1 , x2 , . . . , xn ) ∈ Cn . We set [x]0j = x. Denote by [x]rj the vector with coordinates {xm } for j + r < m j + n and coordinates xm for j < m j + r, where the subscripts are cyclic mod n. We set n−1 1 r f [x]i Si (f (x)) = n r=0
275
and define operations δi : C[x] → C[x, x ] by the formula Si (f (x)) − Si (f ([x]1i−1 ) . δi f (x) = xi − xi With each set f(x) = (f1 (x), f2 (x), . . . , fn (x)) of polynomials, we associate the matrix Φ(f)(x, x ) = {ϕi,j (x, x )}, where ϕi,j (x, x ) = δi (fj (x)). We write H(f)(x, x ) = det Φ(f)(x, x ). Lemma 5.1. (1) The polynomial H(f)(x, x ) is symmetric, that is, H(f)(x, x ) = H(f)(x , x). (2) For any polynomial F (x, x ), we have H(f)(x, x )(F (x, x ) − F (x , x)) ≡ 0 mod (f(x), f(x )). Proof. Assertion (1) follows from the definition. (2) In view of the decomposition F (x, x ) − F (x , x) = (xi − xi )φi (x, x ), φi (x, x ) ∈ C[x, x ], it suffices to prove the lemma for F (x, x ) = x1 . Note that n
ϕi,j (x, x )(xi − xi ) = fj (x) − fj (x ).
i=1
Thus, the first row in the matrix Φ(f)(x, x ) can be replaced by {(fj (x)− fj (x ))/(x1 − x1 )}. Expanding the determinant H(f)(x, x )(x1 −x1 ) by the first row, we obtain j H1,j (fj (x)−fj (x )).
(x ) . Example 5.2. Let n = 1; then f(x) = f (x) and H(f )(x, x ) = f (x)−f x−x Let n = 2; then f(x) = (f1 (x1 , x2 ), f2 (x1 , x2 )) and f1 (x1 ,x2 )−f1 (x1 ,x2 ) f1 (x1 ,x2 )−f1 (x1 ,x2 ) f2 (x1 ,x2 )−f2 (x1 ,x2 ) f2 (x1 ,x2 )−f2 (x1 ,x2 ) + + 1 −x1 x1 −x1 x1 −x1 x1 −x1 H(f1 , f2 ) = f1 (x1 ,xx21)−f (x1 ,x2 ) f1 (x1 ,x2 )−f1 (x1 ,x2 ) f2 (x1 ,x2 )−f2 (x1 ,x2 ) f2 (x1 ,x2 )−f2 (x1 ,x2 ) . 1 4 + + x2 −x x2 −x x2 −x x2 −x 2
2
2
2
Consider the ring A = k[x1 , . . . , xn ]/(f), where k is a commutative ring with unity.
Theorem 5.3. The k-linear mapping D : A → A ⊗k A acting by the formula Dxx (q(x)) = q(x)H(f)(x, x ) determines an associative commutative co-algebra structure on A. Proof. The commutativity is a consequence of Lemma 5.1. The associativity can be verified directly,
(Dxx Dxx − Dxx Dxx )q(x) = q(x)H(f)(x, x ){H(f)(x , x ) − H(f)(x, x )} = 0 mod (f(x), f(x ), f(x )). Example 5.4. Let A = C[x]/(x3 ). Choose the basis (1, x, x2 ) in A. We have D(1) = x2 + xx + x 2 and D(x) = x2 x + xx 2 , D(x2 ) = x2 x 2 . As is shown by the example, there is no co-unity in the above co-algebras. 6. Convolution Matrices Singularity theory provides a wide variety of polynomial matrices V (λ) = {vi,j (λ)} such that the vector fields Li = k vi,k ∂λk are tangent to the hypersurface Σ = {λ | det V (λ) = 0}. These are the so-called convolution matrices of invariants [1]. function germ with an isolated singularity at x = 0 Let f (x) = f (x1 , . . . , xn ) be a holomorphic and Milnor number µ. Denote by E = 1− ni=1 αi xi ∂i the Euler operator such that E(f (x)) = 0. Let Q = C[x]/(∇x f (x)) be the local ring of the singularity f (x) and let e(x) = {eik (x)}, k = 1, . . . , µ, be the monomial basis of Q indexed in the descending order of weights defined by the Euler operator. The weights take on the values in I = {i1 , . . . , iµ = 0}. Let us set d = deg f (x). Then F (x) = f (x) + i∈I λd−i ei is a semi-universal unfolding of the singularity f (x). To each singularity f (x), a reflection group Gf which acts on Cµ is related. The Vi`ete mapping Vµ : Cµ → Cµ /Gf ∼ = Cµ sends the orbit Gf z of a point z = (z1 , . . . , zµ ) to a point with coordinates λ = (λd−i1 , . . . , λd−iµ ), ik ∈ I , i.e., the parameters λ of the semi-universal unfolding are realized as some symmetric functions of z. The Euclidean scalar product of gradients ∇z λi , ∇z λj turns out 276
to be a polynomial in λ, i.e., in the sense of Definition 3.10, the Vi`ete mapping is self-compatible, and it defines the convolution operation λi ∗ λj ∈ P . The matrix V = {vi,j (λ) = λi ∗ λj } is called the complete convolution matrix. We set Fα (x) = αf (x) + (1 − α)F (x). Consider families of bilinear forms hi,j (λ, α)ei (x)ej (x ), Φα (e(x), e(x )) = φi,j (λ, α)ei (x)ej (x ) Hα (e(x), e(x )) = i,j∈I
i,j∈I
given by the formulas Hα (e(x), e(x )) = H(∇x Fα (x))(x, x ) mod (∇x Fα (x), ∇x Fα (x )), Φα (e(x), e(x )) = E(F (x))H(∇x Fα (x))(x, x ) mod (∇x Fα (x), ∇x Fα (x )) = E(F (x))Hα (e(x), e(x )) mod ∇x Fα (x). Theorem 6.1. For a given singularity f (x), the full convolution matrix V (λ) coincides with the matrix of the bilinear form Φ0 , {vi,j (λ)} = {φi,j (λ, 0)}, and the linearized convolution matrix (1) V (1) (λ) coincides with the matrix of the bilinear form Φ1 , {vi,j (λ)} = {φi,j (λ, 1)}. Proof. As is known (e.g., see [12]), in order to calculate the complete convolution matrix, the theorem by V. M. Zakayukin [19] should be used to find the coefficient matrix S = {si,j (λ)} of the vector fields Si = j si,j (λ)∂λj tangent to the discriminant, i.e., the set Σ = {λ ∈ Cµ | ∃ x ∈ Cn : F (x) = 0, ∇x F (x) = 0} = {λ ∈ Cµ | det S(λ) = 0}. The coefficients si,j (λ), i, j ∈ I are found from the unique decomposition n gi,j (x, λ)∂xj F (x) + si,k (λ)ek (x). (6.1) ei (x)E(F (x)) = j=1
k∈I
Furthermore, one has to symmetrize the matrix S taking the linear combinations of rows with polynomial coefficients that do not break the homogeneity of the matrix entries with respect to the grading deg λk = k. The resulting symmetric matrix V is precisely the complete convolution matrix. It follows from (6.1) that S is the matrix of the bilinear form ei (x)ei1 −i (x )E(F (x)) mod ∇x F (x). i∈I
(Recall that i1 = max I .) By Lemma 5.1, the bilinear forms Φα (e(x), e(x )) = Hα (e(x), e(x ))E(F (x)) mod ∇x Fα (x) are symmetric. It remains to show that deg Hα (e(x), e(x )) = i1 . Note that i1 =
n
deg(∂xi f (x)) − deg xi ,
i=1
which, by definition, coincides with the weight of the polynomial H(∇x Fα (x))(x, x ), and its highest monomials have constant coefficients, H0 (e(x), e(x )) = i∈I ei (x)ei1 −i (x ). From the above result and Theorem 5.3, we deduce the following assertion Corollary 6.2. (1) The relation (d − m)λd−m dm vr,s (λ) = r,s (λ) m∈I
holds, where r, s ∈ I and dki,j (λ) are the structure polynomials of the co-multiplication on the ring P [x]/(∇F (x)) in the basis e(x) = (ei1 (x), . . . , eiµ (x)), that is, for k ∈ I , we have Dxx ek (x) = k k k i,j∈I di,j (λ)ei (x)ej (x ) with di,j (λ) = dj,i (λ). (1) k (2) The relation vr,s (λ) = m∈I (d − m)λd−m dm r,s (0) holds, where r, s ∈ I and di,j (0) are the structure constants of the co-multiplication on the ring P [x]/(∇f (x)) in the chosen basis. 277
Consider the case n = 1 in more detail. Let f (x) = xd ; then the Milnor number is µ = d − 1, i the Euler operator has the form E = 1 − d1 x∂x , and F (x) = xd + d−2 i=0 λd−i x . These are simple singularities related to the reflection groups Aµ . By the formula in Example 5.2 for n = 1, we have (x ) H(F (x))(x, x ) = F (x)−F . Let us calculate the complete convolution matrix. The calculation x−x x of Dx E(F (x)) is carried out explicitly, d−2
F (x)E(F (x )) − F (x )E(F (x)) . (x − x ) i,j=0 Using the resulting matrix V (λ), we construct the vector fields i = dk=2 vi−2,k ∂λk and define the d−2 d−2−i i . Then generating function L(x) = i=0 x vd−i,d−j (λ)xj x = i
F (x)F (x ) − F (x)F (x ) 1 − F (x)F (x ). x − x d Let us compare the above result with the one in Sec. 4, namely, with formula (4.4) for v(ξ, ξ ). We carry out the change of variables (x, x ) = (1/ξ, 1/ξ ) and set L(x) = ξ 2−d (ξ) and F (x ) = ξ −d λ(ξ ). In accordance with (4.4), we have ξ −d ξ 2−d v(ξ, ξ ) on the right-hand side, where β = 1/d, α = 1 and λ1 = 0, and, on the left-hand side, we have ξ −d ξ 2−d [(ξ), λ(ξ )]. Corollary 6.3. To the reflection group Aµ , there corresponds the graded p. Lie algebra aµ (c, v) whose structure polynomials are given by formulas (4.4) with λ1 = 0, α = 1, and β = 1/(µ + 1). lim To the group A∞ = − →Aµ , there corresponds the graded p. Lie algebra a∞ (c, v) = a1,0 (c, v)|λ1 =0 . [L(x), F (x )] =
For the group Aµ , the expressions of λ2 , . . . , λµ+1 as functions of z1 , . . . , zµ are given by the relation F (x) = (x − (z1 + · · · + zµ )) µi=1 (x + zi ). Denote by Vµ the mapping thus defined. By construction, V (λ) = JVt µ (z)JVµ (z). Corollary 6.4. The p. Lie algebra a(JVt µ (z)) is strictly polynomial.
Proof. Set Z = {zij }, i, j = 1, . . . , µ. Note that JVt µ (z) = KZ , where K is a polynomial matrix with det K = const whose entries are symmetric functions of z. Thus, the p. Lie algebra a(JVt µ (z)) is left-equivalent to the p. Lie algebra a(Z). As is shown in Example 2.10, the p. Lie algebra a(Z) is strictly polynomial. The desired assertion follows from the transformation formula (3.2) for the Christoffel symbol. In the case of the groups Aµ , the formula for λi ∗ λj was obtained by D. B. Fuks (see [1]). expressions for λi as functions of z1 , . . . , zµ are For the type Bµ and Cµ singularities, the the same and are defined by the relation 1 + µi=1 λ2i x2i = µj=1 (1 + zj2 x2 ) (see [12], where explicit convolution formulas are also given). Let us describe the corresponding p. Lie algebra + : C[x, y] → C[x, y] by the with the help of the methods in Sec. 4. We define the operator πx,y 1 + formula πx,y (φ(x, y)) = 4 (φ(x, y) + φ(x, −y) + φ(−x, y) + φ(−x, −y)) for an arbitrary polynomial + ◦ π + ( · ) = π + ( · ). It is easy to verify that the operators φ(x, y) ∈ C[x, y]. Note that πx,y x,y x,y + + (1) + + + (1) + ◦ Rx,y ◦ πx,y ( · ) and Sx,y ( · ) = 4πx,y ◦ Sx,y ◦ πx,y ( · ), ( · ) = 4πx,y Rx,y (1)
(1)
where Rx,y and Sx,y are the operators in Theorem 4.2, satisfy the Jacobi identities (4.2). In verifying + (λ(x)) = λ(x 2 ), etc. at the very beginning. The pair the identities, it is convenient to set πx,y (2)
(2)
(2)
(2)
+ and S + (Rx,y , Sx,y ) does not lead to a new algebra since the range of the operators Rx,y ◦ πx,y x,y ◦ πx,y + . In the graded p. Lie algebra a+ (c, v) defined by the pair lies in the kernel of the operator πx,y + + (Rx,y , Sx,y ) according to formulas (4.1), the center Z is formed by the linear combinations q1 + + j∈N qj 2j−1 with coefficients qj ∈ C[λ1 , λ3 , . . . ]. The graded p. Lie algebra b∞ (c, v) = a (c, v)/Z lim lim corresponds to the inductive limits B∞ = − →Bµ and C∞ = − →Cµ . The quotient p. Lie algebras bµ (c, v) of the p. Lie algebra b∞ (c, v) modulo the relations {2k−2 = 0, λ2k = 0|k > µ} correspond to the reflection groups Bµ and Cµ .
278
For the groups of Dµ series, the basis is formed by λ2 , λ4 , . . . , λ2µ−2 , which are the same as in µ = z1 · · · zµ . The corresponding p. Lie algebra dµ ( the case of the group Bµ , and λ c, v) is obtained 2 and the left from the p. Lie algebra bµ (c, v) by the composition of the substitution λ2µ = λ µ µ )). transformation (3.1) defined by the matrix X = diag(1, . . . , 1, 1/(2λ For the other simple singularities, the situation is more complicated. Consider an Eµ -type singularity with Milnor number µ = 6. We have f (x, y) = y 3 + x4 . The Euler operator E = 1− 13 y∂y − 14 x∂x defines the grading deg x = 3, deg y = 4. Let us fix the basis of the local ring Q in the form e(x, y) = {yx2 , yx, x2 , y, x, 1}. Then F (x, y) = y 3 +x4 +λ2 yx2 +λ5 yx+λ6 x2 +λ8 y +λ9 x+λ12 . The set of weights of the basis monomials I = {10, 7, 6, 4, 3, 0} contains the number i1 − i = 10 − i along with i. The convolution matrix V (λ) is calculated using the formula in Example 5.2 for = k∈I vi,k ∂λk+2 , i ∈ I , associated with this matrix and n = 2. We construct the vector fields i define the generating function L(x, y) = i∈I e10−i (x, y)i . Set Fy (x, y)F (z, y)L(x, w) − Fw (x, w)F (z, w)L(x, y) (y − w)(x − z) Fw (z, w)F (x, w)L(z, y) − Fy (z, y)F (x, y)L(z, w) + , (y − w)(x − z) Fx (x, y)F (z, y)L(x, w) − Fx (x, w)F (z, w)L(x, y) η(x, y, z, w) = (y − w)(x − z) Fz (z, w)F (x, w)L(z, y) − Fz (z, y)F (x, y)L(z, w) . + (y − w)(x − z) ξ(x, y, z, w) =
The formula below gives the structure polynomials C = {cki,j (λ)}, i, j, k ∈ I , of the p. Lie algebra e6 (C, V ) corresponding to the reflection group E6 , [L(x, y), L(z, w)] = (∂x − ∂z )ξ(x, y, z, w) + (∂y − ∂w )η(x, y, z, w) mod ∇F (x, y), ∇F (z, w) . The complete convolution matrix for an E6 -type singularity was calculated in [12]. References 1. V. I. Arnold, Singularities of Caustics and Wave Fronts, Kluwer Academic Publishers Group, Dordrecht, 1990. 2. V. M. Buchstaber, “Semigroups of maps into groups, operator doubles and complex cobordisms,” Amer. Math. Soc. Transl. (2), Vol. 170, 1995, pp. 9–35. 3. V. M. Buchstaber, “Groups of polynomial transformations of a line, informal symplectic manifolds, and the Landweber–Novikov algebra,” Usp. Mat. Nauk, 54, No. 4, 161–162 (1999). 4. V. M. Buchstaber and D. V. Leykin, “Lie algebras associated with σ-functions and versal deformations,” Usp. Mat. Nauk, 57, No. 3, 145–146, 2002. 5. V. M. Buchstaber and D. V. Leykin, “Graded Lie algebras that define hyperelliptic sigma functions,” Dokl. RAS, 385, No. 5, 2002. 6. V. M. Buchstaber, V. Z. Enolskii, and D. V. Leykin, “Hyperelliptic Kleinian functions and applications,” In: Solitons, Geometry, and Topology: on the Crossroad, Amer. Math. Soc. Transl., Ser. 2, Vol. 179, Amer. Math. Soc., Providence, 1997, pp. 1–34. 7. V. M. Buchstaber, V. Z. Enolskii, and D. V. Leykin, “Kleinian functions, hyperelliptic Jacobians and applications,” Rev. Math. Math. Phys., 10, No. 2, 3–120 (1997). 8. B. Dubrovin, “Geometry of 2D topological field theories,” In: Integrable systems and quantum groups (Montecatini Terme, 1993), Lect. Notes in Math., Vol. 1620, Springer-Verlag, Berlin, 1996, pp. 120–348. 9. B. A. Dubrovin, S. P. Novikov, and A. T. Fomenko, Modern Geometry. Methods and applications, Part I, Graduated Text in Math., Vol. 93, Springer-Verlag, New York–Berlin, 1984. 10. B. A. Dubrovin and S. P. Novikov, “Hydrodynamics of weakly deformed soliton lattices. Differential geometry and Hamiltonian theory,” Usp. Mat. Nauk, 44, No. 6, 29–98 (1989). 279
11. B. Dubrovin and Y. Zang, “Frobenius manifold and Virasoro constraints,” Selecta Math. (N.S.), 5, No. 4, 423–466 (1999). 12. A. B. Givental, “Convolution of invariants of groups generated by reflections and associated with simple singularities of functions,” Funkts. Anal. Prilozhen., 14, No. 2, 4–14 (1980). 13. A. B. Givental, “Gromov–Witten invariants and quantization of quadratic Hamiltonians,” MMJ, 1, No. 4, 551–568 (2001). 14. M. V. Karasev and V. P. Maslov, Nonlinear Poisson Brackets. Geometry and Quantization, Amer. Math. Soc., Providence, RI, 1993. 15. S. P. Novikov, “Various doublings of Hopf algebras. Algebras of operators on quantum groups, complex cobordisms,” Usp. Mat. Nauk, 47, No. 5, 189–190 (1992). 16. K. Saito, “Theory of logarithmic differential forms and logarithmic vector fields,” J. Fac. Sci. Univ. Tokyo Sect. Math., 27, 263–291 (1980). 17. M. A. Semenov-Tian-Shansky, “Poisson Lie groups, quantum duality principle and the quantum double,” Contemp. Math., 175, 219–248 (1994). 18. K. Weierstraß, Zur Theorie der elliptischen Funktionen, Mathematische Werke, Vol. 2, Teubner, Berlin, 1894, pp. 245–255. 19. V. M. Zakalyukin, “Reconstructions of wave fronts depending on one parameter,” Funkts. Anal. Prilozhen., 10, No. 2, 69–70 (1976). 20. New Developments in Singularity Theory (D. Siersma, C. T. C. Wall, V. Zakalyukin, eds.), NATO Science Series II, Vol. 21., 2001. Steklov Mathematical Institute e-mail:
[email protected] Institute of Magnetism, Kiev, Ukraine e-mail:
[email protected],
[email protected]
Translated by D. V. Leykin
280
