The Erwin Schrodinger International Boltzmanngasse 9. Institute for Mathematical Physics A-1090 Wien, Austria

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1 ESI The Erwin Schrodinger International Boltzmanngasse 9 Institute for Mathematical Physics A-1090 Wien, Austria Hypersymmetries of Extremal Equations D.P. Zhelobenko Vienna, Preprint ESI 391 (1996) October 7, 1996 Supported by Federal Ministry of Science and Research, Austria Available via

2 HYPERSYMMETRIES OF EXTREMAL EQUATIONS D.P.Zhelobenko Independent University, Moscow Abstract. A general scheme is proposed, for a study of extremal equations associated with regular Cartan type algebras, in the sence of [Z1]. The theory of extremal equations is mentioned here as a generalization of highest weight theory for semisimple Lie algebras. We obtain a description of a hypersymmetry algebra T associated with a given algebra A, together with a study of corresponding functor from the category of A-modules to the category of T -modules. Some examples are considered concerning with classical and quantum (super)algebras, including certain equations of mathematical physics (Laplace, Dirac, Maxwell equations, etc.). x1. General denitions 1.1. Denition. Let A be an associative algebra with unity over a eld k. Let us x a subset A. Then for any left A-module V we may dene the corresponding null - subspace. V = v 2 V j v = 0 : (1:1) Here the symbol v = 0 means the system of linear equations av = 0 for any a 2. On the other hand, we set A = a 2 A j a 0 mod J ; (1:2) where J = A is a left ideal of the algebra A generated by the set. It is easy to verify that A V V : (1:3) Moreover, A is a largest subalgebra of A acting on V for any A-module V. It is clear that A contains the commutant C of the set (in A). Recall that the action of C in V is associated usually with a notion of symmetry of the system V = 0. Correspodingly, we say that A is the hypercommutant of the set (in A), and the action of A in V we associate with a notion of hypersymmetry of the system V = Noting that J is a two-sided ideal of the algebra A, we dene the corresponding quotient algebra A = A =J (1:4) Note also that J V = 0. Hence the action (1.3) induces the corresponding action of A in V. The algebra T = A is usually called translator algebra of the space V. 1

3 Of course, it is interesting haw many hypersymmetries exist for a given system V = 0.More general question is a study of the map : (A; V )! (A ; V ); (1:5) as a covariant functor from the category of A-modules to the category of A -modules. For example, let we know that the functor is exact. Then it maps any simple A{module V to simple A -module V. In this case, we have V = A v 0, for any 0 6= v 0 2 V.In any words, we may construct all solutions of the system v = 0 from single (nonzero) one. Another interesting example is when the functor is injective. In this case, the action of A in V denes the original A-module V uniquely, up to an isomorphism Example. Let H(n) be the space of harmonic polynomials in C[x 1 ; : : : ; x n ], i.e. the space of solutions of Laplace equation u = 0. Setting = fg we obtain H(n) = V in the notations of 1.1. It is natural to consider as a member of the Weyl algebra A = W (n) acting on C[x 1 ; : : : ; x n ] as the algebra of dierential operators with polynomial coecients. It is known [Z2] that the algebra A in this case is generated by the following system of i i (i = 1; : : : ; n); e = nx i=1 x i a i = x i? x i (the Euler operator); (i = 1; : : : ; n); where = 2e + n? 2. On the other hand, the commutant C in this case is generated by the operators a ij = x j? x i (1 i < j n) It is clear that C conserves all homogeneous components of the space H(n). Contrastly, the algebra A acts irreducible on H(n) for n 3. In other words, H(n) is a simply A -module. In particular, we have H(n) = A 1 =< a 1 ; : : : ; a n > 1; (1:6) wrere < a 1 ; : : : ; a n > means a subalgebra with unity generated by the elements a i ; (i = 1; : : : ; n). Hence we may construct any harmonic polinomials starting from a single one, namely u 0 (x) = 1. Moreover, it is easy to prove that the elements a i ; (i = 1; : : : ; n) are mutually permutable. In particular, we have H(n) =< a n > H(n? 1); (1:7) for n 3. The relation (1.7) may be considered as a fundamental step for a description of harmonic polynomials [Z2]. 2

4 1.4. Remark. Passing from A to its appropriate localization we may replace the elements a i to the quotients b i = a?1, namely b i = x i? x i?1 : (1:8) It is clear also that these elements may be concidered as operators in C[x 1 ; : : : ; x n ], for n 3. The elementary example 1.3 will be included later to more general scheme associated with contragredient associative algebras (x2). In particular, the results of 1.3 may be deduced also from general categorial approach given in 1.2. x2. CONTRAGREDIENT ALGEBRAS 2.1. Denition. Let A be an associative algebra with unity. We say that A is locally triangular [Z1] if the following conditions are fullled. () The algebra A is Z-graded, by a family of homogeneous subspaces A n (n 2 Z): A = M n2z A n ; A n A m A n+m ; (2:1) for any n; m 2 Z. () The algebra A possesses a "triangular decomposition" A = A? HA + ; (2:2) where H (resp. A + ; A? ) is a subalgebra with unity generated by homogeneous elements of zero (resp. positive, negative) degree. () The algebra H satises the following conditions: HA n = A n H; (2:3) for any n 2 Z. Example. Let G be a Z-graded Lie algebra. Then the universal enveloping algebra A = U(G) equipped with extended (uniquelly determined) Z-gradation is locally triangular Denition. Let A be locally triangular. We say that A is contragredient if it possesses an antiinvolution x 7! x 0 such that h 0 = h for any h 2 H, A 0 n = A?n for any n 2 Z. We will assume, for a simplicity, that the algebra A + (resp. A? ) is nitely generated, by a homogeneous family (resp. 0 ). Assume also that H has zero intersection with "augmentation subspace" N = 0 A + A. Then from (2.2) we nd A = H N, and we may dene the following analogue of a known Shapovalov form: '(x; y) = (x 0 y) H for x; y 2 A; (2:4) 3

5 where x 7! x H is the projection of x 2 A onto H parallel to N. It is clear that ' : A A! H is a symmetric H-bilinear form, with respect to right action of H on A. Moreover, the gradation (2.1) is orthogonal with respect to ', i.e. A n? A m for n 6= m. Recalling that J is contained in the kernel ker' of the form ', we may consider ' as a bilinear form on the "universal Verma module" M = A=J : (2:5) The algebra A is called nonsingular [Z1] if J = ker', i.e. the form ' is nonsingular on the space (2.5) Denition. Let A be contragredient. Then for any A-module V the subspace V (resp. the system v = 0) is called extremal, with respect to the given contragredient structure of A. 2.4.Example. Let A = W (n) be the Weyl algebra with canonical generators a i ; b i (i = 1; : : : ; n) and dening relations where ij is the Kronecker symbol. Setting [a i ; a j ] = [b i ; b j ] = 0; [a i ; b j ] = ij ; (2:6) dega i =?degb i = 1; a 0 i = b i ; (2:7) we nd that A is contragredient. As is known, A is nonsingular. The subspace V in this case is called vacuum subspace of V, with respect to usual interpretation of a i (resp. b i ) as annihilation (creation) operators. Remark that the universal Verma module M in this case is nothing but C[x 1 ; : : : ; x n ] with usual action a i i, b i = x i (i = 1; : : : ; n) Example. Let A = U(G) where G is a symmetrizable Kac-Moody algebra equipped with usual Z-gradation [Kc]. Then A is contragredient and nonsingular. The subspace V in this case is considered in the theory of weighted G-modules as a subspace of extremal (or highest) weigts. Alternatively, let A = U q (G) be the corresponding Drinfeld-Jimbo algebra, over the eld C(q) (see [Ks], for example).then A is nonsingular, and the subspace V conserves the previous interpretation. Remark. The Laplace equation u = 0 (see 1.3) is extremal with respect to the contragredient algebra A =< ; x 2 > U(sl(2)) Proposition [Z1]. Let A be nonsingular.then the triangular decomposition (2.2) is free, i.e. we have: A A? H A + ; (2:8) with respect to the usual map abc 7! abc (such that (2.8) is an isomorphism of vector spaces ). 4

6 Let us x homogeneous basis u i (i 2 N) of the vector space A +, setting u 0 = 1. Then the assertion (2.8) means that any element f 2 A may be written uniquely in the following form: f = X i;j u 0 i f iju j ; (2:9) with coecients f ij 2 H. Note that f H = f 00 is the projection onto H dened in 2.2. Let A be the set of innite formal series (2.9) satisfying the following niteness condition: the summation in (2.9) is bounded by the rule jn i? n j j const (depending on f), where n i = degu i Proposition [Z1]. Let A a be nonsingular. Then the space A is an algebra with respect to the multiplication of formal series (2.9). Remark. The niteness condition in 2.6 is essential. For example, let A = W (1) with canonical generators a; b and dening relation [a; b] = 1. Setting x = 1X n=0 a n ; y = 1X n=0 b n ; (2:10) and computing the zero element (xy) 0 we nd that a n b n n! mod Aa, hence the product xy is not dened. Denition. The algebra A is called the -extension of a given (nonsingular) algebra A. It is clear that A inherits (uniquely) the Z-gradation, the antiinvolution, and the projection x 7! x H of the algebra A. Hence the bilinear form ' also may be extended onto the algebra A. Remark. The construction of A is proposed in [Z3], in a frame of the theory of reductive Lie algebras A left A-module V is called admissible if the set acts on V locally nilpotently, i.e. for any v 2 V there exists n 1 such that n v = 0. It is clear that any element f 2 A may be considered as an operator in the category of admissible A-modules. In other words, any admissible A-module V admits a unique extension to the structure of A -module. Example. The universal Verma module (2.5) is admissible. x3. W T -algebras 3.1. Denition. Let A be contragredient, with triangular decomposition A = A? HA +. We say that A is a Weyl type (or W T -) algebra if H is a eld. Example. The Weyl algebra A = W (n) (see 2.4), as well as its usual quantum (super) analogues. The quantum boson algebra of M. Kashiwara, associated with a Cartan matrix [Ks], [Z2]. 5

7 More complicated examples of W T -algebras will be given in Theorem [Z1]. Let A be a nonsingular W T -algebra. Then we have: (i) There exists unique element p 2 A satisfying equations p = p 0 = 0 together with normalizing condition p H = 1. Moreover we have p 2 = p = p 0. (ii) The element p 2 A projects any A -module V onto V parallel to 0 V: In particular, we have V = V 0 V: (3:1) (iii) The category of admissible A-modules is semisimple (i.e. any admissible module is a direct sum of simple modules). Moreover it is easy to describe all the simple admissible A-modules, as a "twisted" analogues of universal Verma module [Z1]. Example. The projector p for A = W (n) is of the form p = p 1 p n, where p i is the corresponding projector for the subalgebra A i W (1) generated by a i ; b i. Setting A = W (1) with canonical generators a; b (see 2.7) we nd p = 1X n=0 (?1) n b n a n : (3:2) n! The category of admissible W (n)-modules admits only one (up to an automorphism) simple object, namely M = C[x 1 ; : : : ; x n ] (see 2.4). Remark. The last assertion is well known and may be considered as an algebraic version of the known euristic principle of the quantum mechanics, on the uniqueness of "annihilation" and "creation" operators Setting to a study of extremal equations v = 0; we will consider a general situation A B; where B is an appropriate extension of a given contragredient algebra A. Then we will consider the corresponding translator algebra as a part of the following B-module: T = T (B) = B =J ; (3:3) M = M(B) = B=J ; (3:4) where J = B : Hence we are interesting on a general properties of the functor = (B) : (B; V ) 7! (T; V ); (3:5) from the category of B-modules to the category of T -modules. A left B-module V is called admissible if it is admissible with respect to A (i.e. the set is locally nilpotent on V ). An embedding A B is called admissible if the module M = M(B) is admissible. We begin from the simplest case when A is a W T -algebra. 6

8 3.4. Theorem. Let A be a nonsingular W T -algebra, and let the embedding A B is admissible. Then we have: (i) T = pm. (ii) The algebra T is generated by any set of the form p where is a set of generators of the algebra A mod J. (iii)the functor ' is exact and injective. Remark. The denition of in (ii) means that the space M coincides with < > =J \ < > Example. Let A = W (n) B, where B is the extension of A by an elements a; b; as a subjects of the following dening conditions: [a; b] = 1; [a; W (n? 1)] = [b; W (n? 1)] = 0; a n a 2? 2aa n a + a 2 a n = 0; [a n ; b] = 0; with respect to the natural embedding W (n? 1) W (n): It is easy to prove that the algebra T = T (B) in this case is isomorphic to W (1); with canonical generators pa; b; where pa = a? b n a n a: x4. CT -algebras 4.1. Let AutH be the group of automorphisms of the algebra H (dened in 2.1), with the action h 7! h of 2 AutH onto an element h 2 H, such that (h ) = h for any 2 AutH. An element x 2 A is called weighted with respect to H if x 6= 0 and hx = xh for any h 2 H: (4:1) We denote by A the corresponding weight subspace (i.e. the set of all x 2 A satisfying (4.1)). The algebra A is called Q-graded, for Q AutH; if A = M 2Q A : (4:2) Remark that A A A for any ; 2 AutH: In particular, for any commutative subgroup Q 2 AutH we obtain an usual Q-gradation of the algebra A Denition. Let A be contragredient. We say that A is a Cartan type (or CT -) algebra [Z4] if the following conditions are fullled. () The algebra H is commutative and without zero divizors. () The algebra A possesses a Q-gradation (4.2), as a renement of given Z- gradation, where Q is an additive subgroup of AutH: We clain that the term "renement" in this context means that any A n (n 2 Z) is Q-graded, and the term "additive" means that we consider a commutative subgroup Q AutH with additive low of multiplication. 7

9 The algebra H in this case is called the Cartan subalgebra of A Let F = F ract H(= the eld of fractions of the algebra H). Remark that F inherits any automorphism of AutH. Setting A F = A H F (4:3) and using the rule (4.1) for an arbitrary element h 2 F, we equipe the space A F with structure of associative extension of the algebra A. It is easy to see that A F is a Cartan type algebra, with triangular decomposition A F = A? F A +, such that F is a Cartan subalgebra of A F. Hence A F is a W T -algebra. Setting A F = (A F ) = (A ) F, with respect to a natural indentication, we nd that there exists the unique "extremal projector" p 2 A F describing by the Theorem 3.2. For applying these results of the theory of A-modules, we need to dene a corresponding "specialization" procedure, for an appropriate category of A-modules Denition. Let A be CT -algebra, with Cartan subalgebra H, and let V be an arbitrary A H-bimodule. A vector v 2 V is called weighted, of a weight 2 AutH, if v 6= 0 and hv = vh for any h 2 H: (4:4) A module V is called Q-graded if we have V = M 2Q V ; (4:5) for the corresponding weight subspaces V (dened in (4.4)). A module V is called constructive if it is admissible (as the left A-module) and Q-graded in the sence (4.5). It is essential that for any left H-module K we may construct the corresponding left A-module V K setting V K = V H K: (4:6) In particular, if V is constructive and K is Q-graded then V K is admissible and Q-graded. Setting K = F = F ract H, we obtian the corresponding "F -envelope" V F of the module V. It is clear that V F may be naturally equipped by the structure of A F F - bimodule. Moreover using admissibility we nd that V F is equipped with the corresponding structure of A F F -bimodule. On the other hand, let P (H) be the set of characters of the algebra H, and let k() be one-dimensional H-module dened by a character 2 P (H). The left A-module V () = V H k() (4:7) is called a specialization of the module V at the point 2 P (H). Example. The universal Verma module M (see 2.2) is equipped with a natural structure of A H-bimodule (induced from A). The F -envelope M F is nothing but the universal Verma module for the algebra A F. The module M() is called a Verma module induced by the character 2 P (H). 8

10 4.5. Recall the usual notion of P -gradation for P P (H), namely V = M 2P V ; (4:8) where P is a H-module and V is a set of v 2 V satisfying the condition hv = (h)v for any h 2 H. In particular, if V is a left A-module then we nd the following rule of weights: A V V for any 2 Q; 2 P; (4:9) where 7! is the usual action of Q on P, namely ()(h) = (h ). Hence we may assume in (4.8) that P is a Q-module (i.e. a set with a given action of the group Q) Denition. Let A be a CT -algebra. We say that A is regular if the following conditions are fullled. (a) A is nonsigular. (b) H is regular, i.e. the set P (H) is total in H. In other words, the condition (b) means that the equalty (h) = 0 for any 2 P (H) implies h = 0. It is clear that the regularity of H is equivalent to the condition that the algebra H is isomorphic to a function algebra, on arbitrary total subset P 2 P (H). Namely any element h 2 H may be identied with a function h() = (h) for 2 P Example. The algebras A = U(G); U q (G) (see 2.5) are regular CT -algebras. In the rst case, we have H = U(H), where H is a Cartan subalgebra of G, such that H C[ 1 ; : : : ; n ] for n = rankg. x5. The category O 5.1. Denition. Let A be a regular CT -algebra. Let us x a total Q-module P P (H) (see 4.6). We dene the category O = O(P ) as a full category of admissible P -graded modules. As is known from the classical examples (see 2.5) the category O is not semisimple in general. We will propose a canonical way for contracting the category O to a smaller semisimple subcategory Recall that H may be considered as a function algebra over P (see 4.7). Hence for any ' 2 F we may dene the corresponding set of singularities (') 2 P. Moreover, writting any element f 2 A F f ij 2 F, we dene the set (f) = [ i;j as a formal series (2.9), with coecient (f ij ) (5:1) as a singular set of the element f. It is easy to see that this denition is actully not depend of a choice of decomposition (2.9). 9

11 In particular, let p 2 A F be the extremal projector dened in 4.3. The set (p) is called a singular set of the algebra A [Z1]. Correspondingly, we set P reg = P n (p): (5:2) 5.3. Denition. Let V be a module of the category O. A weighted vector v 2 V is called primitive if there exists a submodule N V such that v 62 N. In other words, the vector v denes an extremal vector of the quotient module V=N. Let P (V ) be the set of all 2 P such that V 6= 0, and let P 0 (V ) be the subset of the primitive vectors v 2 V, i.e. the weights of primitive vectors v 2 V. A module V is called regular [Z1] if P 0 (V ) P reg. We dene the category O reg as a full subcategory of O consisting of regular modules Theorem [Z1]. Let A be a regular CT -algebra. Then the category O reg is semisimple. Any simple object of the category O reg is of the following form: V N () = M()=N; (5:3) for 2 P reg, where M() is a Verma module of A (see 4.4) and N is a maximal submodule of M() non containing the weight subspace M(). Remark. Let e Q be the semigroup of Q generated by the elements 2 Q such that (A? ) 6= 0. Assume that e Q acts eectively on, i.e. the equality = for 2 e Q implies = 1. Then the denition (5.3) may be written in more convenient form, namely V () = M()=N(); (5:4) where N() is a largest submodule of M() non containing the one-dimensional subspace M() We return to the general problem of description of translator algebras (3.3), in the case when A is a regular CT -algebra. An embedding A B is called constructive if the space M = B=J is constructive A-module (see 4.4). A left B-module is V called regular if it is regular with respect to A (see 5.3). We denote by R the category of regular B-modules Theorem. Let A be a regular CT -algebra, and let A B be a constructive embedding. Then the functor : V 7! V is exact and injective on the category R Note that the algebra T (resp. the extremal space V ) inherits the weight gradation of B (resp. V ). In particular, let T 0 be the zero component of T (0 = neutral element of the additive group Q). Using the weight rule (4.9) we nd that the algebra T 0 acts on any homogeneuos component V. Hence we may consider the induced map () : (A; V ) 7! (T 0 ; V ) (5:5) 10

12 as a covariant functor from the category of A-modules to the category of T 0 -modules. Let R() be the full subcategory of regular B-modules satisfying the condition V 6= 0. Let S() be the subcategory of simple objects of R() Theorem. Let A be a regular CT -algebra. then the functor () is exact on the category R() and injective on the subcategory S(). The last assertion means that any A-module of the category S() is determined, up to an isomorphism, by the corresponding T 0 -module V Example. An embedding K G in the category of nite dimensional Lie algebras is called reductive if the adjoint action of K is reductive in G. In particular, the algebra K is reductive, and we have G = K M, where M is a complementary K-module. Setting A = U(K), B = U(G) we nd that A is a regular CT -algebra and the embedding A B is constructive. The corresponding translator algebra T = T (G; K) is called a Mickelsson algebra associated with a pair (G; K). The algebra T F in this case (F = F ract H admits a complete description [Z2] as a quadratic algebra generated by the set p(h [ M) = H [ pm, where H is a Cartan subalgebra of K (such that H = U(H)) and p 2 A F is the corresponding extremal projector. The Theorem 5.8 in this case is intensively used for a classication of (generalized) Harish-Chandra modules associated with a pair (G; K) [Z2] Example. Let G be a connected semisimple Lie group, and let N be the nilpotent radical of a xed Borel subgroup of G. The space X = G=N (5:6) as a left G-module (with respect to the action x 7! gx on G is called usually the principal ane space of the group G. Let F (G) (resp. F (X)) be the algebra of regular functions on the algebraic manifold G (resp. on X). Note that we have F (X) = f 2 F (G) j f = 0 ; (5:7) where G =Lie G is a set of generators of the nilpotent algebra N = Lie N, with respect to the right action of N on G. Hence we may consider (5.7) as an extremal subspace of the form F (G). Setting B = Diff G (= the algebra of dierential operators on G) we nd [Z2] that T = Diff X, i.e. the translator algebra T in this case coincides with the algebra of dierential operators on the space X. As a consequence of the Theorem 5.4 we nd that X is a simple Diff X-module. Moreover using the Theorem 3.4 one may obtain a complete description of the extended algebra T F (=F = F ract U(H) in the notation of 5.9) as a quadratic algebra, with appropriate generators and relations. x6. Concluding remarks 11

13 6.1. Let A be regular (see 4.6). We remark that the set (p) is closely connected with certain cohomological characteristics of the algebra A. Recall that P is a xed total Q-submodule of P (H) (see 4.6). We dene the following subsets of the set P. = the set of those 2 P for which the Verma module M() is not projective.? = the set of those 2 P for which there exists a nonzero homomorphism M()! M(), for some 2 Q.? 0 = the set of those 2 P for which the simple module V () (see 5.4) is a factor of M() for some 2 Q. Then it is easy to see (using general properties of the extremal projector p) that P reg and?;? 0 (p). It is known [Z1] that for some additional conditions of "strong regularity" we have the corresponding equalities, namely? =? 0 and P reg = = P n?: 6.2. The construction of 1.3 concerning with the Laplace equation u = 0 may be included in a series of examples concerning with similar equations of mathematical physics. For example, the Dirac equation on n variables x 1 ; : : : ; x n may be considered as an extremal equation with respect to contragredient algebra A = U(osp(2; 1)). Similarly, the Maxwell equations are connected with the algebra A = U(osp(2; 2)). Here osp(m; n) is a standard notation for a series of ortho-symplectic Lie superalgebras [Z2] A nonstandard situation, for non admissible modules, arises in the case of natural quantization of examples exposed in 6.2. Let us x a complex parameter q 6= 0; 1, and let C q (n) be a complex associative algebra with generators x 1 ; : : : ; x n and dening relations x i x j = qx j x i for 1 i < j n: (6:2) Let a i 2 AutC q (n) be dened by the rule a i (x j ) = q ij x j. We denote by x (resp. x 0 ) the left (resp. right) multiplication on x 2 C 1 (n) on the spac C q (n). Setting b i = a?1 1 a?1 i?1 a i+1 a n ; (6:3) we nd that x i = x 0 i b i. Setting = q? q?1 we dene the following quantum analogues of partial derivations on x 1 ; : : : ; x n i = (x 0 i )?1?1 (a i? a?1 i ): (6:4) Then we may dene the operators = nx i=1 q 2 i ;! = nx i=1 q 1?i x 2 i : (6:5) 12

14 It is easy to verify that the operators (6.5) generate the contragredient algebra A U q 2(sl(2)). Hence we may consider the quantum Laplace equation u = 0 as a particular case of extremal equations. It is essential that the module V = C q (n) is not admissible (i.e. the operator is not locally nilpotent on V ). On the other hand, the general scheme of 1.3 may be extended to the considered case. Namely, we set z n = (x n?!@ n )a?1 n ; (6:6) where belongs to the algebra H generated by the elements a i, a?1 i (i = 1; : : : ; n). It is easy to prove that there exists unique 2 H such that (6.6) is hypersymmetry of the equation u = 0. Let H q (n) be the space of all quantum harmonic polynomials (= solutions of the equation u = 0 on C q (n)). Then we have: H q (n) =< z n > H q (n? 1); (6:7) for n 3 [Z5]. Hence we nd a starting point for a developement of a theory of q-harmonic polynomials. A similar picture is hold for quantized Dirac and Maxwell equations. 13

15 References [Kc]. Kac V.G. Innite Dimensional Lie Algebras. Cambridge Univ. Press. Cambridge, [Ks]. Kashiwara M. Crystalizing the quantum analog of universal enveloping algebra. Duke Math. J. 63 (1991),no 2, 465{516. [Z1]. Zhelobenko D.P. Constructive modules and the problem of reductivity in the category O. Adv. in Math. Sci (to appear). [Z2]. Representations of reductive Lie Algebras (Russian). Nauka. Moscow, [Z3]. S-algebras and Verma modules on reductive Lie algebras. Dokl. AN SSSR, 283 (1985), no 6, 1306{1308. [Z4]. Cartan type algebras. Dokl. RAN, 339 (1994), 137{140. [Z5]. Harmonic polynomials on quantum spaces. Docl. RAN (to appear) [Z6]. On extremal equations associated with Cartan type algebras (to appear). 14