aa r X i v : . [ m a t h . C V ] F e b A(cid:30)ne and wrap algebras over o tonionsS.V. Ludkovsky10.12.2008Abstra tThe arti le is devoted to a(cid:30)ne and wrap algebras over quaternions and o tonions.Residues of fun tions of quaternion and o tonion variables are studied. They are usedfor onstru tion of su h algebras. Their stru ture is investigated.1 Introdu tionMention that Lie algebras are asso iative while the Cayley-Di kson algebras A r with r ≥ are non asso iative [1, 11, 13, 27℄. This indu es spe i(cid:28) features of algebras over A r . Inthis arti le the developed earlier te hnique of residues of meromorphi fun tions of Cayley-Di kson variables from [17, 18℄ is used. It is ne essary to mention that theory of fun tions ofCayley-Di kson variables di(cid:27)er drasti ally from that of omplex and ertainly has many spe- i(cid:28) features. It is useful not only for mathemati s, but also for theoreti al physi s, in ludingquantum me hani s, quantum (cid:28)eld theory, partial di(cid:27)erential equations, non ommutativegeometry, et . [4, 8, 9, 16, 26℄.Over the omplex (cid:28)eld loop algebras are known. They are de(cid:28)ned for meromorphi fun tions in an open domain U with one singular marked point z ∈ U ⊂ C [10℄. But inthe ase of Cayley-Di kson algebras it is possible to onsider meromorphi fun tion withsingularities in a losed subset W of odimension not less than 2. This W may already beof dimension greater than zero and winding around W may be in any plane ontaining R .Thus there are winding surfa es around W , so that the loop interpretation is lost. Therefore,analogs of loop algebras over A r are alled here wrap algebras.In this arti le a(cid:30)ne and wrap algebras over quaternions and o tonions are introdu edand studied. For this residues of fun tions of quaternion and o tonion variables are de(cid:28)nedand their properties are des ribed. They are used for onstru tion of su h algebras. Theirstru ture is investigated. All main results of the paper are obtained for the (cid:28)rst time.2 Algebras over o tonionsTo avoid misunderstandings we (cid:28)rst introdu e our notations and de(cid:28)nitions.1. Remark. Let V be a ve tor spa e over the Cayley-Di kson algebra A r . This means byour de(cid:28)nition that V = V i ⊕ ... ⊕ r − V i r − , where V, ..., r − V are pairwise isomorphi real ve tor spa es, while { i , ..., i r − } is the set of the standard generators of A r , i = 1 , i j = − , i i j = i j = i j i and i j i k = − i k i j for ea h ≤ j = k ≤ r − , ≤ r , A = H is the quaternion skew (cid:28)eld, A = O is the o tonion algebra. If in addition ea h j V is anasso iative real algebra and ( ax )( by ) = ( ab )( xy ) for ea h x, y ∈ V and all a, b ∈ A r , then we all V the super-algebra over A r . For short we all it also algebra.Let g be a Lie super-algebra over the Cayley-Di kson algebra A r , r ≥ . By our de(cid:28)nitionthis means that g = g i ⊕ g i ⊕ ... ⊕ r − g i r − , where g , ..., r − g are pairwise isomorphi real Lie algebras, { i , i , ..., i r − } are the standard generators of A r . The multipli ation in g is su h that (1) [ xi k , yi j ] = ( − χ ( k,j )+1 [ yi j , xi k ] for ea h pure states xi k ∈ g k i k and yi j ∈ g j i j , where χ ( k, j ) = 0 if either k = 0 or j = 0 or k = j , while χ ( k, j ) = 1 for k = j with k ≥ and j ≥ . The Ja obi identity takes the form: (2) [ xi k , [ yi j , zi s ]] + ( − ξ ( k,j,s ) [ yi j , [ zi s , xi k ]] + ( − ξ ( k,j,s )+ ξ ( j,s,k ) [ zi s , [ xi k , yi j ]] = 0 for ea h pure states xi k ∈ k g i k , yi j ∈ j g i j and zi s ∈ s g i s , where ξ ( k, j, s ) ∈ { , } is su hthat i k ( i j i s ) = ( − ξ ( k,j,s ) i j ( i s i k ) . Moreover, the multipli ation is real R bilinear: (3) [ ax, by ] = ab [ x, y ] for ea h x, y ∈ g and a, b ∈ R ; [ x + x , y ] = [ x , y ] + [ x , y ] and [ y, x + x ] = [ y, x ] + [ y, x ] for all x , x , y ∈ g .For short instead of Lie super-algebra we shall also write Lie algebra or (Lie) algebra over A r .Consider the family M at n ( A r ) of n × n matri es with entries in A r .If X and Y are ve tor spa es over A r then we say that a mapping A : X → Y is left A r linear if it is R linear and A ( z x ) = zA x for ea h pure ve tor x ∈ X and everyCayley-Di kson number z . The spa e of all left A r linear operators from X into Y we denoteby L al ( X, Y ) . Suppose that h is a ve tor spa e over A r . Then we denote by h ∗ l the spa e L al ( h , A r ) of all left A r linear fun tionals. Evidently h ∗ l is the A r ve tor spa e.We all ve tors v , ..., v n in a ve tor spa e X over A r with ≤ r ≤ ve tor independent,if for ea h non zero onstants a , ..., a n , b , ..., b n ea h ve tor { a v b } q (3) + ... + { a n v n b n } q n (3) is non zero for ea h asso iators { α ...α n } q ( n ) indi ating on an order of the multipli ation,when r = 3 . For r = 2 these asso iators an be dropped, sin e the quaternion skew (cid:28)eld isasso iative.Consider a n × m matrix B with entries in A r . If ≤ r ≤ , then A r is alternative andea h equation ax = b in it has the solution, whi h for a = 0 is x = a − b . Therefore, theGauss algorithm of redu ing a matrix to the step form is appli able in this ase. It is seenfrom the step form of the matrix B that the rank of B by rows and olumns is the same.Hen eforth we onsider ≤ r ≤ for algebras, if another is not spe i(cid:28)ed.For example, Conditions (1 − are satis(cid:28)ed, when g = g ⊗ A r and [ xa, yb ] = [ x, y ] ab and xa = ax for ea h a, b ∈ A r and all x, y ∈ g , where g is the real Lie algebra.2. De(cid:28)nitions. A matrix A is alled a generalized Cartan matrix if ( C a j,j = 2 for ea h j , 2 C a j,k are non positive integers for all j = k , ( C a j,k = 0 implies a k,j = 0 .By a realization of a matrix A ∈ M at n ( A r ) with ≤ r ≤ we all the triple ( h , Υ , Υ ∨ ) ,where h is a ve tor spa e over A r , Υ = { β , ..., β n } ⊂ h ∗ l , Υ ∨ = { γ , ..., γ n } ⊂ h so that1 Υ and Υ ∨ are A r ve tor independent;2 < γ k , β j > = a k,j ∈ A r , for all k, j = 1 , ..., n ;3 n − l = dim A r h − n , where l denotes the rank of A by rows over A r ;4 < a, b > := b ( a ) ∈ A r for any a ∈ h and b ∈ h ∗ l and < y, β j > ∈ R for ea h j = 1 , ..., n and y ∈ h ; < s yi s , k βi k > = ( − χ ( s,k ) < k βi k , s yi s > for all pure states.Two realizations ( h , Υ , Υ ∨ } and ( h , Υ , Υ ∨ } are alled isomorphi , if there exists anisomorphism of ve tor spa es φ : h → h so that φ (Υ ∨ ) = Υ ∨ and φ ∗ (Υ) = Υ , where < φ ∗ ( β ) , γ > := < β, φ ( γ ) > for all β ∈ Υ and γ ∈ Υ ∨ .3. Proposition. For ea h n × n matrix A there exists a unique up to an isomorphismits realization. Realizations of matri es A and B are isomorphi if and only if B an beobtained from A by inter hanging its rows or olumns.Proof. We an enumerate rows and olumns of the matrix and onsider that A = (cid:0) A A (cid:1) , where A is the l × n matrix of rank l . Compose the following matrix C = (cid:0) A A I n − l (cid:1) and put h = A n − lr , where I n denotes the unit n × n matrix. Take β j ( x ) = x j for j = 1 , ..., n , where x = ( x , ..., x n ) , x j ∈ A r , x ∈ A nr . Then as γ j take rows of the matrix C . This gives therealization of A .Vise versa, if for a realization ( h , Υ , Υ ∨ ) we omplete Υ up to a basis with the help of β n +1 , ..., β n − l ∈ h ∗ l . Then for suitable l × ( n − l ) matrix B and ( n − l ) × ( n − l ) matrix D ofrank ( n − l ) we get ( < γ j , β k > ) = (cid:0) A BA D (cid:1) . Adding to β n +1 suitable A r linear ombinations of β , ..., β l we an get, that B = 0 , where ≤ r ≤ , sin e the o tonion algebra is alternativeand ea h equation either ax = b or xa = b has the solution with non zero a either x = a − b or x = ba − respe tively. Then substitute β n +1 , ..., β n − l on their A r linear ombinationsthat to get D = I . This means the uniqueness of the realization up to inter hanging of rowsand olumns.If B is obtained from the matrix A by inter hanging its rows and olumns, then tworealizations are evidently isomorphi , sin e x ( x σ (1) , ..., x σ ( n ) ) is the isomorphism of A nr ,where σ is a transposition, that is a bije tive surje tive mapping of the set { , , ..., n } .3.1. Remark. Instead of using left A r linear fun tionals it is possible to use right A r linear fun tionals in the realization of a matrix. Denote by h ∗ r the spa e of right A r linearfun tionals on h , Υ r ⊂ h ∗ r . If ( h , Υ , Υ ∨ ) is the realization of the matrix A , then ( h ∗ r , Υ ∨ , Υ r ) is the realization of the transposed matrix A T .For two matri es A k , k = 1 , , and their realizations ( h k , Υ k , Υ ∨ k ) we an get the realizationof their dire t sum (cid:0) A A (cid:1) of these matri es ( h ⊕ h , Υ × { } ∪ { } × Υ , Υ ∨ × { } ∪ { } × Υ ∨ ) it is alled the dire t sum of their realizations.3he matrix A is alled de omposable, if after enumeration of its olumns and rows A de omposes into a non trivial dire t sum. Clearly that A an be presented as a dire t sumof inde omposable matri es and hen e its representation as a dire t sum of inde omposablerealizations. We all Υ the root basis, while Υ ∨ the dual root basis. Elements of Υ or Υ ∨ are alled simple roots or dual simple roots respe tively. We put Q = P nj =1 Z β j , Q + = P nj =1 Z + β j and all Q = Q ( A ) the root latti e, where Z + denotes the set of all positive integers.For β = P j k j β j ∈ Q the number ht β := P j k j is alled the hight of the element β .Introdu e the partial ordering ≥ on h ∗ l putting a ≥ b if a − b ∈ Q + .4. De(cid:28)nitions. Take the matrix A ∈ M at n ( A r ) and its realization ( h , Υ , Υ ∨ ) . Weintrodu e the auxiliary algebra η ( A ) over A r with generators e j , f j , j = 1 , ..., n and h ∈ h and with de(cid:28)ning relations (1) [ e k , f j ] = δ k,j γ j ; (2) [ h, h ′ ] = 0 for all h , h ′ ∈ h ; (3) [ h, e j ] = < β j , h > e j ; (4) [ h, f j ] = − < β j , h > f j for all j = 1 , ..., n and h ∈ h ; (5) [ au, bv ] = ( ab )[ u, v ] = [ u, v ]( ab ) for any a, b ∈ A r and u, v ∈ η ( A ) .Denote by η + and η − the subalgebra in η ( A ) generated by elements e ,..., e n and f ,..., f n respe tively.A matrix A is alled de omposable, if after an enumeration of its rows and olumns itbe omes the non-trivial dire t sum of two matri es.5. Theorem. Let η ( A ) be as in De(cid:28)nition 4. Then (1) η ( A ) = η − ⊕ h ⊕ η + is the dire t sum of ve tor spa es over A r ; (2) η + and η − are freely generated by e ,..., e n and f ,..., f n orrespondingly; (3) the mapping e j
7→ − f j , f j
7→ − e j for ea h j = 1 , ..., n , h
7→ − h for all h ∈ h has aunique extension up to an involution ω in the algebra η ( A ) ; (4) there exists the de omposition into root spa es relative to h of the form η ( A ) =( L β ∈ Q + η − β ) ⊕ h ⊕ ( ⊕ β ∈ Q + η β ) , where η β = { x ∈ η ( A ) : [ h, x ] = β ( h ) x ∀ h ∈ h } , more-over, dim A r η ( A ) < ∞ and η β ⊂ η ± for ± β ∈ Q + , β = 0 ; (5) among ideals of η ( A ) having the void interse tion with h a maximal ideal τ exists and τ = ( τ ∩ η − ) ⊕ ( τ ∩ η + ) is the dire t sum of ideals.Proof. Let V be a ve tor spa e over A r with a basis v , ..., v n so that v , ..., v n ∈ V and β ∈ h ∗ l . De(cid:28)ne the tensor algebra T ( V ) of V as onsisting of all elements { ( b x ) ... ( b k x k ) } q ( k ) and their (cid:28)nite sums, where b , ..., b k ∈ A r , x , ..., x k ∈ { v , ..., v n } , k ∈ N , q ( s ) is a ve tor indi ating on an order of the tensor multipli ation analogously to[17, 18℄. This means that T ( V ) = T ( V ) ⊕ T ( V ) ⊕ ... ⊕ T k ( V ) ⊕ ... , where T ( V ) = A r , T ( V ) = V , T k ( V ) = T k − ( V ) ⊗ V + T k − ⊗ T ( V ) + ... + V ⊗ T k − ( V ) , where the naturalequivalen e relation in T ( V ) is aused by the alternativity of the o tonion algebra O andasso iativity of the quaternion skew (cid:28)eld H . For produ ts of elements here the identitiesarising from the fa t that R is the enter Z ( A r ) of the algebra A r an be taken into a ount.4f r = 2 , then the quaternion skew (cid:28)eld H = A r is asso iative and in this parti ular aseasso iators {∗} q ( ∗ ) are not ne essary.De(cid:28)ne the a tion of generators of the algebra η ( A ) on the tensor algebra T ( V ) of thespa e V by (6) f j ( a ) = v j ⊗ a for ea h a ∈ T ( V ) ; (7) h (1) = < β, h > and by indu tion h ( v j ⊗ a ) = − < β j , h > v j ⊗ a + v j ⊗ h ( a ) for ea h a ∈ T s − ( V ) , j = 1 , ..., n , h ∈ η , h ∈ h , v j ∈ V ; (8) e j (1) = 0 and by indu tion e k ( v j ⊗ a ) = δ k,j γ k ( a ) + v j ⊗ e k ( a ) for every a ∈ T s − ( V ) , j = 1 , ..., n , v j ∈ V .Verify now that Conditions 4 (1 − are satis(cid:28)ed. Condition 4 (5) is satis(cid:28)ed, sin e R isthe enter of the algebra A r . Relation 4 (2) is satis(cid:28)ed for ea h h, h ′ ∈ h , sin e < β, h > ∈ R for ea h h ∈ h . In view of 1 (3) it is ful(cid:28)lled for all h, h ′ ∈ h as well. Take up to anisomorphism f j ∈ η − and e k ∈ η + . Then from (7 , we infer that ( e k f j − f j e k )( a ) = e k ( v j ⊗ a ) − v j ⊗ e k ( a ) = δ k,j γ k ( a ) + v j ⊗ e k ( a ) − v j ⊗ e k ( a ) = δ k,j γ k ( a ) , that is 4 (1) is satis(cid:28)ed.Then due to (6 , we get ( hf j − f j h )( a ) = h ( v j ⊗ a ) − v j ⊗ h ( a ) = − < β j , h > v j ⊗ a + v j ⊗ h ( a ) − v j ⊗ h ( a ) = − < β j , h > f j ( a ) this implies 2 (8) .Mention that from (6 , for s = 0 Condition 4 (3) follows. For s > we take a = v k ⊗ b and h ∈ h , where b ∈ T s − ( V ) , so we dedu e ( he j − e j h )( v k ⊗ b ) = h ( δ j,k γ j ( b )) + h ( v k ⊗ e j ( b )) − e j ( − < β k , h > v k ⊗ b + v k ⊗ h ( b )) = δ j,k γ j ( h ( b )) − < β k , h > v k ⊗ e j ( b ) + v k ⊗ he j ( b )+ < β k , h > δ j,k γ j ( b )+ < β k , h > v k ⊗ e j ( b ) − δ j,k γ j h ( b ) − v k ⊗ e j h ( b ) = < β j , h > δ j,k γ j ( b ) + v k ⊗ ( he j − e j h )( b ) . Apply the indu tionhypothesis to the se ond term, then we get 4 (3) for h , sin e e j ∈ η + . Taking into a ount2 (4) we get Condition 4 (3) for ea h h ∈ h .In view of Conditions 4 (1 − by indu tion prove that produ ts of elements from theset { e j , f j : j = 1 , ..., n ; h } belong to η − + h + η + . Consider an element u = u − + u h + u + ,where u − ∈ η − , u + ∈ η + , u h ∈ h . If u = 0 , then it a ts in T ( V ) su h that u (1) = u − (1)+ <λ, u h > = 0 , hen e < λ, u h > = 0 for ea h λ ∈ h ∗ l , onsequently, u h = 0 .Now use the mapping f j v j , whi h makes the algebra T ( V ) as the enveloping algebraof the algebra η − . The tensor algebra T ( V ) is free and is the universal enveloping algebra U ( η − ) of η − . This algebra is non-asso iative for r = 3 , but it is asso iative for r = 2 . Themapping u − u − (1) is the anoni al embedding η − ֒ → U ( η − ) . Thus u − = 0 and the (cid:28)rststatement is demonstrated.If A is an algebra over A r , then by [ A ] we denote the algebra obtained from A by supplyingit with the bra ket multipli ation as above.An algebra U ( L ) over the Cayley-Di kson algebra A r with the unit is alled the universalenveloping algebra for an algebra L over A r if there exists a homomorphism ǫ : L → [ U ] su hthat for ea h homomorphism g : L → [ A ] there exists a unique homomorphism f : U → A su h that g = f ◦ ǫ .By our onstru tion the algebra T ( V ) has the de omposition T ( V ) = T ( V ) i ⊕ ... ⊕ r − T ( V ) i r − , where T ( V ) , ..., r − T ( V ) are real pairwise isomorphi algebras. Sin e v , ..., v n ∈ V , then T ( V ) is the universal enveloping algebra of η − . By the Poin are-Birkho(cid:27)-Witt theorem and our hoi e of f , ..., f n above the algebra η − is free generatedby f , ..., f n . This makes Statement (2) evident.Applying the involution ω we get, that η + is free generated by e , ..., e n . Using 4 (3 , we get the de omposition η ± = L β ∈ Q + ,β =0 η ± β . At the same time there is the estimate dim A r η β ≤ (2 n ) | ht β | for the dimension of η β over A r . The latter implies (4).If τ is an ideal of η ( A ) , then it has the de omposition τ = τ i ⊕ ... ⊕ r − τ i r − withpairwise isomorphi real algebras τ, ..., r − τ . Over R it appears that τ is the ideal of η ( A ) . For real algebras (5) is known. The ideal τ has the de omposition τ = L β ( η β ∩ τ ) ,hen e τ = L β [( τ ∩ η − ∩ η β ) ⊕ ( τ ∩ η + ∩ η β )] = ( τ ∩ η − ) ⊕ ( τ ∩ η + ) is the dire t sum of A r ve torspa es. Then [ f j , τ ∩ η + ] ⊂ η + and [ e j , τ ∩ η + ] ⊂ η + , onsequently, [ η ( A ) , τ ∩ η + ] ⊂ τ ∩ η + ,also [ η ( A ) , τ ∩ η − ] ⊂ τ ∩ η − . Thus the sum in (5) is the dire t sum of ideals.6. Note. Consider the quotient algebra g ( A ) = η ( A ) /τ . It exists, sin e the pairwiseisomorphi real quotient algebras ( η ( A )) j /τ j exist for ea h j = 0 , , ..., r − . Ea h elementof g ( A ) is of the form b + τ , where b ∈ η ( A ) . The matrix A is alled the Cartan matrix of the(Lie super-) algebra g ( A ) , while n is alled the rank of g ( A ) . The olle tion ( g ( A ) , h , Υ , Υ ∨ ) we all the quadruplet asso iated with the matrix A .Two quadruplets ( g ( A ) k , h k , Υ k , Υ ∨ ,k ) , k = 1 , , are alled isomorphi , if there exists anisomorphism of algebras φ : g ( A ) → g ( A ) su h that φ ( h ) = h , φ ∗ (Υ ) = Υ , φ (Υ ∨ , ) =Υ ∨ , . We keep the notation e j , f j , h for their images in g ( A ) The subalgebra h is alled theCartan subalgebra and its elements e j , f j are alled Chevalley generators. They generatethe derivative sub-algebra whi h is by our de(cid:28)nition g ′ ( A ) so that g ′ ( A ) = g ′ ( A ) i ⊕ ... ⊕ r − g ′ ( A ) i r − with pairwise isomorphi real algebras g ′ ( A ) ⊕ ... ⊕ r − g ′ ( A ) , where g ′ ( A ) = [ g ( A ) , g ( A )] so that g ( A ) = g ′ ( A ) + h .Mention that g ( A ) = g ′ ( A ) if and only if rank ( A ) = n is maximal.Then we put h ′ = P nj =1 A r γ j , hen e g ′ ( A ) ∩ h = h ′ , g ′ ( A ) ∩ g β = g β , if β = 0 .In view of Theorem 5 there exists the de omposition into root spa es relative to h as g ( A ) = L β ∈ Q g β ,where g β = { x ∈ g ( A ) : [ h, x ] = β ( h ) x ∀ h ∈ h } is the root sub-spa e orresponding to β .Parti ularly, g = h . The number mult β := dim A r g β is alled the dimension of the element β . In view of Ÿ5 there is the estimate mult β ≤ (2 n ) | ht β | .An element β ∈ Q is alled a root, if β = 0 and mult β = 0 . If either β < or β > ,then the root is alled negative or positive respe tively. As usually denote by ∆ = ∆( A ) , ∆ − and ∆ + the family of all elements, of all positive elements, of all negative elements, then ∆ is the disjoint union of ∆ − and ∆ + .So we get that g β ⊂ η + if β > , g β ⊂ η − if β < . Moreover, g β is A r ve tor spa espanned either on ve tors [ ... [[ e j , e j ] , e j ] ...e j s ] with β + ... + β s = β for β > , or on ve tors [ ... [[ f j , f j ] , f j ] ...f j s ] with β + ... + β s = − β for β > , sin e e j , f j ∈ g . Thus g β j = A r e j β j > , g β j = A r f j for β j < , g sβ j = 0 for | s | > . Ea h root is either positive or negativeand this implies the following.7. Lemma. If β ∈ ∆ + \ { β j } , then ( β + Z β j ) ∩ ∆ ⊂ ∆ + .8. Remark. In a ordan e with Theorem 5 (3 , the ideal τ in η ( A ) is ω invariant.Thus it indu es the automorphism of the algebra g ( A ) alled its Chevalley involution. Thismeans that ω ( e j ) = − f j , ω ( f j ) = − e j , ω ( h ) = − h for ea h h ∈ h . Moreover, ω ( g β ) = g − β ,hen e mult β = mult ( − β ) and ∆ − = − ∆ + .9. De(cid:28)nitions. Suppose that G is a subset in the group ring Gi ⊕ R ... ⊕ R r − Gi r − ,where j G are pairwise isomorphi ommutative groups and ab = P j,k j a k b for all a, b ∈ G , a = ai + ... + r − ai r − , j a ∈ j G for every j . Let the multipli ation in G be su h that (1) there exists the unit element e = ei , where e is the unit element in G ; (2) for ea h a ∈ G there exists a − ∈ G with aa − = a − a = e , (3) if ≤ r ≤ , then the multipli ation is asso iative, if r = 3 , then the multipli ation isalternative (weakly asso iative) ( aa ) b = a ( ab ) , b ( aa ) = ( ba ) a for all a, b ∈ G . Then we all G for short the quasi- ommutative group instead of algebrai asso iative quasi- ommutativegroup for r = 2 or alternative quasi- ommutative group for r = 3 .Let G be a group. The de omposition V = L g ∈ G V g of a ve tor spa e V into the dire tsum of subspa es V g over A r is alled a G gradation. Elements of V g are alled homogeneousof degree g .An algebra A having the de omposition A = L g ∈ G A g into subspa es A g su h that A g A h ⊂ A gh for ea h g, h ∈ G is alled G graded. It is not supposed that A is asso iative ora Lie algebra.An algebra A over A r we all quasi- ommutative, if its de omposition A = Ai ⊕ ... ⊕ r − Ai r − with pairwise isomorphi real algebras A ,..., r − A is su h that ea h j A is ommutative.10. Proposition. Suppose that h is a quasi- ommutative Lie algebra over A r , while V is its diagonalizable module, whi h means (1) V = L b ∈ Λ V b , Λ ⊂ h ∗ l , where V b := { v ∈ V : f ( k v ) = b ( f ) k v ∀ f ∈ h , ∀ k = 0 , ..., r − } , Λ is a multipli ativesubgroup in h ∗ l .Then ea h submodule U in V is graded relative to Gradation (1) .Proof. Consider an element v ∈ V and write it in the form v = P nj =1 v j with v j ∈ V b j and v j = v j i + ... + r − v j i r − with k v j ∈ k V j . There exists f ∈ h so that the numbers b j ( f ) are pairwise distin t. Then for v ∈ U we get f p ( k v ) = P nj =1 b j ( f ) p k v j ∈ U for ea h p = 0 , , ..., n − and k = 0 , ..., r − . We get thesystem of linear algebrai equations over o tonions with the non-degenerate matrix. Usingthe gaussian algorithm we get, that all elements k v j belong to k U and hen e v j ∈ U for ea h j . The produ t of left linear fun tionals is also left linear. Mention that ( k vi k )( j vi j ) ∈ s V i s for pure states k vi k ∈ k V i k , where i s = i k i j . Thus U = L b ∈ Λ ( U ∩ V b ) and this means thatthe submodule U is h graded also.11. Note. If V is a graded ve tor spa e, then it an be supplied with the formal (or in7nother words dire t sum) topology having a fundamental system of neighborhoods of zero onsisting of all open subsets in V F , where V F := L g ∈ G \ F V g , F is a (cid:28)nite subset in G ,when ea h V g is a topologi al ve tor spa e. The ompletion of V relative to su h topologyis Q g ∈ G V g and the latter spa e is alled its formal (or dire t produ t) ompletion.A gradation an be introdu ed by ounterpoising to generators a j elements b j ∈ G andputting by the de(cid:28)nition that their degree is deg ( a j ) = b j . This de(cid:28)nes a unique G gradedLie algebra g if and only if the ideal of relations between a j is also G graded. This is the ase for a free system of generators a j , j = 1 , ..., s .Parti ularly let s , ..., s n be integers and put deg e j = − deg f j = s j , deg h = 0 ,then we get the Z graded algebra g ( A ) = L j ∈ Z g j ( s ) , where s = ( s , ..., s n ) , g j ( s ) = L β = P m k m β m ∈ Q ; P m k m s m = j g β . Clearly if s j > for ea h j , then g ( s ) = h and dim A r g j ( s ) < ∞ . If s = (1 , ...,
1) = , then g j ( ) = L β : ht ( β )= j g β and g ( ) = h , g − ( ) = P m A r f m , g ( ) = P m A r e m so that η ± = L j ≥ g ± j ( ) .12. Lemma. Let a ∈ η + and either [ a, f j ] = 0 for ea h j = 1 , ..., n or [ a, e j ] = 0 for all j = 1 , ..., n , then a = 0 .Proof. Suppose that a ∈ η + su h that [ a, g − ( )] = 0 , then P k,m ≥ ( ad g ( )) k ( ad h ) m a is the subspa e in η + over A r invariant relative to ad g ( ) , ad h and ad g − ( ) . Thus if a = 0 we will get the ideal in g ( A ) whi h has with h the trivial interse tion. This ontradi tsthe de(cid:28)nition of g ( A ) .13. Proposition. The enter of the algebra g ( A ) or g ′ ( A ) onsidered over the real (cid:28)eld R is Z = { h ∈ h : < β j , h > = 0 ∀ j = 1 , ..., n } , moreover, dim R Z = n − l .Proof. The enter of the Cayley-Di kson algebra A r is the (cid:28)eld R of real numbers. Weuse the relations 1 (1 − . Suppose that p ∈ Z and p = P j p j its de omposition relative tothe main gradation. Then [ p, g − ( )] = 0 implies [ p j , g − ( )] = 0 for ea h j > . By Lemma12 we get that p j = 0 for all j > . From [ p, g ( )] = 0 we infer that p j = 0 for ea h j < .Thus p ∈ h and we have [ p, e j ] = < β j , p > e j = 0 , hen e < β j , p > = 0 for ea h j = 1 , ..., n .Vi e versa if p ∈ h and < β j , p > = 0 for ea h j = 1 , ..., n , then p ommutes with allChevalley generators and hen e belongs to the enter of the algebra. Mention also that Z ⊂ h , sin e in the ontrary ase dim R Z > n − l and Υ will not be the A r ve torindependent set.14. Proposition. If a matrix A is in M at n ( A r ) , then the algebra g A r +1 ( A ) over A r +1 is the smashed produ t of two opies of the algebra g A r ( A ) over A r for ≤ r ≤ .Proof. Sin e a j,k ∈ A r for ea h j, k , then using onstant multipliers we an hooseelements β j and γ j belonging to ( h A r ) ∗ l and h A r respe tively. The Cayley-Di kson algebra A r +1 is the smashed produ t of two opies of A r with the help of the doubling pro edure[1℄. Therefore, ea h element v ∈ η A r +1 ( A ) an be de omposed in the form v + vi r with v and v ∈ η A r ( A ) , where i r is the doubling generator of A r +1 from A r . This gives thede omposition of η A r +1 ( A ) as the ve tor spa e over A r into the dire t sum η A r ( A ) ⊕ η A r ( A ) i r ,where i , ..., i r +1 − are the standard generators of A r +1 .On the other hand ea h pure state of η A r +1 ( A ) is either a pure state of η A r ( A ) or of8 A r ( A ) i r so that all multipli ation rules are the same in η A r +1 ( A ) and in η A r ( A ) ⊕ η A r ( A ) i r .Using the R bilinearity of the multipli ation and the de omposition of ea h element into asum of pure states j vi j with j v is belonging to the real algebra j η A r +1 ( A ) , j = 0 , , ..., r +1 − , we get that the algebra η A r +1 ( A ) over A r +1 is the smashed produ t η A r ( A ) ⊗ s η A r ( A ) oftwo opies of η A r ( A ) over A r with the help of the doubling generator i r .The maximal ideal τ in η A r +1 ( A ) having with h A r +1 the trivial interse tion possesses thesame de omposition τ A r +1 = τ A r ⊗ s τ A r so that real algebras j τ A r +1 and k τ A r are pairwiseisomorphi for ea h j, k . This implies that j g A r +1 ( A ) = j η A r +1 ( A ) / j τ A r +1 for ea h j .Therefore, g A r +1 ( A ) = η A r +1 ( A ) /τ A r +1 = ( η A r ( A ) /τ A r ) ⊗ s ( η A r ( A ) /τ A r ) = g A r ( A ) ⊗ s g A r ( A ) .15. Lemma. Let J and J be two noninterse ting subsets in { , ..., n } so that a j,k = a k,j = 0 when j ∈ J and k ∈ J . Suppose that δ s = P j ∈ J s k j,s β j , s = 1 , and β = δ + δ isa root of algebra g ( A ) over A r , ≤ r ≤ . Then either δ or δ is zero.Proof. Let j ∈ J and k ∈ J . Then [ γ j , e k ] = 0 , [ γ k , e j ] = 0 , [ e j , f k ] = 0 , [ e k , f j ] = 0 , onsequently, [ e j , e k ] = 0 and [ f j , f k ] = 0 in a ordan e with Lemma 12. Thus algebras g ( A ) and g ( A ) ommute, where g s ( A ) denotes the algebra generated by { e j , f j : j ∈ J s } . Thealgebra g β ( A ) is ontained in the subalgebra generated by g ( A ) and g ( A ) , hen e g β ( A ) is ontained either in g ( A ) or in g ( A ) .16. Proposition. The algebra g ( A ) over A r with ≤ r ≤ is simple if and only if (1) rank A r A = n and (2) for ea h ≤ j, k ≤ n there exist indi es j , ..., j s so that ( ... ( a j,j a j ,j ) ... ) a j s ,k = 0 ,where n is the order of a matrix A .Proof. If either (1) or (2) is not satis(cid:28)ed, then by Lemma 15 g ( A ) will ontain a nontrivial proper ideal ξ , ξ = 0 , ξ = g ( A ) and g ( A ) will not be simple.If Conditions (1 , are satis(cid:28)ed and δ is a non zero ideal in g ( A ) , then δ ontains a nonzero element h ∈ h . Sin e rank A r = n , then by Proposition 13 Z = 0 , hen e [ h, e j ] = ae j = 0 for some j and inevitably e j ∈ δ and γ j = [ e j , f j ] ∈ δ . From Condition (2) and from the fa tthat the norm in the o tonion algebra is multipli ative we get that e j , f j , γ j ∈ δ for all j . Onthe other hand, Condition (1) implies that h is the A r ve tor span of elements γ j . Thus weget δ = g ( A ) .17. De(cid:28)nition. A n × n matrix with entries in A r is alled symmetrizable, if thereexists a diagonal matrix D = ( d , ..., d n ) , d j = 0 for ea h j and a symmetri matrix B , thatis B T = B , so that (1) A = DB ,where B T denotes the transposed matrix B . In this ase B is alled the symmetrization of A and an algebra g ( A ) is alled symmetrizable.For a symmetrizable matrix A with a given de omposition (1) and its realization ( h , Υ , Υ ∨ ) we (cid:28)x a omplemented A r ve tor spa e h to h = span A r { γ j : j } in h a symmetri A r valuedform ( ∗|∗ ) on h su h that (2) it is left A r linear by the left argument and right A r linear by the right argumentsu h that (3) ( γ j | h ) = d j < β j , h > for ea h h ∈ h , 9
4) ( h | h ) = 0 for all h ∈ h and h ∈ h , (5) ( x | y ) ∈ R for ea h x, y ∈ g , (6) ( γ j | γ k ) = d j b j,k d k with b j,k ∈ R for ea h j and k .18. Proposition. 1. The kernel of the restri tion of the form ( ∗|∗ ) on h oin ides with Z . 2. The form ( ∗|∗ ) is non degenerate on h .Proof. The (cid:28)rst statement follows from Proposition 13. To prove the se ond statement onsider the ondition P j c j γ j | h ) for all h ∈ h . Sin e ( P j c j γ j | h ) = < P j c j d j β j , h > for ea h h ∈ h , then P j c j d j β j = 0 , onsequently, c j = 0 for ea h j = 1 , ..., n .19. Note. The form ( ∗|∗ ) is non degenerate and there is the isomorphism ν : h → h ∗ l having the natural extension up to the isomorphism ν : h → h ∗ l and ν r : h → h ∗ r so that (1) < ν ( h ) , p > = ( h | p ) for all h, p ∈ h and with the indu ed form ( ∗|∗ ) on h ∗ l so that (2) ( β j | β k ) = b j,k = d − k a k,j for ea h j, k and (3) < p, ν r ( h ) > = ( p | h ) for ea h h, p ∈ h and with the indu ed form ( ∗|∗ ) on h ∗ r . Clearly (4) ν ( γ j ) = d j β j and (5) ν r ( γ j ) = β j d j .20. Theorem. Let g ( A ) be a symmetrizable Lie algebra and let 17 (1) be its pres ribedde omposition. Then there exists its non degenerate symmetri A r valued form ( ∗|∗ ) on g ( A ) satisfying Conditions 17 (2 − and (1) this form ( ∗|∗ ) is invariant on g ( A ) , that is ([ x, y ] | z ) = ( x | [ y, z ]) for all x , y and z ∈ g ( A ) , (2) ( g β , g δ ) = 0 if β + δ = 0 , (3) the restri tion ( ∗|∗ ) | g β + g − β is non degenerate for β = 0 , (4) [ x, y ] = ( x | y ) ν − ( β ) for ea h x ∈ g β and y ∈ g − β , β ∈ ∆ .Proof. Take the prin ipal Z gradation g ( A ) = L j ∈ Z g j and put g ( m ) = L mj = − m g j for m = 0 , , , ... . De(cid:28)ne the form ( ∗|∗ ) on g (0) = h with the help of 17 (2 − . We extend iton g (1) as (5) ( e j | f k ) = δ j,k d j , j, k = 1 , ..., n , (6) ( g | g ± ) = 0 , ( g | g ) = 0 , ( g − | g − ) = 0 .In view of 17 (2) ([ e j | f k ] | h ) = ( e j | [ f k , h ]) for ea h h ∈ h , or equivalently δ j,k ( γ j | h ) = δ j,k d j < β j , h > , where we an take e j , f k ∈ g . Then the form ( ∗|∗ ) satis(cid:28)es Condition (1) ,when both elements [ x, y ] and [ y, z ] belong to g (1) . Now we an extend the form ( ∗|∗ ) dueto Rule 17 (2) on g ( m ) with the help of indu tion by m ≥ so that ( g j | g k ) = 0 , if | j | , | k | ≤ m and j + k = 0 . Thus Condition (1) is satis(cid:28)ed when [ x, y ] and [ y, z ] belong to g ( m ) .Suppose that this extension is done for g ( m − , then we should de(cid:28)ne ( x | y ) for x ∈ g ± m and y ∈ g ∓ m only. Write y in the form y = P j [ u j , v j ] , where u j and v j are homogeneouselements of non zero degree, belonging to g ( m − . Then [ x, u j ] ∈ g ( m − and for u j , v j ∈ g ( m − we put (6) ( x | y ) = P j ([ x, u j ] | v j ) . Then by Rule 17 (2) we extend it on g ( m ) . It remains to verifythat the de(cid:28)nition of Formula (6) is orre t on g . Suppose that x j ∈ g j for j ∈ Z , take k, j, s, t ∈ Z with | k + j | = | s + t | = m and k + j + s + t = 0 ; | k | , | j | , | s | and | t | < m . Then10
7) ([[ x k , x j ] , x s ] | x t ) = ( x k | [ x j , [ x s , x t ]]) ,sin e ([[ x k , x j ] , x s ] | x t ) = (([[ x k , x s ] , x j ] | x t ) − (([[ x j , x s ] , x k ] | x t )= ([ x k , x s ] | [ x j , x t ]) + ( x k | [[ x j , x s ] , x t ]) = ( x k | [ x s , [ x j , x t ]] + [[ x j , x s ] , x t ]) = ( x k | [ x j , [ x s , x t ]]) .If now x = P k [ p k , w k ] , then from (6 , we get ( x | y ) = P k ([ x, u k ] | v k ) = P k ( p k | [ w k , y ]) .Hen e this value does not depend on a hoi e of expressions for x and y .From de(cid:28)nitions it follows that Condition (1) is ful(cid:28)lled as soon as [ x, y ] and [ y, z ] belongto g ( m ) . Thus we have onstru ted the form ( ∗|∗ ) on g satisfying Conditions (1 , . Itsrestri tion on h is non degenerate due to Proposition 18.The form ( ∗|∗ ) also satis(cid:28)es (3) , sin e h ∈ h , x ∈ g α and y ∈ g β . By the invarian eproperty and 17 (2) we infer h, x ] | y ) + ( x | [ h, y ]) = ( < β, h > + < γ, h > )( x | y ) .For x ∈ g α and y ∈ g − α with α ∈ ∆ and h ∈ h we get ([ x, y ] − ( x | y ) ν − ( α ) | h ) = ( x | [ y, h ]) − ( x | y ) < α, h > = 0 . Thus (5) follows from (2) . From (2 , , it follows that the bilinear form ( ∗|∗ ) is symmetri .If (4) is not satis(cid:28)ed, then by (3) the form ( ∗|∗ ) is degenerate. Put t = Ker ( ∗|∗ ) . Thisis ideal and by (2) we have t ∩ h = 0 . But this ontradi ts to the de(cid:28)nition of g ( A ) .21. Note. Suppose that A = ( a k,j ) is the symmetrizable generalized Cartan matrix.Equation 17 (1) is equivalent with the system of homogeneous linear equations and inequali-ties d j = 0 so that diag ( d − , ..., d − n ) A = ( b j,k ) with b j,k = b k,j for all j, k , sin e the o tonionalgebra O = A is alternative. But in the generalized Cartan matrix all entries are integer,hen e solutions an be hosen in the (cid:28)eld Q of rational numbers. Thus we an hoose itsde omposition 17 (1) with d j > so that d j and b j,k are rational numbers.We an suppose that A is inde omposable. In view of Proposition 16 for ea h < j ≤ n there exists a sequen e j < j < ... < j k − < j k = j su h that a j k ,j k +1 < . Therefore, a j s ,j s +1 d j s +1 = a j s +1 ,j s d j s for ea h s = 1 , ..., k − . Hen e d j d > for ea h j . Therefore,we an hoose d j > for ea h j . If A is inde omposable, then the matrix diag ( d , ..., d n ) isde(cid:28)ned by 17 (1) uniquely up to the multipli ation on a onstant. Now we (cid:28)x a symmetri form satisfying Conditions 17 (2 − related with the de omposition 17 (1) . In a ordan ewith Lemma 18 we get (1) ( β j | β j ) > for ea h j = 1 , ..., n ; (2) ( β j | β k ) ≤ for ea h j = k ; (3) γ j = 2 ν − ( β j ) / ( β j | β j ) . Thus a j,k = 2( β j | β k ) / ( β j | β j ) for ea h j, k . Then we take theextension of the form ( ∗|∗ ) from h onto g ( A ) in a ordan e with Theorem 20. This form willbe alled the standard invariant form.If hoose for a root β dual bases { e jβ } and { e j − β } in g β and in g − β so that ( e jβ | e k − β ) = δ j,k for ea h j, k = 1 , ..., mult β , then for x ∈ g β and y ∈ g − β the identity (4) ( x | y ) = P j ( x | e j − β )( y | e jβ ) is satis(cid:28)ed.22. Lemma. If α, β ∈ ∆ and z ∈ g β − α , then in g ( A ) ⊗ g ( A ) the identity (1) P s e s − α ⊗ [ z, e sα ] = P s [ e s − β , z ] ⊗ e sβ is satis(cid:28)ed.11roof. We de(cid:28)ne the form ( ∗|∗ ) satisfying 17 (2 − by the formula ( k xi k ⊗ j yi j | s wi s ⊗ t zi t ) = ( − ζ ( k,j,s,t ) ( k xi k | s wi s )( j yi j | t zi t ) for pure states and extend it by R bi-linearityon g ( A ) ⊗ g ( A ) , where ζ ( k, j, s, t ) ∈ { , } is su h that ( i k i j )( i s i t ) = ( − ζ ( k,j,s,t ) ( i k i s )( i j i t ) .Then we take e ∈ g α and f ∈ g − β . Therefore, P s ( e s − α ⊗ [ z, e sα ] | e ⊗ f ) = P s ( e s − α | e )([ z, e sα ] | f ) = P s ( e s − α | e )( e sα | [ f, z ]) = ( e | [ f, z ]) also P s ([ e s − β , z ] ⊗ e sβ | e ⊗ f ) = P s ( e s − β | [ z, e ])( e sβ | f ) = ([ z, e ] | f ) by Theorem 20 and Formula 21 (4) . In view of 17 (2) the last two formulas imply (1) , sin ethe o tonion algebra is alternative.23. Corollary. Let onditions of Lemma 22 be satis(cid:28)ed, then (1) P s [ e s − α , [ z, e sα ]] = P s [[ e s − β , z ] , e sβ ] in g ( A ) ; (2) P s e s − α [ z, e sα ] = P s [ e s − β , z ] e sβ in U ( g ( A )) .Proof. This follows from the appli ation of mappings g ( A ) ⊗ g ( A ) ∋ x ⊗ y [ x, y ] ∈ g ( A ) and g ( A ) ⊗ g ( A ) ∋ x ⊗ y xy ∈ U ( g ( A )) to Formula 22 (1) .24. Remark. Let g ( A ) be a Lie algebra orresponding to a matrix A and let h be itsCartan subalgebra, g = L β g β be its de omposition into root subspa es relative to h . A g ( A ) module (or g ′ ( A ) module) V is alled bounded if for ea h v ∈ V we have g β ( v ) = 0 forall roots besides a (cid:28)nite number of positive roots β .Introdu e the fun tional ρ ∈ h ∗ r by the formula: (1) < ρ, γ j > = a jj / , j = 1 , ..., n , where γ j ∈ Υ ∨ . If rank A < n , then this does notde(cid:28)ne ρ uniquely, so we an take any fun tional satisfying these relations. In a ordan ewith Formulas 19 (2 , we get (2) ( ρ | β j ) = ( β j | β j ) / for ea h j = 1 , ...., n .For ea h positive root β we hoose a basis { e jβ } in the spa e g β and take the dual basis { e j − β } in g − β so that they belong to g β and g − β orrespondingly. Then we put (3) Ω = 2 P β ∈ ∆ + P j e j − β e jβ and this operator does not depend on a hoi e of the dualbasis. For ea h v ∈ V only (cid:28)nite number of additives e j − β e jβ ( v ) is non zero, hen e Ω is de(cid:28)ned orre tly on V . Let u , u , ... and u , u , ... be dual bases of the subalgebra h belonging to h . We de(cid:28)ne the generalized Casimir operator as (4) Ω = 2 ν − r ( ρ ) + P j u j u j + Ω . Choose ρ ∈ h ∗ r .We onsider now the de omposition into root subspa es for U ( g ( A )) relative to h : U ( g ( A )) = L β ∈ Q U β , where U β = { x ∈ U ( g ( A )) : [ h, x ] = < β, h > x ∀ h ∈ h } and put U ′ β = U ( g ′ ( A )) ∩ U β so that U ( g ′ ( A )) = L β U ′ β .25. Theorem. Let g ( A ) be a symmetrizable algebra over A r , ≤ r ≤ . If V is abounded g ′ ( A ) module and u ∈ U ′ β , then (1) [Ω , u ] = − (2( ρ | β ) + ( β | β ) + 2 ν − r ( β )) u .If V is a bounded g ′ ( A ) module, then (2) Ω ommutes with the a tion of g ( A ) on V .Proof. Sin e λ = P j < λ, u j > ν r ( u j ) = P j < λ, u j > ν ( u j ) for ea h λ ∈ g ( A ) , then (3) P j < λ, u j >< u j , µ > = ( λ | µ ) . Moreover,12
4) [ P j u j u j , x ] = (( β | β ) + 2 ν − r ( β )) x for ea h x ∈ g β , sin e [ P j u j u j , x ] = P j < β, u j > xu j + P j u j < u j , β > x = P j < β, u j >< u j , β > x + ( P j u j < u j , β > + < β, u j > u j ) x .Thus Statement (2) follows from (1) and Formula (4) .Take now elements e β j and e − β j with j = 1 , ..., n , whi h generate the algebra g ′ ( A ) . Ifeither u = e β j or u = e − β j , then due to Lemmas 7 and 23 we infer: [Ω , e β j ] = 2 P β ∈ ∆ + P s ([ e s − β , e β j ] e sβ + e s − β [ e sβ , e β j ]= 2[ e − β j , e β j ] e β j + 2 P β ∈ ∆ + \{ β j } ( P s [ e s − β , e β j ] e sβ + P s e s − β + β j [ e sβ − β j , e β j ])= − ν − r ( β j ) e β j = − β j | β j ) e β j − e β j ν − r ( β j ) .Analogously we have [Ω , e − β j ] = 2 e − β j [ e β j , e − β j ] = 2 e − β j ν − r ( β j ) . Thus we have got (1) for u = e β j and u = e − β j . If u ∈ U ′ α and v ∈ U ′ β , then [Ω , uv ] = [Ω , u ] v + u [Ω , v ]= −{ (2( ρ | α ) + ( α | α ) + 2 ν − r ( α )) u } v − u { (2( ρ | β ) + ( β | β ) + 2 ν − r ( β )) v } = − ((2( ρ | α ) + ( α | α ) + 2 ν − r ( α ) + 2( α | β ) + 2( ρ | β ) + ( β | β ) + 2 ν − r ( β )) uv = (2( ρ | α + β ) + ( α + β | α + β ) + 2 ν − r ( α + β )) uv ,sin e α, β ∈ h ∗ , e β ∈ g β , ρ ∈ h ∗ r and hen e ν − r ( ρ ) ∈ h , while R is the enter of theCayley-Di kson algebra A r .Thus (1) is proved in general also.26. Corollary. If suppositions of Theorem 25 (2) are satis(cid:28)ed and there exists a ve tor v ∈ V so that e j ( v ) = 0 for all j = 1 , ..., n and h ( v ) = < b, h > v for some b ∈ h ∗ r and all h ∈ h , then (1) Ω( v ) = ( b + 2 ρ | b ) v .Moreover, if U ( g ( A )) v = V , then (2) Ω = ( b + 2 ρ | b ) I V .Proof. Formula (1) follows from the de(cid:28)nition of Ω and Formula 25 (3) . Then Formula (2) follows from Formula (1) and Theorem 25.27. Proposition. Let g be an algebra over A r , ≤ r ≤ , with an invariant form ( ∗|∗ ) satisfying Conditions 17 (2 − , let also { x j : j } and { y j : j } be dual bases belonging to g ,that is ( x j | y k ) = δ j,k for ea h j, k . Suppose that V is an g module so that for ea h pair ofelements u, v ∈ V either x j ( u ) = 0 or y j ( v ) = 0 for all j besides a (cid:28)nite number of j . Thenthe operator Ω := P j x j ⊗ y j is de(cid:28)ned on V ⊗ V and ommutes with a tions of all elements of g on V .Proof. Consider ommutators [ z, x j ] = P k c j,k x k , [ z, y j ] = P k p k,j y k , where c j,k , p k,j ∈A r . Taking the s alar produ ts we dedu e that c j,k = ([ z, x j ] | y k ) and p k,j = ([ z, y j ] | x k ) . Fromthe invarian e of ( ∗|∗ ) it follows that c j,k = ( z | [ x j , y k ]) and p k,j = ( z | [ y j , x k ]) , onsequently, c j,k = − p k,j for all j, k and inevitably P j ([ z, x j ] ⊗ y j + x j ⊗ [ z, y j ]) = 0 , sin e x j , y j ∈ g forea h j .28. Example. Consider the n × n zero matrix A = 0 with either n ∈ N or n = ∞ .This means that [ e j , e k ] = 0 , [ f j , f k ] = 0 , [ e j , f k ] = δ j,k γ j for all j, k = 1 , ..., n , hen e13 (0) = h ⊕ P j A r e j ⊕ P j A r f j . The enter of g (0) is Z = P j R γ j . Moreover, dim A r h = 2 n and we an hoose elements d , ..., d n ∈ h so that h = A r ⊗ R Z + P j A r d j and [ d j , e k ] = δ j,k e k , [ d j , f k ] = − δ j,k f k for ea h j, k = 1 , ..., n .A non degenerate symmetri invariant form satisfying Conditions 17 (2 − is de(cid:28)ned as ( e j | f j ) = 1 , ( γ j | d j ) = 1 for ea h j and all others s alar produ ts are zero. In the onsideredalgebra ρ = 0 and the Casimir operator takes the form Ω = 2 P j γ j d j + 2 P j f j e j .We put q = P A r ( γ j − γ k ) ⊂ A r ⊗ R Z and the algebra H := g ′ (0) /q we all theHeisenberg algebra over the Cayley-Di kson algebra A r so that H = g ′ (0) / q , H = Hi ⊕ ... ⊕ r − Hi r − with pairwise isomorphi real algebras H, ..., r − H , where g ′ = g ′ i ⊕ ... ⊕ r − g ′ i r − .3 Residues of o tonion meromorphi fun tionsFor subsequent pro eedings we need residues of (super) di(cid:27)erentiable fun tions of Cayley-Di kson variables. In this se tion they are studied in more generality and with new detailsas in previous papers.1. De(cid:28)nitions.Let f : V → A r be a fun tion, where V is a neighborhood of z ∈ ˆ A r , where ˆ A r is the one-point ompa ti(cid:28) ation of A r with the help of the in(cid:28)nity point or as the non ommutativeanalog of the r dimensional Riemann sphere [17, 18℄. Then f is said to be meromorphi at z if f has an isolated singularity at z and f is A r -holomorphi in V \ { z } . If U is an opensubset in ˆ A r , then f is alled meromorphi in U if f is meromorphi at ea h point z ∈ U . If U is a domain of f and f is meromorphi in U , then f is alled meromorphi on U .This de(cid:28)nition has the natural generalization. Let W be a losed onne ted subset in ˆ A r and its odimension codim ( W ) = 2 r − dim ( W ) ≥ and W ∩ ( z + \ R ⊕ M R ) is a set onsistingof isolated points for ea h purely imaginary Cayley-Di kson number M ∈ A r with | M | = 1 and every z ∈ W ∪ { } , where dim ( W ) is the topologi al overing dimension of W (see Ÿ7.1[5℄). If there exists an open neighborhood V of W so that a fun tion f is A r -holomorphi in V \ W and may have singularities at points of W , then f is alled meromorphi at W .If U is an open subset in ˆ A r and di(cid:27)erent subsets W k in U are isolated from ea h other,that is inf {| ζ − η | : j = 1 , , k − , k + 1 , ... ; ζ ∈ W k , η ∈ W j } > for ea h marked k and f is meromorphi at ea h W k and A r -holomorphi in U \ [ S k W k ] and S k W k is losed in A r ,then f is alled meromorphi in U .Mainly we shall onsider meromorphi fun tions with point isolated singularities if an-other will not be spe i(cid:28)ed and denote by M ( U ) the set of all meromorphi fun tions on U with singleton singularities W = { z } . Let f be meromorphi on a region U in the set ˆ A r . Apoint c ∈ \ V ⊂ U,V is losed and bounded cl ( f ( U \ V ))
14s alled a luster value of f .Let V be an open subset in A r . De(cid:28)ne the residue of a meromorphi fun tion f at W with a singularity at a point a ∈ W ⊂ A r as ( i ) Res ( a, f ) .M := (2 π ) − lim y → ( Z γ y f ( z ) dz ) whenever this limit exists,where γ y ( t ) = a + yρ exp(2 πtM ) ⊂ V \ W,ρ > , | M | = 1 , M ∈ I r , t ∈ [0 , , < y ≤ , γ := γ en ompasses only one singular point a of f in the omplex plane ( a + R ⊕ M R ) , I r := { z : z ∈ A r , Re ( z ) = 0 } , Re ( z ) := ( z + ˜ z ) / .Here as usually we suppose that a fun tion ( R f is A r -holomorphi on V \ W , ( R W is a losed onne ted subset in ˆ A r of odimension not less than , ( R the interse tion W ∩ ( z + \ R ⊕ N R ) onsists of isolated points for ea h purelyimaginary Cayley-Di kson number N ∈ I r , | N | = 1 , and every z ∈ W ∪ { } .Extend Res ( a, f ) .M by Formula ( i ) on I r as ( ii ) Res ( a, f ) .M := [ Res ( a, f ) . ( M/ | M | )] | M | , ∀ M = 0 ; Res ( a, f ) . , when Res ( a, f ) .M is (cid:28)nite for ea h M ∈ I r , | M | = 1 .2. De(cid:28)nition. For a metri spa e X with a metri ρ let dist ( x, A ) := inf { ρ ( x, y ) : y ∈ A } denotes a distan e from a point x to a subset A in X . Let z be a marked point in the Cayley-Di kson algebra and γ ( t ) := z + R exp(2 πM t ) be a ir le with enter at z of radius < R < ∞ in the plane z + ( R ⊕ M R ) embedded into A r , where t ∈ [0 , . Then γ en ompasses z inthe usual sense.We say that a loop ψ ( t ) in the Cayley-Di kson algebra A r en ompasses the point z relative to γ , if there exists a ontinuous monotonously in reasing fun tion φ : [0 , → [0 , being pie ewise ontinuously di(cid:27)erentiable so that | ψ ( t ) − γ ( φ ( t )) | < min( | ψ ( t ) − z | , R ) forea h t ∈ [0 , .If A m is the subalgebra of A r , m < r , then there exists the proje tion P m : A r → A m asfor real linear spa es.3. Theorem. Let f be an A r -holomorphi fun tion on an open domain U in A r , ∞ ≥ r ≥ . If ( γ + z ) and ψ are presented as pie ewise unions of paths γ j + z and ψ j with respe t to parameter θ ∈ [ a j , b j ] and θ ∈ [ c j , d j ] respe tively with a j < b j and c j < d j for ea h j = 1 , ..., n and S j [ a j , b j ] = S j [ c j , d j ] = [0 , homotopi relative to U j \ { z } , where U j \ { z } is a (2 r − - onne ted open domain in the Cayley-Di kson algebra A r su h that π s,p,t ( U j \ { z } ) is simply onne ted in C for ea h s = i k , p = i k +1 , k = 0 , , ..., r − − ( ∀ ≤ k ∈ Z and P m ( U j \ { z } ) is (2 m − - onne ted for ea h ≤ m ∈ N if r = ∞ ), ea h t ∈ A r,s,p and u ∈ C s,p for whi h there exists z = t + u ∈ A r . If ( γ + z ) and ψ are losedre ti(cid:28)able paths (loops) in U su h that γ ( θ ) = ρ exp(2 πθM ) with θ ∈ [0 , and a marked15 ∈ I r , | M | = 1 and < ρ < ∞ , also z / ∈ ψ and M = [ R ψ dLn ( ζ − z )] / (2 π ) and ψ en ompasses z relative to γ + z . Then (1) (2 π ) f ( z ) M = Z ψ f ( ζ ) dLn ( ζ − z ) for ea h z ∈ U su h that | z − z | < inf ζ ∈ ψ ([0 , | ζ − z | . If either A r is alternative, that is, r = 2 , , or f ( z ) ∈ R for ea h z , then (3 . f ( z ) = (2 π ) − ( Z ψ f ( ζ ) dLn ( ζ − z ) − ) M ∗ . Proof. The logarithmi fun tion is ζ -di(cid:27)erentiable so that dLn ( ζ − z ) = [ D ζ Ln ( ζ − z )] .dζ for a marked z and the variable ζ with ζ = z , onsequently, the onsidered integrals exist.Take a Cayley-Di kson number z satisfying the onditions of the theorem. Put ψ y ( θ ) := z + y ( ψ ( θ ) − z ) and γ y ( θ ) = yρ exp(2 πθM ) for ea h θ ∈ [0 , and every < y ≤ . Up toa notation we an onsider φ ( θ ) = θ for ea h θ denoting γ ◦ φ for simpli ity by γ . Then | ψ y ( θ ) − ( γ y ( θ ) + z ) | < | ψ y ( θ ) − z | for ea h < y ≤ .Join paths γ y + z and ψ y by a re ti(cid:28)able path ω y su h that z / ∈ ω , whi h is going inone dire tion and the opposite dire tion, denoted ω − y , su h that ω y,j ∪ ψ y,j ∪ γ y,j ∪ ω y,j +1 is homotopi to a point relative to U j \ { z } for suitable ω y,j and ω y,j +1 , where ω y,j joins γ y ( a j ) + z with ψ y ( c j ) and ω y,j +1 joins ψ y ( d j ) with γ y ( b j ) + z su h that z and z / ∈ ω j forea h j .Then the equality Z ω y,j f ( ζ ) dLn ( ζ − z ) = − Z ω − y,j f ( ζ ) dLn ( ζ − z ) is a omplished for ea h j .Mention that [ D ζ Ln ( ζ − z )] . ζ − z ) − for ea h ζ = z and the fun tion f ( ζ ) is A r -holomorphi in U . The bran hing of the logarithmi fun tion Ln ( ζ − z ) by the variable ζ with ζ = z for a marked Cayley-Di kson number z is independent from | ζ − z | > as showsits non- ommutative Riemannian surfa e des ribed in [17, 18, 21℄, sin e Ln ( y ( ζ − z )) =ln y + Ln ( ζ − z ) for ea h y > and ea h ζ = z , so that bran hes of Ln ( ζ − z ) are indexed bypurely imaginary Cayley-Di kson numbers M ∈ I r . Consider now sub-domains U j \ { z } andloops there omposed of fragments ψ ([ c j , d j ]) and ψ y ([ c j , d j ]) and joining their orrespondingends re ti(cid:28)able paths gone in a de(cid:28)nite dire tion, as well as loops omposed of fragments γ ([ a j , b j ]) + z and γ y ([ a j , b j ]) + z and joining their respe tive ends re ti(cid:28)able paths gone in ade(cid:28)nite dire tion. Therefore, due to the homotopy Theorem 2.15 [17, 18℄ and the onditionsof this theorem we infer, that Z ψ f ( ζ ) dLn ( ζ − z ) = Z ψ y f ( ζ ) dLn ( ζ − z ) and16 γ + z f ( ζ ) dLn ( ζ − z ) = Z γ y + z f ( ζ ) dLn ( ζ − z ) for ea h < y ≤ .Sin e γ y + z is a ir le around z its radius yρ > an be hosen so small, that f ( ζ ) = f ( z ) + α ( ζ , z ) , where α is a ontinuous fun tion on U su h that the limit lim ζ → z α ( ζ , z ) = 0 exists, then Z γ y + z f ( ζ ) dLn ( ζ − z ) = Z γ y + z f ( z ) dLn ( ζ − z ) + δ ( yρ ) = 2 πf ( z ) M + δ ( yρ ) , where | δ ( yρ ) | ≤ | Z γ y + z α ( ζ , z ) dLn ( ζ − z ) | ≤ π sup ζ ∈ γ y | α ( ζ , z ) | C exp( C ( yρ ) m ) , where C and C are positive onstants, m = 2 + 2 r (see Inequality (2 . . [17, 18℄), hen ethere exists lim y → ,y> δ ( yρ ) = 0 . Analogous estimates are for ψ y instead of γ y + z .We have that M = [ R ψ y dLn ( ζ − z )] / (2 π ) and also M = [ R γ y + z dLn ( ζ − z )] / (2 π ) for ea h < y ≤ due to onditions of this theorem, sin e Ln ( y ( ζ − z )) = ln( y ) + Ln ( ζ − z ) for ea h ζ = z and y is the positive parameter independent from z , where ln is the standard naturallogarithmi fun tion for positive numbers.On the other hand, Ln (1 + z ) = z − z / z / − z / ... + ( − n +1 z n /n + ... for ea hCayley-Di kson number of absolute value | z | < less than one, sin e ea h z an be written inthe form z = Re ( z ) + Im ( z ) , where Re ( z ) = ( z + ˜ z ) / , Im ( z ) = z − Re ( z ) , Im ( z ) is a purelyimaginary number so that ( Im ( z )) = −| Im ( z ) | . The latter series uniformly onverges in aball of a given radius < R < with the enter at zero. Therefore, the winding numbers of ψ y and γ + z around z are the same for ea h < y ≤ and equal to R γ + z dLn ( ζ − z ) / (2 πM ) .Taking the limit while y > tends to zero yields the on lusion of this theorem, sin e lim y → Z γ y + z f ( ζ ) dLn ( ζ − z ) = lim y → Z ψ y f ( ζ ) dLn ( ζ − z ) . If either r = 2 , , or f ( z ) ∈ R for ea h z , then ((2 π ) f ( z ) M ) M ∗ = 2 πf ( z ) .4. De(cid:28)nitions. This de(cid:28)nition 1 of the residue spreads also on loops ψ not ne essarily ina de(cid:28)nite plane like ψ in ŸŸ2 and 3. But then the purely imaginary Cayley-Di kson numbersubstitutes on the mean value of M omputed with the help of the line integral for dLn ( z ) along the loop ψ divided on πn , that is M = R ψ dLn ( z ) / (2 πn ) , where n ∈ N is the windingnumber of the loop ψ . Due to ŸŸ3 we have that this operator Res at a marked point a = z is the same for γ + z , γ y + z , ψ and ψ y loops for ea h < y ≤ , when the loops γ y + z en ompass only one singular point z in the omplex plane ( z + R ⊕ M R ) for γ y + z , while ψ y en ompasses z relative to γ y + z in the sense of Ÿ2.For the in(cid:28)nite point a = ∞ the de(cid:28)nitions of the index and the residue hange so thattake instead of a ir le γ or a loop ψ it with the opposite orientation γ − or ψ − respe tively.Thus taking the mapping z /z redu e the pole at the in(cid:28)nity into the pole at zero.17n addition in the residue's de(cid:28)nition we take a su(cid:30) iently large radius ρ > su h thatin the omplex plane R ⊕ M R for a marked purely imaginary Cayley-Di kson number the ir le γ − with the enter at zero en ompasses only one singular point ∞ from W in thenon- ommutative analog ˆ A r of the Riemann sphere. We also onsider ir les γ − y with y ≥ and take the limit ( i ) Res ( ∞ , f ) .M := (2 π ) − lim y →∞ Z γ − y f ( z ) dz whenever it exists.Mainly we shall onsider residues at isolated singular points, when another will not bespe i(cid:28)ed.If f has an isolated singularity at a ∈ ˆ A r , then oe(cid:30) ients b k of its Laurent series (seeŸ3 [17, 18℄) are independent from ρ > . The ommon series is alled the a -Laurent series.If a = ∞ , then g ( z ) := f ( z − ) has a -Laurent series c k su h that c − k = b k . Let β := sup b k =0 η ( k ) , where η ( k ) = k + ... + k m , m = m ( k ) for a = ∞ ; β = inf b k =0 η ( k ) for a = ∞ . We say that f has a removable singularity, pole, essential singularity at ∞ a ording as β ≤ , < β < ∞ , β = + ∞ . In the se ond ase β is alled the order of thepole at ∞ . For a (cid:28)nite a the orresponding ases are: β ≥ , −∞ < β < , β = −∞ . If f has a pole at a , then | β | is alled the order of the pole at a .A value of a fun tion ∂ f ( a ) := inf { η ( k ) : b k = 0 } is alled a divisor of f at a = ∞ , ∂ f ( a ) := inf {− η ( k ) : b k = 0 } for a = ∞ , where b k = 0 means that b k, = 0 ,...., b k,m ( k ) = 0 . Then ∂ f + g ( a ) ≥ min { ∂ f ( a ) , ∂ g ( a ) } for ea h a ∈ dom ( f ) ∩ dom ( g ) and ∂ fg ( a ) = ∂ f ( a ) + ∂ g ( a ) . For a fun tion f meromorphi on an open subset U in ˆ A r the fun tion ∂ f ( p ) by the variable p ∈ U is alled the divisor of f .5. Example and Remark. Consider another example of a multi-valued lo ally analyti fun tion. Let ξ be a Cayley-Di kson number and onsider the power fun tion z = ξ n , where ≤ n ∈ N , n is a natural number. Ea h Cayley-Di kson number has the polar form z = ρ exp(2 πθM ) , where θ ∈ R , M is a purely imaginary Cayley-Di kson number of the18nit absolute value | M | = 1 , ρ := | z | . For de(cid:28)niteness we an onsider M of the form M = M i + ... + M k i k with ≤ k ≤ r − , M , ..., M k ∈ R , M k > , that to ex luderepeating aused by the identity ( − θ )( − M ) = θM . For ea h marked M as above thefun tion exp(2 πθM ) is periodi so that exp(2 π ( θ + n )) M ) = exp(2 πθM ) for ea h integernumber n , n ∈ Z .Therefore, the inverse fun tion ξ = z /n on A r \ { } is multi-valued with n bran hes.Ea h bran h is ξ j = ρ /n exp(2 π ( θ + ( j − M/n ) , where M = M ( z ) , b /n > denotes thepositive value of the n -th root for ea h b > , j = 1 , ..., n . When a loop, for example, a ir le with the enter at zero in the plane R ⊕ M R is gone around on π , then there isthe transition from the j -th bran h into j + 1 for ea h ≤ j < n , while the n -th bran htransits into the (cid:28)rst bran h. That is, the point z = 0 is the bran hing point of the fun tion z /n . Bran hes of this fun tion are indexed by j = 1 , ..., n and depend on purely imaginaryCayley-Di kson numbers M ∈ I r . Sin e z /n = exp[( Ln ( z )) /n ] , then z /n is A r -holomorphi on A r \ { } . This also follows from the inverse fun tion theorem: if f is A r -holomorphi onan open subset V and it has an inverse fun tion f − , then its inverse fun tion f − is also A r -holomorphi [17, 18, 21℄.The non- ommutative Riemannian surfa e for it is onstru ted from n opies of theCayley-Di kson algebra A r ut by the subset Q as in Ÿ3.7 [17, 18℄ su h that they are embeddedinto A r . Ea h ut opy Y j of A r is slightly bend and ea h edge Q j is glued with Q j +1 for ea h j = 1 , ..., n − and Q n with Q by straight rays with initially parallel dire tingve tors. Then for su h non- ommutative Riemannian r -dimensional surfa e R = R ( z /n ) the fun tion z /n : ( A r \ { } ) → R is already univalent.Generally suppose that f : ( U \ W ) → A r is a A r -holomorphi fun tion on an open set ( U \ W ) in the one-point ompa ti(cid:28) ation ˆ A r of the Cayley-Di kson algebra A r , so that the ompa ti(cid:28) ation is relative to the weak topology for card ( r ) ≥ ℵ . We have ˆ A r \ A r = {∞} .Here we onsider W a losed onne ted subset of odimension at least , codim ( W ) ≥ , inthe Cayley-Di kson algebra A r su h that W ∩ ( z + \ R ⊕ M R ) is a set onsisting of isolatedpoints for ea h purely imaginary M ∈ I r and ea h z ∈ W ∪ { } , U is open in A r . We alsosuppose that f is multi-valued and ea h its bran h is de(cid:28)ned on ( U \ W ) .If z ∈ W and along ea h loop in ( z + \ R ⊕ M R ) en ompassing z and neither en ompassingnor ontaining another points from W the omplete ir uit around the loop on π leads to atransition from the j -th bran h of f into a de(cid:28)nite bran h k ( j ) for ea h j , then we say that z is a bran hing point of the fun tion f .A non- ommutative surfa e R = R ( f ) omposed from opies of Y k properly embeddedinto a suitable spa e A sr , card ( s ) ≥ , and with suitable gluing of Y j with Y k ( j ) by the orresponding edges so that f : ( U \ W ) → R be omes univalent and A r -holomorphi , then R is alled the non- ommutative Riemannian surfa e of f .If f is su h fun tion, then for suitable U satisfying onditions of Theorem 3 with z repla ed on W , due to the monodromy Theorem 2.41 [21℄ and the homotopy Theorem 2.15[17, 18℄ the bran hing of f depends only on M ∈ I r and is independent from | ζ − z | for z ∈ W . This follows from the onsideration of loops γ in planes z + ( \ R ⊕ M R ) with z ∈ W γ ⊂ (( U j \ W ) ∩ [ z + ( \ R ⊕ M R )]) for ea h j .Mention that generally onditions of Theorem 2.15 [17, 18℄ an not be repla ed on lo alhomotopies within ea h U j \ W . This is aused by several reasons. At (cid:28)rst, onsider two ir les γ and γ of radii < R < R < ∞ with the enter at a point z in planes z + ( R ⊕ M R ) and z + ( R ⊕ N R ) , where N = aM + bN , a, b ∈ R \ { } , Re ( M N ) = 0 , | M | = | N | = | N | = 1 , M and N and N are purely imaginary marked Cayley-Di ksonnumbers. Let U and z satisfy onditions of Theorem 3. Consider γ ( t ) = z + R exp(2 πtM ) and γ − ( t ) = z + R exp( − πtN ) , t ∈ [0 , . Divide them on ar s respe tive to the partition [( j − /n, j/n ] with ≤ n ∈ N and j = 1 , ..., n . Take the orresponding ar s of these ir lesand join their respe tive ends γ ( j/n ) with γ ( j/n ) for ea h j by segments w j of straightlines. We get loops η j from su h segments and ar s of γ and γ − ir uit in a ordan e with γ and γ − .Now onsider two ir les γ and γ embedded into z + ( R ⊕ M R ⊕ N R ) . Take a newsystem of oordinates in the latter three dimensional manifold with the origin at z . Theaxis e take parallel to i , the axis e parallel to M and e parallel to N . Therefore, γ is ina half-spa e above γ and in the other half-spa e below γ . Then join γ ( t ) with γ ( t ) by thesegment of the straight line for ea h t ∈ [0 , . So we get a set L whi h is homeomorphi tothe M(cid:4)obius band, whi h is a non-oriented two dimensional surfa e. Suppose that L ⊂ U \ W .There exists k for whi h w k is gone twi e in the same dire tion, be ause L is non-oriented. Then the sum of the integrals by all j -th loops η j generally does not vanish for an A r -holomorphi fun tion f in ( U \ { z } ) . Generally it has the rest of the type c R w k f ( z ) dz ,where c = 1 or c = − .The surfa e Q utting A r has the unit odimension, codim ( Q ) = 1 , onsequently, theloops γ and ψ − z from Ÿ3 both interse t Q , Q ∩ γ ([0 , = ∅ and Q ∩ ψ ([0 , = ∅ . So theintegral by the loop η j may be di(cid:27)erent from zero for some j , when γ and ψ are not in oneplane R ⊕ M R and z is a bran hing point of f .If f , U , z , ψ and γ + z are as in Theorem 3 su h that ψ ([0 , does not interse t γ ([0 , z , ψ ([0 , is not ontained in the plane z + ( R ⊕ M R ) , then from ψ and γ onstru t a loop η taking their ar s and joining their ends by re ti(cid:28)able paths in U so that η ⊂ U \ W . Then there exists a purely imaginary Cayley-Di kson number N ∈ I r su hthat it is not ontained in M R for whi h a proje tion of η on the plane z + ( R ⊕ N R ) en ompasses the point z . If f has a singularity at z , whi h may be a bran hing point, thengenerally R γ + z f ( z ) dz is di(cid:27)erent from R ψ f ( z ) dz , sin e a bran hing of f generally dependson all purely imaginary Cayley-Di kson numbers N ∈ I r , | N | = 1 . This is easily seen onthe examples of the fun tions Ln ( ζ − z ) and ( ζ − z ) /n for ≤ n ∈ N . Moreover, forthe domain U satisfying onditions of Theorem 3 generally onditions of the monodromyTheorem 2.41 in [21℄ are not satis(cid:28)ed.In the parti ular ase of γ and ψ ontained in the same plane z + ( R ⊕ M R ) the abovesituation with N ⊥ M is already ex luded. On the other hand, the bran hing of f isindependent from | ζ − z | (see above). If there are not another singular points in U besides20 and U satis(cid:28)es onditions of Theorem 3, and when γ + z and ψ are homotopi relativeto U ∩ ( z + ( R ⊕ M R )) , then due to the homotopy Theorem 2.15 [17, 18℄ and sin e M is(cid:28)xed we an on lude as in the omplex ase that R γ + z f ( z ) dz = R ψ f ( z ) dz .Therefore, the onditions of the homotopy Theorem 2.15 [17, 18℄ are substantially stronger,than in the omplex ase.6. Theorem. Let U be an open region in the set ˆ A r , ≤ r ≤ ∞ , with n distin t markedpoints p , ..., p n , and let f be an A r -holomorphi fun tion on U \ { p , ..., p n } =: U and ψ bea re ti(cid:28)able losed urve lying in U su h that U satis(cid:28)es the onditions of Theorem 3 forea h z ∈ { p , ..., p n } . Then Z ψ f ( z ) dz = 2 π n X j =1 Res ( p j , f ) . ˆ In ( p j , ψ ) and Res ( p j , f ) .M is the R -homogeneous I r -additive (of the variable M in I r ) A r -valuedfun tional for ea h j .Proof. In the onsidered ase ea h singular point p j is isolated, hen e in the De(cid:28)nitions1, 4 of the residue operator Res ( p j , f ) the limit by y an be omitted, sin e all integrals withdi(cid:27)erent < y ≤ are equal, when γ y are in the same omplex plane (see Ÿ3 above andŸ3.9.3 [17, 18℄).For ea h p j onsider the prin ipal part T j of a Laurent series for f in a neighborhood of p j , that is, T j ( z ) = X k,η ( k ) < { ( b k , ( z − p j ) k ) } q ( m ( k )+ η ( k )) , where η ( k ) = k + ... + k n for k = ( k , ..., k n ) (see Theorem 3.21 [17, 18℄). Therefore, h ( z ) := f ( z ) − X j T j ( z ) is a fun tion having an A r -holomorphi extension on U . In view of Theorem 3 for an A r -holomorphi fun tion g in a neighborhood V of a point p and a re ti(cid:28)able losed urve ζ wehave g ( p ) ˆ In ( p, ζ ) = (2 π ) − ( Z ζ g ( z )( z − p ) − dz ) (see Ÿ3.22 [17, 18℄). Let loops γ and ea h ζ j be lying in planes ea h of whi h is parallel tothe omplex plane R ⊕ M R with a marked purely imaginary M so that M and ψ satisfy onditions of Theorem 3.We may onsider small loops ζ j around ea h p j with ˆ In ( p j , ζ j ) = ˆ In ( p j , γ ) for ea h j = 1 , ..., n . Then we infer that Z ζ j f ( z ) dz = Z ζ j T j ( z ) dz for ea h j . Representing U as a (cid:28)nite union of open regions U j and joining ζ j with γ bypaths ω j going in one and the opposite dire tion as in Theorem 3 we get Z γ f ( z ) dz + X j Z ζ − j f ( z ) dz = 0 , Z γ f ( z ) dz = X j Z ζ j f ( z ) dz = X j πRes ( p j , f ) ˆ In ( p j , γ ) , where ˆ In ( p j , γ ) and Res ( p j , f ) are invariant relative to homotopies satisfying onditions ofTheorem 3 within a given omplex plane.Sin e the integral R ζ j g ( z ) dLn ( z − p j ) is R -homogeneous and I r -additive relative to adire ting ve tor M ∈ I r of a loop ζ j , then Res ( p j , f ) M de(cid:28)ned by Formulas 1 ( i, ii ) , 4 ( i ) is R -homogeneous I r -additive of the argument M in I r .7. Theorem. Let f be a meromorphi fun tion in V (see Ÿ1), so that f ( ζ ) = g ( ζ )[1 /v ( ζ )] and /f ( ζ ) = v ( ζ )[1 /g ( ζ )] for ea h ζ ∈ V , where g and v are two A r -holomorphi fun tionson an open set V , g has not zeros in V , v − (0) = W , W ⊂ V , ≤ r < ∞ , codim ( W ) ≥ ,then Res ( z, f ) is an operator ontinuously depending on z ∈ W .Proof. The set W is losed and onne ted in ˆ A r and the interse tion W ∩ ( z + \ R ⊕ M R ) onsists of isolated points for ea h z ∈ W ∪ { } and purely imaginary M ∈ I r . Therefore, ˆ A r \ W is open in ˆ A r . Sin e v is ontinuous and W = v − (0) and g has not zeros in V ,then ea h point z in W is singular for f . Making a shift z z + z we an onsider that and ∞ do not belong simultaneously to W . To treat a singularity at ∞ we an use thetransformation z /z under whi h we an onsider instead a singularity at zero. So we an suppose without loss of generality that ea h z ∈ W is a (cid:28)nite singular point.We an take a neighborhood U of z satisfying onditions of Theorem 3 and hoose thefamily γ y + z of loops in U en ompassing only one singular point z of f in the omplex plane z + R ⊕ M R for ea h M ∈ I r with | M | = 1 and ea h < y ≤ and every z ∈ W ∩ U . Theintegral Z γ y + z f ( ζ ) dζ = Z γ y f ( ζ + z ) dζ is the ontinuous fun tional relative to f and z , sin e f is holomorphi in U \ W , γ y ([0 , z ⊂ U \ W for ea h z ∈ W ∩ U and every < y ≤ .If z ∈ W is another either singular point di(cid:27)erent from z or any other point z ∈ U \ { z } ,then | z − z | > . Choose < y < | z − z | /ρ , where < ρ < ∞ is a radius of the ir le γ , then γ y + z does not en ompass z in the entire U as well. Sin e π Res ( z, f ) .M =lim y → R γ y + z f ( z ) dz , when M is (cid:28)xed, hen e Rez ( z, f ) .M may depend on z and the residue Res ( z, f ) .M is independent from ea h z not equal to z , z = z .The fun tions g and v being A r -holomorphi are lo ally ζ -analyti in V and for ea h z ∈ W there exists a ball with enter at z of some radius < δ < ∞ su h that g and v havenon- ommutative power series expansions in it. Sin e g ( ζ ) = 0 for ea h ζ ∈ V , then g ( z ) = 0 and v ( z ) = 0 for ea h z ∈ W , onsequently, f ( z ) = ∞ and /f ( z ) = 0 , and inevitably /f ( ζ ) = v ( ζ )[1 /g ( ζ )] is A r holomorphi in V and hen e in a neighborhood V z of z ∈ W , V z ⊂ V . Thus the fun tion /f ( ζ ) has the power series expansion in a ball with the enter at z ∈ W of some radius < ǫ < ∞ . Let its expansion oe(cid:30) ients be A m = ( a m,m , ..., a m,m k ) ,22 m,j ∈ A r for ea h m, j , m = ( m , ..., m k ) , m j ≥ for ea h j = 1 , ..., k , k ∈ N so that (1) 1 /f ( ζ ) = X m { A m , ( ζ − z ) m } q ( m ) , where { A m , z m } q ( m ) := { a m,m z m ...a m,m k z m k } q ( m ) , q ( m ) is the ve tor indi ating on an orderof the multipli ation. Let us seek the fun tion f in the form: (2) f ( ζ ) = X p { B p , ( ζ − z ) p } q ( p ) , where p = ( p , ..., p k ) , k = k ( p ) ∈ N , p j ∈ Z for ea h j , for ea h p either all p j ≥ or all p j ≤ simultaneously. Put η ( m ) := m + ... + m k , where k = k ( m ) , denote Q n ( ξ ) := X η ( p )= n { B p , ξ p } q ( p ) and P n ( ξ ) := X η ( m )= n { A m , ξ m } q ( m ) homogeneous terms so that Q n ( tξ ) = t n Q n ( ξ ) and P n ( tξ ) = t n P n ( ξ ) for ea h = t ∈ R andevery ξ ∈ A r \ { } . When n > , then we have P n (0) = 0 and Q n (0) = 0 . On the otherhand, for a negative number n the term Q n (1 /ξ ) is de(cid:28)ned for ea h (cid:28)nite Cayley-Di ksonnumber ξ .For the fun tion /f we have η ( m ) = n ≥ , sin e /f is holomorphi and /f ( z ) = 0 .If ea h P n ( ξ ) would be an identi ally zero polynomial, then /f would be identi ally zero,that is not the ase. Thus there exists the maximal α > for whi h P α ( ξ ) is the nontrivialpolynomial, while P n ( ξ ) is identi ally zero for ea h n < α . That is α is the order of zero z of the fun tion /f .We have the equation (3) f ( ζ )[1 /f ( ζ )] = 1 identi ally in the set B ( A r , z, ǫ ) \ W and by the ontinuity this equation extends on theentire ball. Sin e (4) Q n ( tξ ) P n ( tξ ) = t n + n Q n ( ξ ) P n ( ξ ) for ea h = t ∈ R and every ξ ∈ A r \ { } , then we infer the inequality n ≥ − α . In viewof (4) Equation (3) gives the system of equations (5) X n + n = l Q n ( ξ ) P n ( ξ ) = δ l, for ea h < | ξ | < ǫ , where ≤ l ∈ Z , δ i,j = 0 for ea h i = j ∈ Z , δ j,j = 1 for ea h j ∈ Z , onsequently, α ≤ n ≤ l + α in ea h l -th equation (5) .The Cayley-Di kson algebra A r has the (cid:28)nite dimension r over the real (cid:28)eld R , hen efor ea h n the number of di(cid:27)erent R -linearly independent terms { B p , ξ p } q ( p ) with η ( p ) = n is(cid:28)nite, as well as a number of di(cid:27)erent R -linearly independent terms { A m , ξ m } q ( m ) is (cid:28)nite for23 ( m ) = n . Thus ea h term Q n and P n is a (cid:28)nite sum and a number of expansion oe(cid:30) ientsin them is (cid:28)nite.Now using the multipli ation table in the Cayley-Di kson algebra A r , where ≤ r < ∞ ,and exploiting the de omposition w = wi + ... + r − wi r − for ea h w ∈ A r , where w, ..., r − w ∈ R , it is possible by indu tion on l = 0 , , ... resolve the system (5) relativeto B p through A m . The pro edure is the following. For l = 0 we get the power relative to ξ, ..., r − ξ Equation (5) expressing ve tors B p with η ( p ) = − α through ve tors A m with η ( m ) = α . For l = 1 Equation (5) together with the previous equation expresses ve tors B p with η ( p ) = − α + 1 through ve tors A m with η ( m ) ∈ { α, α + 1 } . By indu tion equations (5) with l = 0 , ..., β express ve tors B p with η ( p ) = β through ve tors A m with η ( m ) ∈{ α, ..., α + β } . Thus ea h B p is a ontinuous fun tion of a (cid:28)nite number of A m = A m ( z ) .At the same time ea h ve tor A m ( ξ ) is a ontinuous fun tion of ξ in B ( A r , z, ǫ/ , sin e thefun tion /f ( ζ − z ) is A r -holomorphi by ζ , z ∈ A r with ζ − z ∈ V , where V is open in theCayley-Di kson algebra A r by the onditions of this Theorem. Consequently, ea h B p ( ξ ) isa ontinuous fun tions of ξ in the ball B ( A r , z, ǫ/ .Evidently we have Z γ y { B p , ( ζ − z ) p } q ( p ) dζ = 0 for ea h η ( p ) = − and ea h < y ≤ , so that z here is equal to a in the De(cid:28)nition of theresidue in ŸŸ1, 4. Therefore, the limit π Res ( z, f ) .M := lim y → Z γ y f ( ζ ) dζ = lim y → X p,η ( p )= − Z γ y { B p , ( ζ − z ) p } q ( p ) dζ = X p,η ( p )= − Z γ { B p , ( ζ − z ) p } q ( p ) dζ exists, sin e R γ y is the additive fun tional by integrands and it is ontinuous on the spa eof bounded ontinuous fun tions (see Theorem 2.7 [17, 18℄) and due to the homotopy of γ y with γ in the same omplex plane z + \ R ⊕ M R relative to ( V \ W ) ∩ ( z + \ R ⊕ M R ) (see Ÿ3above and 3.9.3 [17, 18℄).This implies that Res ( z, f ) is ompletely de(cid:28)ned by the power series expansion of f ( z ) ,namely by ve tor oe(cid:30) ients B p with η ( p ) = − only. The family of oe(cid:30) ients with η ( p ) = − is (cid:28)nite for a (cid:28)nite ≤ r < ∞ . Therefore, we get that Res ( z, f ) .M is the ontinuous operator-valued fun tion by the variable z ∈ W .8. Corollary. Let U be an open region in the set ˆ A r , ≤ r ≤ ∞ , with n distin t points p , ..., p n , let also f be an A r -holomorphi fun tion on U \ { p , ..., p n } =: U , p n = ∞ , and U satis(cid:28)es onditions of Theorem 3 with at least one ψ , γ and ea h z ∈ { p , ..., p n } . Then X p j ∈ U Res ( p j , f ) M = 0 . γ is a losed urve en ompassing points p ,..., p n − , then γ − ( t ) := γ (1 − t ) ,where t ∈ [0 , , en ompasses p n = ∞ with positive going by γ − relative to p n . Sin e Z γ f ( z ) dz + Z γ − f ( z ) dz = 0 , we get from Theorem 6, that X p j ∈ U Res ( p j , f ) M = 0 for ea h M ∈ I r , hen e X p j ∈ U Res ( p j , f ) M = 0 is the zero R -homogeneous I r -additive A r -valued fun tional on I r .9. Remark. If a re ti(cid:28)able loop ψ is as in Ÿ1, then there exists a sequen e ψ p ofre ti(cid:28)able loops omposed of ar s of ir les entered at z and segments of straight lines su hthat straight lines ontain z , so that ψ p onverges uniformly to ψ with p tending to thein(cid:28)nity. Therefore, if f is the A r di(cid:27)erentiable fun tion in an open set U so that ψ ⊂ U and ψ p ⊂ U for ea h p , then lim p →∞ R ψ p f ( z ) dz = R ψ f ( z ) dz .There are useful Moufang identities in the o tonion algebra: ( M
1) ( xyx ) z = x ( y ( xz )) , ( M z ( xyx ) = (( zx ) y ) x , ( M
3) ( xy )( zx ) = x ( yz ) x for ea h x, y, z ∈ O (see page 120 in [9℄). For al ulations it is also worth that a ˜ a = ˜ aa = | a | and the real valueds alar produ t has the symmetry properties Re ( a ˜ b ) = Re (˜ ba ) = Re (˜ ab ) = Re ( b ˜ a ) , where a ∗ = ˜ a denotes the onjugated Cayley-Di kson number.Mention also that ( de z /dz ) .h = P ∞ n =1 P n − j =0 z j hz n − j − /n ! for all z, h ∈ A = O , hen egenerally ( de z /dz ) .h is not simply e z h besides the ase of Im ( h ) ∈ Im ( z ) R , where Im ( z ) = z − Re ( z ) and the order of the multipli ation in ea h additive of the series orresponds tothe order of multipli ation from right to left (right order of bra kets). Due to the Moufangidentities an order of multipli ations either left or right or (( z j ) h )( z n − j − ) or ( z j )( h ( z n − j − )) gives the same result.The inverse operator ( de z /dz ) − to de z /dz provides dLn ( y ) /dy for y = e z . The operator de z /dz is A r additive and R homogeneous, hen e its inverse operator is su h also. Parti u-larly, if Im ( h ) ∈ Im ( y ) R , where y = 0 , then ( dLn ( y ) /dy ) .h = y − h . Therefore, R γ k dLn ( y ) = R γ k y − dy for ea h γ k either ar of a ir le with the enter at zero or a segment of a straightline su h that the straight line ontains , onsequently, R ψ p − z dLn ( y ) = R ψ p − z y − dy forea h p (see also Ÿ12).Thus due to De(cid:28)nition 1 we dedu e that Theorems 3 and 7, Corollary 8 spread on moregeneral re ti(cid:28)able loops ψ , but then M in formulas there substitutes on the mean value of M omputed as 25 M = lim p →∞ R ψ p − z dLn ( z ) / (2 πj ) = R ψ − z dLn ( z ) / (2 πj ) , where j ∈ N is the windingnumber of the loop ψ (see also De(cid:28)nitions 1, 2 and Theorem 3). This formula for M isessential, sin e generally the logarithmi fun tion over the Cayley-Di kson algebra has more ompli ated non- ommutative Riemannian surfa e, than in the ommutative omplex ase.If a, b, c and e are onstants in A r so that { abM ce } q (5) = { abce ˜ M } q (5) , then (2) R ψ d { abLn ( z ) ce } q (5) / (2 πn ) = { abce ˜ M } q (5) ,where ve tors q (5) and q (5) pres ribe an order of the multipli ation in bra kets, ˜ M = − M for purely imaginary Cayley-Di kson number M . On the other hand, if f is a (super) di(cid:27)er-entiable fun tion in a neighborhood of γ , then R γ { abf ( z ) ce } q (5) dz = { ab ( R γ f ( z ) dz ) ce } q (5) due to the de(cid:28)nition of the line integral [17, 18℄. If N and M are two orthogonal purelyimaginary o tonions, Re ( M ˜ N ) = 0 , also Im ( z ) ∈ M R , then N ( zN ∗ ) = ( N z ) N ∗ = ˜ z .Therefore, if for onstants a , ..., b k +1 ∈ A r the identity (3) { a z n ...a k z n k a k +1 } q (2 k +1) = { b ...b k +1 ( z s ˜ z m ) } q ( k +2) is satis(cid:28)ed in an open neighbor-hood of a re ti(cid:28)able loop ψ for z = 0 and with the winding number j , where k ∈ N , n , ..., n k , s, m ∈ Z , then (4) (2 πj ) − R ψ { a z n ...a k z n k a k +1 } q (2 k +1) dz = { b ...b k +1 M } q ( k +2) for m − s = 1 with n = − ; (5) (2 πj ) − R ψ { a z n ...a k z n k a k +1 } q (2 k +1) dz = { b ...b k +1 ˜ M } q ( k +2) for s − m = 1 with n = − ; (6) (2 πj ) − R ψ { a z n ...a k z n k a k +1 } q (2 k +1) dz = 0 for either | m − s | 6 = 1 or n = − , where n = n + ... + n k . Sin e s + m = n , then Case (4) implies m = 0 and s = − , Case (5) implies s = 0 and m = − .10. De(cid:28)nitions. Let f be an A r -holomorphi fun tion, ≤ r ≤ ∞ , on a neighborhood V of a point z ∈ A r . Then the in(cid:28)mum: η ( z ; f ) := inf { k : k ∈ N , f ( k ) ( z ) = 0 } is alled a multipli ity of f at z . Let f be an A r -holomorphi fun tion on an open subset U in the set ˆ A r , ≤ r ≤ ∞ . Suppose w ∈ ˆ A r , then the valen e ν f ( w ) of the fun tion f at w is by the de(cid:28)nition ν f ( w ) := ∞ , when the set { z : f ( z ) = w } is in(cid:28)nite,and otherwise ν f ( w ) := X z,f ( z )= w η ( z ; f ) .
11. Theorem. Let γ and ψ be two re ti(cid:28)able paths in A r ontained in open sets U γ and U ψ respe tively and let a di(cid:27)eomorphism ξ : U ψ → U γ exists with (super) di(cid:27)erentiable ξ and ξ − su h that ξ ( ψ ([0 , γ ([0 , . If f is a ontinuous A r valued fun tion on U γ , then (1) R γ f ( z ) dz = R ψ f ( ξ ( y )) . ( ξ ′ ( y ) .dy ) .Proof. For f in a anoni al losed bounded neighborhood V of γ so that V ⊂ U takea sequen e of (super) di(cid:27)erentiable fun tions f n onverging uniformly on V to f with the26orresponding phrases η n (cid:28)xing a z -representation of f n and of f (see Ÿ2 in [17, 18℄). Thenthe limit lim n →∞ R γ f n ( z ) dz = R γ f ( z ) dz exists (see Theorem 2.7 [17, 18℄). Therefore, it issu(cid:30) ient to prove this theorem for f n or in the ase, when f is (super) di(cid:27)erentiable.In a ordan e with Proposition 2.6 [17, 18℄ take a fun tion g ( z ) and its phrase ν given byeither left or right algorithm so that g ′ ( z ) . f ( z ) and ν ′ ( z ) . η ( z ) for ea h z ∈ V , where g ′ ( z ) = dg ( z ) /dz , η ( z ) is the phrase of f . This spe i(cid:28)es the bran h of the non ommutativeline integral (2) R γ f ( z ) dz = lim δ ( P ) → P mj =1 ( dg ( z ) /dz ) | z = z j . ∆ j z ,where P denotes a partition of [0 , with points t = 0 < t < ... < t m = 1 and τ j ∈ [ t j − , t j ] for ea h j = 1 , ..., m , ∆ j z := γ ( t j ) − γ ( t j − ) , z j := γ ( τ j ) , δ ( P ) := max j =1 ,...,m ( t j − t j − ) .Sin e by the hain rule ( dg ( ξ ( y )) /dy ) .h = ( dg ( z ) /dz ) | z = ξ ( y ) . (( dξ ( y ) /dy ) .h ) for ea h y ∈ U ψ and h ∈ A r (see also Ÿ2 [17, 18℄), then ( dg ( z ) /dz ) | z = ξ ( y j ) . (( dξ ( y ) /dy ) . ∆ j y ) = ( dg ( z ) /dz ) | z = z j . ∆ j z + o (∆ j z ) ,where y j = ψ ( τ j ) , ∆ j y = ψ ( t j ) − ψ ( t j − ) , z j = ξ ( y j ) , ∆ j z = ξ ( ψ ( t j )) − ξ ( ψ ( t j − )) . Therefore, (3) lim δ ( P ) → P mj =1 ( dg ( z ) /dz ) | z = z j . ∆ j z = lim δ ( P ) → P mj =1 ( dg ( z ) /dz ) | z = ξ ( y j ) . ( ξ ′ ( y j ) . ∆ j y ) ,sin e paths γ and ψ are re ti(cid:28)able. Thus Formula (1) follows from (2 , .12. Remark. Mention that R γ f ( z ) dz is independent from parametrization of γ in thefollowing sense. Two paths γ : [ a , b ] → A r and γ : [ a , b ] → A r are alled equivalent, γ ∼ γ , if there exists a ontinuous monotonously in reasing fun tion φ : [ a , b ] → [ a , b ] so that γ ( t ) = γ ( φ ( t )) for ea h t ∈ [ a , b ] , where a < b , a < b . This relation is re(cid:29)exive( γ ∼ γ ), symmetri (if γ ∼ γ , then γ ∼ γ ), transitive (if γ ∼ γ and γ ∼ γ , then γ ∼ γ ). A lass of equivalent paths is alled a urve.If γ ∼ γ are two re ti(cid:28)able paths, then the mapping φ gives the bije tive orresponden ebetween partitions of γ and γ and between the orresponding integral sums. From thede(cid:28)nition of the non ommutative line integral it follows, that if γ ∼ γ , then R γ f ( z ) dz = R γ f ( z ) dz for a ontinuous fun tion f . Thus the non ommutative line integral depends on urves.13. Theorem. If γ is a re ti(cid:28)able path γ : [ a, b ] → A r in the Cayley-Di kson algebra A r and f : U → A r is a ontinuous fun tion, where U is an open subset in A r and γ ⊂ U , −∞ < a < b < ∞ , then the non ommutative line integral redu es to the non ommutativeLebesgue-Stieltjes integral: (1) R γ f ( z ) dz = ( L − S ) R ba ˆ f ( γ ( t )) .dγ ( t ) .Moreover, if γ ( t ) is absolutely ontinuous, then the non ommutative line integral redu esto the non ommutative Lebesgue integral: (2) R γ f ( z ) dz = ( L ) R ba ˆ f ( γ ( t )) .γ ′ ( t ) dt .Proof. Re all that a fun tion f : [ a, b ] → A r is with (cid:28)nite variation, if there exists C > so that V ba := sup P mj =1 | f ( t j ) − f ( t j − | ≤ C ,where the supremum is taken by all (cid:28)nite partitions of the segment [ a, b ] , m ∈ N , V ba ( f ) is alled the variation of f on [ a, b ] . This implies that ea h f j is also with the (cid:28)nite variation,where f ( t ) = f ( t ) i + ... + r − f ( t ) i r − , j f ( t ) ∈ R for ea h j = 0 , ..., r − and every27 ∈ [ a, b ] .A path γ : [ a, b ] → A r is alled re ti(cid:28)able, if it is ontinuous and of (cid:28)nite variation.A fun tion g : [ a, b ] → A r is alled absolutely ontinuous, if for ea h δ > there exists ǫ > so that for ea h system of pairwise disjoint intervals ( a j , b j ) in [ a, b ] , j = 1 , ..., m , m ∈ N , the inequality P mj =1 | f ( b j ) − f ( a j ) | < ǫ is satis(cid:28)ed. A ontinuous fun tion is alled singular if its derivative is almost everywhererelative to the Lebesque measure equal to zero.In view of ŸVI.4 [12℄ ea h fun tion f of (cid:28)nite variation an be presented as f = H + ψ + χ ,where H is the jump fun tion, ψ is the absolutely ontinuous fun tion and χ is the singularfun tion. For a re ti(cid:28)able path γ we get H = 0 . By Theorem VI.4.2 [12℄ the fun tion F ( x ) = ( L ) R xa p ( t ) dt of a Lebesgue integrable fun tion p ( t ) on the segment [ a, b ] is absolutely ontinuous, where ( L ) R ba p ( t ) dt is the Lebesgue integral. In view of the Lebesgue TheoremIV.4.3 [12℄ the derivative p ′ = P of an absolutely ontinuous fun tion p on the segment [ a, b ] is Lebesgue integrable on it and ( L ) R xa p ( t ) dt = P ( x ) − P ( a ) for ea h a ≤ x ≤ b . Thus γ is almost everywhere on [ a, b ] di(cid:27)erentiable.The operator ˆ f is de(cid:28)ned for ea h (super) di(cid:27)erentiable fun tion with the help of eitherthe left or the right algorithm, where ˆ f .h := g ′ ( z ) .h for ea h z ∈ V , h ∈ A r . Moreover, ˆ f is(super) di(cid:27)erentiable, if f is su h. Then we use the ontinuous extension of the ontinuousfun tional R γ on the spa e of ontinuous fun tions on V spe ifying the bran h of the integraland get the operator ˆ f for a hosen sequen e of (super) di(cid:27)erentiable fun tions f n and theirphrases η n onverging uniformly on V to f (see [17, 18℄ and Ÿ11 above).Ea h A r additive R homogeneous operator A on A r an be written in the form: A.h = P r − k =0 A. [ k hi k ] = P r − j,k =0 j,k A. [ k h ] i j ,where ea h j,k A is a linear fun tional on R for all j, k . The analogous de omposition is foran operator A r additive R homogeneous valued fun tion A ( t ) .h , t ∈ [ a, b ] . If ea h j,k A ( t ) fun tion is Lebesgue-Stieltjes integrable on [ a, b ] relative to ea h fun tion k P of boundedvariation, then there is de(cid:28)ned the non ommutative Lebesgue-Stieltjes integral ( L − S ) R ba A ( t ) .dP = P j,k [ R ba j,k A ( t ) .d k P ( t )] i j . On the the other hand, if v ( t ) = vi + ... + r − vi r − is a fun tion so that ea h j v ( t ) is Lebesgue integrable on [ a, b ] , then there isde(cid:28)ned the non ommutative Lebesgue integral ( L ) R ba v ( t ) dt = P r − j =0 R ba j v ( t ) dt .Consequently, R γ f ( z ) dz = ( L − S ) R ba ˆ f ( γ ( t )) .dγ ( t ) ,where ˆ f is an A r additive R homogeneous operator orresponding to f , ( L − S ) R ba ˆ f ( γ ( t )) .dγ ( t ) denotes the non ommutative Lebesgue-Stieltjes integral. If γ ( t ) is absolutely ontinu-ous, then the non ommutative Lebesgue-Stieltjes integral redu es to the non ommutativeLebesgue integral ( L − S ) R ba ˆ f ( γ ( t )) .dγ ( t ) = ( L ) R ba ˆ f ( γ ( t )) .γ ′ ( t ) dt , sin e28 L − S ) R ba j,k A ( t ) d k γ ( t ) = ( L ) R ba j,k A ( t ) k γ ′ ( t ) dt for ea h j, k (see also ŸVI.6.2 [12℄).14. Parti ular ases of integrals and residues.For onvenien e we an hoose the parametrization of the urve so that its path γ :[ α, β ] → A r has [ α, β ] = [0 , if another is not spe i(cid:28)ed, where γ (0) = a , γ (1) = b areCayley-Di kson numbers.While al ulation of line integrals of A r di(cid:27)erentiable fun tions f : U → A r , where U isa domain in A r satisfying onditions of the homotopy Theorem 2.15 [17, 18℄, for a re ti(cid:28)ablepath γ in U ontained in a omplex plane \ R ⊕ M R with a marked purely imaginary Cayley-Di kson number M , | M | = 1 , it is possible to make simpli(cid:28) ations in algorithms. For this hoose purely imaginary Cayley-Di kson numbers N ,..., N r − being generators in A r , ≤ r ∈ N , su h that N = M , | N | = 1 ,..., | N r − | = 1 , N j ⊥ N k for ea h ≤ j = k ≤ r − , thatis Re ( N j N k ) = 0 , and hen e ertainly N j N k = − N k N j for all ≤ j = k ≤ r − ,..., N = 1 , N N j = N j N for all j . This an be done standardly by indu tion using doubling generators N , N ,..., N r − , so that N = N N , ..., N r − + p = N p N r − for ea h p = 1 , ..., r − − .With the doubling pro edure of A s +1 from A s , ≤ s , the multipli ation rule is given by theformula: (1) ( u + vl )( w + xl ) = ( uw − ˜ xv ) + ( xu + v ˜ w ) l ,for ea h u, v, w, x ∈ A s , where l = N s , u + vl and w + xl ∈ A s +1 (see [1, 11, 13℄).With this new basis of generators write f in the form (2) f ( z ) = g ( z ) N + g ( z ) N + ... + r − − g ( z ) N r − ,where p g ( z ) ∈ C M for ea h z ∈ U and all p = 0 , ..., r − − , while C M := R ⊕ M R denotesthe omplex plane embedded into A r . If z is an arbitrary Cayley-Di kson number z ∈ A r ,then it an be written as (3) z = x + yN z ,where x = x z ∈ C M , y = y z ∈ R , N z is a purely imaginary Cayley-Di kson number may bedependent on z , | N z | = 1 , N z ⊥ M , whi h follows also from (2) . Sin e N z ⊥ M and they arepurely imaginary, then N z M = − M N z . Thus z k = x k + y k N z k for any integer k ∈ Z with x k ∈ C M and y k ∈ R , with purely imaginary N z k , N z k ⊥ M .Ea h p g is the C M valued fun tion, onsequently, up to the isomorphism it is the omplexlo ally analyti fun tion. Therefore, its restri tion on C M an be written in the form (4) p g ( x ) | C M = P k c k,p ( x − x ) k , where x ∈ C M is a marked point and c k,p ∈ C M forea h k . Suppose that ν is some phrase of f in the z -representation of f on U pres ribed byEquations (2 , on the entire domain U with a ommon x ∈ C M independent from p sothat the series onverge uniformly on U . Using the translation z z − z we an onsiderfor simpli ity, that x = 0 . Then (5) R γ P k,p c k,p x k N p dx = P k,p [ c k,p / ( k + 1)]( b k +1 − a k +1 ) N p when f does not ontain anysingularities in U , that is A r di(cid:27)erentiable in U , where a = γ (0) , b = γ (1) , γ ( t ) with t ∈ [0 , is the re ti(cid:28)able path in C M .If z ∈ C M ∩ U is an isolated pole of f en ompassed by γ , then (6) Res ( z , ν ) .M = P p ( c − ,p M ) N p . 29ndeed, ea h Cayley-Di kson number an be written in the polar form also (7) z = | z | exp( Arg ( z )) , where Arg ( z ) is a purely imaginary Cayley-Di kson number, sothat Arg ( z ) = αM + βN z with α, β ∈ R , onsequently, z k = | z | k exp( kArg ( z )) for every k ∈ Z . Therefore, z k = | z | k exp( kM φ ) for ea h z ∈ C M , where φ = φ ( z ) ∈ R . The lineintegral is additive, hen e Z γ X p p g ( z ) N p dz = X p ( Z γ p g ( z ) dz ) N p . Then ea h term of the form η ( z ) = { az k b } q (3) on U with integer k = − has the fun tion v ( z ) given by any either left or the right algorithm as v ( z ) = { az k +1 b } q (3) with ( dv ( z ) /dz ) . { az k b } q (3) , sin e dv ( z ) /dz = P kj =0 { a (( z j ) z k − j ) b } q (3) , where denotes the unit operator on A r , while q (3) indi ates on the order of multipli ation for r ≥ , for r = 2 the quaternionalgebra is asso iative. In ea h C M for η with k = − the fun tion v has the restri tion v | C M = { a Ln ( z ) b } q (3) for z = 0 , sin e ( de z /dz ) .h = e z h and ( dLn ( z ) /dz ) .h = z − h forea h = z ∈ C M , h ∈ C M and the logarithmi fun tion Ln ( z ) is the inverse fun tionof the exponential fun tion (see also [17, 18, 25℄). Thus Res ( z , { a ( z − z ) − b } q (3) ) .M = { aM b } q (3) for any purely imaginary Cayley-Di kson number M . Another valuable identitiesfor al ulating residues follow from Formulas 9 (3 − .Take the word η ( z ) = { a z n ...a k z n k a k +1 } q (2 k +1) the restri tion of whi h on C M has theform { b ...b k +1 ( z s ˜ z m ) } q ( k +2) , where onstants a , ..., b k +1 belong to A r , ≤ r ≤ , k ∈ N , n , ..., n k , s, m ∈ Z . Then for the re ti(cid:28)able loop γ en ompassing zero in C M with thewinding number j we infer, that (8) (2 πj ) − R γ { a z n ...a k z n k a k +1 } q (2 k +1) dz = { b ...b k +1 M } q ( k +2) for m − s = 1 with n = − ; (9) (2 πj ) − R γ { a z n ...a k z n k a k +1 } q (2 k +1) dz = { b ...b k +1 ˜ M } q ( k +2) for s − m = 1 with n = − ; (10) (2 πj ) − R ψ { a z n ...a k z n k a k +1 } q (2 k +1) dz = 0 for either | m − s | 6 = 1 or n = − , where n = n + ... + n k , s + m = n ,sin e exp( − πM t ) M exp(2 πM t ) = M in A r , also ( de z /dz ) .h = e z h for Im ( h ) ∈ Im ( z ) R .If z = a + bM is some Cayley-Di kson number with a, b ∈ R and a purely imaginarynumber M , then any other z ∈ A r an be written in the form z = α + βM + φN , where α, β, φ ∈ R , N is a purely imaginary number orthogonal to M , N ⊥ M . Therefore, (11) z z = ( α + βM ) z + φN ˜ z , sin e N M = − M N .This identity and the formulas given above together with the additivity and R homoge-neousity of the non ommutative line integral over A r an be used for al ulations of integralsalong paths in planes su h as C M and the orresponding residues Res ( z , µ ) .M .Generally if there is given f and its phrase µ is spe i(cid:28)ed, then su h transformations to theform (2 , may hange the phrase, so µ may be not equal to ν . If R γ ν ( z ) dz or Res ( z , ν ) .M is al ulated, then using transition formulas from µ to ν and vise versa one may al ulate thesequantities for µ if these transition formulas do not hange su h integrals and residues, thatgenerally may be not a ase. Also mention that K exp( M t ) = K cos( t ) + KM sin( t ) =: N ( t ) K and M with | M | = 1 and the real variable t .If K and M are perpendi ular, K ⊥ M , and | K | = 1 , then K exp( M t ) = exp( πN ( t ) / andhen e Ln ( K exp( M t )) = πN ( t ) / πN ( t ) k , k ∈ Z depending on the bran h of Ln , sin e | N ( t ) | = 1 .There is also an interesting parti ular ase of terms η ( z ) = { c ,n z n ...c k,n k z n k c k +1 ,n k +1 } q (2 k +1) ,when there exist K and M purely imaginary numbers and onstants c j,n j in the Cayley-Di kson algebra A r , ≤ r ≤ , so that Im ( c j,n j ) ⊥ K C M for ea h j , where | K | = | M | = 1 , K ⊥ M , Im ( z ) := ( z − ˜ z ) / , n j ∈ N = { , , , ... } for ea h j , n = n + ... + n k > . The real(cid:28)eld R is the enter of A r and ea h purely imaginary number S orthogonal to K C M anti- ommutes with ea h z ∈ K C M . Then using multipli ations of generators and distributivityof the multipli ation ( a + b ) z = az + bz in A r one (cid:28)nds the restri tion v ( z ) | K C M of a fun tion v ( z ) whi h an be redu ed to the form αz n +1 / ( n + 1) and it has the extension on A r su hthat ( dv ( z ) /dz ) .h = P nj =0 α (( z j h ) z n − j ) / ( n + 1) for ea h z and h ∈ A r ,where a onstant α is in A r , Im ( α ) ⊥ K C M , sin e the o tonion algebra is alternative andthe quaternion skew (cid:28)eld is asso iative. Therefore, ( dv ( z ) /dz ) . η ( z ) for ea h z ∈ K C M .On the other hand, ea h fun tion given by either the left or right algorithm of integration isof total degree by z equal to n + 1 and an be redu ed to the form αz n +1 / ( n + 1) on K C M .Evidently ( v ( b ) + w ( x )) − ( v ( a ) + w ( x )) = v ( b ) − v ( a ) for ea h fun tion w ( x ) with values in A r and the argument x ∈ A r ⊖ K C M ⊖ R , Im ( x ) ⊥ K C M . This implies that (12) R γ η ( z ) dz = α ( b n +1 − a n +1 ) / ( n + 1) , when n ∈ N , a and b ∈ K C M and either onditions of the homotopy Theorem 2.15 [17, 18℄ are satis(cid:28)ed or γ ([0 , ⊂ K C M .Another example is of a fun tion f whi h an be written in the form f ( z ) = ( a ( z )(( b ( z )1 / ( z − y )) c ( z ))) e ( z ) in a neighborhood of y ∈ A r , where a ( z ) , b ( z ) , c ( z ) and e ( z ) are A r -holomorphi and a ( y ) = 0 , b ( y ) = 0 , c ( y ) = 0 and e ( y ) = 0 , ≤ r ≤ . Then the residue operator is: (13) Res ( y, f ) .N = (2 π ) − lim <β → Z γ β ( a ( z )(( b ( z )(1 / ( z − y )) c ( z ))) e ( z ) dz = ( a ( y )(( b ( y ) N ) c ( y )) e ( y ) , where to a in Formula 1 ( i ) here orresponds y , while to y in 1 ( i ) here orresponds β .In relation with words and phrases there are also some useful identities. Let q ( n + 2) bea ve tor indi ating an order of the multipli ation of n + 2 multipliers, then (14) ( d { az n b } q ( n +2) /dz ) .h = P n − j =0 { az j hz n − j − b } q ( n +2) for ea h h ∈ A r , where z n is treated as the produ t of n multipliers z . Sin e the Cayley-Di kson algebra A r , r ≥ , is power asso iative, then (15) ( d { z n } q ( n ) /dz ) .h = P n − j =0 { z j hz n − j − } q ( n ) = P n − j =0 { z j hz n − j − } q l ( n ) for ea h q ( n ) , but q ( n ) is the same for all additives in the sum independently from j , where q l ( n ) orresponds to the left order of bra kets, { a ...a n } q ( n ) = (( ... ( a a ) ... ) a n − ) a n . In theo tonion algebra an order of multipli ations in the term { z j hz n − j − } q ( n ) is not so important(see Ÿ9) due to the Moufang identities. 31he omplete di(cid:27)erential is ( Dη ( z, ˜ z )) .h = ( ∂η ( z, ˜ z ) /∂z ) .h + ( ∂η ( z, ˜ z ) /∂ ˜ z ) .h for a ( z, ˜ z ) (super) di(cid:27)erentiable phrase η ( z, ˜ z ) . Parti ularly for η ( z, ˜ z ) = z n ˜ z m the omplete di(cid:27)erentialis the same for ˜ z m z n , sin e z and ˜ z ommute. This implies, that (16) (( dz n /dz ) .h )˜ z m ) + z n ( d (˜ z m ) /d ˜ z ) .h ) = ˜ z m (( dz n /dz ) .h ) + (( d ˜ z m /d ˜ z ) .h ) z n for all z, h ∈ A r .Thus Formulas (1 − of this se tion and Formulas 9 ( M − M , − an serve forpra ti al al ulations of residues.In the ase of Cayley-Di kson algebras al ulations of line integrals are more ompli atedbe ause of non ommutativity, non asso iativity and possible behavior of fun tions aroundbran hing points z bran hes of whi h generally depend on M/ | M | for M = Im ( z − z ) = 0 .Evaluate some useful integrals. In the notation of [17, 18℄ ˆ f is the operator su h that ˆ f ( z ) .dz = ( dg ( z ) /dz ) .dz for a (super) di(cid:27)erentiable fun tion f ( z ) with ( dg ( z ) /dz ) . f ( z ) in an open subset U and g is al ulated with the help of either left or right algorithm forea h power series. Then ( z n ) ˆ . . ∆ z = ∆( z n +1 / ( n + 1)) + o (∆ z ) with n ∈ N and ( e z ) ˆ . . ∆ z = ∆( e z ) + o (∆ z ) for ea h z ∈ A r , (1 / (1 + y )) ˆ . . ∆ y = ∆( Ln (1 + y )) + o (∆ y ) for | y | < , y ∈ A r , sin e the power series Ln (1 + y ) = P ∞ n =1 ( − n +1 y n /n has real expansion oe(cid:30) ients and uniformly onverges in ea h ball of radius < ρ < withthe enter at zero, where ∆ y is a su(cid:30) iently small in rement of y . Thus (17) R γ z n dz = ( b n +1 − a n +1 ) / ( n + 1) , (18) R γ e z dz = e b − e a for ea h re ti(cid:28)able path γ su h that γ (0) = a and γ (1) = b ; (19) R γ dLn (1 + y ) = R γ (1 + y ) − dy = Ln (1 + b ) − Ln (1 + a ) for γ su h that | γ ( t ) | < for ea h t ∈ [0 , . On the other hand, if Im ( z ) and Im ( y ) ∈ M R for some marked purely imaginary Cayley-Di kson number M , then z and y ommute, onsequently, Ln ( z (1 + y )) = Ln ( z ) + Ln (1 + y ) for su h y and z . Therefore, formula (19) isvalid also for ea h re ti(cid:28)able γ in the ( C M \ { z ∈ A r : z ≤ } ) − , where C M = R ⊕ M R is the omplex plane embedded into the Cayley-Di kson algebra A r .Then for any given re ti(cid:28)able path γ in A r \ Q , where Q is the slit ( ut) sub-manifoldused for onstru tion of the non ommutative analog of the Riemann surfa e over A r , hoosea sequen e γ n of re ti(cid:28)able paths in A r \ Q so that γ n is the ombination of paths γ n,j :[ b j − , b j ] → A r in C M n,j , γ n ( t ) = γ n,j ( t ) for ea h t ∈ [ b j − , b j ] , j = 1 , ..., m ( n ) ∈ N , n ∈ N , b = 0 < b ... < b m ( n ) = 1 . Then for ea h γ n,j and hen e for ea h γ n Formula (19) is alsovalid in ( A r \ Q ) − . The limit by n then gives (20) R γ dLn ( z ) = R γ z − dz = Ln ( b ) − Ln ( a ) for ea h re ti(cid:28)able path γ in A r \ Q with a hosen bran h of Ln . This is natural, sin e ( dLn ( z ) /dz ) . /z for ea h z = 0 in A r .If f ( z ) is a univalent fun tion in an open domain U in A r having a power series de om-position f ( z ) = P ∞ n =0 a n z n uniformly onverging in U with real expansion oe(cid:30) ients a n ,then due to (17) we dedu e, that (21) R γ f ( z ) dz = P ∞ n =0 a n ( b n +1 − a n +1 ) / ( n + 1) ,32here γ : [0 , → U is a re ti(cid:28)able path in U with γ (0) = a and γ (1) = b . In parti ular thisgives: (22) R γ sin( z ) dz = cos( a ) − cos( b ) , (23) R γ cos( z ) dz = sin( b ) − sin( a ) ,where sin( z ) = P ∞ n =0 ( − n z n +1 / (2 n + 1)! , cos( z ) = P ∞ n =0 ( − n z n / (2 n )! .If f is a (super) di(cid:27)erentiable fun tion of the o tonion variable not equal to zero on a openset U , then f ( z )(1 /f ( z )) = 1 on U and (( df ( z ) /dz ) .h )(1 /f ( z )) + f ( z )(( d [1 /f ( z )] /dz ) .h ) forea h z ∈ U and h ∈ O , onsequently, ( d [1 /f ( z )] /dz ) .h = − [1 /f ( z )](( df ( z ) /dz ) .h )[1 /f ( z )] onthe orresponding domain (see Proposition 3.8.2 [18℄). The Cayley-Di kson algebra is powerasso iative, so repeating this for z n with n ∈ N instead of f we get, that ( dz − n /dz ) .h = − z − n (( dz n /dz ) .h ) z − n for ea h z = 0 and h in A r with Im ( z ) and Im ( h ) ∈ M R for somemarked purely imaginary number M . In parti ular, ( dz n /dz ) . nz n − for ea h integernumber n with z = 0 , when n < , dz /dz = 0 . Thus (24) R γ z n dz = ( b n +1 − a n +1 ) / ( n + 1) for ea h integer n = − an for every re ti(cid:28)able path γ with a = γ (0) , b = γ (1) , so that γ ([0 , does not ontain ,when n < .If α ∈ A r then we de(cid:28)ne z α := exp( αLn ( z )) for ea h z = 0 . So we al ulate its derivative ( dz α /dz ) .h = ( de y /dy ) | y = αLn ( z ) . ( α [( dLn ( z ) /dz ) .h )]) for ea h h ∈ A r . Parti ularly, ( dz α /dz ) . de y /dy ) | y = αLn ( z ) . ( αz − ) . Take α ∈ R , then ( dz α /dz ) . αz α − , hen e (25) R γ z α dz = ( b α +1 − a α +1 ) / ( α + 1) for ea h real α = − and the re ti(cid:28)able path in A r \ Q .If α ∈ R , then we infer the power series de omposition (1+ z ) α = P ∞ n =0 (cid:0) αn (cid:1) z n for ea h | z | < , where (cid:0) αn (cid:1) = α ( α − ... ( α − n +1) /n ! . Parti ularly,for α = − / and z = − y we dedu e, that (26) R γ (1 / p − y ) dy = arcsin( b ) − arcsin( a ) for γ ([0 , ⊂ [ A r \ { z ∈ A r : Re ( z ) = 0 } ] , where the square root bran h is taken √ x > for ea h x > , | γ ( t ) | < for ea h t ∈ [0 , .For the tangent fun tion its power series has the form: tan( z ) = P ∞ n =1 n (2 n − B n z n − / (2 n )! andthe otangent fun tion is: z cot( z ) = 1 − P ∞ n =1 n B n z n / (2 n )! ,where B n are the Bernoulli numbers and these power series absolutely onverge in the balls | z | < π/ and | z | < π respe tively. The Bernoulli numbers appear from the generatingfun tion x/ ( e x −
1) = 1 + P ∞ n =1 β n x n /n ! for ea h x ∈ ( − δ, δ ) ⊂ R with su(cid:30) iently small δ > so that β n = ( − n − B n , β n +1 = 0 for ea h n ≥ (see ŸXII.4 (445, 449) in [6℄). Due toFormula (25) and the periodi ity of the trigonometri fun tions we get: (27) R γ cos − ( z ) dz = tan( b ) − tan( a ) , when | γ ( t ) − πm | < π/ for ea h t ∈ [0 , for somemarked m ∈ Z , (28) R γ sin − ( z ) dz = cot( a ) − cot( b ) , when | γ ( t ) − πm − π/ | < π/ for ea h t ∈ [0 , forsome marked m ∈ Z . On the other, hand tan( z ) and cot( z ) are meromorphi fun tions so33hat ( d tan( z ) /dz ) . − ( z ) for ea h z ∈ V := { y ∈ A r : Re ( y ) = π ( m + 1 / , m ∈ Z } ,also ( d cot( z ) /dz ) . − sin − ( z ) for ea h z ∈ U := { y ∈ A r : Re ( y ) = πm, m ∈ Z } .Therefore, Formulas (27 , extend on any re ti(cid:28)able path γ in V or U orrespondingly.Mention also that Formula (18) implies: (29) R γ cosh( z ) dz = sinh( b ) − sinh( a ) , (30) R γ sinh( z ) dz = cosh( b ) − cosh( a ) , where cosh( z ) = ( e z + e − z ) / , sinh( z ) = ( e z − e − z ) / .For coth( z ) = cosh( z ) / sinh( z ) the power series z coth( z ) = 1 + P ∞ n =1 ( − n − n B n z n / (2 n )! absolutely onverges in the ball | z | < π , hen e (31) R γ sinh − ( z ) dz = coth( a ) − coth( b ) for < | γ ( t ) | < π for ea h t ∈ [0 , . We an mention that if y is written in the polarform y = ρe S , where ρ = | y | ≥ and S is a purely imaginary Cayley-Di kson number, then y = − if and only if ρ = 1 and | S | = π ( m + 1 / with m = 0 , , , , ... .Sin e tanh( z ) and coth( z ) are lo ally analyti and ( d tanh( z ) /dz ) . − ( z ) for ea h z ∈ V := A r \{ y : Re ( y ) = 0 , | Im ( y ) | = π ( m +1 / , m = 0 , , , , ... } and ( d coth( z ) /dz ) . − sinh − ( z ) for ea h z ∈ U := A r \ { y : Re ( y ) = 0 , | Im ( y ) | = πm, m = 0 , , , , ... } , then (31) extends on γ ⊂ U and (32) R γ cosh − ( z ) dz = tanh( b ) − tanh( a ) for ea h γ ⊂ V .The series / (1+ z ) = P ∞ n =0 (cid:0) − / n (cid:1) z n /n ! absolutely onverges for | z | < and its integralgives arctan in this ball. The fun tion arctan( z ) is lo ally analyti and ( d arctan( z ) /dz ) . / (1+ z ) for z = − , sin e ( d tan( z ) /dz ) . / cos ( z ) and ( df − ( z ) /dz ) | z = f ( y ) = ( df ( y ) /dy ) − on the orresponding domains (see Proposition 3.8.1 in [18℄). Thus (33) R γ [1 / (1 + z )] dz = arctan( b ) − arctan( a ) for γ ⊂ { y ∈ A r : y = − } .In the o tonion algebra due to its alternativity ( dLn ( z + √ z + α ) /dz ) . / √ z + α for a non zero real number α and ea h z = − α in O , where Ln ( z + √ z + α ) and √ z + α are lo ally analyti . Therefore, with the help of Formula (20) we infer: (34) R γ [1 / √ z + α ] dz = R γ dLn ( z + √ z + α )= Ln ( b + √ b + α ) − Ln ( a + √ a + α ) for ea h path γ in the set U := { y ∈ O : y = − α, Re ( y ) = 0 } , sin e in it bran hes offun tions / √ z + α and Ln ( z + √ z + α ) are spe i(cid:28)ed. One mentions that y = − α if andonly if ρ = | α | and | S | = π ( m + κ ( α ) / for m ∈ Z , where y = ρe S , ρ = | y | , Re ( S ) = 0 , κ ( α ) = 1 for α > , κ ( α ) = 0 for α ≤ .Consider now the general algorithm in more details. Let f and f be two analyti fun -tions of z on U . Denote f = f , f − n = f ( n ) , f ( n ) ( z ) := ( d n f ( z ) /dz n ) . ⊗ n , f n is su h that ( df n ( z ) /dz ) . f n − ( z ) for ea h n ∈ N . Parti ularly, ( { az k b } q (3) ) n = { az k + n b } q (3) [( k +1) ... ( k + n )] − for onstants a, b ∈ A r , k = 0 , , , ... , where the symbol z is also integratedfor onvenien e so that ea h phrase of a z (super) di(cid:27)erentiable fun tion is a series of ad-dends having obligatory omponents z n or e = (1) (see also Ÿ2.14 [21℄). In view of this ( df ( z ) /dz ) .h = ˆ f .h and f f l is the anti-derivation operation of order l , l ∈ N . Then theleft algorithm is (35) ( f f ) = P n s =0 ( − s f s f − s , where n is the least natural number su h that34 − n − = 0 . The right algorithm is symmetri to the left. The work with phrases andpower series redu es to produ ts of polynomials and their sums, so onsider this algorithmfor ( { f ...f k } ) l , where l ∈ N , f ,..., f k are polynomials by z of degrees n , ..., n k respe tively.Then (36) ( f f ) = P n s =0 ( − s ( f s f − s ) = P n s =0 P n − s s =0 ( − s + s f s + s f − s − s = P n s =0 ( − s ( s + 1) f s f − s .Continue these al ulations by indu tion using that P ss =0 P s − s s =0 ... P s − s − ... − s p − s p =0 (cid:0) s + p − s (cid:1) ,where (cid:0) nm (cid:1) = n ! / ( m !( n − m )!) denotes the binomial oe(cid:30) ient. Then (37) ( f f ) l = P n s =0 ( − s (cid:0) s + l − s (cid:1) f s + l f − s .Applying Formula (37) to produ t of k terms we dedu e that (38) ( { f f ...f k } q ( k ) ) l = P n k s k =0 ( − s k (cid:0) s k + l − s k (cid:1) { [ f ...f k − ] s k + l f − s k k } q ( k ) = P n k s k =0 P n k − s k − =0 ... P n s =0 ( − s + ... + s k (cid:0) s k + l − s k (cid:1)(cid:0) s k − + s k + l − s k − (cid:1) ... (cid:0) s + ... + s k + l − s (cid:1) { f l + s + ... + s k f − s ...f − s k k } q ( k ) ,where the notation [ f ...f k − ] s k + l does not mean any order of multipli ations but only thatthe anti-derivation operator is of order s k + l for a olle tion of these terms in square bra kets.For the skew (cid:28)eld of quaternions urled bra kets an be omitted due to the asso iativity.Thus either left or right algorithm spe i(cid:28)es the bran h of the non ommutative line integralfor f (see also [17, 18, 21℄) and for this Formulas (35 − are helpful.If f ( z ) is an analyti fun tion on a ball B with the enter at zero and the uniformly onverging power series of f has real expansion oe(cid:30) ients, also α ∈ O , α = 0 , then themapping z αzα − in O indu es the mapping f ( z ) αf ( z ) α − so that αf ( z ) α − = f ( αzα − ) . But the line integral hanges as (39) α ( R γ f ( z ) .dz ) α − = R γ f ( αzα − ) . ( αdzα − ) = R ψ f ( y ) .dy for su h fun tion f , sin e the o tonion algebra is alternative and R is its enter, where ψ ( t ) = αγ ( t ) α − for ea h t ∈ [0 , . If f is a multi-valued fun tion, for example, Ln ( z ) or n √ z , n ≥ , then the mapping z αz or z zα may ause a transition from one bran h of f into another.15. Proposition. Let f and f be two (super) di(cid:27)erentiable univalent fun tions on anopen set U in the Cayley-Di kson algebra A r and γ be a re ti(cid:28)able path in U . Then R γ f ( z )[( df ( z ) /dz ) . dz = f ( z ) f ( z ) | ba − R γ [( df ( z ) /dz ) . f dz .Proof. Consider the (super) derivative ( df ( z ) f ( z ) /dz ) .h = [( df ( z ) /dz ) .h ] f ( z ) + f ( z )[( df ( z ) /dz ) .h ] ,where z ∈ U and h ∈ A r . Then we take the integral sums P nj =1 orresponding to partitions P of γ . Therefore, in the notation of Ÿ11 we get R γ [( df ( z ) f ( z ) /dz ) . dz = lim δ ( P ) → P nj =1 ( df ( z ) f ( z ) /dz ) | z = z j . ∆ j z = f ( z ) f ( z ) | ba = lim δ ( P ) → P nj =1 \ ([( df ( z ) /dz ) . f ( z )) . ∆ j z + lim δ ( P ) → P nj =1 \ ( f ( z )[( df ( z ) /dz ) . . ∆ j z = R γ [( df ( z ) /dz ) . f ( z ) dz + R γ f ( z )[( df ( z ) /dz ) . dz , onsequently, R γ f ( z )[( df ( z ) /dz ) . dz = f ( z ) f ( z ) | ba − R γ [( df ( z ) /dz ) . f dz .35 A(cid:30)ne algebras over o tonions1. De(cid:28)nition. If a generalized Cartan matrix A has positive all proper main minorsand det ( A ) = 0 , then one says that A is of the a(cid:30)ne type. The algebra g ( A ) (see Ÿ2.6) orresponding to the generalized Cartan matrix A of the a(cid:30)ne type is alled the a(cid:30)nealgebra over A r .2. Remarks. Let f and g be two meromorphi fun tions on an open set V in theCayley-Di kson algebra with singularities at a set W satisfying onditions 3.1 ( R − R .Then ( df ( z ) g ( z ) /dz ) .h = [( df ( z ) /dz ) .h ] g ( z ) + f ( z )[( dg ( z ) /dz ) .h ] for ea h h ∈ A r and every z ∈ V \ W =: U . In parti ular, take h = 1 . On the other hand, R γ [( df ( z ) g ( z ) /dz ) . dz = 0 for ea h re ti(cid:28)able loop γ in U and we get (1) Res ( z , ( f ′ ( z ) . g ( z )) .M = − Res ( z , f ( z )( g ′ ( z ) . .M for ea h z ∈ V and every purely imaginary Cayley-Di kson number M (see also ŸŸ3.1, 3.6and 3.15). In the notation ψ z ( f ′ , g ) := Res ( z , ( f ′ ( z ) . g ( z )) Formula (1) takes the form (2) ψ z ( f ′ , g ) = − ψ z ( f, g ′ ) for ea h z ∈ U , where ψ z ( f ′ , g ) .M is A r additive by f andby g also, by purely imaginary Cayley-Di kson number M , moreover, it is R homogeneousby f and by g and by M , whi h follows from the properties of the residue operator (seeTheorem 3.6 also).Consider the set theoreti al omposition of mappings f ◦ f ( y ) = f ( f ( y )) . In view ofProposition 2.2.1 and Theorems 2.11, 2.15, 2.16 and 3.10 [18℄ we an reformulate the result.2.1. Proposition. Let g : U → A mr , r ≥ , and f : W → A nr be two di(cid:27)erentiablefun tions on U and W respe tively su h that g ( U ) ⊂ W , U is open in A kr , W is open in A mr , k, n, m ∈ N , where f and g are simultaneously either ( z, ˜ z ) , or z , or ˜ z -di(cid:27)erentiable. Thenthe omposite fun tion f ◦ g ( z ) := f ( g ( z )) is di(cid:27)erentiable on U and (3) ( Df ◦ g ( z )) .h = ( Df ( g )) . (( Dg ( z )) .h ) for ea h z ∈ U and ea h h ∈ A kr , and hen e f ◦ g is of the same type of di(cid:27)erentiability as f and g .2.2. Multipli ations of operators. The set theoreti omposition serves as the for-getful fun tor on an order of the derivatives: (4) ( d ( f ◦ f ) ◦ f ( x ) /dx ) .h = ( df ◦ f ( y ) /dy ) | y = f ( x ) . [( df ( x ) /dx ) .h ] =( df ◦ f ( y ) /dy ) | y = f ( x ) .v | v =( df ( x ) /dx ) .h = [( df ( z ) /dz ) | z = f ( y ) . [( df ( y ) /dy ) | y = f ( x ) .v ] | v =( df ( x ) /dx ) .h = ( df ( z ) /dz ) | z = f ( y ) . [( df ( y ) /dy ) | y = f ( x ) . [( df ( x ) /dx ) .h ]] = ( df ( f ( f ( x ))) /dx ) .h for every h ∈ A n r , for open domains U j in A n j r for j = 1 , , so that f j : U j → U j +1 for j = 1 , , f : U → A nr , n , n , n , n ∈ N , f j is z di(cid:27)erentiable on U j for ea h j = 1 , , . If onsider A r additive R homogeneous operators A j on ve tor spa es X j over A r , A j : X j → X j +1 , with another type of omposition A ⋄ A indu ed by that of the matrix multipli ationof matri es with entries in A r , also Cayley-Di kson numbers and also by the multipli ationof matri es on the orresponding ve tors, then it will be ne essary to des ribe an order ofsu h new multipli ation indi ating a ve tor q ( m ) pres ribing su h order { A m ⋄ ... ⋄ A } q ( m ) ,36here A j ⋄ h j := A j ( h j ) for ea h h j ∈ X j . The set theoreti omposition of operators (5) ( A ◦ A ) ◦ A ( h ) = ( A ◦ A )( v ) | v = A ( h ) = A ( A ( v )) | v = A ( h ) = A ( A ( A ( h ))) for ea h h ∈ X is always asso iative and they orrespond as (6) A m ◦ ... ◦ A ( h ) = A m ⋄ ( A m − ⋄ ... ( A ⋄ ( A ⋄ h )) ... ) for ea h h ∈ X , parti ularly, (7) ( A ◦ A )( h ) = A ⋄ ( A ⋄ h ) su h that the set theoreti omposition of operators an be onsidered as the parti ular aseof more general multipli ation of operators with the right omposition on the right side ofFormulas (6 , . Mention that generally ( A ⋄ A ) ⋄ h may be not equal to A ⋄ ( A ⋄ h ) over O . Over the quaternion skew (cid:28)eld H bra kets an be omitted, sin e H is asso iative.2.3. Take the algebra M ( V, W ) of all meromorhi fun tions in V with singularities ontained in W , where V is open in ˆ A r and W satisfy Conditions 3.1 ( R − R , W ⊂ V . Inparti ular, for a singleton { } instead of W this gives the algebra L ( V ) of all Loran serieson the set V \ { } open in A r : (8) η ( z ) = P m,j,p { A m,j , z m } q (2 p ) with enter at zero, where (9) { A m,j , z m } q (2 p ) := { a m,m , z m ...a m,m p ,p z m p } q (2 p ) , a m,k,p ∈ A r and m k ∈ Z for ea h k ∈ N , j, p ∈ N , m = ( m , ..., m p ) (see [17, 18℄). To ea hphrase η ( z ) the fun tion f η ( z ) = ev z ( η ) = η ( z ) orresponds, where ev z denotes the valuationoperation at z ∈ U .There is the natural equivalen e relation of phrases pres ribed by the rules: (10) bη η = η bη = η η b for ea h real number b , (11) ( z l z n ) = ( z n + l ) for all integers l, n , (12) ( η + η ) − η = η for all phrases η and η , sin e A r is the power asso iative algebrawith the enter R . Another equivalen es are aused by asso iativity of the quaternion skew(cid:28)eld H and the alternativity of the o tonion algebra O . Over H bra kets in phrases an beomitted.If W is not a singleton, then the restri tion of f ∈ M ( V, W ) on z + C M has the Loranseries (9) by z − z ∈ C M instead of z for ea h z ∈ W and ea h purely imaginary Cayley-Di kson number M ∈ A r . This follows from Formulas (3 . , [17, 18℄ applied to re ti(cid:28)ableloops γ and γ in z + C M and Conditions 3.1 ( R − R .2.3. De(cid:28)nitions. For a generalized Cartan matrix A take the matrix (cid:6) A obtained from A by deleting its zero olumn and zero row and put ˇ g = g ( (cid:6) A ) . De(cid:28)ne the wrap algebra M (ˇ g ) := M ( V, W ) ⊗ A r ˇ g with the multipli ation: [ P ⊗ x, Q ⊗ y ] := P Q ⊗ [ x, y ] for all P, Q ∈ M ( V, W ) and x, y ∈ ˇ g .Extend the form ( x | y ) from ˇ g onto M (ˇ g ) by the formula: ( P ⊗ x | Q ⊗ y ) z = P Q ( x | y ) .3. Proposition. There exists an A r valued graded - o y le on the wrap algebra M (ˇ g ) over A r , ≤ r ≤ .Proof. The di(cid:27)erentiation 37 D P ( z ) := ( dP ( z ) /dz ) . of the algebra M ( V, W ) for z ∈ V \ W we extend to the di(cid:27)erentiation of M (ˇ g ) by the formula (2) D ( P ⊗ x ) := ( D P ) ⊗ x .De(cid:28)ne the mapping (3) ω ( z ; P ⊗ x, Q ⊗ y ) := Res ( z , ( D P ) Q ) . ( x | y ) for ea h x, y ∈ ˇ g and all P, Q ∈ M ( V, W ) and every z ∈ W , where the form ( ∗|∗ ) is givenby Theorem 2.20. Its restri tion ( ∗|∗ ) | g on g is real valued. We have (4) Res ( z , f ) .b = 0 for ea h real number b and every f ∈ M ( V, W ) with z ∈ V , sin e in this ase the line integral is al ulated alonga re ti(cid:28)able loop γ ontained in R , hen e (5) Res ( z , f ) .h = Res ( z , f ) . ( Im ( h )) for ea h Cayley-Di kson number h ∈ A r . For the de(cid:28)niteness of residues we onsider themfor spe i(cid:28)ed phrases of fun tions. Consider the ordered produ t { f f f } q (3) (usual point-wise) of z di(cid:27)erentiable fun tions f , f , f on the open set U in A r . For ea h z ∈ U wehave Res ( z , ( { f f f } q (3) ) ′ .
1) = 0 , onsequently, (6)
Res ( z , { ( f ′ . f f } q (3) ) + Res ( z , { f ( f ′ . f } q (3) )+ Res ( z , { f f ( f ′ . } q (3) ) = 0 .There is the de omposition (7) P = P i + ... + r − P i r − ,where j P is the R valued fun tion for ea h j , so that they are related by the formulas (8) P = ( P + (2 r − − {− P + P r − j =1 i j ( P i ∗ j ) } ) / , (9) k P = ( i k (2 r − − {− P + P r − j =1 i j ( P i ∗ j ) } − P i k ) / for ea h k = 1 , ..., r − .We have the natural identity (10) k P i k ⊗ k v = k P ⊗ k vi k .On the other hand, (11) ω ( P ⊗ x, Q ⊗ y ) = ψ (( DP ) , Q ) . ( x | y ) , onsequently, for pure states k ai k = P k ⊗ k xi k ∈ k M (ˇ g ) i k and bi j = Q j ⊗ j yi j ∈ j M (ˇ g ) i j we get (12) ω ( k P ⊗ k xi k , j Q ⊗ j yi j ) = ( − η ( k,j )+1 ω ( j Q ⊗ j yi j , k P ⊗ k xi k ) ,sin e the form ( x | y ) is symmetri on g ( A ) and hen e on ˇ g and satis(cid:28)es 17 (2) .At the same time we have the identities: (13) Res ( z , (( k P j Q ) ′ . s R ) + Res ( z , (( j Q s R ) ′ . k P ) + Res ( z , (( s R k P ) ′ . j Q )= 2( Res ( z , (( k P ) ′ . j Q s R ) + Res ( z , k P (( j Q ) ′ . s R )+ Res ( z , k P j Q (( s R ) ′ . Res ( z , ( k P j Q s R ) ′ .
1) = 0 and for pure states k ai k = k P ⊗ k xi k , j bi j = j Q ⊗ j yi j , s ci s = s R ⊗ s zi s we dedu etaking into a ount Theorem 2.20 and Properties 17 (1 − and Identity (13) : (14) ω ( z ; [ k ai k , j bi j ] , s ci s ) + ( − ξ ( k,j,s ) ω ( z ; [ j bi j , s ci s ] , k ai k )+( − ξ ( k,j,s )+ ξ ( j,s,k ) ω ( z ; [ s ci s , k ai k ] , j bi j )= Res ( z , ( D ( k P j Q )) s R ) . ([ k xi k , j yi j ] | s zi s )+( − ξ ( k,j,s ) Res ( z , ( D ( j Q s R )) k P ) . ([ j yi j , s zi s ] | k xi k )+( − ξ ( k,j,s )+ ξ ( j,s,k ) Res ( z , ( D ( s R k P )) j Q ) . ([ s zi s , k xi k ] | yi j )= 2 Res ( z , D ( k P j Q s R )) . ([ k xi k , j yi j ] | s zi s ) = 0 .38hus Formulas (12 , mean that ω is the graded - o y le over the Cayley-Di ksonalgebra A r .4. De(cid:28)nitions. Put ˇ M (ˇ g ) := M (ˇ g ) ⊕ F ( W, A r ) K with the multipli ation (1) [ a ⊕ α ( z ) K, b ⊕ β ( z ) K ] := [ a, b ] + ω ( z ; a, b ) K for ea h a, b ∈ L (ˇ g ) and α, β ∈ F ( W, A r ) , z ∈ W where K is the additional generator, F ( W, A r ) denotes the A r ve tor spa e of all fun tions α, β : W → A r .We introdu e the operator d l P ( z ) := − ( dP ( z ) /dz ) .z l +1 for ea h P ∈ M ( V, W ) and all z ∈ V \ W , also d l K := 0 , d l ( P ⊗ x ) = d l ( P ) ⊗ x , where l ∈ Z .De(cid:28)ne new algebra ˆ M (ˇ g ) = ˇ M (ˇ g ) ⊕A r d with the di(cid:27)erentiation d P ( z ) := ( dP ( z ) /dz ) .z = − d P ( z ) for ea h z ∈ U \ { } and every P ∈ M ( U ) and d K := 0 . We put as the multipli- ation (2) [ k P ⊗ k xi k ⊕ αi k K ⊕ ei k d , j Q ⊗ j yi j ⊕ βi j K ⊕ ti j d ] :=( k P j Q ⊗ [ k xi k , j yi j ] ⊕ ei k ( d j Q ) ⊗ j yi j − ( − η ( k,j ) ti j ( d k P ) ⊗ k xi k ) ⊕ ω ( z ; k P ⊗ k xi k , j Q ⊗ j yi j ) for pure states k P ⊗ xi k ⊕ αi k K ⊕ ei k d ∈ k ˆ M (ˇ g ) i k and j Q ⊗ yi j ⊕ βi j K ⊕ ti j d ∈ j ˆ M (ˇ g ) i j ,where α ( W ) ⊂ R , β ( W ) ⊂ R , e, t ∈ R . Extend this multipli ation on all elements of ˆ M (ˇ g ) by R bi-linearity (see Formulas 2.1 (3) ).5. Corollary. The set ˇ M (ˇ g ) is the Lie super-algebra over A r .Proof. By the onstru tion ˇ M (ˇ g ) is the A r ve tor spa e. Sin e M (ˇ g ) is the Lie super-algebra and ω satis(cid:28)es Conditions (12 , , then the multipli ation in ˇ M (ˇ g ) de(cid:28)ned byFormula 4 (1) satis(cid:28)es Identities 2.1 (1 − .6. Proposition. The operator d l from De(cid:28)nition 4 is the di(cid:27)erentiation of the algebra ˇ M (ˇ g ) .Proof. From the de(cid:28)nition of d l we get (1) d l ([ P ⊗ x, Q ⊗ y ] ) = d l ( P Q ⊗ [ x, y ])= ( d l ( P Q )) ⊗ [ x, y ] = −{ (( dP ( z ) /dz ) .z l +1 ) Q + P (( dQ ( z ) /dz ) .z l +1 ) } ⊗ [ x, y ]= (( d l P ) Q + P ( d l Q )) ⊗ [ x, y ] = [( d l P ) ⊗ x, Q ⊗ y ] +[ P ⊗ x, ( d l Q ) ⊗ y ] . By the bi-additivitywe get (2) d l [ a, b ] = [ d l a, b ] + [ a, d l b ] , onsequently, d l [ a + αK, b + βK ] = [ d l a, b ] + [ a, d l b ] for all a, b ∈ ˇ M (ˇ g ) and ea h α, β ∈ F ( W, A r ) .On the other hand, [ d l a, b ] = [ d l a, b ] + ω ( d l a, b ) K .Sin e Res ( z , ( d d l P ) Q )+ Res ( z , ( d l P )( d Q )) = Res ( z , d (( d l P ) Q )) = 0 and symmetri ally Res ( z , P d d l Q ) + Res ( z , ( d P )( d l Q )) = Res ( z , d ( P d l Q )) = 0 , then ( Res ( z , ( d d l P ) Q ) + Res ( z , ( d l P )( d Q ))) . ( x | y )= ( − Res ( z , ( d l P )( d Q )) + Res ( z , ( d l P )( d Q ))) . ( x | y ) = 0 and inevitably (3) ω ( d l a, b ) + ω ( a, d l b ) = 0 for ea h a = P ⊗ x and b = Q ⊗ y . Using R linearity of Res ( z , ( d P ) Q ) . ( x | y ) by P , Q , x and y and Equations (1 − we get the statement of thisproposition. 39. Proposition. The ommutator of di(cid:27)erentiations d k and d j on the spa e M ( V, W ) is (1) [ d k , d j ] = ( k − j ) d j + k .Proof. Sin e d k d j P Q = d k (( d j P ) Q + P d j Q ) = ( d k d j P ) Q + ( d j P )( d k Q ) + ( d k P )( d j Q ) + P ( d k d j Q ) , then [ d k , d j ] P Q = ( d k d j − d j d k ) P Q = ([ d k , d j ] P ) Q + ( d j P )( d k Q ) − ( d k P )( d j Q ) +( d k P )( d j Q ) − ( d j P )( d k Q ) + P ([ d k , d j ] Q ) = ([ d k , d j ] P ) Q + P ([ d k , d j ] Q ) .For ea h term αz l with a onstant α ∈ A r we have [ d k , d j ] αz l = l ( j + l ) αz j + l + k − l ( k + l ) αz k + l + j = ( j − k ) lαz j + l + k = ( k − j ) d j + k αz l on V \ W for ea h l ∈ Z . We also have (2) d j { α m ,j z m ...α m p ,j z m p } q (2 p ) = P ps =1 { α m ,j z m ...α m s − ,j z m s − ( α m s ,j d l z m s ) α m s +1 ,j z m s +1 ...α m p ,j z m p } q (2 p ) .By the non ommutative analog of the Stone-Weierstrass theorem (see Ÿ2 [17, 18℄) forea h ompa t anoni al losed subset J in V \ W the set of polynomials over A r is dense inthe set of ontinuous A r valued fun tions on J . For ea h z di(cid:27)erentiable f fun tion on J there exists a sequen e of polynomials f n onverging to f with f ′ n onverging to f ′ uniformlyon J and J × { h ∈ A r : | h | ≤ } respe tively. Applying identities dedu ed above (3) [ d k , d j ] P Q = ([ d k , d j ] P ) Q + P ([ d k , d j ] Q ) and (4) [ d k , d j ] αz l = ( k − j ) d j + k αz l to ordered produ ts of terms α m k ,j z m k with asso iators q (2 p ) , where α m k ,j are Cayley-Di kson onstants, p ∈ N , m , ..., m p ∈ Z , we get that (4) [ d k , d j ] f = ( k − j ) d k + j f for ea h f ∈ M ( V, W ) on the domain V \ W .8. Remark. From the de(cid:28)nition of operators d j it follows, that (1) [ pβd j , d k ] = [ pd j , βd k ] = pβ [ d j , d k ] for ea h real number p and every Cayley-Di ksonnumber β . Certainly (2) [ αd s + βd j , d k ] = [ αd s , d k ] + [ βd j , d k ] and [ d k , αd s + βd j ] = [ d k , αd s ] + [ d k , βd j ] forea h α, β ∈ A r . De(cid:28)ne the algebra ∂ := L j ∈ Z A r d j of ve tor (cid:28)elds on A r \ { } . The non ommutative analog of the Virasoro algebra is V r := ∂ ⊕ A r c , where c denotes the additionalgenerator and with the multipli ation (3) [ d j + sc, d k + tc ] = ( j − k ) d j + k + [( j − j ) δ j, − k / c , where s, t ∈ A r .Then it is possible to de(cid:28)ne the semidire t produ t V r ⊗ s ˇ M (ˇ g ) with the multipli ationgiven by Formulas (1 − , 4 (1) and (4) [ c, ˇ M (ˇ g )] = 0 .This algebra may be useful in the quantum (cid:28)eld theory over quaternions and o tonions.Referen es[1℄ J.C. Baez. "The o tonions". Bull. Amer. Mathem. So . 39: 2 (2002), 145-205.[2℄ Yu.A. Ba hturin. "Basi stru tures of modern algebra" (Mos ow: Nauka, 1990).[3℄ C. Barto i, U. Bruzzo, D. Hern(cid:1)andes-Ruip(cid:1)erez. "The Geometry of supermanifolds"(Dordre ht: Kluwer A ad. Publ., 1991).404℄ G. Em h. "M ` e hanique quantique quaternionienne et Relativit ` ee