Noncommutative geometry on central extension of U(u(2))
aa r X i v : . [ m a t h . QA ] S e p Noncommutative geometry on central extension of U ( u (2)) Dimitri Gurevich ∗ Universit´e Polytechnique Hauts-de-France, LMIF-59313 Valenciennes, FranceandInterdisciplinary Scientific Center J.-V.PonceletMoscow 119002, Russian Federation
Pavel Saponov † National Research University Higher School of Economics,20 Myasnitskaya Ulitsa, Moscow 101000, Russian FederationandInstitute for High Energy Physics, NRC ”Kurchatov Institute”Protvino 142281, Russian Federation
September 15, 2020
Abstract
In our previous publications we have introduced analogs of partial derivatives on thealgebras U ( gl ( N )). In the current paper we compare two methods of introducing theseanalogs: via the so-called quantum doubles and by means of a coalgebraic structure. Inthe case N = 2 we extend the quantum partial derivatives from U ( u (2)) (the compact formof the algebra U ( gl (2))) on a bigger algebra, constructed in two steps. First, we definethe derivatives on a central extension of this algebra, then we prolongate them on someelements of the corresponding skew-field by using the Cayley-Hamilton identities for certainmatrices with noncommutative entries. Eventual applications of this differential calculus arediscussed. Attempts to construct a Noncommutative (NC) geometry, related to Quantum Groups (QG)have been undertaken since the creation of the QG theory. Initiated in [W], this study wasdeveloped in [IP1, FP] and other papers. The authors of these papers aimed at construct-ing a differential calculus in which the role of the function algebra on the group GL ( N ) wasplayed by an RTT algebra and that of the one-sided vector fields was played by elements of thecorresponding Reflection Equation (RE) algebra.In [GPS2] we have constructed a different version of the calculus in which the role of functionalgebra was attributed to another copy of the RE algebra. By combining the generating matrices ∗ [email protected] † [email protected] Note that all algebras, we are dealing with, are Quantum Matrix (QM) algebras, i.e. they are introduced viasystems of relations imposed onto the entries of some matrices. We call them the generating matrices .
1f these two copies of the RE algebra, we constructed a new matrix, whose entries ∂ ji playedthe role of partial derivatives in entries of the matrix L = k l ji k ≤ i,j, ≤ N , which is the generatingmatrix for the second copy of the RE algebra.This scheme can be expressed in terms of a Quantum Double (QD) ( A, B ), where the algebra B is generated by the entries l ji of the matrix L and the algebra A is generated by the entries ofthe matrix D = k ∂ ji k ≤ i,j, ≤ N . The action of the ∂ ji on l sk is defined via the so-called permutationmap σ : A ⊗ B → B ⊗ A, and a counit ε : A → C . The ground field is assumed to be C .The RE algebra, we are dealing with, is defined by entries of a generating matrix L subjectto the following relation R L R L − L R L R = 0 , (1.1)where R is a Hecke symmetry. (The reader is referred to [GPS1] for the terminology.) Notethat by a linear shift of generators it is possible to transform the RE algebra into a modifiedRE algebra, which is defined by the system R L R L − L R L R = R L − L R. (1.2)The latter algebra tends to U ( gl ( N )), provided the Hecke symmetry R tends to the usual flip P as q → D turn at this limit into operators whichare treated to be patrial derivatives in l ji . Below, we call them the quantum partial derivatives (QPD). Similarly to the classical case, the QPD commute with each other and thus generate acommutative algebra, but the Leibniz rule for them differs from the classical one. Nevertheless,the new Leibniz rule can be also expressed via a coproduct ∆, acting on this algebra (by contrastwith the braided versions of U ( gl ( N )), considered in [GPS2]).The problem of defining analogs of partial derivatives on the algebra U ( g ), where g is anarbitrary Lie algebra, was considered in [MS]. Partial derivatives constructed in that paper wereintroduced by the methods of deformation quantization. The Leibniz rule for the correspondingpartial derivatives differs from ours even if g = gl ( N ). The coproduct ∆, entering the Leibnizrule from [MS], being applied to the partial derivatives, gives rise to infinite series in thesederivatives. Whereas, our Leibniz rule is very simple and can be presented in a group-like form.We do not know other examples of algebras U ( g ) admitting partial derivatives with a similarproperty.Our global aim is to extend the action of QPD onto a central extension A of the algebra U ( gl ( N )) and furthermore on the skew-field B = A [ A − ] = { a/b | a, b ∈ A , b = 0 } . Once this extension is constructed, we become able to transfer a large number of usualdifferential operators, defined on the algebra
Sym ( gl ( N )) onto the algebra B . We call thisprocedure quantization with non-commutative configuration space . However, since the Leibnizrule for our QPG does not admit any analog of the formula for derivatives of composed functions,the results of application of the QPD to elements of B become more difficult to compute.In the current paper we consider the central extension of the compact form U ( u (2)) ofthe algebra U ( gl (2)), by adding the ”eigenvalues” of its generating matrix L to the algebra U ( u (2)). By these eigenvalues we mean the roots of the characteristic polynomial arising fromthe Cayley-Hamilton (CH) identity for the matrix L . Note that a central extension of the REalgebra (playing the role of the algebra A in the related Heisenberg double ( A, B )) was used in[IP2] in order to study the dynamics of a q -isotropic top, introduced in [AF]. By contrast with2IP2], we construct a central extension of the algebra B in the corresponding quantum double( A, B ). It should be emphasized that the CH identities for the matrices with non-commutativeentries is the main tool enabling us to extend the action of the QPD onto some elements of B .In the last section we compare this method with the Gelfand-Retakh method based on the useof quasideterminants.The paper is organized as follows. In the next section we comparer two methods of definingQPD on the algebra U ( gl ( N )): one is based on using a quantum double with properly definedpermutation relations, the other one is based on constructing the corresponding coproduct de-fined on the algebra, generated by the QPD. In section 3 we consider the case N = 2 and for thecompact form U ( u (2)) of the algebra U ( gl (2)) we express the Leibniz rule in terms of a matrixˆΘ, composed of the QPD, on which the coproduct acts in a multiplicative or group-like way.In section 4 we define a central extension of the algebra U ( u (2)) by using the eigenvalues ofthe generating matrix L and prolongate on them the actions of the QPD. Finally, in section 5we extend our QPD onto some elements from the corresponding skew-field, in particular, thosewhich could be useful for finding a non-commutative counterpart of the Dirac vector-potentialfor the magnetic monopole field. Acknowledgement
The work of P.S. was partially supported by the RFBR grant 19-01-00726.
Consider the algebra U ( gl ( N ) h ), where h is a numerical factor, introduced in the Lie bracket .It is generated by the unit 1 U ( gl ( N ) h ) and elements l ji , 1 ≤ i, j ≤ N , subject to the followingsystem of relations: l ji l sk − l sk l ji = h ( l si δ jk − l jk δ si ) ⇔ L L − L L = h ( L P − P L ) , (2.1)where L = k l ji k ≤ i,j ≤ N , L = L ⊗ I, L = I ⊗ L . Hereafter, I stands for the identity operatoror its matrix and P stands for the usual flip or its matrix.The generating matrix L satisfies the Cayley-Hamilton identity. Namely, there exists a monicpolynomial p ( t ) = t N + a t N − + ... + a N − t + a N , a k ∈ Z ( U ( gl ( N ) h )) , called characteristic , such that p ( L ) = 0. Hereafter Z ( A ) stands for the center of a given algebra A . So, it is natural to introduce the notion of eigenvalues of the generating matrix L as rootsof the characteristic polynomial. Remark 1
Note that a similar CH identity is valid for the generating matrix of the RE algebra(modified or not), associated with any involutive or Hecke symmetry R , provided R is skew-invertible and even symmetry. The reader is referred to [GPS1] for details.In the quantum double ( A, B ), which we are going to construct, the role of the A -module B will be played by the algebra U ( gl ( N ) h ). As for the algebra A , it is a unital associative algebra,generated by elements ∂ ji which are entries of the N × N generating matrix D = k ∂ ji k ≤ i,j ≤ N .The algebra A is commutative. This property, expressed in a matrix form, reads D D = D D . (2.2) This procedure enables us to treat the algebra U ( gl ( N )) as the result of a quantization (deformation) of thecommutative algebra Sym ( gl ( N )). permutation relations between generators of the algebras A and B by therule D L = L D + P + hD P . (2.3)The system (2.3) enables us to define a map σ : A ⊗ B → B ⊗ A by sending any element ∂ ji l mk from the left hand side of (2.3) to the corresponding element from the right hand side. Byconsecutive applying this rule and by assuming that σ (1 A ⊗ b ) = b ⊗ A ∀ b ∈ B and σ ( a ⊗ B ) = 1 B ⊗ a ∀ a ∈ A, we can send any element from A ⊗ B to an element from B ⊗ A . We call σ the permutation map .Note that the above algebras A and B are quotient-algebras: A = T (span( ∂ ji )) / h J i , B = T (span( l ji )) / h J i , where the notation T ( V ) = ⊕ k ≥ V ⊗ k stands for the free tensor algebra, generated by a space V and h J i i , i = 1 , T ( V ). We say that the permutation relations are compatiblewith the defining relations of the algebras A and B if the map σ preserves these ideals, i.e. σ ( J ⊗ span( l ji )) ⊂ span( l ji ) ⊗ J and σ (span( ∂ ji ) ⊗ J ) ⊂ J ⊗ span( ∂ ji ) . Proposition 2
The permutation map σ is compatible with the systems of relations (2.1) and (2.2) . A verification of the compatibility of the systems (2.1), (2.2) and the permutation relations(2.3) is straightforward: it suffices to check that the following relation D ( L L − L L − hL P + hP L ) = ( L L − L L − hL P + hP L ) D is valid in virtue of the permutation relations. Remark 3
There exists another system of permutation relations compatible with systems (2.1)and (2.2), namely, the following one D L = L D + P − hP D . (2.4)However, the system (2.1), (2.2), and (2.3) is equivalent to that (2.1), (2.2), and (2.4). In orderto show this equivalence it suffices to apply the operation of transposition to all matrices enteringthe first system and to replace h by − h .Given a quantum double ( A, B ), let us assume that there exists a counit (an algebra homo-morphism) ε A : A → C . Then we are able to construct an action of the algebra A onto B bysetting a ⊲ b = ( I ⊗ ε A ) σ ( a ⊗ b ) , a ∈ A, b ∈ B. (2.5)Here, we identify b ⊗ C and b .Let us go back to the above quantum double. We define the counit ε A in the algebra A bysetting ε A (1 A ) = 1 C , ε A ( ∂ ji ) = 0 ∀ i, j, ε A ( ab ) = ε A ( a ) ε A ( b ) . Then formula (2.5) leads to the action ∂ ji ⊲ B = 0, ∀ i, j . Also, by taking into account (2.3),we find ∂ ji ⊲ l sk = δ si δ jk ⇔ D ⊲ L = P . Sym ( gl ( N )).Now, we apply the QPD to a second order monomial. We have D ⊲ ( L L ) = P L + L P + hP P . (2.6)A general formula of applying the QPD to an arbitrary monomial can be found in [GS1].Let us observe that there exists a coproduct ∆ : A → A ⊗ A , enabling us to express theabove action as follows. We put ∆(1 A ) = 1 A ⊗ A and∆( ∂ ji ) = ∂ ji ⊗ A + 1 A ⊗ ∂ ji + h X k ∂ jk ⊗ ∂ ki := ∂ ji (1) ⊗ ∂ ji (2) . (2.7)Here, in the last equality we use the standard Sweedler’s notation for the coproduct ∆( ∂ ji ). Onthe whole algebra A the map ∆ is extended by the homomorphism property∆( ∂ ji ∂ nm ) = ∂ ji (1) ∂ nm (1) ⊗ ∂ ji (2) ∂ nm (2) and so on. Proposition 4
This coproduct is coassociative. Also, the QPD applied according to the usualrule ∂ ji ⊲ ( a b ) = ( ∂ ji (1) ⊲ a ) ( ∂ ji (2) ⊲ b ) , send the elements l sk l qp − l qp l sk − h ( δ sp l qk − δ qk l sp ) (2.8) to zero. As follows from this proposition, the ideal h J i is invariant subspace with respect to theaction of the QPD ∂ ji . Otherwise stated, the QPD extended via the above coproduct are well-defined on the algebra U ( gl ( N ) h ). Note that the coproduct ∆ together with the counit ε A definea bi-algebra structure in the algebra A .Let us perform a linear shift of the generating matrix D = k ∂ ji k of the algebra A as followsˆ D = D + Ih ⇔ ˆ ∂ ji = ∂ ji + δ ji h A . (2.9)The permutation relations (2.3) being expressed in terms of the matrix ˆ D becomeˆ D L = L ˆ D + h ˆ D P . (2.10)The action of the copproduct ∆ (2.7) on the generators ˆ ∂ ji reads∆( ˆ ∂ ji ) = h X k ˆ ∂ jk ⊗ ˆ ∂ ki ⇔ ∆( ˆ D t ) = h ˆ D t . ⊗ ˆ D t , (2.11)where t stands for the matrix transposition, and notation M . ⊗ N denotes the matrix withentries m ki ⊗ n jk for any two square matrices M = k m ji k and N = k n ji k of the same size.This coproduct leads to the following form of the Leibniz ruleˆ D ( ab ) t = h ˆ D ( a ) t ˆ D ( b ) t , (2.12)where ˆ D ( a ) denotes the matrix with entries ˆ ∂ ji ( a ). Below, we use the matrix D = h ˆ D t , which ismultiplicative (group-like) with respect to the coproduct:∆(D) = D . ⊗ D . (2.13)In conclusion, we want to emphasize that the method of defining analogs of partial derivativesby permutation relations is valid for the modified RE algebra, associated with any skew-invertiblesymmetry (involutive or Hecke), whereas the corresponding coproduct can be constructed onlyif R is involutive (i.e. R = I ). Thus, the first method is more general.5 Quantum partial derivatives on U ( u (2) h ) Let us consider the case N = 2 in more detail. Below, we use the following notations a = l , b = l , c = l , d = l d . Thus, we have L == l l i l l ! = a bc d ! , ˆ D = ˆ ∂ ∂ ∂ ˆ ∂ ! = ˆ ∂ a ∂ c ∂ b ˆ ∂ d ! . Hereafter, we use the “hat” notation only for the diagonal elements of the matrix ˆ D sincethe off-diagonal partial derivatives are actually not changed under the shift (2.9): ˆ ∂ ji ≡ ∂ ji forany i = j .Let us exhibit the permutation relations (2.10) in the case under considerationˆ ∂ a a = a ˆ ∂ a + h ˆ ∂ a ˆ ∂ a b = b ˆ ∂ a + h ∂ c ∂ c a = a ∂ c ∂ c b = b ∂ c ˆ ∂ a c = c ˆ ∂ a ˆ ∂ a d = d ˆ ∂ a ∂ c c = c ∂ c + h ˆ ∂ a ∂ c d = d ∂ c + h ∂ c ∂ b a = a ∂ b + h ∂ b ∂ b b = b ∂ b + h ˆ ∂ d ˆ ∂ d a = a ˆ ∂ d ˆ ∂ d b = b ˆ ∂ d ∂ b c = c ∂ b ∂ b d = d ∂ b ˆ ∂ d c = c ˆ ∂ d + h ∂ b ˆ ∂ d d = d ˆ ∂ d + h ˆ ∂ d . Now, we pass to the compact form u (2) h of the algebra gl (2) h . Namely, we introduce thefollowing generators t = 12 ( a + d ) , x = i b + c ) , y = 12 ( c − b ) , z = i a − d ) . The corresponding Lie brackets read:[ x, y ] = hz, [ y, z ] = hx, [ z, x ] = hy, [ t, x ] = [ t, y ] = [ t, z ] = 0 . Since the change of the generators is linear, we apply the classical formula for computingthe QPD in new generators: ∂ t = ∂ a + ∂ d and so on. In these generators the ”shifted version”is used only for the derivative in t : ˆ ∂ t = ∂ t + h id. The permutation relations becomeˆ ∂ t t − t ˆ ∂ t = h ˆ ∂ t ˆ ∂ t x − x ˆ ∂ t = − h ∂ x ˆ ∂ t y − y ˆ ∂ t = − h ∂ y ˆ ∂ t z − z ˆ ∂ t = − h ∂ z ∂ x t − t ∂ x = h ∂ x ∂ x x − x ∂ x = h ˆ ∂ t ∂ x y − y ∂ x = h ∂ z ∂ x z − z ∂ x = − h ∂ y ∂ y t − t ∂ y = h ∂ y ∂ y x − x ∂ y = − h ∂ z ∂ y y − y ∂ y = h ˆ ∂ t ∂ y z − z ∂ y = h ∂ x ∂ z t − t ∂ z = h ∂ z ∂ z x − x ∂ z = h ∂ y ∂ z y − y ∂ z = − h ∂ x ∂ z z − z ∂ z = h ˆ ∂ t . (3.1)These permutation relations play the role of the Leibniz rule for the QPD acting on thealgebra U ( u (2)). Another form of this rule can be expressed via a coproduct acting on theQPD. In a matrix form this coproduct reads∆( ˆΘ) = ˆΘ . ⊗ ˆΘ , ˆΘ = i ~ ˆ ∂ t ∂ x ∂ y ∂ z − ∂ x ˆ ∂ t − ∂ z ∂ y − ∂ y ∂ z ˆ ∂ t − ∂ x − ∂ z − ∂ y ∂ x ˆ ∂ t , (3.2)Hereafter, we use a new parameter ~ = h/ i . Also, the symbol ˆΘ( a ) denotes the matrix whoseentries result from applying the corresponding partial derivatives to an element a ∈ U ( u (2) h ).6hus, we get a linear map U ( u (2) h ) → Mat ( U ( u (2) h )) which will be also denoted by the symbolˆΘ: ˆΘ : a ˆΘ( a ) ∀ a ∈ U ( u (2) h ) . (3.3)We state that the map ˆΘ preserves the multiplication of elements of the algebra U ( gl (2) h )ˆΘ( ab ) = ˆΘ( a ) ˆΘ( b ) , ∀ a, b ∈ U ( u (2) h ) (3.4)Thus, the matrix ˆΘ enables us to present the Leibniz rule in a multiplicative or group-like form.Consequently, the map ˆΘ defines a representation of the algebra U ( u (2) h ):ˆΘ( x ) ˆΘ( y ) − ˆΘ( y ) ˆΘ( x ) = 2 i ~ ˆΘ( z ) = h ˆΘ( z )and so one.The images of the U ( u (2) h ) generators under the map ˆΘ are as follows:ˆΘ(1 U ( u (2) h ) ) = I, ˆΘ( t ) = ( t + i ~ ) I, ˆΘ( x ) = x I + i ~ A, ˆΘ( y ) = y I + i ~ B, ˆΘ( z ) = z I + i ~ C, (3.5)where A = − −
10 0 1 0 , B = − − , C = − − . (3.6)The numerical matrices A , B and C possess the following multiplication table A = B = C = − I, AB = − BA = C, BC = − CB = A, CA = − AC = B. (3.7)Below, we extend the action of the QPD onto some lager algebras with preserving theproperty (3.4) of the map ˆΘ. U ( u (2) h ) First, we present the CH identity for the generating matrix of the algebra U ( gl (2) h ) expressedin terms of the compact generators L = a bc d ! = t − iz − ix − y − ix + y t + iz ! This identity reads p ( L ) = 0, where the characteristic polynomial p ( τ ) is of the form p ( τ ) = τ − (2 t + h ) τ + ( t + x + y + z + 2 i ~ t ) I. Let us denote µ and µ the roots of the characteristic polynomial p ( τ ): µ + µ = 2 t + 2 i ~ , µ µ = t + Cas + 2 i ~ t, where we use the notation Cas = x + y + z . So, the elements µ i belong to a central extensionof U ( u (2)) h .Consider the algebra A = U ( u (2) h )[ µ , µ ] and compute the quantities ˆΘ( µ i ) , i = 1 ,
2. Withthe use of (3.5) it is not difficult to find their sums:ˆΘ( µ ) + ˆΘ( µ ) = ˆΘ( µ + µ ) = ˆΘ(2 t + 2 i ~ ) = (2 t + 4 i ~ ) I, (4.1)7ow, we find the difference ˆΘ( µ ) − ˆΘ( µ ). Consequently, it becomes possible to calculateeach ˆΘ( µ i ). For this purpose let us first compute the matrix ˆΘ( µ ), where µ = µ − µ . Takinginto account that µ = ( µ + µ ) − µ µ = − ~ ) , and with the use of (3.5) we haveˆΘ( µ ) = ˆΘ( − x + y + z + 4 ~ )) = ( µ − ~ ) I − i ~ M, where M = ( xA + yB + z C ).Now, to find the matrix ˆΘ( µ ) we have to calculate the square root ˆΘ( µ ) = q ˆΘ( µ ). Thisrelation is motivated by our wish to preserve the multiplicative property of the map ˆΘ.Since the matrix M = xA + yB + z C satisfies the CH identity M − i ~ M + Cas I = 0its eigenvalues are easy to find: λ = i ~ + µ , λ = i ~ − µ . Consequently, the eigenvalues of the matrix ˆΘ( µ ) are as follows ν = µ − ~ − i ~ µ = ( µ − i ~ ) , ν = µ − ~ + 4 i ~ µ = ( µ + 2 i ~ ) . As for the matrix ˆΘ( µ ), it can be found via the spectral decomposition of the matrix ˆΘ( µ ):ˆΘ( µ ) = ˆΘ( µ ) − ν Iν − ν √ ν + ˆΘ( µ ) − ν Iν − ν √ ν . Thus, we have 4 candidates to the role of the matrix ˆΘ( µ ) in dependence of the sign choices forthe roots √ ν = ǫ ( µ − i ~ ) , √ ν = ǫ ( µ + 2 i ~ ) , ǫ , ǫ ∈ {± } . But a straightforward verification shows that the only choice ǫ = ǫ = 1 leads to the resultcompatible with the classical limit (i.e. corresponding to ~ = 0) for the action of the partialderivatives: ∂ t µ = 0 and ∂ x µ = − xµ . (4.2)The second relation in (4.2) follows from the equality µ = − µ ) = ( µ + 2 i ~ ) + ( µ − i ~ ) µ I + ( µ − i ~ ) − ( µ + 2 i ~ ) µ M = µ − ~ µ I − i ~ µ M. (4.3)In a similar way we can get the action of the QPD on an arbitrary power of µ :ˆΘ( µ p ) = ( µ + 2 i ~ ) p +1 + ( µ − i ~ ) p +1 µ I + ( µ − i ~ ) p − ( µ + 2 i ~ ) p µ M. Now, by using (4.1) and (4.3) we getˆΘ( µ ) = t + ( µ + 2 i ~ ) µ ! I − i ~ µ M, ˆΘ( µ ) = t − ( µ − i ~ ) µ ! I + 2 i ~ µ M. ∂ t ( µ ) = − i ~ t + ( µ + 2 i ~ ) µ ! , ∂ x ( µ ) = − xµ , and so on.Thus, we have computed the action of the QPD onto the elements µ i , i = 1 ,
2. We completethis section by recalling the notion of the quantum radius and by finding the result of QPDaction on this element.In our previous papers we introduced the so-called quantum radius r ~ = √ Cas + ~ . Interms of µ it is written as r ~ = ± µ/ i . The sign here is not fixed because our ordering of µ and µ is arbitrary.By using the above relation between µ and r ~ (with any sign) we get from (4.3)ˆΘ( r ~ ) = r ~ + ~ r ~ I + i ~ r ~ M. (4.4)This formula enables us to find the action of all QPD on the quantum radius: ∂ t r ~ = − i ~ r ~ , ∂ x r ~ = xr ~ , ∂ y r ~ = yr ~ , ∂ z r ~ = zr ~ . (4.5)Note that in the classical limit ~ = 0 the derivative ∂ t r vanishes and other formulae turn intosimilar ones (with r ~ replaced by r ).Now, consider the algebra A = ( U ( su (2) h ) ⊗ C [ t, r ~ ]) / h x + y + z + ~ − r ~ i . (4.6)It is easy to see that the map ˆΘ sends the element x + y + z + ~ − r ~ to 0. Consequently,this map is well-defined on the algebra A .We assume the quantum radius to be real and positive provided that the parameter ~ isreal and the generators x , y and z are represented by Hermitian operators. Observe that ifthe algebra U ( u (2) h ) is represented in the space of spin n the quantum radius takes the value r ~ = (2 n + 1) ~ .Recall that in our treatment of the algebra U ( u (2) h ) as a non-commutative analog of thepolynomial algebra on the Minkowski space, the generators x, y, z play the role of spacial vari-ables and t is interpreted as the time (see [GS2]). B via CH identities In this section we extend the QPD onto some elements of the skew-field B = A [ A − ] withpreserving the Leibniz rule in its multiplicative form. Note that this Leibniz rule can be expressedvia the matrix ˆΘ or D, introduced at the end of the section 2. Since the size of the matrix D issmaller (this fact becomes more significant in the higher dimensions), we deal with the matrixD in this section.Let b be an arbitrary nontrivial element of the algebra A . If we can extend the map D ontothe element b − ∈ B with preserving the Leibniz rule, we should haveD( b − ) = D( b ) − . Thus, in order to compute D( b − ) we have to invert the matrix D( b ) with non-commutativeentries. In principle, this procedure can be performed by means of the Gelfand-Retakh method9sing the so-called quasideterminants. In order to present the entries of the matrix D( b − ) aselements of B we need the Ore property of the algebra A . Hopefully, the algebra A has thisproperty since it is so for the algebra U ( u (2) h ). Nevertheless, practically, reduction of any ”left”fraction to a ”right” one and vice versa in this algebra is difficult indeed.Below, we deal with some elements b ∈ A for which the computation of the matrix D( b ) − can be performed by means of the CH identities for matrices with non-commutative entries. Forthese elements b we succeeded in finding the matrices D( b ) − with entries from the skew-field B .First, we calculate the matrix D( b ) for some basic b ∈ A . Taking into account the explicitform of D D = i ~ ˆ ∂ t + i∂ z i∂ x − ∂ y i∂ x + ∂ y ˆ ∂ t − i∂ z ! , (5.1)we findD( t ) = ( t + i ~ ) I, D( x ) = x − ~ − ~ x ! , D( y ) = y − i ~ i ~ y ! , D( z ) = z − ~ z + ~ ! D( r ~ ) = 1 r ~ r ~ + ~ − ~ z − ~ ( x + iy ) − ~ ( x − iy ) r ~ + ~ + ~ z ! . The matrix D( r ~ ) obeys the CH identity:D( r ~ ) − r ~ D( r ~ ) + ( r ~ − ~ ) I = 0 , (5.2)which can be easily rewritten in the factorized form:(D( r ~ ) − ( r ~ + ~ ) I )(D( r ~ ) − ( r ~ − ~ ) I ) = 0 . (5.3)Now, consider the following example. Let b be a linear combination of generators withcoefficients α i ∈ C : b = α t + α x + α y + α z, α + α + α = 0 . (5.4)Since D( b ) = ( b + iα ) I − ~ N, N = α α + iα α − iα − α ! , then taking into account that N = ( α + α + α ) I , we find the matrix D − ( b ):D( b ) − = ( b + iα ) I + ~ N ( b + iα ) − ( α + α + α ) ~ . Note that the meaning of the fraction here is not ambiguous since its numerator and denominatorcommute with each other.Introduce an element c = r ~ − b , where b is given by (5.4) with the following restriction onthe coefficients α = 0 , α + α + α = 1 . The matrices D( b ) and D( r ~ ) commute with each other and obey the second order CH identities(D( b ) − b I ) = ( α + α + α ) ~ I, (5.5)and (5.2) respectively. Thus, it is reasonable to look for the inverse matrix D( c ) − of the form:D( c ) − = I a + D( r ~ ) a + D( b ) a + D( r ~ )D( b ) a . (5.6)10n multiplying (5.6) by the matrix D( c ) = D( r ~ ) − D( b ) from the left and demanding theresult to be the unit matrix I we find a system of linear equations for coefficients a i : − ( r ~ − ~ ) a + ( b − ~ ) a = 1 a + 2 r ~ a + ( b − ~ ) a = 0 a + 2 b a + ( r ~ − ~ ) a = 0 a − a − r ~ − b ) a = 0Here, we used the relations (5.5) and (5.2).The coefficients of the system above commute with each other, so the solution can be foundby the standard methods of linear algebra. Applying the Cramer’s rules we get the followingresult a = 2( r ~ + b − br ~ − ~ )( r ~ − b )(( r ~ − b ) − ~ ) , a = 3 b − r ~ ( r ~ − b )(( r ~ − b ) − ~ ) ,a = 3 r ~ − b ( r − b )(( r ~ − b ) − ~ ) , a = − r ~ − b )(( r ~ − b ) − ~ ) . Now, we are able to extend the action of the QPD on the element c − = ( r ~ − b ) − . Namely,from (5.1) we get(D + D )( c − ) = 2 i ~ ˆ ∂ t c − , (D + D )( c − ) = − ~ ∂ x c − , etc . To write down the general answer it is convenient to introduce the following vectors ~α = ( α , α , α ) , ~r ~ = ( x, y, z ) , ~ ∇ = ( ∂ x , ∂ y , ∂ z ) . Then, in virtue of the relation D( c − ) = D( c ) − we come to the final results: ∂ t (cid:18) r ~ − b (cid:19) = − i ~ r ~ (( r ~ − b ) − ~ ) ,~ ∇ (cid:18) r ~ − b (cid:19) = r ~ ~α − ~r ~ r ~ r ~ − b ) − ~ ) − ( ~ ~α + i [ ~r ~ × ~α ]) 2 ~ r ~ ( r ~ − b )(( r ~ − b ) − ~ ) , where [ · × · ] stands for the vector product of two vectors. We point out that the components ofthe vector ~r ~ do not commute with the element r ~ − b entering the denominator. So, we haveto explicitly fix the order of factors in the formula for ~ ∇ ( r ~ − b ) − .Also, note that in the limit ~ → ~ ∇ (cid:18) r − b (cid:19) = r~α − ~rr ( r − b ) , where r = | ~r | and b = ~α · ~r is the scalar product.Thus, by applying the QPD to elements of the form ac k , a ∈ A , k ∈ Z , we can represent theresulting elements as those from B .Our interest in applying the QPD to the element ( r ~ − b ) − is motivated by our wish toconstruct a non-commutative analog of the Dirac potential. Let us recall that in the classicalsetting this potential is a solution of the following equation rot A = H , where H is the vectorof the magnetic field, namely, a stationary solution of the Maxwell systemdiv H = 0 , rot H = 4 gπδ ( r ) , (5.7)11here g is a nontrivial constant factor. The electric field is assumed to vanish. As usual, thenotation div and rot stand for respectively the divergence and rotor of a given vector field, and δ ( r ) is the delta-function. Dirac found a solution to the system (5.7) in the form H = g r r .Besides, he found a family of vector-potentials of this model: A = gr [ ~r × ~α ]( r − ~r · ~α ) , (5.8)where, ~α is a unit vector, and · is the scalar product of two vectors. Each of these vector-potentials is singular on a half-line.In [GS2] we have found a non-commutative counterpart of the Dirac monopole, i.e. a solutionto the system (5.7). It is of the form H = g ~r ~ r ~ ( r ~ − ~ ) . (5.9)Unfortunately, we have not succeeded in finding a non-commutative counterpart of the po-tential A , i.e. a solution to the equation rot A = H , where H is defined by (5.9). It should be em-phasized that the problem of finding such a potential in the framework of our non-commutativesetting is much more complicated, than in the classical setting. 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