Doubles of associative algebras and their applications
aa r X i v : . [ m a t h . QA ] J a n Doubles of associative algebras and their applications
Dimitri Gurevich ∗ LMI, UPHF, 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
January 22, 2020
Abstract
For a couple of associative algebras we define the notion of their double and give aset of examples. Also, we discuss applications of such doubles to representation theoryof certain quantum algebras and to a new type of Noncommutative Geometry.
In this letter by a double of associative algebras we mean an ordered couple (
A, B ) ofassociative unital algebras A and B such that their tensor products B ⊗ A can be alsoendowed with an associative product by means of a permutation map σ : A ⊗ B → B ⊗ A . Ifthe algebra A is equipped with a counit (an algebra homomorphism) ε : A → C , then undersome natural conditions on σ and ε the algebra A can be represented in the algebra B .The simplest example of such a double is a Heisenberg-Weyl (HW) algebra. The smash-product of a bi-algebra A and an A -module M is another example of a double. In this casethe role of the algebra B can be played by the free tensor algebra T ( M ) = ⊕ k M ⊗ k or bysome of its quotient algebras. We are mainly interested in doubles related to braidings.Let V be a finite dimensional complex vector space, dim V = N . An invertible operator R : V ⊗ → V ⊗ is called a braiding , if it is subject to the braid relation R R R = R R R , R = R ⊗ I, R = I ⊗ R. ∗ [email protected] † [email protected] I stands for the identity operator or its matrix. A braiding R is called respectivelyan involutive or a Hecke symmetry if it is subject to a supplementary condition R = I or ( qI − R )( q − I + R ) = 0 q
6∈ { , ± } . The best known examples of Hecke symmetries come from the Drinfeld-Jimbo QuantumGroups (QG) U q ( sl ( N )). They are deformations of the usual flips. Nevertheless, there existinvolutive and Hecke symmetries, which are neither deformations of the flips nor super-flips.All symmetries, we are dealing with, are assumed to be skew-invertible (see [1]). Wemainly deal with Hecke symmetries R = R ( q ) at a generic value of the parameter q . Toany such a Hecke symmetry R we associate R -analogs of the symmetric and skew-symmetricalgebras of the space V by respectively settingSym R ( V ) = T ( V ) / h Im( qI − R ) i , Λ R ( V ) = T ( V ) / h Im( q − I + R ) i . Besides, we consider the so-called RTT and Reflection Equation (RE) algebras defined re-spectively by R T T − T T R = 0 , (1.1) R L R L − L R L R = 0 , (1.2)where T = k t ji k ≤ i,j ≤ N , L = k l ji k ≤ i,j ≤ N , T = T ⊗ I and T = I ⊗ T etc.In section 3 we exhibit examples of doubles ( A, B ), where the algebra A is an RTT or anRE algebra. In a number of papers there were considered doubles with RE algebras playingthe role of A and the corresponding RTT algebras, playing the role of B . By contrast, in[2, 3] we considered doubles, where B was another copy of the RE algebra. Combining thegenerating matrices of the algebras A and B we constructed other doubles, giving rise to thenotion of partial derivatives in the noncommutative generators of B .Since the RE algebra in its modified form tends to the algebra U ( gl ( N )), provided theHecke symmetry R = R ( q ) tends to the usual flip P as q →
1, we obtained partial derivativeson the algebra U ( gl ( N )) . These partial derivatives gives rise to a new noncommutative (NC)differential calculus which is GL ( N )-covariant and which turns into the usual calculus onthe algebra Sym ( gl ( N )) as h → N = 2 we treat the compact form U ( u (2) h ) of the algebra U ( gl (2) h )as an NC version of the polynomial algebra on the Minkowski space. Given a differentialoperator with polynomial coefficients on the classical Minkowski space, we quantize thecoefficients and replace the usual partial derivatives with their ”quantum counterparts”. Inthis way we get an operator, defined on the algebra U ( u (2) h ) which turns into the initial oneas h →
0. We call this procedure the quantization with an NC configuration space . In [4, 5]we extend this procedure on some operators with non-polynomial coefficients.In the present letter we reproduce some elements of this NC calculus. However, themain our objective is comparing the doubles (
A, B ) where A is an RTT or RE algebra,and constructing the corresponding representations of these algebras. In the last section weconsider an example of the mentioned quantization with an NC configuration space. Acknowledgement
The work of P.S. was partially funded the RFBR grant 19-01-00726. Sometimes we will deal with the algebra U ( gl ( N ) h ), where h is a numerical multiplier introduced in thebracket of the Lie algebra gl ( N ). This rescaling of the bracket enables us to treat the algebra U ( gl ( N ) h ) asa quantization of the commutative algebra Sym ( gl ( N )) with respect to the linear Poisson bracket. Representations via doubles, first examples
Let A and B be two associative unital algebras endowed with a linear map σ : A ⊗ B → B ⊗ A ,such that σ ◦ ( µ A ⊗ id B ) = µ A ◦ σ ◦ σ on A ⊗ A ⊗ B,σ ◦ (id A ⊗ µ B ) = µ B ◦ σ ◦ σ on A ⊗ B ⊗ B,σ (1 A ⊗ b ) = b ⊗ A , σ ( a ⊗ B ) = 1 B ⊗ a ∀ a ∈ A, ∀ b ∈ B, where µ A : A ⊗ A → A is the product in the algebra A , 1 A is its unit, and similarly for B .Under these assumptions the space B ⊗ A can be equipped with a bilinear map ∗ :( B ⊗ A ) ⊗ ∗ → B ⊗ A : ( b ⊗ a ) ∗ ( b ′ ⊗ a ′ ) := ( µ B ⊗ µ A ) ◦ (id B ⊗ σ ⊗ id A )( b ⊗ a ⊗ b ′ ⊗ a ′ ) . Proposition 1
The map ∗ endows the space B ⊗ A with the structure of a unital associativealgebra with the unit element B ⊗ A . We call the corresponding algebra the double of associative algebras A and B and denote itas B ⊗ σ A .If the algebra A is equipped with a counit (an algebra homomorphism) ε A : A → C , thenwe define an action of the algebra A onto B by the rule a ⊲ b = (id B ⊗ ε A ) ◦ σ ( a ⊗ b ) , ∀ a ∈ A, ∀ b ∈ B. Identifying b ⊗ C and b , we get that a ⊲ b ∈ B , so each element a ∈ A defines a linear operator Op ( a ) : B → B. Proposition 2
The map a Op ( a ) defines a representation of the algebra A in the algebra B : Op ( ab ) = Op ( a ) Op ( b ) , Op (1 A ) = I. Note that if σ = P , this representation becomes trivial a ε ( a ) I .1. As an example we consider an HW algebra, generated by two polynomial subalgebras A = C [ x , ..., x m ] and B = C [ x , ..., x m ]. Introduce the following permutation relations x j x i = x i x j + δ ji B ⊗ A . (2.1)Then, by setting ε ( x j ) = 0, we get a double ( A, B ), such that the corresponding operators ∂ i = Op ( x i ) are the partial derivatives defined on the polynomial algebra B . Below, we omitthe factors 1 A and 1 B in the permutation relations similar to (2.1). Remark 3
In the particular case N = 1 by slightly modifying the permutation relations as y x = q x y + 1 , q ∈ C , q
6∈ { , ± } , we get the well-known Jackson derivative. By permutation relations we mean equalities a ⊗ b = σ ( a ⊗ b ), a ∈ A , b ∈ B . All the doubles ( A, B )below are defined via relations on generators of each component and the permutation relations.
3. Especially, we are interested in a matrix version of the permutation relations (2.1).Consider N × N matrices M = k m ji k and D = k d ji k . Define the algebra B ⊗ σ A , where A = A ( D ) , B = B ( M ) (in the brackets we put the generating matrix of the algebra) by thefollowing system D D = D D , M M = M M , D M = M D + P . Two first equalities of this system mean that the algebras A and B are commutative.The last equality (the permutation relations) together with the counit ε ( D ) = 0 leads to theaction D ⊲ M = P , ⇔ ∂ ji ⊲ m lk = δ li δ jk , where ∂ ji = Op ( d ji ) . The Leibniz rule for the matrix Op ( D ) = k Op ( d ji ) k can be expressed via the coproduct∆( Op ( D )) = Op ( D ) ⊗ I + I ⊗ Op ( D ) .
3. Now, consider the double (
A, B ), with A = U ( gl ( N )) and B = T ( V ), where V is thespace of the covariant representation of the algebra A . Let { x , ..., x N } be a basis of V and { l ji } be the corresponding basis of U ( gl ( N )), i.e. such that l ji ⊲ x k = x i δ jk . Then the relationsbetween the generators l ji can be cast in the following matrix form L L − L L = L P − L P, L = k l ji k . (2.2)We impose no relation on the generators x i . The permutation relations are defined as follows L x = x L + P x ⇔ l ji x k = x k l ji + x i δ jk . Hereafter, x stands for the column ( x , ..., x N ) t . Note that in this double the algebra B = T ( V ) can be replaced by one of the algebras Sym( V ) or Λ ( V ). Also, there exist similardoubles with the dual space V ∗ instead of V .4. Let B be another copy of the algebra U ( gl ( N )) with a similar basis m ji and M = k m ji k be the corresponding generating matrix. It meets the system of relations similar to (2.2).We define two types of the permutation relations by the following formulae( i ) : L M = M L + M P − M P or ( ii ) : L M = M L + M P . Then, taking the counit ε ( L ) = 0, we get the corresponding actions( i ) : L ⊲ M = M P − M P or ( ii ) : L ⊲ M = M P . The above algebra B = B ( M ) can be replaced by the commutative algebra Sym( gl ( N )).Then the relations in the algebra B become M M = M M . All other relations remainunchanged. In this case the algebra A = U ( gl ( N )) is respectively represented by the adjointand left vector fields onto the algebra B = Sym( gl ( N )).5. The following double was constructed in [3] as a limit case of a double, considered inthe next section. Namely, introduce a double ( A ( D ) , B ( N )), where the generating matrices D = k d ji k and N = k n ji k satisfies the the following systems D D = D D N N − N N = h ( N P − N P ) , D N = N D + P + h D P . A = A ( D ) is commutative and B = U ( gl ( N ) h ).The algebra B ⊗ σ A is an NC analog of the HW algebra from the example 2 above.Namely, this algebra is the main ingredient of our NC GL ( N )-covariant calculus.It is convenient to introduce the matrix ˜ D = D + h − I and simplify the permutationrelations to the form: ˜ D N = N ˜ D + h ˜ D P . By setting ε ( D ) = 0 (and therefore ε ( ˜ D ) = h − I ), we get the action of operators ∂ ji = Op ( d ji ) on all elements of the algebra U ( gl ( N ) h ). Note that this action is classical on thegenerators of the algebra B ( N ): ∂ ji ⊲ n lk = δ li δ jk . Its extension on the higher monomials canbe done by means of the coproduct∆( ∂ ji ) = ∂ ji ⊗ ⊗ ∂ ji − h X k ∂ ki ⊗ ∂ jk . Observe that our partial derivatives turn into the usual ones on Sym( gl ( N )) as h →
1. Let A = A ( T ), T = k t ji k and B = B ( M ), M = k M ji k be two RTT algebras, correspondingto a Hecke symmetry R . Let us define the permutation relations by the rule R T M = M T R ⇔ T M = R − M T R . Defining the counit ε ( T ) = I , we get to the following action T ⊲ M = R − M R .
2. Now, we set B = T ( V ) and define the permutation relations as follows: T x = R P x T ⇔ t ji x k = R mnik x m t jn , where the summation over repeated indices is understood. With the same counit, we havethe action T ⊲ x = R P x ⇔ t ji ⊲ x k = R mjik x m . Computing the action of the elements t ji on higher elements from T ( V ), we arrive to therepresentations, which can be constructed via the fusion procedure. If a Hecke symmetry R = R ( q ) → P at q →
1, in this limit we get the trivial representations t ji → ε ( t ji ) I in theboth examples above. Note that the algebra B = T ( V ) in this construction can be replacedby R -symmetric or R -skew-symmetric algebras of the space V .3. The differential calculus from [6] (see section 7), which is a generalization of the Wess-Zumino calculus on the quantum planes [7], can be also presented in terms of a double.Consider a double ( A, B ) where B = Sym R ( V ) and A = Sym R ( V ∗ ). Here the space V ∗ isendowed with the right dual basis { x , ..., x N } , i.e. such that < x i , x j > = δ ji . Let us puttogether all defining relations of this double: q x i x j = R klij x k x l , q x i x j = R jikl x l x k , x j R iljk x i = h δ lk + q − x k x l . (3.1)Setting ε ( x j ) = 0, we get R -analogs ∂ i = Op ( x i ) of the partial derivatives multiplied by h . The above permutation relations together with the counit ε ( x i ) = 0 play the role of the5eibniz rule for the operators ∂ i . The algebra Sym R ( V ) endowed with these operators is an R -counterpart of the bosonic Fock space. In a similar manner an R -analog of the fermionicFock space can be constructed.4. Let us consider elements k ji = x i x j and compose the matrix K = k k ji k . Proposition 4
In virtue of (3.1) the matrix K is subject to the following relation: R K R K − K R K R = h ( R K − K R ) . (3.2)We call the algebra defined by (3.2) the modified RE algebra. If R is an involutive symmetry,the claim above is still valid. However, only if R is a Hecke symmetry, this algebra isisomorphic to the RE algebra defined by (1.2). This isomorphism can be defined as follows L = h I − ( q − q − ) K. (3.3)Now, consider a double ( A, B ), where A = A ( K ) is a modified RE algebra (3.2) and B isone of the algebras T ( V ), Sym R ( V ), Λ R ( V ). Taking into account (3.1) and the identification k ji = x i x j we get the following permutation relations between these algebras R K R x = x K + h R x . The counit ε ( K ) = 0 leads to the action R K R ⊲ x = h R x . Assuming that R = R ( q ) → P as q →
1, we get the limit action l ji ⊲ x k = h x i δ jk whichcoincides with the covariant representation of the algebra U ( gl ( N ) h ).In a similar manner it is possible to define a double with the space V ∗ instead of V andthus to get the contravariant representation of the modified RE algebra A .In [1] there was described a way of constructing a category of finite dimensional A -modules similar to U ( gl ( N ) h )-module. In that construction we used the ”braided bi-algebrastructure” of the modified RE algebra and the categorical morphisms transposing the objectsspan( k ji ) ∼ = V ⊗ V ∗ and M , where M is an arbitrary object of the mentioned category. Moreprecisely, the corresponding permutation relations are σ ( a ⊗ b ) = ( ⊲ ⊗ id) ◦ R ( a ⊗ a ⊗ b ) , where a ⊗ a = ∆( a ) , and ∆ is the coproduct in the modified RE algebra defined in [1], while R stands for thebraiding (a categorical morphism), transposing the objects span( k ji ) and M . Note that inthe case related to the quantum group U q ( sl ( N )), R is the product of the usual flip andthe image of the corresponding universal R -matrix. However, in general, the mentionedcategorical morphism can be constructed via the initial symmetry R without any quantumgroup.5. Let A be again a modified RE algebra, defined by (3.2). The role of B is often attributedto the corresponding RTT algebra. We consider two doubles where the role of B = B ( M ), M = k m ji k is also played by another copy of the RE algebra (in its non-modified form).Define the two types of permutation relations:( i ) : R K R M = M R K R + h ( R M − R M ) , (3.4)6 ii ) : R K R M = M R K R + h R M . (3.5)The first system of permutation relations defines braided analogs of the adjoint vector fields.The second one defines braided analogs of the left vector fields (see [2]).Turn to the double, defined by (3.5). As was shown in [3], the matrix D = M − K (the matrix M − can be found via the Cayley-Hamilton identity) and K generate a double( A ( D ) , B ( M )), where D = k d ji k and M = k m ji k , with the following defining system R − D R − D = D R − D R − , R M R M = M R M R ,D R M R = R M R − D + R . Now, in this double we replace the matrix M by N where M = h I − ( q − q − ) N and geta double ( A ( D ) , B ( N )). The matrix N = k n ji k generates the modified RE algebra and thepermutation relations are as follows D R N R − R N R − D = R + h D R . Note that if R = R ( q ) tends to P , in the limit we get the double, exhibited at the end ofthe previous section.It is possible to construct similar doubles associated with generalized (braided) Yangiansintroduced in [6]. We plan to consider them elsewhere. Consider the last example from section 2 for the case N = 2 and pass by a change of basis tothe algebra B = U ( u (2) h ). Making the corresponding change of basis in the algebra A , weget a double ( A, B ), where B is generated by the elements t, x, y, z , subject to the relations[ t, x ] = [ t, y ] = [ t, z ] = 0 , [ x, y ] = h z, [ y, z ] = h x, [ z, x ] = h y. The commutative algebra A is generated by ∂ t , ∂ x , ∂ y , ∂ z and the permutation relations read[ ∂ t , t ] = h ∂ t + 1 [ ∂ t , x ] = − h ∂ x [ ∂ t , y ] = − h ∂ y [ ∂ t , z ] = − h ∂ z [ ∂ x , t ] = h ∂ x [ ∂ x , x ] = h ∂ t + 1 [ ∂ x , y ] = h ∂ z [ ∂ x , z ] = − h ∂ y [ ∂ y , t ] = h ∂ y [ ∂ y , x ] = − h ∂ z [ ∂ y , y ] = h ∂ t + 1 [ ∂ y , z ] = h ∂ x [ ∂ z , t ] = h ∂ z [ ∂ z , x ] = h ∂ y [ ∂ z , y ] = − h ∂ x [ ∂ z , z ] = h ∂ t + 1Introducing the generator ˜ ∂ t = ∂ t + 2 /h A , we can treat the double B ⊗ σ A as theenveloping algebra of the Lie algebra with 8 generators t , x , y , z , ˜ ∂ t , ∂ x , ∂ y and ∂ z .Now, we introduce the so-called quantum radius by the formula r ν = q x + y + z + ν , ν = h/ i. In a series of papers we extend the above partial derivatives onto any rational functions in r ν and some rational functions in x , y , z and r ν . This enabled us to extend the procedure Note that this extension is not straightforward, since the usual Leibnitz rule for the derivatives on thealgebra U ( u (2) h ) is not valid.
7f quantization with NC configurational space onto a larger algebra. We exhibit two exam-ples of dynamical models obtained by such a quantization. First, consider an equation ofSchr¨odinger type ( a∂ t + b ( ∂ x + ∂ y + ∂ z ) + qr ) ψ = 0 , (4.1)where a, b, q are some constants. In this model the mentioned quantization is reduced toreplacing the usual radius by its quantum counterpart r ν and similarly for the derivatives.Note that in the limit ν → U ( u (2) h ).Another example is the model of the Dirac monopole in the classical electrodynamics. Ina particular case corresponding to the static Dirac monopole the system of Maxwell equationstake the form rot H = div H = 4 gπδ ( r ) , where H = ( H , H , H ) is the magnetic field, r = ( x, y, z ), rot and div are the curl anddivergence respectively. We succeeded in finding an NC solution of this system: H = gr ν ( r ν − ν ) r . Note that it tends to the usual Dirac monopole as ν → References [1] Gurevich D., Pyatov P., Saponov P. Representation theory of (modified) reflectionequation algebra of GL ( m | n ) type // St. Petersburg Math. J. 2008. V. 20. P. 213-253.[2] Gurevich D., Pyatov P., Saponov P. Braided differential operators on quantum alge-bras// J. of Geometry and Physics. 2011. V. 61. P. 1485-1501.[3] Gurevich D., Pyatov P., Saponov P. Braided Weyl algebras and differential calculuson U ( u (2)) // J. of Geometry and Physics. 2012. V. 62. P. 11751188.[4] Gurevich D., Saponov P. Noncommutative Geometry and dynamical models on U ( u (2)) background // J. of Generalized Lie theory and Applications. 201. V. 9:1[5] Gurevich D., Saponov P. Quantum geometry and quantization on U ( uu