Derivatives in Noncommutative Calculus and Deformation Property of Quantum Algebras

Home , Weyl algebra

Calculus on U(gl(m) ) ~ Example: calculus on the algebra U(u(2) ) ~ Extension of the Weyl algebra q-Witt algebra: deformation property Deformation property of Reﬂection Equation algebra

Derivatives in noncommutative calculus and deformation property of quantum algebras

Dimitri Gurevich with P.Saponov and (partially) P.Pyatov

JINR Dubna, 7 August 2015

1 Calculus on U(gl(m)~)

2 Example: calculus on the algebra U(u(2)~)

3 Extension of the Weyl algebra

4 q-Witt algebra: deformation property

5 Deformation property of Reﬂection Equation algebra

Consider the algebra U(gl(m)). In order to represent it as result of a quantization of the commutative algebra Sym(gl(m)) we introduce a quantization parameter ~ in the front of the Lie bracket. The corresponding algebra is denoted U(gl(m)~). Below, we deﬁne analogs of partial derivatives on this algebra in such a way that for ~ = 0 we recover the usual partial derivatives in generators of Sym(gl(m)).

Also, we deﬁne a quantum analog of the diﬀerential algebra Ω(Sym(gl(m)) and this of the de Rham operator. All objects are deformations of their classical counterpart.

Emphasize that given a NC algebra A, the differential algebra Ω(A) on A is usually defined via the de Rham operator satisfying the classical Leibniz rule d(a b) = da b + a db without transposing elements a ∈ A and their differentials db via the relation a(db) = (db)a, which plays the central role in the classical case. This approach leads to the universal differential algebra which is much bigger than the classical one is, if A is commutative. In our construction we retrieve the classical differential algebra as ~ → 0.

Dimitri Gurevich with P.Saponov and (partially) P.Pyatov Derivatives in noncommutative calculus and deformation property of quantum algebras Calculus on U(gl(m) ) ~ Example: calculus on the algebra U(u(2) ) ~ Extension of the Weyl algebra q-Witt algebra: deformation property Deformation property of Reﬂection Equation algebra j Fix in the Lie algebra gl(m)~ the standard basis {ni }, 1 ≤ i, j ≤ m and the usual Lie bracket j l l j j j [ni , nk ] = ~(ni δk − nk δi ), 1 ≤ i, j, k, l, ≤ m.

Now, deﬁne "partial derivatives" on the algebra U(gl(m)~) . Observe that in the algebra Sym(gl(m)) the partial de derivatives l ∂ = ∂ k are deﬁned via the action on the generators k nl l j l j ∂k (ni ) = δi δk (i.e. the partial derivatives span the space dual to j that span(ni )) and the coproduct l l l ∆(∂k ) = ∂k ⊗ 1 + 1 ⊗ ∂k . This coproduct means that if we apply a derivative to a product a b ∈ Sym(gl(m)) we have l l l ∂k (ab) = ∂k (a)b + a∂k (b).

This property is called Leibniz rule.

By passing to the algebra U(gl(m)~) we do not change the ﬁrst property (the pairing) and deﬁne the new Leibniz rule by means of the following coproduct

j j j X j k ∆(∂i ) = ∂i ⊗ 1 + 1 ⊗ ∂i + ~ ∂k ⊗ ∂i . k So, we have

j j j X j k ∂i (ab) = ∂i (a)b + a∂i (b) + ~ ∂k (a)∂i (b). k Observe that the partial derivatives commute with each other. Denote D commutative algebra generated by the partial derivatives j ∂i .

Our next aim is to deﬁne an analog of the Weyl algebra

W(U(gl(m)~). The simplest classical Weyl algebra is pq − qp = 1. (Note that physicists call it Heisenberg one.)

Our NC Weyl algebra (denoted W(U(gl(m)~)) is generated by two subalgebras U(gl(m)~) and D, also subject to some permutation relations.

How to deﬁne these permutation relations? We proceed in the standard way. Namely, we put

j l j l j j j j ∂i ⊗ nk → (∂i )1(nk ) ⊗ (∂i )2, where ∆(∂i ) = (∂i )1 ⊗ (∂i )2 in Sweedler’s notation. Let us exhibit these permutation relations:

j l l j l j l j j l ∂i ⊗ nk → nk ⊗ ∂i + δi δk + ~(∂i δk − ∂k δi ).

Note that for ~ = 0 we get the usual Weyl algebra generated by the algebra Sym(gl(m)) and the usual partial derivatives.

Observe that if g ⊂ gl(m) is a Lie subalgebra it is possible to deﬁne partial derivatives on its enveloping algebra as operators but in general neither the corresponding Weyl algebra nor the coproduct similar to that above can be deﬁned. It is the case of the Lie subalgebra g = sl(m).

Now, we deﬁne an analog of the de Rham complex on U(gl(m)~) as follows. j Introduce "pure diﬀerentials" dni by assuming that they anti-commute with each other.

Let ^ ^ ^ ω = dnj1 dnj2 ... dnjk ⊗ f , f ∈ U(gl(m) ) i1 i2 ik ~ be a k-diﬀerential. Then by deﬁnition we put ^ ^ ^ ^ X d ω = dnj1 dnj2 ... dnjk dnj ⊗ ∂i (f ). i1 i2 ik i j i,j

Theorem d2 = 0 Proof It is so since the pure diﬀerentials anticommute and the partial derivatives commute.

Let us emphasize that the above calculus is covariant w.r.t. the usual group GL(m). It is possible to deform all algebras and operators so that they become covariant w.r.t. the corresponding quantum group Uq(sl(m)) but we do not consider this construction.

Now, consider the particular case m = 2 in more detail.

Denote a, b, c, d the standard generators of the algebra U(gl(2)~) such that

[a, b] = ~ b, [a, c] = −~ c, [a, d] = 0, ....., [d, c] = ~ c.

Now, pass to generators of the compact form, namely, U(u(2)~) 1 i 1 i t = (a + d), x = (b + c), y = (c − b), z = (a − d) 2 2 2 2

we get the standard u(2)~ table of commutators

[x, y] = ~ z, [y, z] = ~ x, [z, x] = ~ y, t is central.

The generator t is called time, x, y, z play the role of spacial variables.

On this algebra the coproduct mentioned above becomes

∆(∂ ) = ∂ ⊗ 1 + 1 ⊗ ∂ + ~(∂ ⊗ ∂ − ∂ ⊗ ∂ − ∂ ⊗ ∂ − ∂ ⊗ ∂ ), t t t 2 t t x x y y z z

∆(∂ ) = ∂ ⊗ 1 + 1 ⊗ ∂ + ~(∂ ⊗ ∂ + ∂ ⊗ ∂ + ∂ ⊗ ∂ − ∂ ⊗ ∂ ), x x x 2 t x x t y z z y

∆(∂ ) = ∂ ⊗ 1 + 1 ⊗ ∂ + ~(∂ ⊗ ∂ + ∂ ⊗ ∂ + ∂ ⊗ ∂ − ∂ ⊗ ∂ ), y y y 2 t y y t z x x z

∆(∂ ) = ∂ ⊗ 1 + 1 ⊗ ∂ + ~(∂ ⊗ ∂ + ∂ ⊗ ∂ + ∂ ⊗ ∂ − ∂ ⊗ ∂ ). z z z 2 t z z t x y y x

Also, the corresponding permutation relations are

[∂ , t] = ~∂ + 1, [∂ , x] = −~∂ , [∂ , y] = −~∂ , [∂ , z] = −~∂ , t 2 t t 2 x t 2 y t 2 z

[∂ , t] = ~∂ , [∂ , x] = ~∂ + 1, [∂ , y] = ~∂ , [∂ , z] = −~∂ , x 2 x x 2 t x 2 z x 2 y

[∂ , t] = ~∂ , [∂ , x] = −~∂ , [∂ , y] = ~∂ + 1, [∂ , z] = ~∂ , y 2 y y 2 z y 2 t y 2 x

[∂ , t] = ~∂ , [∂ , x] = ~∂ , [∂ , y] = −~∂ , [∂ , z] = ~∂ + 1. z 2 z z 2 y z 2 x z 2 t

Besides, the generators ∂t ..., ∂z commute with each other and generate a commutative algebra D. Thus, we get a Weyl algebra

W(U(u(2)~)) generated by two subalgebras U(u(2)~) and D and the above permutation relations.

Given permutation relation it is possible to deﬁne the partial derivatives as operators. To this end we also need the counit on D

ε(∂t ) = ... = ε(∂z ) = 0, ε(1) = 1

extended in the multiplicative way. Then we deﬁne ∂(a) by permutating ∂ and a and by applying ε to the right factor from D. For instance, in virtue of the permutation relations we have

∂ yz = (y∂ + ~∂ ) z = y(z∂ − ~∂ ) + ~(z∂ + ~∂ + 1). x x 2 z x 2 y 2 z 2 t

~ Now, by applying the counit we conclude that ∂x (yz) = 2 . This result turns into the classical one as ~ = 0.

Note that sometimes it is more convenient to use the shifted derivative ∂˜ = ∂ + 2 in the time t. Thus, the above coproduct t t ~ becomes

∆(∂˜ ) = ~(∂˜ ⊗ ∂˜ − ∂ ⊗ ∂ − ∂ ⊗ ∂ − ∂ ⊗ ∂ ), t 2 t t x x y y z z

∆(∂ ) = ~(∂˜ ⊗ ∂ + ∂ ⊗ ∂˜ + ∂ ⊗ ∂ − ∂ ⊗ ∂ ), x 2 t x x t y z z y

∆(∂ ) = ~(∂˜ ⊗ ∂ + ∂ ⊗ ∂˜ + ∂ ⊗ ∂ − ∂ ⊗ ∂ ), y 2 t y y t z x x z

∆(∂ ) = ~(∂˜ ⊗ ∂ + ∂ ⊗ ∂˜ + ∂ ⊗ ∂ − ∂ ⊗ ∂ ). z 2 t z z t x y y x

Now, we want to extend the Weyl algebra. Since the algebra

U(u(2)~) has the Ore property, we can consider the ﬁeld of fractions a−1 b. However, it is not clear what are permutation relations of the derivatives and elements a−1 and what is the action of the derivatives on these elements. In the classical case if V is a vector ﬁeld and consequently, it is subject to the classical Leibniz rule, we have

0 = V (1) = V (aa−1) = V (a)a−1 + aV (a−1).

Thus, we get V (a−1) = −a−1 V (a)a−1. In our present setting we are dealing with another coproduct.

Dimitri Gurevich with P.Saponov and (partially) P.Pyatov Derivatives in noncommutative calculus and deformation property of quantum algebras Calculus on U(gl(m) ) ~ Example: calculus on the algebra U(u(2) ) ~ Extension of the Weyl algebra q-Witt algebra: deformation property Deformation property of Reﬂection Equation algebra As an example we compute the permutation relations and the element x−1 by using this new coproduct. First, present the permutation relations between the partial derivatives and the element x under the following form

   ~    ∂˜ x − 0 0 ∂˜ t 2 t          ~    ∂x  x 0 0  ∂x   x =  2       ~    ∂y  0 0 x −  ∂y    2     ~  ∂z 0 0 x ∂z 2 In order to compute the permutation relations of the partial derivatives and x−1, we have to invert the matrix entering this relation. It is not diﬃcult to do this. However, we do not know to invert similar matrices corresponding to other nontrivial elements.

This calculus can be used for noncanonical quantization of dynamical model. Given an operator in partial derivatives, describing such a model, we first quantize its coefficients (assumed to be elements of Sym(u(2)) or its skew-field) and second we treat all partial derivatives as above.

By representing the algebra U(u(2)~) in a family of Verma modules, we realize the "NC observables" as operators acting in this family.

Now, consider deformation property of some quantum algebras. First, consider a q-analog of Witt algebra.

Let ∂q be q-derivative (Jackson derivative) deﬁned by

f (qt) − f (t) ∂ (f (t)) = . q t(q − 1)

k k−1 Let us precise that ∂q(x ) = kqx , k ∈ Z. Hereafter, we use the qk −1 notation kq = q−1 . The Leibniz rule for it is

∂q(f (x)g(x)) = (∂qf (x))g(x) + f (qx)∂qg(x).

Also exhibit the permutation relation between ∂q and the generator x: ∂q x − q x ∂q = 1. (1)

Now, similarly to the usual Witt algebra, consider the operators

k+1 ek = x ∂q, k = −1, 0, 1, 2, ...

−1 acting on the algebra K[x, x ]. These operators act on the elements xl as follows

l k+l ek (x ) = lq x , l = 0, ±1, ±

and are subject to the relations

m+1 n+1 q emen − q enem − ((n + 1)q − (m + 1)q)em+n = 0. (2)

These relations are usually considered (see [Hu] and the references threin) as a motivation for introducing the following "q-Lie bracket"

U ⊗U → U : em ⊗en 7→ [em, en] = ((n+1)q −(m+1)q)em+n, (3)

where U = span(ek ) is the space of all ﬁnite linear combinations of the elements ek . Then by q-Witt algebra one means the space U endowed with the q-Lie bracket (3). We denote this q-Witt algebra Wq. Its enveloping algebra U(Wq) is deﬁned to be the quotient of the free tensor algebra of the space U over the ideal generated by the l.h.s. of (2).

Emphasize that the bracket (3) is well-deﬁned on the whole space U⊗2. This bracket has the following properties: "q-skew-symmetry" and "q-Jacobi identity"

[em, en] = −[en, em],

k l m (1+q )[ek , [el , em]]+(1+q )[el , [em, ek ]]+(1+q )[em, [ek , el ]] = 0

The ﬁrst relation entails that the element em ⊗ en + en ⊗ em is killed by the bracket. Consequently, in the space U⊗2 we have two subspaces

m+1 n+1 I+ = hem ⊗ en + en ⊗ emi, I− = hq emen − q enemi, (4)

which are analogs of symmetric and skew-symmetric subspaces (in fact, the symmetric one is classical).

Below, we deal with the PBW theorem under the form of [Polishchuk-Positselski]. Let us recall it. Let U be a vector space ⊗2 over the ﬁeld K and I ⊂ U be a subspace. Consider an operator [ , ]: I → U satisfying two conditions \ [ , ]12 − [ , ]23 : I ⊗ U U ⊗ I ⊂ I ,

\ [ , ]([ , ]12 − [ , ]23) = 0 on I ⊗ U U ⊗ I If moreover, the quadratic algebra A = T (U)/hI i is Koszul then the associated graded algebra GrA[ , ] where A[ , ] = T (U)/hI − [ , ]I i is isomorphic to that A. Here hI i stands for the ideal generated by a set I and by I − [ , ]I we mean the family of elements u − [ , ]u, u ∈ I .

Below, the call the above conditions the Jacobi-PP condition.

Emphasize that the subspace I ⊗ U T U ⊗ I ⊂ U⊗3 is an analog of the space of third degree skew-symmetric elements. Also, observe that the bracket is deﬁned only on the subspace I . Thus, the ﬁrst of the above conditions (which means that the bracket maps I ⊗ U T U ⊗ I ⊂ I into I ), ensures the possibility to apply the bracket once more.

Let us also show that the ﬁrst condition above (without assuming the algebra T (U)/hI i to be Koszul) is necessary for the isomorphism. Consider the element

[ ,, ]12Z − [ ,, ]23Z (5)

where Z is an arbitrary element belonging to I ⊗ U T U ⊗ I . Since the element Z − Z equals 0 in the algebra A[ , ], its image under replacing factors from I ⊗ U (resp., U ⊗ I ) by the terms [ ,, ]12Z (resp., [ ,, ]23Z) is also trivial in the algebra A[ , ]. If nevertheless, the term (5) does not belong to I , we have that in the graded algebra GrA[ , ] its second degree component is less than this component in A. Consequently, the isomorphism of the algebras GrA[ , ] and A is impossible.

Now, go back to the q-Witt algebra. This algebra is inﬁnite dimensional. However, if by U we mean all ﬁnite linear ⊗k combinations of the generators {ei }, and by U , k = 2, 3, ... we

also mean ﬁnite linear combinations of ei1 ⊗ ei2 ⊗ ... ⊗ eik , then we can extend our reasoning to this case. Namely, denote the vector space of ﬁnite linear combinations of k+1 l+1 elements q ek el − q el ek by I and consider an element belonging to the space I ⊗ U T U ⊗ I

l+1 m+1 l+1 m+1 Z = q q (q el em − q emel )ek +

m+1 k+1 k+1 q q (qm+1emek − q ek em)el + k+1 l+1 l+1 2(m+1) k+1 l+1 q q (qk+1ek el −q el ek )em = q em(q ek el −q el ek )+ 2(k+1) l+1 m+1 2(l+1) m+1 k+1 q ek (q el em−q emel )+q el (q emek −q ek em).

Compute the images of this element under the maps [ , ]12 and [ , ]23 correspondingly. We have

l+1 m+1 [ , ]12Z = q q ((m + 1)q − (l + 1)q)el+mek +

m+1 k+1 q q ((k + 1)q − (m + 1)q)em+k el + k+1 l+1 q q ((l + 1)q − (k + 1)q)ek+l em, 2(m+1) [ , ]23Z = q ((l + 1)q − (k + 1)q)emek+l + 2(k+1) q ((m + 1)q − (l + 1)q)ek el+m+ 2(l+1) q (((k + 1)q − (m + 1)q)el em+k .

Let us assume that the numbers k, l, m, k + l, k + m, l + m are pairwise distinct. Then for a generic q the diﬀerence [ , ]12Z − [ , ]23Z does not belong to the space I . Consequently, the algebras

k+1 l+1 T (U)/hq ek el − q el ek i and Gr(U(Wq)) (6)

are not isomorphic to each other, or in other words, the PBW property fails.

Remark. By introducing another parameter ~ in front of the bracket of the q-Witt algebra we get a two parameter analog of the usual Witt one. By putting ~ = 0 we get a quadratic algebra (the ﬁrst algebra from (6)) which has a good deformation property. Since this quadratic algebra is inﬁnite dimensional, we mean by this the k1 k2 kl property that the ordered monomials e1 e2 ...el form a basis of its degree l component. It can be considered as a quantization of the corresponding Poisson structure. Nevertheless, the q-Witt algebra is not a two-parameter quantization of a Poisson pencil. This is due to the failure of the PBW property for the q-Witt algebra.

Though the enveloping algebra of the q-Witt algebra does not have a good deformation property, it contains a subalgebra which does. Consider the subalgebra generated by three elements 2 e−1 = ∂q, e0 = x∂q and e1 = x ∂q. They are subject to the following relations

2 e−1e0−qe0e−1 = e−1, e−1e1−q e1e−1 = (1+q)e0, e0e1−qe1e0 = e1. (7) This algebra was called in [Larsson-Silvestrov] Jackson sl(2). However, it was treated in terms of so-called Hom-Lie algebras. This notion is based on the modiﬁed Leibniz rule above. By contrast, our approach is only based on the permutation relations between derivative(s) and generators of a given algebra. All related algebras are associative. And the PBW property for it is valid.

Dimitri Gurevich with P.Saponov and (partially) P.Pyatov Derivatives in noncommutative calculus and deformation property of quantum algebras Calculus on U(gl(m) ) ~ Example: calculus on the algebra U(u(2) ) ~ Extension of the Weyl algebra q-Witt algebra: deformation property Deformation property of Reﬂection Equation algebra The second type algebra for which we consider deformation property is so-called reﬂection equation algebra. Let R : V ⊗2 → V ⊗2 be a braiding, i.e. an invertible operator satisfying the braid relation

R12 R23 R12 = R23 R12 R23, R12 = R ⊗ I , R23 = I ⊗ R.

If R comes from the QG Uq(sl(m)) it meets a second degree equation (qI − R)(q−1I + R) = 0 and is called Hecke symmetry. If R comes from GQ of other classical series, it meets some more complected conditions and is called BMW symmetry. In all cases it depends on a parameter q and is covariant w.r.t. the action of the corresponding QG. Let us associate to it the so-called RE algebra

RL1 RL1 − L1 RL1 R = 0.

The problem we want to discuss is: whether this algebra has a good deformation property? This means that for a generic q dimension of any homogeneous component of this quadratic algebra is classical (corresponding to the value q = 1)? Note that for q = 1 we get the commutative algebra Sym(gl(m). We’ve shown that it is so if R comes from the QG Uq(sl(m)). For other QG the answer is negative. Moreover, if g is a Lie algebra belonging to one of the series Bn, Cn, Dn there is no any quadratic algebra which is Uq(g)-covariant deformation of the algebra Sym(g) with good deformation property. It was shown by J.Donin.

If R is a Hecke symmetry, the further deformation of the RE algebra is possible. Namely, consider the so-called modiﬁed RE algebra

RL1 RL1 − L1 RL1 R = ~(RL1 − L1 R) = 0.

It is a two-parameter deformation of of the algebra of the algebra Sym(gl(m). Its Poisson counterpart is a Poisson pencil generated by the quadratic bracket corresponding to the RE algebra and the linear one which is Poisson-Lie bracket { , }gl(m). Their simultaneous quantization is just the modiﬁed RE algebra.

Unfortunately, we do not know any similar two parameter deformation of the loop aﬃne algebras.

Many Thanks

Dimitri Gurevich with P.Saponov and (partially) P.Pyatov Derivatives in noncommutative calculus and deformation property of quantum algebras