On the quantization of zero-weight super dynamical r-matrices
aa r X i v : . [ m a t h . QA ] F e b ON THE QUANTIZATION OF ZERO-WEIGHT SUPER DYNAMICAL r -MATRICES GIZEM KARAALI
Abstract.
Solutions of the classical dynamical Yang-Baxter equation on a Lie superalgebraare called super dynamical r -matrices . A super dynamical r -matrix r satisfies the zero weightcondition if: [ h ⊗ ⊗ h, r ( λ )] = 0 for all h ∈ h , λ ∈ h ∗ . In this note we explicitly quantize zero-weight super dynamical r -matrices with zero couplingconstant. We also answer some questions about super dynamical R -matrices. In particularwe offer some support for one particular interpretation of the super Hecke condition. Introduction
Overview.
One of the major breakthroughs in the theory of quantum groups in thelast decade was the main quantization result from [2], the general explicit quantization of allclassical dynamical r -matrices which fit Schiffmann’s classification [11]. This complementedthe categorical quantization results of Etingof-Kazhdan [1] and provided a fully constructivemethod to quantize a given r -matrix.In this note we initiate an analogous program of constructing explicit quantizations inthe context of Lie superalgebras. In particular we explicitly quantize zero-weight superdynamical r -matrices with zero coupling constant. We next discuss the classification problemfor super dynamical R -matrices and provide some partial answers. Then we use our resultsto weigh in on the question of what the correct graded analogue should be for the Heckecondition. Thus the note overall contributes to the theory of super quantum groups, whichis still widely incomplete.1.2. Results.
Our main quantization result is the following theorem, proved in Section 3:
Theorem 1.
Let h be a finite dimensional commutative Lie superalgebra over C and let V bea finite dimensional semisimple h -module whose weights make up a basis for h ∗ . Then everysuper dynamical r -matrix r : h ∗ → End( V ⊗ V ) with zero weight and zero coupling constant,holomorphic on an open polydisc U ⊂ h ∗ , can be quantized to a super dynamical R -matrix R on U . (See §§ §§ r -matriceswith no spectral parameters were classified by the author in [8] in a manner which generalizedthe analogous non-graded results of [3]. In the proof of the above theorem, we make extensiveuse of this result, as well as results from [4].The quantum theme of this note is developed mostly in Section 4. There we briefly studythe classification problem for super dynamical R -matrices. and then focus on the super Heckecondition. The (non-graded) Hecke condition, introduced in [4] as a desirable property ofdynamical R -matrices, is a quantum analogue of the generalized unitarity condition. We roposed a super version of it in our [9]. In Section 5, we use our work here and some otherconsiderations to weigh in on this issue of the correct super version.The extension to the graded world of the general constructive quantization [2] of all clas-sical dynamical r -matrices which fit Schiffmann’s classification [11] is still an open problem,and work on it is still ongoing [5]. Part of the difficulty comes from the fact that there is notyet a complete classification result analogous to [11]; see [6, 7, 8, 9] for partial results andcounterexamples in this direction.1.3. The organization of this note.
This note is organized as follows: In Section 2 weprovide the basic definitions and summarize the result from [8] that we will need. In Section3, we prove Theorem 1. In Section 4, we sketch the development of a super analogue for theclassification of super dynamical R -matrices given in [4]. Section 5 concludes the note witha discussion of the implications of our work to the problem of determining the correct wayto superize the Hecke condition.2. Definitions and relevant earlier results
Basic notation and terminology.
Let g be a simple Lie superalgebra with non-degenerate Killing form ( · , · ). Let h ⊂ g be a Cartan subsuperalgebra, and let ∆ ⊂ h ∗ bethe set of roots associated to h . Fix a set of simple roots Γ or equivalently a Borel b . Wewill say that a set X ⊂ ∆ of roots of g is closed if it satisfies the following:(1) If α, β ∈ X and α + β is a root, then α + β ∈ X , and(2) If α ∈ X , then − α ∈ X. For any positive root α fix e α ∈ g α and pick e − α ∈ g − α dual to e α i.e.( e α , e − α ) = 1 for all α ∈ ∆ + . Note that we can do this uniquely up to scalars because all the g α are one-dimensional,(which follows from the nondegeneracy of the Killing form). Define: A α = (cid:26) ( − | α | if α is positive1 if α is negative (2.1)It is easy to see that A − α = ( − | α | A α . We can use A α for instance to write the duals of ourbasis vectors in terms of one another: e ∗ α = A − α e − α or equivalently: ( e α , e − α ) = A − α . Finally let Ω be the quadratic Casimir element, i.e. the element of g ⊗ g corresponding tothe Killing form.The super twist map T s : V ⊗ V → V ⊗ V is defined on the homogeneous elements of agiven super vector space V as T s ( a ⊗ b ) = ( − | a || b | b ⊗ a. Similarly the super symmetrizing map
Alt s : V ⊗ V ⊗ V → V ⊗ V ⊗ V is defined on homo-geneous elements by:Alt s ( a ⊗ b ⊗ c ) = a ⊗ b ⊗ c + ( − | a | ( | b | + | c | ) b ⊗ c ⊗ a + ( − | c | ( | a | + | b | ) c ⊗ a ⊗ b. .2. The classical dynamical Yang-Baxter equation.
The classical dynamical Yang-Baxter equation for a meromorphic function r : h ∗ → g ⊗ g is the equation:Alt s ( dr ) + [ r , r ] + [ r , r ] + [ r , r ] = 0 . (2.2)Here, for a fixed (even) basis { x i } for h , the differential of r is defined as: dr : h ∗ −→ g ⊗ g ⊗ g λ P i x i ⊗ ∂r∂x i ( λ )Thus, we can see that for r = P r (1) ⊗ r (2) , Alt s ( dr ) may be rewritten as: X i x (1) i (cid:18) ∂r∂x i (cid:19) (23) + X i x (2) i (cid:18) ∂r∂x i (cid:19) (31) + X i ( − | r (1) || r (2) | x (3) i (cid:18) ∂r∂x i (cid:19) (12) . We will say that a meromorphic function r : h ∗ → g ⊗ g is a super dynamical r -matrix withcoupling constant ǫ if it is a solution to Equation (2.2) and satisfies the generalized unitaritycondition : r ( λ ) + T s ( r )( λ ) = ǫ Ω . (2.3)A super dynamical r -matrix r satisfies the zero weight condition if:[ h ⊗ ⊗ h, r ( λ )] = 0 for all h ∈ h , λ ∈ h ∗ . Classification of super dynamical r -matrices of zero weight. In [8] we proved:
Theorem 2.
Let g be a simple Lie superalgebra with non-degenerate Killing form ( · , · ) , h ⊂ g a Cartan subsuperalgebra, and ∆ ⊂ h ∗ the set of roots associated to h . (1) Let X be a closed subset of the set of roots ∆ of g . Let ν ∈ h ∗ , and let D = P i Quantization of zero weight r -matrices In this section we prove Theorem 1. .1. The quantum dynamical Yang-Baxter equation. Let h be a finite dimensionalcommutative Lie superalgebra over C , V a finite dimensional super vector space over C witha diagonal(izable) h action, and let V = ⊕ ω ∈ h ∗ V [ ω ] be V ’s h -weight decomposition. In otherwords, for every v ∈ V [ ω ] and x ∈ h , we have x · v = ω ( x ) v .In this context, the quantum dynamical Yang-Baxter equation with step γ for a function R : h ∗ → End( V ⊗ V ) is the equation: R ( λ − γh (3) ) R ( λ ) R ( λ − γh (1) ) = R ( λ ) R ( λ − γh (2) ) R ( λ ) . (3.1)Here the operator R ij is interpreted to be acting nontrivially on the i th and the j th compo-nents of a given 3-tensor, and the notation h ( k ) is to be replaced by the weight of the k th com-ponent of the same. For instance R ( λ − γh (3) )( v ⊗ v ⊗ v ) = ( R ( λ − γω )( v ⊗ v )) ⊗ v whenever v ∈ V [ ω ].We will say that an invertible function R : h ∗ → End( V ⊗ V ) is a super dynamical R -matrix if it is a solution to Equation (3.1) and satisfies the zero weight condition :[ h ⊗ ⊗ h, R ( λ )] = 0 for all h ∈ h , λ ∈ h ∗ . The quantization problem. Let R γ : h ∗ → End( V ⊗ V ) be a smooth family of solu-tions to Equation (3.1) such that: R γ ( λ ) = 1 − γr ( λ ) + O ( γ ) . Then the function r ( λ ) satisfies Equation (2.2) and is called the semi-classical limit of R γ ( λ ).In the same setup R γ ( λ ) is called a quantization of r ( λ ).Alternatively we can begin with a super dynamical r -matrix r : h ∗ → End( V ⊗ V ) definedon an open subset U of h ∗ . We then call r quantizable if there is a power series in γ of theform: R γ ( λ ) = 1 − γr ( λ ) + ∞ X n =2 γ n r n ( λ )satisfying Equation (3.1). The quantization problem for us must now be obvious: Given asuper dynamical r -matrix construct a power series R γ ( λ ) of the form above (or prove theimpossibility of such a construction).3.3. Multiplicative forms. In the following we will make use of multiplicative k -forms ala [4, §§ V = V ⊕ V be a super vector space with a homogeneous linear coordinate system λ , · · · , λ N . We define a multiplicative k -form on V to be a collection: ϕ = { ϕ i ,...,i k ( λ , · · · , λ N ) } of meromorphic functions, where the ordered k -tuples ( i , . . . , i k ) run through all k -elementsubsets of { , · · · , N } , and we require that: ϕ τ ( I ) ϕ I = 1whenever I = ( i , . . . , i k ) is some ordered k -tuple and τ ( I ) is a transposition ( i s i s +1 ) switchingthe consecutive indices i s , i s +1 for some 1 ≤ s < k . Let Ω k ( V ) = Ω k be the set of allmultiplicative k -forms on V . There is a natural abelian group structure on Ω k . ow we fix a complex number γ . We define, for each i = 1 , · · · , N , an operator δ i on thespace of all meromorphic functions on the N variables λ , · · · , λ N : δ i : f ( λ , · · · , λ N ) f ( λ , · · · , λ N ) /f ( λ , · · · , λ i − γ, · · · , λ N ) . We next define an operator d γ : Ω k → Ω k +1 mapping ϕ to d γ ϕ given by:( d γ ϕ ) i ,...,i k +1 ( λ , · · · , λ N ) = k +1 Y s =1 (cid:0) δ i s ϕ i ,...,i s − ,i s +1 ,...,i k +1 ( λ , · · · , λ N ) (cid:1) ( − s +1 . A multiplicative k -form ϕ is γ -closed if d γ ϕ = 0. Obviously, d γ = 0 because the zero elementof Ω k is the form { ϕ i ,...,i k ( λ , · · · , λ N ) ≡ } .Still following [4, §§ ϕ ( γ ) = { ϕ i ,i ,...,i k ( λ , λ , · · · , λ N , γ ) } of multiplicative k -forms with ϕ I ( λ, γ ) = 1 − γC I ( λ ) + O ( γ ) for each I = ( i , i , . . . , i k )is a quantization of the differential form C = X i
Every closed holomorphic differential k -form C defined on an open polydisc isquantizable to a holomorphic multiplicative closed k -form ϕ ( γ ) . The proof is included in [4] and will not be repeated here.3.4. R -matrices of gl ( m, n ) type. Now let h be a finite dimensional commutative Liesuperalgebra over C and let V = V ⊕ V be a finite dimensional semisimple h -modulewhose weights W = { ω , ω , · · · , ω N } make up a basis for h ∗ . We label the elements ofthe dual basis for h by x i ; clearly the x i are all even, and dim C V = dim C h = N . Let { v , v , · · · , v N } be an h -eigenbasis for V with x i v j = δ ij v j . By relabeling as needed, we canassume that v , v , · · · v m is a basis for V , the even part of V , while v m +1 , v m +2 · · · , v N isa basis for V , the odd part of V . Let n = N − m . We will say that a super dynamical R -matrix R : h ∗ → End( V ⊗ V ) for such h and V is an R -matrix of gl ( m, n ) type . The superdynamical R -matrices in this paper will all be of this kind unless explicitly noted otherwise.In this setup V ⊗ V has the following weight decomposition: V ⊗ V = N M i =1 V ii ! ⊕ M i 5) The transformation: r ( λ ) r ( λ ) + c Idfor a nonzero complex number c ∈ C .Each of these transformations corresponds to a specific quantum gauge transformation al-lowed for super dynamical R -matrices (cf. [4]). We will briefly study these quantum gaugetransformations in §§ R -matrices.It is easy to show that the transformations (1-5) map a given super dynamical r -matrix toanother. We omit the proofs here since they are straightforward modifications of those in [3].We will say that two super dynamical r -matrices are gauge equivalent (or simply equivalent when the context is unambiguous) if one can be obtained from the other by a sequence ofgauge transformations.We can now simplify the expression in Theorem 2 using the above. Let X ⊂ { , , · · · , N } be a subset of indices and write it as a disjoint union of subintervals X = X ⊔ X ⊔ · · · ⊔ X n .In other words, every subinterval X k should be of the form X k = [ i k , i k +1 , i k +2 , · · · , j k ], and j k < i k +1 for each k . Define: A ij = (cid:26) ( − σ ( i )+ σ ( j ) if i < j, i > j, (cf. Equation (2.1)). Now applying the above transformations and using the { E ij } basis, wecan show that the super dynamical r -matrix in Equation (2.4) is (gauge-)equivalent to: r rat ( λ ) = n X k =1 X i,j ∈ X k ,i = j A ij λ ij E ij ⊗ ( E ij ) ∗ ! . Since ( E ij ) ∗ = A ji ( − σ ( i ) E ji , this further reduces to: r rat ( λ ) = n X k =1 X i,j ∈ X k ,i = j ( − σ ( j ) λ ij E ij ⊗ E ji ! . (3.3)3.6. The construction. We are finally ready to construct the quantization necessary forTheorem 1. Let h and V be as in §§ { E ij } for End( V )and we will write X ⊂ { , , · · · , N } as a disjoint union of subintervals X = X ⊔ X ⊔· · ·⊔ X n Consider: R rat ( λ, γ ) = Id + n X k =1 X i,j ∈ X k ,i = j γλ ij (cid:0) E ii ⊗ E jj + ( − σ ( i ) E ji ⊗ E ij (cid:1) Then R rat ( λ, γ ) satisfies Equation (3.1) (cf. Theorem 4), and its semi-classical limit is: r ′ rat ( λ ) = n X k =1 X i,j ∈ X k ,i = j − λ ij (cid:0) E ii ⊗ E jj + ( − σ ( i ) E ji ⊗ E ij (cid:1) . Using the gauge transformation of type (1) with the closed form: D = n X k =1 X i,j ∈ X k ,i In this section we define the super Hecke condition ( §§ §§ R -matrices satisfying the super Hecke condition. It turns out that the super Hecke conditionencodes the constraint on the coupling constant in the classical case.4.1. Some initial computations. Let h and V be as in §§ { E ij } for End( V ) and throughout this section we will once again restrict ourselves tothe study of R -matrices of gl ( m, n ) type. Recall that this means, in particular, that thesuper vector space V ⊗ V has the weight decomposition given in (3.2).More specifically, a super dynamical R -matrix R : h ∗ → End( V ⊗ V ) of gl ( m, n ) type canbe written in the form: R ( λ ) = N X i,j =1 α ij ( λ ) E ii ⊗ E jj + X i = j β ij ( λ ) E ji ⊗ E ij for some meromorphic functions α ij , β ij : h ∗ → C . If we now assume for simplicity (and forother reasons which will become clearer in §§ R -matrices allsatisfy α ii = 1 for all i , we can rewrite the above as: R ( λ ) = N X i =1 E ii ⊗ E ii + X i = j α ij ( λ ) E ii ⊗ E jj + X i = j β ij ( λ ) E ji ⊗ E ij (4.1)for some meromorphic functions α ij , β ij : h ∗ → C .In this subsection we list a few conditions on these α and β functions. We limit ourselvesto simply summarizing the results of necessary computations; the explicit derivations can befound in Appendix ?? .By applying the two sides of Equation (3.1) for an R of the form (4.1) to a basis element v i ⊗ v i ⊗ v k of V ⊗ with i = k and setting the coefficients of like terms equal to one another,we obtain: α ki ( λ − γω i ) β ik ( λ ) α ik ( λ − γω i ) + ( β ik ( λ − γω i )) = β ik ( λ − γω i ) (4.2)and ( − σ ( i )+ σ ( k ) β ki ( λ − γω i ) β ik ( λ ) α ik ( λ − γω i ) + α ik ( λ − γω i ) β ik ( λ − γω i )= β ik ( λ ) α ik ( λ − γω i ) . (4.3)Note that Equation (4.2) is identical to [4, Eqn.1.8.4] while Equation (4.3) is a signed versionof [4, Eqn.1.8.5].Similarly we can derive the following equations by applying the two sides of Equation (3.1)to a basis element v i ⊗ v j ⊗ v k with i, j, k all distinct: α ij ( λ − γω k ) α ik ( λ ) α jk ( λ − γω i ) = α jk ( λ ) α ik ( λ − γω j ) α ij ( λ ) (4.4)which is precisely the same as [4, Eqn.1.8.6]; α ik ( λ − γω j ) α ij ( λ ) β jk ( λ − γω i ) = β jk ( λ ) α ik ( λ − γω j ) α ij ( λ ) (4.5) hich is precisely the same as [4, Eqn.1.8.7]; β ij ( λ − γω k ) α ik ( λ ) α jk ( λ − γω i ) = α ik ( λ ) α jk ( λ − γω i ) β ij ( λ ) (4.6)which is precisely the same as [4, Eqn.1.8.8];( − σ ( k ) β kj ( λ − γω i ) β ik ( λ ) α jk ( λ − γω i ) + ( − σ ( j ) α jk ( λ − γω i ) β ij ( λ ) β jk ( λ − γω i )= ( − σ ( i ) β ik ( λ ) α jk ( λ − γω i ) β ij ( λ ) (4.7)which is a signed analogue of [4, Eqn.1.8.9]; α kj ( λ − γω i ) β ik ( λ ) α jk ( λ − γω i ) + β jk ( λ − γω i ) β ij ( λ ) β jk ( λ − γω i ) = α ji ( λ ) β ik ( λ − γω j ) α ij ( λ ) + ( − σ ( i )+ σ ( j ) β ij ( λ ) β jk ( λ − γω i ) β ij ( λ ) (4.8)which is a signed analogue of [4, Eqn.1.8.10]; and β ik ( λ − γω j ) α ij ( λ ) β jk ( λ − γω i ) = β ji ( λ ) β ik ( λ − γω j ) α ij ( λ ) + α ij ( λ ) β jk ( λ − γω i ) β ij ( λ ) (4.9)which is precisely the same as [4, Eqn.1.8.11].4.2. The Super Hecke Condition. Let p = − q be two complex numbers. Set ˇ R = P s R where P s ∈ End( V ⊗ V ) is the element corresponding to T s . In a way analogous to [4] wewill say that a function R : h ∗ → End( V ⊗ V ) satisfies the strong super Hecke condition if ithas the following properties:(1) The function preserves the weight decomposition given in (3.2).(2) For any i = 1 , , · · · , N , and λ ∈ h ∗ , ˇ R ( λ )( v i ⊗ v i ) = p ( v i ⊗ v i ).(3) For any i = j , and λ ∈ h ∗ , the operator ˇ R ( λ ) restricted to V ij has eigenvalues( − σ ( i )+ σ ( j ) p and − ( − σ ( i )+ σ ( j ) q .A function R : h ∗ → End( V ⊗ V ) satisfies the weak super Hecke condition if it has thefollowing properties (cf [4, Eq.1.3.6]) :(1) The function preserves the weight decomposition given in (3.2).(2) For any λ ∈ h ∗ and i, j ≤ N , ( ˇ R ( λ ) − ( − σ ( i )+ σ ( j ) p )( ˇ R ( λ ) + ( − σ ( i )+ σ ( j ) q ) = 0 whenrestricted to V ij .Just as in the non-graded case these two properties are intimately related. In fact whenevera continuous family R t : h ∗ → End( V ⊗ V ), t ∈ [0 , < t < R = Id, satisfies the weak super Hecke condition for all t , then R t satisfiesthe strong super Hecke condition as well. Hence we will simply assume that R satisfies bothwhenever we say that R satisfies the super Hecke condition.Now we consider a super dynamical R -matrix R ( λ ) with step γ = 1 which satisfies thesuper Hecke condition with p = 1 and q arbitrary. Then we can see that α ii = 1 and R hasthe form given by Equation (4.1). Furthermore, whenever i = j , we have:( − σ ( i ) β ij ( λ ) + ( − σ ( j ) β ji ( λ ) = ( − σ ( i )+ σ ( j ) (1 − q ) (4.10)and ( − σ ( i )+ σ ( j ) β ij ( λ ) β ji ( λ ) − α ij ( λ ) α ji ( λ ) = − q (4.11)obtained from the trace and determinant of ˇ R on V ij . Note that these are signed versions of[4, Eqn.1.8.2] and [4, Eqn.1.8.3]. t this point it is easy to notice that if i = j , then assuming α ij ≡ β ij ( λ ) β ji ( λ ) = − ( − σ ( i )+ σ ( j ) q by Equation (4.11), and Equation (4.2) gives us:( β ij ( λ )) = β ij ( λ ) and ( β ji ( λ )) = β ji ( λ ) . These then contradict with Equation (4.10). Therefore α ij cannot be identically zero. Simi-larly we can show that α ij ( λ ) α ji ( λ ) = (( − σ ( i ) β ij ( λ ) + ( − σ ( i )+ σ ( j ) q )(( − σ ( j ) β ji ( λ ) + ( − σ ( i )+ σ ( j ) q ) (4.12)and therefore the quantity ( − σ ( i ) β ij ( λ ) + ( − σ ( i )+ σ ( j ) q is also not identically zero.Finally we consider a super dynamical R -matrix R ( λ ) of the form (4.1) with step γ = 1,and assume that R ( λ ) satisfies the super Hecke property with Hecke parameters p = 1 and q . Then the collection of functions: ϕ = { ϕ ij ( λ ) } where ϕ ij ( λ ) = ( − σ ( i ) β ij ( λ ) + ( − σ ( i )+ σ ( j ) qα ij ( λ ) for i = j (4.13)is a γ -closed multiplicative 2-form with γ = 1. This follows from our earlier computationsand in particular from Equation (4.12); just as in [4], Equations (4.4) and (4.5) are used toshow that d γ ϕ = 0. We will use this ϕ in the next subsection.4.3. Gauge transformations for super dynamical R -matrices. Let us now assumethat we have a super dynamical R -matrix of gl ( m, n ) type and we write it in the form givenby Equation (4.1). The following is a list of the gauge transformations for such R ( λ ) whichwe will need in the rest of this note (cf. [4, §§ R ( λ ) N X i =1 E ii ⊗ E ii + X i = j ϕ ij ( λ ) α ij ( λ ) E ii ⊗ E jj + X i = j β ij ( λ ) E ji ⊗ E ij for some meromorphic s-multiplicative γ -closed multiplicative 2-form { ϕ ij } on h ∗ .(2) The transformation: R ( λ ) ( τ ⊗ τ ) R ( τ − · λ )( τ − ⊗ τ − )for some permutation τ ∈ S N of the coordinates in h ∗ and V .(3) The transformation: R ( λ ) cR ( λ )for a nonzero complex number c ∈ C .(4) The transformation: R ( λ ) R ( cλ + µ )for a nonzero complex number c ∈ C and an element µ ∈ h ∗ .It is easy to see that transformations of type (1-3) transform a super dynamical R -matrixwith step γ to another one with step γ . In particular it suffices to check that the rele-vant equations in §§ §§ α ij ( λ ) and β ij ( λ ) are invariant with respect to them.Transformations of type (4) modify the step γ to γ/c .In all cases the super Hecke property is preserved. While the transformations of type(3) modify the relevant Hecke parameters, the rest preserve them. Moreover, any super ynamical R -matrix R ( λ ) of Hecke type can be shown to be (gauge-)equivalent to a superdynamical R -matrix R ( λ ) with step γ = 1 which satisfies the super Hecke condition with p = 1 and q arbitrary. This requires simply gauge transformations of types (3) and (4).At this point we can specialize (4.1) even further. Once again let R ( λ ) be a super dynamical R -matrix of the form (4.1) with step γ satisfying the super Hecke condition. As justifiedby the above we can assume that the step γ = 1 and the Hecke parameters are p = 1 and q arbitrary. Then if we apply the gauge transformation of type (1) to this R ( λ ) using thereciprocal of the multiplicative 2-form given in (4.13), we obtain a new super dynamical R -matrix (satisfying the super Hecke condition with the same parameters) whose coefficientsnow satisfy α ij ( λ ) = ( − σ ( i ) β ij ( λ ) + ( − σ ( i )+ σ ( j ) q for i = j. (4.14)4.4. Statement of the main quantum theorem. We are now ready to state the mainresult of this section: Theorem 4 (Classification Theorem for Equal Parameters) . Let h be a finite dimensionalcommutative Lie superalgebra over C and let V = V ⊕ V be a finite dimensional semisimple h -module whose weights make up a basis for h ∗ . Let N = dim C V = dim C h . (1) Let X ⊂ { , , · · · , N } be a subset of indices written as a disjoint union of subintervals X = X ⊔ X ⊔ · · · ⊔ X n . Fix a γ -quasiconstant µ : h ∗ → h ∗ with γ = 1 . Define scalarmeromorphic γ -quasiconstant functions µ ij : h ∗ → C by µ ij ( λ ) = x ( µ ( λ )) − x j ( µ ( λ )) .Then the meromorphic function R X : h ∗ → End( V ⊗ V ) defined by: R X ( λ ) = N X i,j =1 ( − σ ( i )+ σ ( j ) E ii ⊗ E jj + n X s =1 X i,j ∈ X s ,i = j λ ij − µ ij ( λ ) [ E ii ⊗ E jj + ( − σ ( i ) E ji ⊗ E ij ] ! is a super dynamical R -matrix of gl ( m, n ) type satisfying the super Hecke conditionwith p = q = 1 and step γ = 1 . (2) Every super dynamical R -matrix of gl ( m, n ) type satisfying the super Hecke conditionwith p = q is equivalent to a super dynamical R -matrix of this form. Proof of Theorem 4. Now we let R ( λ ) be a super dynamical R -matrix satisfying thesuper Hecke condition with parameters p = q . As we showed in the previous subsection, wecan use appropriate gauge transformations to ensure that γ = p = q = 1. Then Equation(4.14) becomes: α ij ( λ ) = ( − σ ( i ) β ij ( λ ) + ( − σ ( i )+ σ ( j ) for i = j. Next look at Equation (4.3) for indices i, j . Clearly β ij ( λ ) = β ji ( λ ) ≡ §§ α ij cannot beidentically zero, we obtain from the two versions (for i, i, j and j, j, i , reading the coefficientsof i, j, i and j, i, j respectively), the following two conditions on β ij :1 β ij ( λ ) − β ij ( λ − ω i ) = 1 for i = j, (4.15)and 1 β ij ( λ ) − β ij ( λ − ω j ) = − ( − σ ( i )+ σ ( j ) for i = j. (4.16)where we are using ( − σ ( i ) β ij ( λ )+( − σ ( j ) β ji ( λ ) = 0 or equivalently A ij β ij ( λ )+ A ji β ji ( λ ) = 0(obtained from Equation (4.10) with q = 1). ewriting these equations as: β ij ( λ − ω i ) = β ij ( λ )1 − β ij ( λ )and β ji ( λ − ω i ) = β ji ( λ )1 + ( − σ ( i )+ σ ( j ) β ji ( λ )and using the description of α ij ( λ ) in terms of the β ij ( λ ) given above, we see that solutions β ij ( λ ) , β ji ( λ ) to the above equations will also be solutions to Equation (4.2) (cf. [4, Lemma1.4].Furthermore defining µ ij ( λ ) = λ ij − ( − σ ( i ) β ij ( λ )we can show that µ ij ( λ − ω i ) = µ ij ( λ − ω j ) = µ ij ( λ ) for all i = j . Thus the meromorphicfunctions β ij ( λ ) = ( − σ ( i ) λ ij − µ ij ( λ ) and β ji ( λ ) = ( − σ ( j ) λ ji − µ ji ( λ )where µ ij ( λ ) = − µ ji ( λ ) and µ ij ( λ ) is a meromorphic function periodic with respect to shiftsof λ by ω i and ω j will be solutions to Equations (4.2) and (4.3) (cf. [4, Lemma 1.4]).Note that Equations (4.5) and (4.6) imply that the function β ij ( λ ) is periodic with respectto shifts of λ by ω k for all k distinct from i and j . This periodicity then holds also for µ ij ( λ ).Next look at Equation (4.7) on functions β ij ( λ ), β jk ( λ ), β ik ( λ ), we note that if any oneof these is identically zero, then at least one more has to be identically zero. This allowsus to define an equivalence relation on the indices { , , · · · , N } : First assert that all i arerelated to themselves. Then for i = j let i be related to j if β ij ( λ ) is not identically zero.The symmetry property follows directly from the trace condition.For the equivalence relation defined above, let Y = Y ∪ Y ∪ · · · ∪ Y n be the union of all n equivalence classes Y i with more than one element. If pairwise distinct i, j, k do not allbelong in the same equivalence class, then at least two of β ij , β jk , β ik will be identically zero,thus the triple will be consistent with Equation (4.7). If all three lie in the same equivalenceclass, then we get:( − σ ( k ) β kj ( λ − ω i ) β ik ( λ ) + ( − σ ( j ) β ij ( λ ) β jk ( λ − ω i ) = ( − σ ( i ) β ik ( λ ) β ij ( λ ) , and by periodicity of β kj and β jk with respect to ω i , we reduce this further to:( − σ ( k ) β kj ( λ ) β ik ( λ ) + ( − σ ( j ) β ij ( λ ) β jk ( λ ) = ( − σ ( i ) β ik ( λ ) β ij ( λ ) . We can rewrite this as:( − σ ( k ) (cid:18) ( − σ ( k ) λ kj − µ kj ( λ ) (cid:19) (cid:18) ( − σ ( i ) λ ik − µ ik ( λ ) (cid:19) + ( − σ ( j ) (cid:18) ( − σ ( i ) λ ij − µ ij ( λ ) (cid:19) (cid:18) ( − σ ( j ) λ jk − µ jk ( λ ) (cid:19) = ( − σ ( i ) (cid:18) ( − σ ( i ) λ ik − µ ik ( λ ) (cid:19) (cid:18) ( − σ ( i ) λ ij − µ ij ( λ ) (cid:19) , which, after sign cancelations, yields µ ik ( λ ) = µ ij ( λ )+ µ jk ( λ ). Therefore as in the non-gradedcase of [4, §§ µ : h ∗ → h ∗ such that µ ij ( λ ) = x i ( µ ( λ )) − x j ( µ ( λ )) for all i, j with µ ij not identically zero,and thus Equation (4.8) is also satisfied. et τ be a permutation of { , . . . , N } that transforms the set Y into a set X which cannow be written as a disjoint union of subintervals X = X ⊔ X ⊔ · · · ⊔ X n . In other words,every subinterval X k should be of the form X k = [ i k , i k +1 , i k +2 , · · · , j k ], and j k < i k +1 foreach k . Finally applying a gauge transformation of type (2) for this τ to the R -matrix R will yield an R -matrix of the form desired. This completes the proof of Theorem 4. (cid:3) Conclusion In this note we proved a quantization theorem for super dynamical r -matrices. More specif-ically we explicitly constructed quantizations for zero weight super dynamical r -matrices withzero coupling constant. We expect that the definitions and constructions here will also behelpful in the proof of an analogous quantization result for nonzero coupling constants, weplan to follow up on this thread in future work.It must be clear that quantization in this note meant finding a solution to the quantumdynamical Yang-Baxter equation whose semi-classical limit was the original super dynamical r -matrix. In particular we have not explicitly constructed algebraic structures which shouldbe the corresponding dynamical quantum groups associated to the resulting R -matrices.However, while working in the quantum picture, we have proposed and used a particularalgebraic condition which we called the super Hecke condition (cf. Subsection 4.2). Findingthe correct super Hecke condition is important because the Hecke condition in the non-graded case turns out to be the right pre-condition for a meaningful description of dynamicalquantum groups in the language of Hopf algebroids (cf. [4]).Studying the proof of our main classification result for super dynamical R -matrices (The-orem 4), one can see that the building blocks fall into their right places when one defines thesuper Hecke condition as we do. In this framework, the super dynamical R -matrices withequal Hecke parameters correspond precisely to the zero weight super dynamical r -matriceswith zero coupling constant. This is exactly analogous to the non-graded picture in [4].This observation may offer some support for our particular definition of the super Heckecondition.The construction of the actual algebraic structures that correspond to the super dynamical R -matrices we study in Section 4 involves the difficult problem of determining what theappropriate super analogue to dynamical quantum groups should be. This is beyond thescope of this note, but we believe that our work here will shed some light to it by contributingsome evidence for the right way to superize the Hecke condition. We intend to address thisissue in depth in our followup work. For various possible approaches to the theory of superdynamical quantum groups and some preliminary results, see [9, 10]. References [1] Etingof, P., Kazhdan, D.; “ Quantization of Lie Bialgebras I ”, Selecta Math. (1996), no.1, pp.1–41.[2] Etingof, P., Schedler, T., Schiffmann, O.; “ Explicit quantization of dynamical r -matrices for finitedimensional semisimple Lie algebras ”; J. Amer. Math. Soc. (2000), no.3, pp.595–609.[3] Etingof, P., Varchenko, A.; “ Geometry and Classification of Solutions of the Classical Dynamical Yang-Baxter Equation ”, Comm. Math. Phys. (1998), no. 1, pp.77–120.[4] Etingof, P.; Varchenko, A.; “ Solutions of the quantum dynamical Yang-Baxter equation and dynamicalquantum groups ”; Comm. Math. Phys. (1998), no. 3, pp.591–640.[5] Geer, N.; Karaali, G.; “ Explicit quantization of super dynamical r-matrices ”; work in progress. 6] Karaali, G.; “ Constructing r-matrices on Simple Lie Superalgebras ”, J. Algebra (2004), no.1, pp.83–102. (preliminary version available at arXiv:math.QA/0303246, 2003).[7] Karaali, G.; “ A New Lie Bialgebra Structure on sl (2 , (2006), pp.101-122.(preliminary version available at arXiv:math.RA/0410473, 2004).[8] Karaali, G.; “ Super Solutions of the Dynamical Yang-Baxter Equation ”, Proc. Amer. Math. Soc. (2006), pp.2521-2531. (preliminary version available at arXiv:math.QA/0503499, 2005).[9] Karaali, G.; “ Dynamical Quantum Groups - The Super Story ”, Contemp. Math. (2007), pp.19-52.(preliminary version available at arXiv:math.QA/0508556, 2005).[10] Karaali, G.; “ On Hopf Algebras and Their Generalizations ”, Comm. Algebra (2008), pp.4341–4367.(preliminary version available at arXiv:math.QA/0703441, 2007).[11] Schiffmann, O.; “ On Classification of Dynamical r-matrices ”, Math. Res. Lett. (1998), pp.13–30. Department of Mathematics, Pomona College, Claremont, CA 91711 E-mail address : [email protected]@pomona.edu