Simple Modules Over The Quantum Euclidean 2n-Space
aa r X i v : . [ m a t h . R T ] N ov SIMPLE MODULES OVER THE QUANTUM EUCLIDEAN SPACE
SNEHASHIS MUKHERJEE Abstract. In this article, the simple modules of the coordinate ring O π ( o K π ) ofthe quantum Euclidean 2 π -space are classiο¬ed. Especially, we use the results ofDe Concini and Processi to comment about the dimension of the simple modulesover quantum Euclidean space and then construct the simple modules assuming theparameter π is a root of unity. Intr oduction
Let K be a ο¬eld and let K β denote the group K \ { } . Let π β K β . The coordinatering O π ( o K π ) of the quantum Euclidean 2 π -space is the algebra generated over theο¬eld K by the variables π₯ , Β· Β· Β· , π₯ π , π¦ , Β· Β· Β· , π¦ π subject to the relations. π¦ π π¦ π = π β π¦ π π¦ π β π < π .π₯ π π¦ π = π β π¦ π π₯ π β π β π .π₯ π π₯ π = ππ₯ π π₯ π β π < π .π₯ π π¦ π = π¦ π π₯ π + Γ π<π ( β π β ) π¦ π π₯ π β π. The algebra O π ( o K π ) ο¬rst arose in [3] and was given a simplier set of relations in[5]. The generators for the even case with π = π are given by π , π β² , Β· Β· Β· , π π , π π β² .To get the above relations set π₯ π = π ( π + β π ) β² and π¦ π = π ( π + β π ) π ( π + β π ) .Musson studied Ring-theoretic properties of the coordinate rings of quantum Eu-clidean space in [5]. Primitive ideals in the coordinate ring of quantum Euclideanspace was studied by Sei Qwon Oh and Chun Gil Park in [6]. Oh also wrote an articleon Quantum and Poisson structures of multi-parameter symplectic and Euclideanspaces [7]. The algebras arising in the quantum mechanics are mostly non commu-tative and it is an really important problem to study their irreducible representations Mathematics Subject Classiο¬cation.
Key words and phrases.
Quantum Euclidean space, Simple modules, Polynomial Identity algebra. i.e. simple modules. In this article we want to give an explicit construction of thesimple modules over O π ( o K π ) assuming π is a primitive π -th root of unity where π is an odd integer. Throughout the paper a module means a right module and K isan algebraically closed ο¬eld.The arrangement of the paper is as follows: In Section 2 we discuss the prelimi-naries about Polynomial Identitiy algebra and quantum Euclidean space. In Section3 we recall the theory developed by De Concini and Processi in [2]. In Section 4 wegive an explicit construction of a class of simple modules over O π ( o K π ) and ο¬nallyin Section 5 we prove that any π₯ π , π π : = Γ π β€ π ( β π β ) π¦ π π₯ π -torsionfree simple moduleover O π ( o K π ) is isomorphic to the module deο¬ned in Section 4. In particular weprove the following theorems in this article. Theorem A (Construction Of Simple Modules) . For πΎ = ( πΌ , Β· Β· Β· , πΌ π , π , Β· Β· Β· , π π ) β( K β ) π , let π ( πΎ ) be the K -vector space with basis π ( π , Β· Β· Β· , π π ) , where β€ π π β€ π β . Then there is an O π ( o K π ) -module structure on π ( πΎ ) deο¬ned as follows: π ( π , Β· Β· Β· , π π ) π₯ = πΌ π β( π +Β·Β·Β·+ π π ) π ( π , Β· Β· Β· , π π ) π ( π , Β· Β· Β· , π π ) π₯ π = πΌ π π ( π +Β·Β·Β·+ π π β ) π ( π , Β· Β· Β· , π π + , Β· Β· Β· , π π ) , β β€ π β€ ππ ( π , Β· Β· Β· , π π ) π¦ = πΌ β π ( β π β ) β π β( π +Β·Β·Β·+ π π ) π ( π , Β· Β· Β· , π π ) π ( π , Β· Β· Β· , π π ) π¦ π = πΌ β π (cid:16) π β ( π π + +Β·Β·Β·+ π π )β( π π β +Β·Β·Β·+ π ) (cid:17) ( β π β ) β ( π π β π β π π π π β ) π ( π , Β· Β· Β· , π π + (β ) , Β· Β· Β· , π π ) , β β€ π β€ π where + is addition in the additive group Z / π Z . Moreover, π ( πΎ ) is a simple O π ( o K π ) -module of dimension π π β . The following result also hold.
Theorem B.
Each simple O π ( o K π ) -module with π₯ π , π π -torsion free for all π , has theform π ( πΎ ) , for some πΎ β ( K β ) π . IMPLE MODULES OVER THE QUANTUM EUCLIDEAN SPACE 3 Preliminaries
In this section we recall some known facts concerning prime Polynomial Identityalgebra as well as the K -algebra O π ( o K π ) that we shall be needing in the developmentof our results. Deο¬nition 2.1.
A ring π is a polynomial identity ring (or PI ring for short) if π satisο¬es a monic polynomial π β Z h π i . Here Z h π i is the free Z -algebra on a ο¬nite set π = { π₯ , Β· Β· Β· , π₯ π } , and to say π satisο¬es π = π ( π₯ , Β· Β· Β· , π₯ π ) means π ( π , Β· Β· Β· , π π ) = π , Β· Β· Β· , π π β π . Theminimal degree of a PI ring π is the least degree of a monic polynomial satisο¬ed by π . Proposition 2.1 ([4], Corollary 13.1.13) . If π is a ring which is ο¬nitely generatedmodule over a commutative subring, then π is a PI ring. Proposition 2.2. O π ( o K π ) is a PI ring.Proof. Let π be the subalgebra of O π ( o K π ) generated by π₯ ππ , π¦ ππ , β€ π β€ π . Then π is contained in the center of the algebra O π ( o K π ) . Also O π ( o K π ) is ο¬nitelygenerated π -module. Hence the assertion follows from proposition (2.1). (cid:3) Fundamental to the theory of PI rings is
Proposition 2.3 ([4], Theorem 13.38, Kaplanskyβs Theorem) . Let π be a primitivePI ring of minimal degree π . Then π is even and π is a central simple algebra ofdimension ( π / ) over its centre. The generalisation of Kaplanskyβs theorem to prime rings is
Proposition 2.4 ([4], Theorem 13.6.5, Ponserβs Theorem) . Ler π be a prime PI ringwith centre π and with minimal degree π . Let π = π \ { } , let π = π π β and πΉ = π π β denote the quotient ο¬eld of π . Then π is a central simple algebra withcentre πΉ , and dim πΉ ( π ) = ( π / ) . Deο¬nition 2.2.
The PI-degree of a prime PI ring π is the square root of the vectorspace dimension of its Goldie quotient ring over the ο¬eld of fractions of its centre. By Ponserβs theorem, PI-deg( π ) = (minimal degree of π ). SNEHASHIS MUKHERJEE
Proposition 2.5 ([1], Theorem I.13.5) . Let π΄ be a prime aο¬ne PI algebra over analgebraically closed ο¬eld K , with PI-deg( π΄ ) = π and π be an simple π΄ -module. Then π is a vector space over K of dimension π‘ , where π‘ β€ π , and π΄ / πππ π΄ ( π ) (cid:27) π π‘ ( K ) . From proposition (2 .
1) and (2 . π over O π ( o K π ) is ο¬nite dimensional and can have dimension at most PI-deg( O π ( o K π ) ).3. The degree of a prime algebra
The main tool used to compute the degree of O π ( o K π ) is the theory developed in[2] by De Concini and Procesi. So we shall ο¬rst recall the set-up from there. Let π΄ be a prime algebra (i.e. π π΄π = π = π =
0) over the ο¬eld K and π be thecenter of π΄ . Then π is a domain and π΄ is a torsion free module over π . Assume that π΄ is a ο¬nite module over π . Then π΄ embeds in a ο¬nite dimensional central simplealgebra π ( π΄ ) = π ( π ) β π π΄ , where π ( π ) is the fraction ο¬eld of π . If π ( π ) denotesthe algebraic closure of π ( π ) , we have that π ( π ) β π π΄ is the full algebra π π ( π ( π )) of π Γ π matrices over π ( π ) . Then π is called the degree of π΄ .Let π΄ be a prime algebra over K generated by π₯ , Β· Β· Β· π₯ π and let π be a central sub-algebra of π΄ . For each π = , Β· Β· Β· , π , denote by π΄ π the subalgebra of π΄ generated by π₯ , Β· Β· Β· , π₯ π and let π π = π β© π΄ π . We assume that the following conditions hold foreach π = , Β· Β· Β· , π . π₯ π π₯ π = π π π π₯ π π₯ π + π π π if π > π , where π π π β K and π π π β π΄ π β .2 . π΄ π is a ο¬nite module over π π .3 . The formulas π π ( π₯ π ) = π π π π₯ π for π < π deο¬ne an automorphism of π΄ π β which isthe identity on π π β .4. The formulas π· π ( π₯ π ) = π π π for π < π deο¬ne a twisted derivation relative to π π .It is easy to see that π΄ then is an iterated twisted Ore extension. Let π΄ be an as-sociative algebra generated by π¦ , Β· Β· Β· , π¦ π with deο¬ning relations π¦ π π¦ π = π π π π¦ π π¦ π for π < π . We call this algebra the associated quasipolynomial algebra of π΄ . In [2] thefollowing was proved: Proposition 3.1.
Under the above assumptions the degree of π΄ is equal to the degreeof the associated quasipolynomial algebra π΄ . Given an π Γ π skew-symmetric matrix π» = ( β π π ) over Z we construct the twistedpolynomial algebra K π» [ π₯ , Β· Β· Β· , π₯ π ] as follows: It is the algebra generated by elements π₯ , Β· Β· Β· , π₯ π with the following deο¬ning relations: IMPLE MODULES OVER THE QUANTUM EUCLIDEAN SPACE 5 π₯ π π₯ π = π β π π π₯ π π₯ π for π, π = , Β· Β· Β· , π The matrix π» is called the deο¬ning matrix of the algebra K π» [ π₯ , Β· Β· Β· , π₯ π ] . It canbe viewed as an iterated twisted polynomial algebra with respect to any ordering ofthe indeterminates π₯ π . Given π = ( π , Β· Β· Β· , π π ) β ( Z + ) π , we write π₯ π = π₯ π Β· Β· Β· π₯ ππ π Let π be a primitive π th root of unity. We consider the matrix π» as a matrix of thehomomorphism π» : Z π β ( Z / π Z ) π and we denote by πΎ the kernel of π» and by β the cardinality of the image of π» . The following was proved in [2]: Proposition 3.2. (a) The monomials π₯ π with π β πΎ β© ( Z + ) π form a basis of the centerof K π» [ π₯ , Β· Β· Β· , π₯ π ] .(b) degree K π» [ π₯ , Β· Β· Β· , π₯ π ] = β β In general the degree is less than or equal to π π where π is half the rank of (here)a 2 π Γ π skew symmetric matrix. Note that π₯ π₯ = π π₯ π₯ , π₯ π π₯ = π β π₯ π₯ π β π = , Β· Β· Β· , π.π¦ π₯ = π π₯ π¦ , π¦ π π₯ = ππ₯ π¦ π β π = , Β· Β· Β· , π. Again π₯ π¦ = π π¦ π₯ , π₯ π π¦ = π β π¦ π₯ π β π = , Β· Β· Β· , π.π¦ π¦ = π π¦ π¦ , π¦ π π¦ = ππ¦ π¦ π β π = , Β· Β· Β· , π. Hence the skew symmetric matrix has two identical rows namely 1st row and ( π + ) th row. Both are equal to ( , β , Β· Β· Β· , β , , , Β· Β· Β· , ) where 0 appears in 1st and ( π + ) th place. So the rank of the matrix is less than or equal to 2 π β
2. Thereforethe degree of O π ( o K π ) β€ π π β .4. Construction of a Simple Module
In this section we wish to construct simple modules over O π ( o K π ) . Our construc-tion proceeds in the following steps.Step 1: (The representation space) For πΎ = ( πΌ , Β· Β· Β· , πΌ π , π , Β· Β· Β· , π π ) β ( K β ) π , let π ( πΎ ) be the K -vector space with basis π ( π , Β· Β· Β· , π π ) , where 0 β€ π π β€ π β β€ π β€ π . SNEHASHIS MUKHERJEE
Step 2: (Module structure) Let us deο¬ne an O π ( o K π ) -module structure on π ( πΎ ) deο¬ned as follows: π ( π , Β· Β· Β· , π π ) π₯ = πΌ π β( π +Β·Β·Β·+ π π ) π ( π , Β· Β· Β· , π π ) π ( π , Β· Β· Β· , π π ) π₯ π = πΌ π π ( π +Β·Β·Β·+ π π β ) π ( π , Β· Β· Β· , π π + , Β· Β· Β· , π π ) , β β€ π β€ ππ ( π , Β· Β· Β· , π π ) π¦ = πΌ β π ( β π β ) β π β( π +Β·Β·Β·+ π π ) π ( π , Β· Β· Β· , π π ) π ( π , Β· Β· Β· , π π ) π¦ π = πΌ β π (cid:16) π β ( π π + +Β·Β·Β·+ π π )β( π π β +Β·Β·Β·+ π ) (cid:17) ( β π β ) β ( π π β π β π π π π β ) π ( π , Β· Β· Β· , π π + (β ) , Β· Β· Β· , π π ) , β β€ π β€ π where + is addition in the additive group Z / π Z . Remark 4.1.
Note that π ( π , Β· Β· Β· , π π ) π π = π π π β ( π π +Β·Β·Β·+ π π + ) π ( π , Β· Β· Β· , π π , Β· Β· Β· , π π ) , β β€ π β€ π Step 3: (Well-deο¬nedness) In order to establish the well-deο¬nedness of the aboverules, we need to check that for 1 β€ π β€ π and 0 β€ π π β€ π β π ( π , Β· Β· Β· , π π ) π¦ π π¦ π = π β π ( π , Β· Β· Β· , π π ) π¦ π π¦ π , β π < π (4.1) π ( π , Β· Β· Β· , π π ) π₯ π π¦ π = π β π ( π , Β· Β· Β· , π π ) π¦ π π₯ π , β π β π . (4.2) π ( π , Β· Β· Β· , π π ) π₯ π π₯ π = ππ ( π , Β· Β· Β· , π π ) π₯ π π₯ π , β π < π . (4.3) π ( π , Β· Β· Β· , π π ) π₯ π π¦ π = π ( π , Β· Β· Β· , π π ) ( π¦ π π₯ π + Γ π<π ( β π β ) π¦ π π₯ π ) , β π = , Β· Β· Β· , π. (4.4) IMPLE MODULES OVER THE QUANTUM EUCLIDEAN SPACE 7
The relations (4.1) - (4.3) are easy to show. For relation (4.4) we have the followingcalculation. As for π = β€ π β€ ππ ( π , Β· Β· Β· , π π ) π₯ π π¦ π = π β ( π π +Β·Β·Β·+ π π + ) ( β π β ) β ( π π β π β π π β π π β ) π ( π , Β· Β· Β· , π π , Β· Β· Β· , π π ) Now for the right hand side we have, π ( π , Β· Β· Β· , π π ) ( π¦ π π₯ π + Γ π<π ( β π β ) π¦ π π₯ π ) = π ( π , Β· Β· Β· , π π ) π¦ π π₯ π + π ( π , Β· Β· Β· , π π ) ( Γ π<π ( β π β ) π¦ π π₯ π ) = π ( π , Β· Β· Β· , π π ) π¦ π π₯ π + π ( π , Β· Β· Β· , π π ) π π β = [ π β ( π π +Β·Β·Β·+ π π + ) ( β π β ) β ( π π β π β π π π π β ) + π π β π β ( π π +Β·Β·Β·+ π π ) ] π ( π , Β· Β· Β· , π π , Β· Β· Β· , π π ) = π β ( π π +Β·Β·Β·+ π π + ) ( β π β ) β ( π π β π β π π π π β + π β π π π π β β π β π π β π π β ) π ( π , Β· Β· Β· , π π , Β· Β· Β· , π π ) = π β ( π π +Β·Β·Β·+ π π + ) ( β π β ) β ( π π β π β π π β π π β ) π ( π , Β· Β· Β· , π π , Β· Β· Β· , π π ) With this we have the following.
Theorem A.
The module π ( πΎ ) is a simple O π ( o K π ) -module of dimension π π β .Proof. Let π be a non-zero submodule of π ( π ) . We claim that π contains a basisvector of the form π ( π , Β· Β· Β· , π π ) . Indeed, any member π β π is a ο¬nite K -linearcombination of such vectors. i.e., π : = Γ ο¬nite π π π (cid:16) π ( π ) , Β· Β· Β· , π ( π ) π (cid:17) for some π π β K . Suppose there exist two non-zero coeο¬cients, say, π π’ , π π£ . Wecan choose the largest index π such that π ( π’ ) π β π ( π£ ) π in Z / π Z . Now the vectors π (cid:16) π ( π’ ) , Β· Β· Β· , π ( π’ ) π (cid:17) and π (cid:16) π ( π£ ) , Β· Β· Β· , π ( π£ ) π (cid:17) are eigenvectors of π₯ π π¦ π associated with theeigenvalues π β ( π ( π’ ) π +Β·Β·Β·+ π ( π’ ) π + ) ( β π β ) β ( π π β π β π ( π’ ) π β π π β ) = π π’ (say)and π β ( π ( π£ ) π +Β·Β·Β·+ π ( π£ ) π + ) ( β π β ) β ( π π β π β π ( π£ ) π β π π β ) = π π£ (say) SNEHASHIS MUKHERJEE respectively. We claim that π π’ β π π£ . Indeed, π π’ = π π£ = β π β ( π ( π’ ) π β π ( π£ ) π ) = = β π | (cid:16) π ( π’ ) π β π ( π£ ) π (cid:17) , which is a contradiction.Now ππ₯ π π¦ π β π π’ π is an non zero element in π of smaller length than π . Hence byinduction it follows that every non zero submodule of π ( πΎ ) contains a basis vector ofthe form π ( π , Β· Β· Β· , π π ) . Thus π ( πΎ ) is simple by the actions of π₯ π , π¦ π β€ π β€ π . (cid:3) Main Results
Theorem B.
Let π be a simple π₯ π , π π torsionfree O π ( o K π ) -module. Then π isisomorphic to π ( πΎ ) as O π ( o K π ) -module for some πΎ β ( K β ) π .Proof. Note that π π π₯ π = π π₯ π π π β€ π < π β€ π (5.1) π π π₯ π = π₯ π π π β€ π β€ π β€ π (5.2) π π π π = π π π π β€ π, π β€ π. (5.3)Hence π₯ , π₯ ππ for π = , Β· Β· Β· , π and π π for π = , Β· Β· Β· , π commute with each other.As π is a ο¬nite dimensional vector space over K (by Section 2), there is a commoneigenvector π£ of π π for π = , Β· Β· Β· , π , π₯ ππ for π = , Β· Β· Β· , π and π₯ . Put π£π π = π π π£ β π = , Β· Β· Β· , π.π£π₯ ππ = π π π£ β π = , Β· Β· Β· , π.π£π₯ = πΌ π£. Since π is both π₯ π and π π -torsionfree π π , πΌ and π π are all nonzero. Let πΌ π be the π -throot of unity of π π for π = , Β· Β· Β· , π . Deο¬ne a linear transformation π : π ( πΎ ) β π by π ( π ( π , Β· Β· Β· , π π )) : = πΌ β π Β· Β· Β· πΌ β π π π π£π₯ π π π Β· Β· Β· π₯ π . (5.4) IMPLE MODULES OVER THE QUANTUM EUCLIDEAN SPACE 9
To prove π is an O π ( o K π ) -module homomorphism, it suο¬ces to check that π ( π ( π , Β· Β· Β· , π π ) π₯ π ) = π ( π ( π , Β· Β· Β· , π π )) π₯ π , β β€ π β€ π. (5.5) π ( π ( π , Β· Β· Β· , π π ) π¦ π ) = π ( π ( π , Β· Β· Β· , π π )) π¦ π , β β€ π β€ π. (5.6)The relations (5.5) and (5.6) can be easily veriο¬ed using the linear map π in (5.4)and O π ( o K π ) -module π ( πΎ ) . To verify (5.6), the following relations are going to beuseful.For π = , Β· Β· Β· , π , if π π >
0, then π£π₯ π π π Β· Β· Β· π₯ π π¦ π = (cid:16) π β ( π π + +Β·Β·Β·+ π π )β( π π β +Β·Β·Β·+ π ) (cid:17) ( β π β ) β ( π π β π β π π π π β ) π£π₯ π π π Β· Β· Β· π₯ π π β π Β· Β· Β· π₯ π , β β€ π β€ π. If π π =
0, then π£π₯ π π π Β· Β· Β· π₯ π π¦ π = π β π (cid:16) π β ( π π + +Β·Β·Β·+ π π )β( π π β +Β·Β·Β·+ π ) (cid:17) ( β π β ) β ( π π β π π β ) π£π₯ π π π Β· Β· Β· π₯ π β π Β· Β· Β· π₯ π , β β€ π β€ π (cid:3) Remark 5.1.
There is a surjective map Ξ¨ from ( K β ) π to the set of all simple π₯ π , π π , π = , Β· Β· Β· , π torsionfree module O π ( o K π ) with dim Ξ¨ ( πΎ ) = π π β . Remark 5.2.
Since the degree of an algebra is equal to the maximal dimension ofthe irreducible representations of that algebra, deg O π ( o K π ) = π π β . Remark 5.3. If π ( β ) is even we can go along the same line of reasoning whileconstructing the simple modules in Section . Here π π will lie between and π β .In this case the dimension of the simple module will be ( π ) π β . Let πΎ = ( πΌ , Β· Β· Β· , πΌ π , π , Β· Β· Β· , π π ) , πΎ = ( π½ , Β· Β· Β· , π½ π , π β² , Β· Β· Β· , π β² π ) be two elementsin ( K β ) π such that π ( πΎ ) and π ( πΎ ) are isomorphic as O π ( o K π ) -module. As π ( π , Β· Β· Β· , π π ) = πΌ β π Β· Β· Β· πΌ β π π π π ( , Β· Β· Β· , ) π₯ π π π Β· Β· Β· π₯ π , the O π ( o K π ) -module isomorphism π : π ( πΎ ) β π ( πΎ ) can be uniquely deter-mined by the image of π ( , Β· Β· Β· , ) , i.e., say(5.7) π ( π ( , Β· Β· Β· , )) = Γ π‘ βI , ο¬nite set π π‘ π (cid:16) π ( π‘ ) , Β· Β· Β· , π ( π‘ ) π (cid:17) , for π π‘ β K β . We claim that I is a singleton set. Since π₯ π π¦ π -eigenvectors of π ( πΎ ) must map to π₯ π π¦ π -eigenvectors of π ( πΎ ) with the same eigenvalue, therefore from(5.7) we have(5.8) π = π β² and for each π = , Β· Β· Β· , π :(5.9) ( π π β π β π π β ) = π β ( π ( π‘ ) π +Β·Β·Β·+ π ( π‘ ) π + ) ( π β² π β π β π ( π‘ ) π β π β² π β ) β π‘ β I But we have already proved that for ο¬xed π with π ( π‘ ) π β π ( π‘ β² ) π , the R.H.S of (5.9) isunequal. Hence it follows that I is a singleton set and the relation (5.7) is of the form π ( π ( , Β· Β· Β· , )) = π π ( π , Β· Β· Β· , π π ) , for some π β K β and π π β Z / π Z . Also using this isomorphism π , we obtain a relationbetween πΎ and πΎ as mention below in ( ?? ). Lemma 5.1.
Let πΎ = ( πΌ , Β· Β· Β· , πΌ π , π , Β· Β· Β· , π π ) , πΎ = ( π½ , Β· Β· Β· , π½ π , π β² , Β· Β· Β· , π β² π ) β( K β ) π such that πΌ = π½ πΌ ππ = π½ ππ π = π β² π π β π β π π β = π β π π ( π β² π β π β π π π β² π β ) Then there is an O π ( o K π ) -module isomorphism π : π ( πΎ ) β π ( πΎ ) given by π ( π ( π , Β· Β· Β· , π π )) = π Γ π = (cid:16) πΌ β π π½ π (cid:17) π π π ( π + π , Β· Β· Β· , π π + π π ) . where π π + Β· Β· Β· + π π + = π π and π π = π β² π β . Thus the above discussion deο¬nes an equivalence relation on ( K β ) π so that eachequivalence class determines a simple O π ( o K π ) -module up to isomorphism. IMPLE MODULES OVER THE QUANTUM EUCLIDEAN SPACE 11 A cknowledgements The author is very grateful to Dr. Ashish Gupta for engaging conversations onthe topic. The author will also like to thank National Board of Higher Mathematics,Department of Atomic Energy, Government of India for funding his research.
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Snehashis Mukherjee School of Mathematical Sciences,Ramakrishna Mission Vivekananda Educational and Research Institute (rkmveri),Belur Math, Howrah, Box: 711202, West Bengal, India.
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