Symmetry, Integrability and Geometry: Methods and Applications SIGMA 5 (2009), 006, 4 pages ⋆ Heisenberg-Type Families in U (sl ) q 2 Alexander ZUEVSKY c Max-Planck Institut fu¨r Mathematik, Vivatsgasse 7, 53111 Bonn, Germany E-mail: [email protected] Received October 20, 2008, in final form January 13, 2009; Published online January 15, 2009 doi:10.3842/SIGMA.2009.006 9 Abstract. Using the second Drinfeld formulation of the quantized universal enveloping 0 0 algebra Uq(sl2) we introduce a family of its Heisenberg-type elements which are endowed 2 with a deformed commutator and satisfy properties similar to generators of a Heisenberg subalgebra.cExplicit expressions for new family of generators are found. n a Key words: quantized universal enveloping algebras; Heisenberg-type families J 5 2000 Mathematics Subject Classification: 17B37;20G42; 81R50 1 ] A 1 Introduction Q . h The purpose of this paper is to introduce a new family of elements in the quantized enveloping t a algebra Uq(sl2) of an affine Lie algebra sl2. This Heisenberg-type families possess properties m similar to ordinary Heisenberg algebras. Heisenberg subalgebras of affine Lie algebras and of [ their q-deforcmed enveloping algebras are bceing actively used in various domains of mathemati- 1 cal physics. Most important applications of Heisenberg subalgebras can be found in the field of v classical and quantum integrable models and field theories. Vertex operator constructions both 5 for affine Lie algebras [5] and for q-deformations of their universal enveloping algebras [4, 6] 3 3 are essentially based on Heisenberg subalgebras. Given a quantized universal enveloping alge- 2 bra U (G) of an affine Kac–Moody Lie algebra G, it is rather important to be able to extract q . 1 explicitly generators of a Heisenberg subalgebra associated to a chosen grading of U (G) which q 0 is not ablways a trivial task. For instance, one cban easily recognize elements of a Heisenberg 9 0 subalgebra among generators in the homogeneous grading of the second Drinfeld realizbation of v: Uq(sl2) [4] while it is not obvious how to extract a Heisenberg subalgebra in the principal grad- i ing. Ideally, one would expect to obtain a realization of the Heisenberg subalgebra associated X to thce principal graiding of U (sl ) which would involve ordinary (rather then q-deformed) com- r q 2 a mutator in commutation relations with certain elements in the family giving central elements. This would lead to many direcct applications both in quantum groups and quantum integrable theories in analogy with the homogeneous graiding case. In [2] the principal commuting subal- gebra in the nilponent part of U (sl ) was constructed. Its elements expressed in q-commuting q 2 coordinates commute with respect to the q-deformed bracket. InthispaperweintroduceanothcerpossibleversionofafamilyelementsinU (sl )whichcould q 2 play a role similar to ordinary Heisenberg subalgebra. Our general idea is to form certain sets of U (sl )-elements containing linear combinations of generators x±, n ∈ Z, multipclied by various q 2 n powers of K and the central element γ. Under certain conditions on corresponding powers we ocbtain commutation relations for a Heisenberg-type family with respect to an integral p-th power of K ∈ U (sl )-deformed commutator. We consider this as some further generalization of q 2 variousq-deformedcommutatoralgebras(inparticular,q-bracketHeisenbergsubalgebras)which c ⋆This paper is a contribution to the Proceedings of the XVIIth International Colloquium on Integrable Sys- tems and Quantum Symmetries (June 19–22, 2008, Prague, Czech Republic). The full collection is available at http://www.emis.de/journals/SIGMA/ISQS2008.html 2 A. Zuevsky find numerous examples in quantum algebras and applications in integrable models. Though the commutation relation we use in order to define a Heisenberg-type family look quite non- standard we believe that these families and their properties deserve a consideration as a new structure inside the quantized universal enveloping algebra of sl even when p 6= 0. 2 The paper is organized as follows. In Section 2 we recall the definition of the quantized universal enveloping algebra U (sl ) in the second Drinfeld recalization. In Section 3 we find q 2 explicit expressions for elements of Heisenberg-type families. Then we prove their commutation relations. We conclude by makingccomments on possible generalizations and applications. 2 Second Drinfeld realization of U (sl ) q 2 Let us recall the second Drinfeld realization [1, 3] of the quantized universal enveloping algebra Uq(sl2). It is generated by the elements {x±k, k ∈ Z; anc, n ∈ {Z\0}; γ±12, K}, subject to the commutation relations c [2k]γk −γ−k [K,a ] = 0, Kx±K−1 = q±2x±, [a ,a ] = δ , k k k k l k,−l k q−q−1 [a ,x±] = ±[2n]γ∓|n2|x± , [x+,x−] = 1 γ12(n−k)ψ −γ−21(n−k)φ , (1) n k n n+k n k q−q−1 n+k n+k x± x±−q±2x±x± = q±2x±x± −x± x±, (cid:0) (cid:1) k+1 l l k+1 k l+1 l+1 k where γ±21 belong to the center of Uq(sl2), and qn−q−n c [n] ≡ . q−q−1 The elements φ and ψ , for non-negative integers k ∈ Z , are related to a by means of the k −k + ±k expressions ∞ +∞ ψ z−m = Kexp (q−q−1) a z−k , m k ! m=0 k=1 X X ∞ +∞ φ zm = K−1exp −(q−q−1) a zk , (2) −m −k ! m=0 k=1 X X i.e., ψ = 0, m < 0, φ = 0, m > 0. m m 3 Heisenberg-type families In this section we introduce a family of U (sl )-elements which have properties similar to a or- q 2 dinary Heisenberg subalgebra of an affine Kac–Moody Lie algebra [5]. We consider families of linearcombinations ofx±-generators ofU (slc)multiplied bypowersofK andcentralelementγ. n q 2 Let us introduce for m, l, η, θ ∈ Z, n ∈ Z , the following elements: + c E±(m,η) = γ±(n+12)x+Km+x− Kη, (3) n n n+1 E± (l,θ)= x+ Kl+γ±(n+21)x− Kθ. (4) −n−1 −n−1 −n Denote also for some p ∈ Z, a deformed commutator [A,B]Kp = AKpB −BKpA. (5) Heisenberg-Type Families in U (sl ) 3 q 2 We then formulate c Proposition. Let p, m ∈ Z, l = m, θ = η = −m−2p. Then the family of elements {E± (m), E± (m), n ∈ Z }, (6) p,n p,−n−1 + where E± (m) ≡ E±(m,−m − 2p), E± (m) ≡ E± (m,−m − 2p), we have denoted p,n n p,−n−1 −n−1 in (3), (4), are subject to the commutation relations with k ∈ Z , + E+ (m),E+ (m) =0, for all n < k, (7) p,n p,−k−1 Kp (cid:2)Ep−,n(m),Ep−,−k−1(m)(cid:3)Kp =0, for all n > k, (8) (cid:2)E±±1,n(m),E±±1,−n−1(m(cid:3) ) K∓1 =c±n(m), (9) where(cid:2) (cid:3) q−2(m−1) q−2(m+1) c+(m) = γ2n+1(γn−γ−n−1), c−(m) = γ−n−1(γ2n+2−γ−n), n q−q−1 n q−q−1 belong to the center Z(U (sl )) of U (sl ). q 2 q 2 We call a subset (6) of U (sl )-elements with all appropriate m, p ∈ Z, n ∈ Z , such that cq 2 c + it satisfies the commutation relations (7)–(9) the Heisenberg-type family. In particular, when p = 0, (7)–(8) reduce to ordinarcy commutativity conditions. Note that if we formally substitute n 7→ −n−1, then, E± (m,η) 7→ γ∓(n+1/2)E∓ (m,η). p,n p,−n−1 Under the action of an automorphism ω of U (sl ) which maps K 7→ K−1, γ 7→ γ−1, x± 7→ x∓ , q 2 n −n a 7→ a , one has ω(E± (m)) = E∓ (m)K2p. n −n p,n p,−n−1 c Proof. The proof is the direct calculation of the commutation relations (7)–(9). Indeed, consider Kp-deformed commutator (5) of E±(m,η) and E± (l,θ) with some m,η,l,θ ∈ Z, n −k−1 n,k ∈Z . Using the commutation relations (1) we obtain + E±(m,η),E± (l,θ) = γ±(n+k+1) q−2m−2px+x− −q2θ+2px− x+ Km+θ+p n −k−1 Kp n −k −k n (cid:2) + q2η+2px− x+(cid:3) −q−2l−2px+ (cid:0) x− Kη+l+p (cid:1) n+1 −k−1 −k−1 n+1 +γ(cid:0)±(n+1/2) q2m+2px+x+ −q2l+2px+ (cid:1) x+ Kη+θ+p n −k−1 −k−1 n +γ±(k+1/2)(cid:0)q−2η−2px− x− −q−2θ−2px− x−(cid:1) Kη+θ+p. n+1 −k −k n+1 Then for m = l, θ = η(cid:0)= −m−2p, from (1) it follows (cid:1) q−2(m+p) E± (m),E± (m) = γ±(n+k+1) γ1/2(n+k)ψ −γ−1/2(n+k)φ p,n p,−k−1 Kp q−q−1 n−k n−k (cid:2) − γ−1/2(n+k+2)ψ(cid:3) −γ1/2(n+k+(cid:2)2)φ K(cid:0)p. (cid:1) n−k n−k Since for n <(cid:0) k, ψ = 0, and the reaming ter(cid:1)m(cid:3)s containing φ cancels, we obtain (7). n−k n−k Similarly, for n> k, φ = 0, the terms containing ψ cancels, and (8) follows. n−k n−k From (2)weseethatψ = K, φ = K−1. TakingK∓1-deformedcommutators (5)ofE± (m), 0 0 1,n E± (m), we then have 1,−k−1 q−2m+2 E+ (m),E+ (m) = γ2n+1(γnK)K−1−(γ−n−1K)K−1 = c+(m), 1,n 1,−n−1 K−1 q−q−1 n (cid:2) (cid:3) q−2m−2(cid:2) (cid:3) E− (m),E− (m) = γ−2n−1K(−γ−nK−1)+K(γn+1K−1) = c−(m), −1,n −1,−n−1 K q−q−1 n for n =(cid:2) k. (cid:3) (cid:2) (cid:3) (cid:4) 4 A. Zuevsky 4 Conclusions In the second Drinfeld realization of U (sl ) we have defined a subset of elements that consti- q 2 tutes a Heisenberg-type family, explicitly constructed their elements, and proved corresponding commutation relations. Properties of a Hceisenberg-type family are similar to ordinary Heisen- berg subalgebra properties. These families might be very useful in construction of special types of vertex operators in U (sl ), and, in particular, might have their further applications in the q 2 soliton theory of non-linear integrable partial differential equations [6]. One of our aims to introduce Heisenberg-typecfamilies is the development of corresponding vertex operator rep- resentation which plays the main role in the theory of quantum soliton operators in exactly solvable field models associated to the infinite-dimensional Lie algebra sl [6]. 2 Finally, we would like also to make some comments comparing present work to [2] where the quantum principal commutative subalgebra in U (sl ) associated tocthe principal grading q 2 of sl was found. Here we introduce Heisenberg-type families of U (sl ) in the principal grading 2 q 2 of U (sl ) [6]. Although we use Kp-deformed commutactors (which for p = 1 can be seen quite q 2 simcilar to q-deformed commutators in [2]) these two approaches arecquite different. We prefer to worck with the explicit set of U (sl ) generators (q-commutative coordinates) in its second q 2 Drinfeld realization [1], and introduce elements of our Heisenberg-type families not involving lattice constructionsortraceinvariantcs asin[2]. Ageneralization ofourresultstoanarbitraryG case does not face any serious technical problems. We assume that Heisenberg-type families introduced which exhibit properties similar to a Heisenberg subalgebra in U (sl ) are not thbe q 2 most general ones. At the same time the construction described in this paper allows further generalization to cases of arbitrary affine Lie algebras [7]. Using formulae fromc[2] we see that even in the q-commutator case there exist more complicated q-commutative elements in U (sl ). q 2 Thus one would expect the same phenomena for Kp-deformed algebras. More advanced examples of Heisenberg-type families associated to various gradings of Uc(G) q intheDrinfeld–Jimboandsecond Drinfeldrealizations as wellas correspondingvertex operators will be discussed in a forthcoming paper [7]. b Acknowledgements We would like to thank A. Perelomov, D. Talalaev and M. Tuite for illuminating discussions and comments. Making use of the occasion, the author would like to express his gratitude to the Max-Planck-Institut fu¨r Mathematik in Bonn where this work has been completed. References [1] Drinfel’d V.G., Anewrealization of Yangiansandquantizedaffine algebras, Soviet Math. Dokl.36(1988), 212–216. [2] Enriquez B., Quantum principal commutative subalgebra in the nilpotent part of Uqsl2 and lattice KdV variables, Comm. Math. Phys. 170 (1995), 197–206, hep-th/9402145. [3] Frenkel I.B., Jing N.H., Vertex representations of quantum affine algebras, Proc. Nat. Acad. Sci. USA 85 (1988), 9373–9377. 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