Heisenberg-Type Families in Uq(sl₂)

Using the second Drinfeld formulation of the quantized universal enveloping algebra Uq(sl₂) we introduce a family of its Heisenberg-type elements which are endowed with a deformed commutator and satisfy properties similar to generators of a Heisenberg subalgebra. Explicit expressions for new family...

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Datum:	2009
1. Verfasser:	Zuevsky, A.
Format:	Artikel
Sprache:	English
Veröffentlicht:	Інститут математики НАН України 2009
Schriftenreihe:	Symmetry, Integrability and Geometry: Methods and Applications
Online Zugang:	http://dspace.nbuv.gov.ua/handle/123456789/149257
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spelling	irk-123456789-1492572019-02-20T01:28:06Z Heisenberg-Type Families in Uq(sl₂) Zuevsky, A. Using the second Drinfeld formulation of the quantized universal enveloping algebra Uq(sl₂) we introduce a family of its Heisenberg-type elements which are endowed with a deformed commutator and satisfy properties similar to generators of a Heisenberg subalgebra. Explicit expressions for new family of generators are found. 2009 Article Heisenberg-Type Families in Uq(sl₂) / A. Zuevsky // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 7 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 17B37; 20G42; 81R50 http://dspace.nbuv.gov.ua/handle/123456789/149257 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	Using the second Drinfeld formulation of the quantized universal enveloping algebra Uq(sl₂) we introduce a family of its Heisenberg-type elements which are endowed with a deformed commutator and satisfy properties similar to generators of a Heisenberg subalgebra. Explicit expressions for new family of generators are found.
format	Article
author	Zuevsky, A.
spellingShingle	Zuevsky, A. Heisenberg-Type Families in Uq(sl₂) Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Zuevsky, A.
author_sort	Zuevsky, A.
title	Heisenberg-Type Families in Uq(sl₂)
title_short	Heisenberg-Type Families in Uq(sl₂)
title_full	Heisenberg-Type Families in Uq(sl₂)
title_fullStr	Heisenberg-Type Families in Uq(sl₂)
title_full_unstemmed	Heisenberg-Type Families in Uq(sl₂)
title_sort	heisenberg-type families in uq(sl₂)
publisher	Інститут математики НАН України
publishDate	2009
url	http://dspace.nbuv.gov.ua/handle/123456789/149257
citation_txt	Heisenberg-Type Families in Uq(sl₂) / A. Zuevsky // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 7 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT zuevskya heisenbergtypefamiliesinuqsl2
first_indexed	2025-07-12T21:42:46Z
last_indexed	2025-07-12T21:42:46Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 5 (2009), 006, 4 pages Heisenberg-Type Families in Uq(ŝl2) ? Alexander ZUEVSKY Max-Planck Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany E-mail: zuevsky@mpim-bonn.mpg.de Received October 20, 2008, in final form January 13, 2009; Published online January 15, 2009 doi:10.3842/SIGMA.2009.006 Abstract. Using the second Drinfeld formulation of the quantized universal enveloping algebra Uq(ŝl2) we introduce a family of its Heisenberg-type elements which are endowed with a deformed commutator and satisfy properties similar to generators of a Heisenberg subalgebra. Explicit expressions for new family of generators are found. Key words: quantized universal enveloping algebras; Heisenberg-type families 2000 Mathematics Subject Classification: 17B37; 20G42; 81R50 1 Introduction The purpose of this paper is to introduce a new family of elements in the quantized enveloping algebra Uq(ŝl2) of an affine Lie algebra ŝl2. This Heisenberg-type families possess properties similar to ordinary Heisenberg algebras. Heisenberg subalgebras of affine Lie algebras and of their q-deformed enveloping algebras are being actively used in various domains of mathemati- cal physics. Most important applications of Heisenberg subalgebras can be found in the field of classical and quantum integrable models and field theories. Vertex operator constructions both for affine Lie algebras [5] and for q-deformations of their universal enveloping algebras [4, 6] are essentially based on Heisenberg subalgebras. Given a quantized universal enveloping alge- bra Uq(Ĝ) of an affine Kac–Moody Lie algebra Ĝ, it is rather important to be able to extract explicitly generators of a Heisenberg subalgebra associated to a chosen grading of Uq(Ĝ) which is not always a trivial task. For instance, one can easily recognize elements of a Heisenberg subalgebra among generators in the homogeneous grading of the second Drinfeld realization of Uq(ŝl2) [4] while it is not obvious how to extract a Heisenberg subalgebra in the principal grad- ing. Ideally, one would expect to obtain a realization of the Heisenberg subalgebra associated to the principal graiding of Uq(ŝl2) which would involve ordinary (rather then q-deformed) com- mutator in commutation relations with certain elements in the family giving central elements. This would lead to many direct 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 Uq(ŝl2) was constructed. Its elements expressed in q-commuting coordinates commute with respect to the q-deformed bracket. In this paper we introduce another possible version of a family elements in Uq(ŝl2) which could play a role similar to ordinary Heisenberg subalgebra. Our general idea is to form certain sets of Uq(ŝl2)-elements containing linear combinations of generators x±n , n ∈ Z, multiplied by various powers of K and the central element γ. Under certain conditions on corresponding powers we obtain commutation relations for a Heisenberg-type family with respect to an integral p-th power of K ∈ Uq(ŝl2)-deformed commutator. We consider this as some further generalization of various q-deformed commutator algebras (in particular, q-bracket Heisenberg subalgebras) which ?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 mailto:zuevsky@mpim-bonn.mpg.de http://dx.doi.org/10.3842/SIGMA.2009.006 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 ŝl2 even when p 6= 0. The paper is organized as follows. In Section 2 we recall the definition of the quantized universal enveloping algebra Uq(ŝl2) in the second Drinfeld realization. In Section 3 we find explicit expressions for elements of Heisenberg-type families. Then we prove their commutation relations. We conclude by making comments on possible generalizations and applications. 2 Second Drinfeld realization of Uq(ŝl2) Let us recall the second Drinfeld realization [1, 3] of the quantized universal enveloping algebra Uq(ŝl2). It is generated by the elements {x±k , k ∈ Z; an, n ∈ {Z\0}; γ± 1 2 , K}, subject to the commutation relations [K, ak] = 0, Kx±k K −1 = q±2x±k , [ak, al] = δk,−l [2k] k γk − γ−k q − q−1 , [an, x ± k ] = ± [2n] n γ∓ \|n\| 2 x±n+k, [x+ n , x − k ] = 1 q − q−1 ( γ 1 2 (n−k)ψn+k − γ− 1 2 (n−k)φn+k ) , (1) x±k+1x ± l − q±2x±l x ± k+1 = q±2x±k x ± l+1 − x±l+1x ± k , where γ± 1 2 belong to the center of Uq(ŝl2), and [n] ≡ qn − q−n q − q−1 . The elements φk and ψ−k, for non-negative integers k ∈ Z+, are related to a±k by means of the expressions ∞∑ m=0 ψmz −m = K exp ( (q − q−1) +∞∑ k=1 akz −k ) , ∞∑ m=0 φ−mz m = K−1 exp ( −(q − q−1) +∞∑ k=1 a−kz k ) , (2) i.e., ψm = 0, m < 0, φm = 0, m > 0. 3 Heisenberg-type families In this section we introduce a family of Uq(ŝl2)-elements which have properties similar to a or- dinary Heisenberg subalgebra of an affine Kac–Moody Lie algebra [5]. We consider families of linear combinations of x±n -generators of Uq(ŝl2) multiplied by powers of K and central element γ. Let us introduce for m, l, η, θ ∈ Z, n ∈ Z+, the following elements: E±n (m, η) = γ±(n+ 1 2 )x+ nK m + x−n+1K η, (3) E±−n−1(l, θ) = x+ −n−1K l + γ±(n+ 1 2 )x−−nK θ. (4) Denote also for some p ∈ Z, a deformed commutator [A,B]Kp = AKpB −BKpA. (5) Heisenberg-Type Families in Uq(ŝl2) 3 We then formulate Proposition. Let p, m ∈ Z, l = m, θ = η = −m− 2p. Then the family of elements {E±p,n(m), E±p,−n−1(m), n ∈ Z+}, (6) where E±p,n(m) ≡ E±n (m,−m − 2p), E±p,−n−1(m) ≡ E±−n−1(m,−m − 2p), we have denoted in (3), (4), are subject to the commutation relations with k ∈ Z+,[ E+ p,n(m), E+ p,−k−1(m) ] Kp = 0, for all n < k, (7)[ E−p,n(m), E−p,−k−1(m) ] Kp = 0, for all n > k, (8)[ E±±1,n(m), E±±1,−n−1(m) ] K∓1 = c±n (m), (9) where c+n (m) = q−2(m−1) q − q−1 γ2n+1(γn − γ−n−1), c−n (m) = q−2(m+1) q − q−1 γ−n−1(γ2n+2 − γ−n), belong to the center Z(Uq(ŝl2)) of Uq(ŝl2). We call a subset (6) of Uq(ŝl2)-elements with all appropriate m, p ∈ Z, n ∈ Z+, such that it satisfies the commutation relations (7)–(9) the Heisenberg-type family. In particular, when p = 0, (7)–(8) reduce to ordinary commutativity conditions. Note that if we formally substitute n 7→ −n− 1, then, E±p,n(m, η) 7→ γ∓(n+1/2)E∓p,−n−1(m, η). Under the action of an automorphism ω of Uq(ŝl2) which maps K 7→ K−1, γ 7→ γ−1, x±n 7→ x∓−n, an 7→ a−n, one has ω(E±p,n(m)) = E∓p,−n−1(m) K2p. Proof. The proof is the direct calculation of the commutation relations (7)–(9). Indeed, consider Kp-deformed commutator (5) of E±n (m, η) and E±−k−1(l, θ) with some m, η, l, θ ∈ Z, n, k ∈ Z+. Using the commutation relations (1) we obtain[ E±n (m, η), E±−k−1(l, θ) ] Kp = γ±(n+k+1) ( q−2m−2px+ n x − −k − q2θ+2px−−kx + n ) Km+θ+p + ( q2η+2px−n+1x + −k−1 − q−2l−2px+ −k−1x − n+1 ) Kη+l+p + γ±(n+1/2) ( q2m+2px+ n x + −k−1 − q2l+2px+ −k−1x + n ) Kη+θ+p + γ±(k+1/2) ( q−2η−2px−n+1x − −k − q−2θ−2px−−kx − n+1 ) Kη+θ+p. Then for m = l, θ = η = −m− 2p, from (1) it follows[ E±p,n(m), E±p,−k−1(m) ] Kp = q−2(m+p) q − q−1 [ γ±(n+k+1) ( γ1/2(n+k)ψn−k − γ−1/2(n+k)φn−k ) − ( γ−1/2(n+k+2)ψn−k − γ1/2(n+k+2)φn−k )] Kp. Since for n < k, ψn−k = 0, and the reaming terms containing φn−k cancels, we obtain (7). Similarly, for n > k, φn−k = 0, the terms containing ψn−k cancels, and (8) follows. From (2) we see that ψ0 = K, φ0 = K−1. Taking K∓1-deformed commutators (5) of E±1,n(m), E±1,−k−1(m), we then have [ E+ 1,n(m), E+ 1,−n−1(m) ] K−1 = q−2m+2 q − q−1 [ γ2n+1(γnK) K−1 − (γ−n−1K)K−1 ] = c+n (m), [ E−−1,n(m), E−−1,−n−1(m) ] K = q−2m−2 q − q−1 [ γ−2n−1K(−γ−nK−1) +K(γn+1K−1) ] = c−n (m), for n = k. � 4 A. Zuevsky 4 Conclusions In the second Drinfeld realization of Uq(ŝl2) we have defined a subset of elements that consti- tutes a Heisenberg-type family, explicitly constructed their elements, and proved corresponding commutation relations. Properties of a Heisenberg-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 Uq(ŝl2), and, in particular, might have their further applications in the soliton theory of non-linear integrable partial differential equations [6]. One of our aims to introduce Heisenberg-type families 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 ŝl2 [6]. Finally, we would like also to make some comments comparing present work to [2] where the quantum principal commutative subalgebra in Uq(ŝl2) associated to the principal grading of ŝl2 was found. Here we introduce Heisenberg-type families of Uq(ŝl2) in the principal grading of Uq(ŝl2) [6]. Although we use Kp-deformed commutators (which for p = 1 can be seen quite similar to q-deformed commutators in [2]) these two approaches are quite different. We prefer to work with the explicit set of Uq(ŝl2) generators (q-commutative coordinates) in its second Drinfeld realization [1], and introduce elements of our Heisenberg-type families not involving lattice constructions or trace invariants as in [2]. A generalization of our results to an arbitrary Ĝ case does not face any serious technical problems. We assume that Heisenberg-type families introduced which exhibit properties similar to a Heisenberg subalgebra in Uq(ŝl2) are not the 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 from [2] we see that even in the q-commutator case there exist more complicated q-commutative elements in Uq(ŝl2). Thus one would expect the same phenomena for Kp-deformed algebras. More advanced examples of Heisenberg-type families associated to various gradings of Uq(Ĝ) in the Drinfeld–Jimbo and second Drinfeld realizations as well as corresponding vertex operators will be discussed in a forthcoming paper [7]. 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 für Mathematik in Bonn where this work has been completed. References [1] Drinfel’d V.G., A new realization of Yangians and quantized affine 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. [4] Jimbo M., Miki K., Miwa T., Nakayashiki A., Correlation functions of the XXZ model for ∆ < −1, Phys. Let. A 168 (1992), 256–263, hep-th/9205055. [5] Kac V.G., Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. [6] Saveliev M.V., Zuevsky A.B., Quantum vertex operators for the sine-Gordon model, Internat. J. Modern Phys. A 15 (2000), 3877–3897. [7] Zuevsky A., Heisenberg-type families of Uq(Ĝ), in preparation. http://arxiv.org/abs/hep-th/9402145 http://arxiv.org/abs/hep-th/9205055 1 Introduction 2 Second Drinfeld realization of U_q(\widehat{sl_2}) 3 Heisenberg-type families 4 Conclusions References

Heisenberg-Type Families in Uq(sl₂)

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