QUANTUM DEFORMATIONS C.K. ZACHOS

The submitted manuscript has been authored by a contractor of the U.S. Government under contract No. W-31-lO9-cNG-38. Accordingly, the U. S- Government retain* a nonexclusive, royalty-irw I ice me to publish or reproduce the publijbed form oJ Jhia contribution, or allow others to do so, *or V. S. Government purpote).

ANL-HEP-CP—90-43

QUANTUM DEFORMATIONS

DE91 006005

C.K. ZACHOS High Energy Physics Division*, Argonne National Laboratory Argonne, IL 6043S-4815, USA zachos@anlhep

I review and illustrate applications of explicit functionals we have found which map SU(2) algebra generators to those of several quantum deformations of this Lie algebra. I indicate how virtually any such quantized algebra can be mapped to any other, and how representations of such algebras can be expressed as simple functions of SU(2) representations. The representation theory and its comultiplication rules are thus systematized and streamlined by direct reference to their SU(2) correspondents, and may be rapidly surveyed. 1 further provide a candidate quantum deformation of the Virasoro algebra.

I will outline and sample work I have done in collaboration with T. Curtright1 on Quantized Universal Enveloping algebras (QUE-algebras; also see [2-4]) pioneered by Drinfeld and Jimbo5. The aim is to illustrate applications of our results, and you are referred to our work and T. Curtright's talk for general statements. These remarkable mathematical structures (noncommutative, noncocommutative Hopf algebras) are emerging in several disparate physics contexts5"7 which they link together: in solutions of the Yang-Baxter equation for integrable models; in spin chains, and in vertex operators of rational conformal field theory in string noncommutative geometry, Wilson lines of the 3-d Chern-Simons action, and knot theory. They are called "quantum deformations", since they generally depend on one or more parameters which yield a conventional Lie algebra in their "classical" limit to special values, analogous to the ft —» 0 limit of the correspondence principle or the Wigner-Inonii contraction. We have assigned more explicit meaning to the term "deformation", by providing simple invertible functionals of the generators of SU(2) which satisfy each QUE-algebra, respectively. These functionals then directly transmute SU(2) continuously and reversibly (except for special values of the deformation parameter) into each QUE-algebraic structure, and thereby interconnect all these structures. Substituting any representation of SU(2) into 'Work supported by the U.S.Department of Energy, Division of High Energy Physics, Contract W-31109-ENG-38. Talk at the Spring Workshop on Quantum Gioups, Argonne, April 18, 1990, to appear in the proceedings, T. Curtright, D. Fairlie, and C. Zachos (eds.), World Scientific, 1990.

DISTRIBUTION

these functionals consequently produces the corresponding representation of these QUEalgebras. Thus their rich and useful representation theory8"9 is codified and systematized, and its comultiplication rules are directly referred to their classical correspondents. For specificity, consider the SU(2) algebra: Uo,j+] = j+

U+J-] = Jo

U-,Jo] = j - •

(1)

C = 2j+j. + joija - i) = 2j_j+ + jo(jb + 1) s j(j + 1) ,

(2)

The Casimir invariant in both its antipodal forms is

where j is the formal operator (\/l + 4C—1)/2. Consequently, —2j'+J- = (jo+j)(jo—l—j), and the antipodal -2j_j' + = (jo - j)(j0 + 1 + j ) . Note j+f(jo) = /(jo - l)j+Also for specificity, consider Kulish-Reshetikhin/Drinfeld/Jimbo's prototype quantum deformation5 : \Jo, J+] = J+

[J+ ,J-] = \ [2Jo],

[J-, M = J- ,

(3)

where the Chebyshev polynomial of the 2nd kind [x]q = (gx — q~x)/(q — q'1) is called the "g-deformation"10 of x. It is evident from [x]g3 - [2x]g/[2]q that the r.h.s. of the middle eq.(3) alternatively amounts to [Jo]q*, provided the generators J± absorb a factor -J2J[2]q in their normalization. The invariant operator of this algebra is C, s 2J+J- + [Jb],[Jb - 1], = W-J+ + [JoWo + 1], 2 \j\,\j + 1], -

(4)

Since this is not a closing Lie algebra, its group structure (exponentiation) is problematic. The classical limit q —* 1 yields SU(2). Can we reverse this starting from SU(2), i.e. can we find functionals of the generators (1) that satisfy the "quommutation" relations (3)? Tha answer is yes,, in analogy to the well-known "group expansions"11. Recall the elementary physics example of inflating a 2-sphere to a plane: S 2 ~ SO(3)/SO(2) —> R.2~ E(2)/SO(2), [jx,jy]=jz

[jy,3z] = 3x = rPy

\jz,jx]=jy

= -rPx,

(5)

by sending the radius r —* oo, which contracts the SU(2) algebra to [P*,PV} = 0

\j,,P*] = PV

\jz,Py]=-PX.

(6)

E.g. this is how the Lorentz group is introduced via the de Sitter group. But conversely, S0(3) may be reconstituted out of the Euclidean operators. Since P% + Py commutes with all three E(2) generators, it may be handled as a constant. Defining, for arbitrary p,

it follows directly that they obey the original SO(3) commutation relations* : \j'x,]']=jz

b'y,Jz]=j'x

\Jz,3x]=/

•

(8)

In the same spirit, we find deformation functional Q which convert the SU(2) generators g to operators G = Q(g) which obey the quommutations (3). To start with, we take in general 5 real, which allows hermitean conjugation O_ = O\. for the generator operator" under discussion: Jo = Q0(jQ) = j

J+ = Q+(g) =

0

(9)

The maps Q± are functionals of all three SU(2) generators jo, j + , j _ , since they depend on the operator j . Note that Cq = \j]q\j + l]q which can be easily solved for j . Thus, these deformation maps are readily invertible (except for special cases when the parameters are roots of unity, which can be analyzed as limits of the above). For generic q then, SU(2) and SU(2), provide nonlinear realizations of each other. The antipodes specified by the map amount to mere sign flips, just as in the classical algebra. Since Cq is a function of C, it and all its functions are also invariant under SU(2)*. Since it is an invertible function, C and all its functions are also invariant under SU(2),. Substituting specific matrix representations in the above connection formulas produces the corresponding representations of the QUE-algebra in a direct, mechanical fashion. For example, the (j = 1/2, Pauli-matrices) 2 representation of SU(2) : 0 \ -l) maps to Jo = jo,

(10)

~ ,M 0 0 /

J+ — j+ j while Cq = 1 — [1/2]^ . The obvious hermitean conjugates are

omitted. The 3: =

maps to Jo = jo,

/ 1 0 0 \ 0 0 0 \ 0 0 -1 /

J+ —

0 0 3/2 0 0 0 0 1/2 0 0 0 -1/2 3 /2; 0 0 0

u =

/ 0 1 0 \ 0 0 1 ) V o o o /

(11)

The 4: ( 0 -^572 0 0 0 V2 0 0 0 \ 0 0 0

0 0 ^/Tj2 0 /

(12)

'This particular map is not without pathologies, implicit in its Casimir invariant 1/4 — ( + Pyy) which argues for renormalizing11 to SO(2,1)- It is only reviewed here to illustrate expansions. *In an extension to spin-chain hatniltonians, Caldi et al. (these proceedings) uncover the SU(2) symmetry of anisotropic spin chains, given complete decomposition to irreducible blocks (see below).

maps to I

fo Jo = jo

J+ =

0 0 0

0 0

^0

0

[2]Jy/2 0 0

0 0

\ (13)

/2 0

J

and so forth. Now observe that [3]g = 0 = [3/2jg for q = exp (2xi/3); the 4 representation J+ now has only one nontrivial entry and •/.£ = 0, while Cq = 0; the middle commutator in (3) breaks up, so the representation reduces: 4 = 1 © 2 © 1. This reduction obtains for roots of unity with period smaller than the dimensionality of the representation9. Searching for zeros in the square-roots of eq.(9) which interrupt raising and lowering within a representation provides a quick survey of such reductions for complex q. For unitary representations, the requisite hermitean pairing of generators obtaining above does not hold automatically, and care must be taken to make the resulting generators hermitean conjugates, in some cases, by changing SU(2)? to SU(1,1),; however, in some cases this is not possible—cf. L. Mezincescu's and R. Nepomecbie's talk and refs.[4,12]: e.g. the 4 for q = exp (2JTI'/5). Parenthetically, 1 *hall consider SU(1,1). Its algebra, normalized slightly differently, [Lo, L1] = -L1 C = o(l-a)

[Xj, i_i] = 2X0 =*

LiL-i

[Xo, X_a] = X_! ,

= {L0 + a)(L0 + l-a),

(14)

maps to SU(1,1)?: Lo + 1- a], T _\ Lo +, 1••— Oj

-"l >

_ /[X o + a -f lj g [X 0 - a ] , °/\~

*»-i ~ w~77 . „ i i v r y (JJO + O + -Ul^O ~

* '

(15) with C, = [a][l — o]. Classically, Bargmann's highest(lowest)-weight infinite-dimensional discrete unitary representations D+ (X^~)have integer or half-integer a > 0 and Xo = a,a + l,o + 2,... (XQ = —o, —a — 1,—a — 2,.- ); the continuous principal series Dv has a = 1/2 - ip, 0 < p < oo, Xo = I,/± 1,/ ± 2,..., / = 0 or 1/2, and C > 1/4; and the continuous supplementary series D, has real o, Xo = 0,±l,±2,..., and C < 1/4 , jl/2 — o| < 1/2. The second of refs.[8] describes the unitary representations of SU(1,1)9 when q is not a root of unity; does breakup occur when o is a root of unity? In analogy to SU(2)? above, e.g. ior q = exp (2;rz/3), the raising and lowering operators C±\ indeed do have zeros with period 3 in the maps of the X)* irreducible representations, which therefore reduce (e.g. one may choose a = 1/2 for hermiticity). The map of the Ds irreducible representation also breaks up, as e.g. one may check for a = 1/3, and q = exp(3iri/4), a reduction to unitary doublets and triplets, and unitarity is easy to arrange in general. It is, however, harder to find finite-dimensional unitary reductions of the map of the Dv representations, since the zeros of £,±i now occur only for complex j's which are not phases, and it is problematic to achieve unitarity in this picture. Similar results hold for the more general projective representations. It may be interesting to ask whether SU(1,1)9 could serve as a spectrum-generating algebra of some "q-solvable" potentials, in analogy to its classical limit.

Refs.fl] similarly handle further familiar deformations of SU(2), such as Witten's deformation6 [Wo, W+\r s rW0W+ - hv+W0

= W+ ,

\W+, W.]1/T,

= Wo ,

[W_, W0]T = W_,

(16) which interpolates between SU(2) for r = 1 and SU(1,1) for r = —1, and whose representations likewise reduce for r a root of unity; or the 2-parameter generalization of Fairlie3, [Io,I+)r = I+

[/+,/_]!/, = Jo

[J-,Jo] r = J _ ,

(17)

2

(going to the previous one upon ,s —* r ; its representations also break up when both s and r are roots of unity), and several others. The structure of the deforming functionals in ref.[l] dictates relations on the representation labels of these deformations and leads to a complete characterization, such as that of ref.[3], even without direct reference to the classical algebra. A particularly appealing deformation is the cyclically symmetric "Fairlie rotation" 3

qXY-q~1YX

=Z

qYZ-q~1ZY=X

qZX - q~lXZ = Y , (18)

for which I find the invariant

C, = (q3 + 2q-*)(XYZ + YZX + ZXY) - (

(22)

which obeys SU(2)g quommutations, since its argument obeys SU{2). That is, the quantum generators are classicized, composed, and requantized. Any similarity transformation U~1QU on the above coproduct will also produce an isomorphic induced tensor coproduct (comultiplication). In particular, the same Clebsch operator C will automatically also reduce the coproduct (18): C-2#(A(ff))C = Gx © G2 © G3 ©... Recall, however, the standard5 comultiplication rule for (3) = J±

A,(J 0 ) = Jo ® I + 1 ® Jo

(23)

which appears different than the induced coproduct (22). It is reduced to a direct sum by the unitary ^-Clebsch operators15 Cq, and consequently Q(A(g)) = CC;1 A,(G') CqC~x = U'1 A,(