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The Pattern Calculus for Tensor Operators in Quantum Groups Mark Gould and L The pattern calculus for tensor operators in quantum groups Mark Gould and L. C. Biedenharn Citation: Journal of Mathematical Physics 33, 3613 (1992); doi: 10.1063/1.529909 View online: http://dx.doi.org/10.1063/1.529909 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/33/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Irreducible tensor operators in the regular coaction formalisms of compact quantum group algebras J. Math. Phys. 37, 4590 (1996); 10.1063/1.531643 Tensor operators for quantum groups and applications J. Math. Phys. 33, 436 (1992); 10.1063/1.529833 Tensor operators of finite groups J. Math. Phys. 14, 1423 (1973); 10.1063/1.1666196 On the structure of the canonical tensor operators in the unitary groups. I. An extension of the pattern calculus rules and the canonical splitting in U(3) J. Math. Phys. 13, 1957 (1972); 10.1063/1.1665940 Irreducible tensor operators for finite groups J. Math. Phys. 13, 1892 (1972); 10.1063/1.1665928 Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 130.102.42.98 On: Thu, 29 Sep 2016 02:42:08 The pattern calculus for tensor operators in quantum groups Mark Gould Department of Applied Mathematics, University of Queensland,St. Lucia, Queensland,Australia 4072 L. C. Biedenharn Department of Physics, Duke University, Durham, North Carolina 27706 (Received 28 February 1992; accepted for publication 1 June 1992) An explicit algebraic evaluation is given for all q-tensor operators (with unit norm) belonging to the quantum group U&(n)) and having extremal operator shift patterns acting on arbitrary U&u(n)) irreps. These rather complicated results are shown to be easily comprehensible in terms of a diagrammatic calculus of patterns. A more conceptual derivation of these results is discussed using fundamental properties of the q-6j coefficients. I. INTRODUCTION Schwinger (“boson operator”) map’ to realize explicitly the required general irreps and elementary tensor opera- Quantum groups are of considerable current interest, tors. These results were later verified using the “vector in both mathematics and physics, and have been the focus coherent state” (VCS) method (an algebraic version of of much recent research.le3 Although quantum groups the Borel-Weil construction) by LeBlanc and Hecht6 are not groups, they share many group-like features. For and, using a more conceptual approach, by Gould.’ The generic values of q-the deformation parameter-the rep- extension to B(n) was also given by Gould.8 resentation theory for compact quantum groups is sur- The approach used by Gould’ constructed the re- prisingly close to that of standard compact Lie groups, quired tensor operator matrix elements by abstract pro- and it is of interest to see how far this similarity extends jection operator techniques, and then evaluated these ma- for more complicated structures, such as tensor operator trix elements explicitly, using a projection operator algebras, built on representation theory. In the present realization, originally developed for angular momentum paper we shall demonstrate that a pattern calculus for [SU( 2)] calculations,g and later extended to a general Lie tensor operators in the U&u(n)) quantum group exists group.lot” To understand the concepts underlying this and is very similar in form-but not detail-to the pat- method by an elementary example, consider the total an- tern calculus known for tensor operators on Lie group gular momentum operator J [for the group SU(2)] as representations. being the sum of two independent angular momenta L It appears helpful, before discussing the modifications and u/2, that is, J=u/2+L. Then 2o9L= J2-L2-i is introduced by quantum groups, to review briefly the ideas an invariant operator (that is, [J,crL)] =O), and this op- and methods used in constructing the pattern calculus for erator may be used to construct (a) the (orbital) Casimir ordinary (compact) Lie groups. The pattern calculus is invariant operator: L2 = tr( ( wLj2) and (b) the projection actually three things: (a) an explicit evaluation of all operators: P,:(J+j=lfi):P+= (a-L+Z+ 1)/(21+ l), matrix elements of all elementary tensor operators be- P- = (I- 0.L )/21+ 1. (This latter step uses the eigenval- tween all (unitary) irreps of a given compact Lie group; ues CPL-P Z, - I - 1. ) Matrix elements of P, then yield the (b) a diagrammatic procedure whereby the diagram cor- desired spin-i tensor operator results. relates (or more precisely, implies) the explicit (alge- The required extension to an arbitrary Lie group con- braic) matrix element associated (by the pattern calculus siders the algebra of tensor products, &: rules) to the diagram; and (c) a procedure for the con- struction of all tensor operators. The elementary tensor a?= U(g) @ U(g), (1.1) operators are given directly by the pattern calculus, and by extending the calculus so that patterns act on patterns, where gcLie (G) and U(g) is the universal enveloping one obtains an algebraic basis for the construction of gen- algebra of Lie (G), and we use the diagonal coproduct eral tensor operators. d(g) = 18 g+g Q 1. (Coproducts are usually denoted by It is the merit of the pattern calculus that it “ex- the symbol A, but for tensor operator theory this symbol plains” and makes accessible structurally the otherwise has a different significance. Hence we use a for the co- difficult-to-comprehend explicit matrix elements (which product to avoid confusion.] The analog of the spin-f re- can be of arbitrarily large complexity and length). alization u/2 is now an irrep /z of G, with the associated The pattern calculus was first constructed [for U(n)] carrier space (module) VA, and Lie algebra generators by Biedenham and Louck4 using the straightforward pro- rA(g) playing the r6le of u/2. Generating elements of the cedure of direct evaluation, exploiting the Jordan- algebra &‘A = End( VA) o U(g) are then given by J. Math. Phys. 33 (1 l), November 1992 0022-2488/92/l 13613-23$03.00 @ 1992 American Institute of Physics 3613 Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 130.102.42.98 On: Thu, 29 Sep 2016 02:42:08 3614 M. Gould and L. C. Biedenharn: Tensor operators in quantum groups Commutation relations, a,dg) = h 8 mw). (1.2) The Casimir element C of G is C= &x,.x’, where {x,] ~wq = 0, (2.1) is a basis for Lie G and {x3 is the dual basis defined by the Killing form. The analog to PL is then [HisET] = fk,ij,Ei’, A&~Q(c) 0 I+IQ(I) 63c-a,(c)). (1.3) 1, i=j, (2.2a) where k,, = -l/2, i=j* 1, (2.2b) It is easily seen that An lies in the center of otherwise, (2.2c) End( Vn) o U(g), that is, [a,(g>,,4An]=O. 0, Just as in the defining example, the invariants of the Lie group G, that is, the center of U(g)-denoted Z-is [Ei+,E~] =sO~-‘~~~‘s4[2H,1 [see (2.8)]; generated as an algebra by the (finitely many) elements --4 z,,,= trA( (AJ “). Similarly, projection operators for irre- (2.3) ducible modules of -CBAcan be constructed as polynomials Quadratic q-Serre relations, in Ah. These projection operators can be given explicitly [as in (b) above] when one has determined the eigenvalue [ET,ET] =O, j#ih 1, l<i, j<n-1; (2.4) spectrum of An. Since this construction has been formulated purely and Cubic q-Serre relations, algebraically, the extension to quantum groups is imme- diate: one simply uses the algebra (and coalgebra) struc- ture relevant to the quantum group, and the quantum (E;)2-“ET(EF)Y=O, j. (--1)f]4 group analog of the trace operation. As we shall show in detail, this general procedure is sufficient to determine the (ij) = (i,i* l), l<i, j<n- 1, (2.5) desired quantum group matrix elements of quantum group tensor operators. The interpretation of these results where the q-binomial coefficient is defined by in terms of a calculus of patterns is then developed. The paper is organized in this way. In Sets. II and III n [nl! (2.6a) we summarize the necessary background, carrying out I m 1qT[m]![n-m]! ’ the construction, sketched above, for determining explicit algebraic matrix elements for all unit q-tensor operators [n]!=[n][n-l]***[l]. (2.6b) of U&u (n )), having extremal shifts, in Sets. IV-X. These algebraic results are shown to be easily comprehended, We use the notation [n] for the q-number structurally, in terms of a pattern calculus given in Sec. XI. In a concluding section (Sec. XII) we develop a more conceptual derivation of these results using properties of q-6j coefficients. We have tried to make the work in this paper essentially self-contained for the convenience of the =qWV2+q(n-3)/2+. +q-W/2), reader. n&5, qER+. (2.7) II. DEFINITION OF THE QUANTUM GROUP U,@(n)) This notation for q numbers is extended to the diagonal The specific results of the present paper are confined operators Hi by defining to the quantum group U,(u(n)). Almost all of the alge- braic manipulations are, however, generic and indepen- dent of the specific quantum group. Accordingly we will (2.8) use a general quantum group U,(g) in the algebraic structure, reverting to the specific quantum group Remark: Note that [n] is symmetric under q-q-’ U4(u( n)) for the detailed results.
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