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The quantum Clebsch-Gordan coefficients. (Russian) Zbl 0684.17003

Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 168, 67-84 (1988).
[For the entire collection see Zbl 0657.00006.]
Let \(V_ i\) be a finite dimensional \({\mathfrak sl}(2)\)-module with highest weight i, \(\{v^ i_ a,-m\leq a\leq m\}\) an orthonormal with respect to a nondegenerate inner product (, ) weight basis of \(V_ i\). (Clearly, (, ) is not necessarily invariant.) As is well-known, as \({\mathfrak sl}(2)\)- module, \(V_ i\otimes V_ j\) has the following decomposition into irreducible \({\mathfrak sl}(2)\)-modules: \(\oplus_{| j-i| \leq k\leq i+j}V_ k\). Schur’s lemma implies that the isometry operator I(i,j,k) intertwining \(V_ i\otimes V_ j\) and \(V_ k\) is defined up to a scalar (of absolute value 1). The matrix elements of I(i,j,k) with respect to the bases \(\{v^ k_ a; v^ i_ a\otimes v^ j_ b\}\) are called Clebsch-Gordan coefficients (CGC). Properties of Clebsch- Gordan coefficients and their generating function are of importance in the theory of atom nuclei.
In the (very lucid) paper under review \({\mathfrak sl}(2)\)-modules or, equivalently, U(\({\mathfrak sl}(2))\)-modules are replaced with their “quantum” q-analogues - modules over the “quantum group” \(U_ q({\mathfrak sl}(2))\), a certain deformation of the Hopf algebra U(\({\mathfrak sl}(2))\). As is shown, the theory of Clebsch-Gordan coefficients has a complete quantum counterpart (van der Waerden and Racah-Fock formulas, etc.). References to recent preprints by V. Pasquier; N. Reshetikhin; the author, and M. Jimbo, A. Kuniba, T. Miwa, M. Okado reveal connections of QCGC with integrable systems of statistical physics (R-matrices, Yang- Baxter equations) and the theory of links.
Reviewer: D.Leites

MSC:

17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
17B35 Universal enveloping (super)algebras
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
17B20 Simple, semisimple, reductive (super)algebras
81T60 Supersymmetric field theories in quantum mechanics

Citations:

Zbl 0657.00006
Full Text: EuDML