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Then c1 acts on the highest–weight module Vλ , and has the form c1 = trq (L+ SL− ) = (trq π ⊗ 1)(R21 R12 ). (1-136) Because it is a Casimir, it is enough to evaluate it on the lowest–weight state |λ− of Vλ , given by λ− = σm (λ) where σm denotes the longest element of the Weyl group. Now the universal R has the form −1 R = q Hi (B )ij ⊗Hj (1 ⊗ 1 + U + ⊗ U − ). (1-137) Here B is the (symmetric) matrix d−1 j Aij where A is the Cartan Matrix, di are the lengths of the simple roots (di = 1 for g = su(N)) and U + , U − stands for terms in the Borel sub-algebras of rising respectively lowering operators.

It is well–known that there is a unique Hilbert space structure on the subspaces FN such that they are unitary irreducible representations of Uq (su(2)). Then the above star is simply the operator adjoint. 3 Reality structure for q a phase When q is a phase, finding the correct star structure is not quite so easy. The difference with the case q ∈ R is that ∆(u∗ ) = (∗ ⊗ ∗)∆′ (u) for |q| = 1 and u ∈ Uq (su(2)), where ∆′ denotes the 2 flipped coproduct. We shall define a star only on the algebra Sq,N generated by the xi , and not + on the full algebra generated by Aα and Aα .

Q q (1-124) (for Uq (sl(N)). N = (Mn )(R (1-125) q q ˆk = R ˆ k−1,k (ignoring a possible constant factor). n , and similarly q q for the s. This in turn implies that indeed det(M) = 1 (fixing the constant factor in the definition of det(M) = 1) in the realization M = L+ SL− . n det(M) = (Mn R (1-128) ) ˆ 12 M2 R ˆ 12 ε12 = −q − 23 M2 R ˆ 12 M2 ε12 , which is For N = 2, this becomes det(M) = M2 R 1 2 2 2 1 proportional to ε12 q times (M1 M2 − q M1 M2 ). Hence we find the quantum determinant detq (M) = (M11 M22 − q 2 M12 M21 ) (1-129) as in [41].

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A guide to making energy-smart purchases by Energy Efficiency and Renewable Energy Clearinghouse (U.S.)

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