Editing Quantum group (section)

====Case 1: ''q'' is not a root of unity====
One important type of representation is a weight representation, and the corresponding [[Module (mathematics)|module]] is called a weight module. A weight module is a module with a basis of weight vectors. A weight vector is a nonzero vector ''v'' such that ''k<sub>λ</sub>'' · ''v'' = ''d<sub>λ</sub>v'' for all ''λ'', where ''d<sub>λ</sub>'' are complex numbers for all weights ''λ'' such that

:<math>d_0 = 1,</math>
:<math>d_\lambda d_\mu = d_{\lambda + \mu},</math> for all weights ''λ'' and ''μ''.

A weight module is called integrable if the actions of ''e<sub>i</sub>'' and ''f<sub>i</sub>''  are locally nilpotent (i.e. for any vector ''v'' in the module, there exists a positive integer ''k'', possibly dependent on ''v'',  such that <math>e_i^k.v = f_i^k.v = 0</math> for all ''i'').  In the case of integrable modules, the complex numbers ''d''<sub>''λ''</sub> associated with a weight vector satisfy <math>d_\lambda = c_\lambda q^{(\lambda,\nu)}</math>,{{Citation needed|date=July 2016}} where ''ν'' is an element of the weight lattice, and ''c<sub>λ</sub>'' are complex numbers such that

:*<math>c_0 = 1,</math>
:*<math>c_\lambda c_\mu = c_{\lambda + \mu},</math> for all weights ''λ'' and ''μ'',
:*<math>c_{2\alpha_i} = 1</math> for all ''i''.

Of special interest are [[highest-weight representation]]s, and the corresponding highest weight modules.  A highest weight module is a module generated by a weight vector ''v'', subject to ''k''<sub>''λ''</sub> · ''v'' = ''d<sub>λ</sub>v'' for all weights ''μ'', and ''e<sub>i</sub>'' · ''v'' = 0 for all ''i''.  Similarly, a quantum group can have a lowest weight representation and lowest weight module, ''i.e.'' a module generated by a weight vector ''v'', subject to ''k<sub>λ</sub>'' · ''v'' = ''d<sub>λ</sub>v'' for all weights ''λ'', and ''f<sub>i</sub>'' · ''v'' = 0 for all ''i''.

Define a vector ''v'' to have weight ''ν'' if <math>k_\lambda\cdot v = q^{(\lambda,\nu)} v</math> for all ''λ'' in the weight lattice.

If ''G'' is a Kac–Moody algebra, then in any irreducible highest weight representation of ''U''<sub>''q''</sub>(''G''), with highest weight ν, the multiplicities of the weights are equal to their multiplicities in an irreducible representation of ''U''(''G'') with equal highest weight.  If the highest weight is dominant and integral (a weight ''μ'' is dominant and integral if ''μ'' satisfies the condition that <math>2 (\mu,\alpha_i)/(\alpha_i,\alpha_i)</math> is a non-negative integer for all ''i''), then the weight spectrum of the irreducible representation is invariant under the [[Weyl group]] for ''G'', and the representation is integrable.

Conversely, if a highest weight module is integrable, then its highest weight vector ''v'' satisfies <math>k_\lambda\cdot v = c_\lambda q^{(\lambda,\nu)} v</math>, where ''c''<sub>''λ''</sub> · ''v'' = ''d''<sub>''λ''</sub>''v'' are complex numbers such that

:*<math>c_0 = 1,</math>
:*<math>c_\lambda c_\mu = c_{\lambda + \mu},</math> for all weights ''λ'' and ''μ'',
:*<math>c_{2\alpha_i} = 1</math> for all ''i'',

and ''ν'' is dominant and integral.

As is the case for all Hopf algebras, the [[tensor product]] of two modules is another module.  For an element ''x'' of ''U<sub>q</sub>(G)'', and for vectors ''v'' and ''w'' in the respective modules, ''x'' ⋅ (''v'' ⊗ ''w'') = Δ(''x'') ⋅ (''v'' ⊗ ''w''), so that <math>k_\lambda\cdot(v \otimes w) = k_\lambda\cdot v \otimes k_\lambda.w</math>, and in the case of coproduct Δ<sub>1</sub>, <math>e_i\cdot(v \otimes w) = k_i\cdot v \otimes e_i\cdot w + e_i\cdot v \otimes w</math> and <math>f_i\cdot(v \otimes w) = v \otimes f_i\cdot w + f_i\cdot v \otimes k_i^{-1}\cdot w.</math>

The integrable highest weight module described above is a tensor product of a one-dimensional module (on which ''k''<sub>λ</sub> = ''c''<sub>''λ''</sub> for all ''λ'', and ''e<sub>i</sub>'' = ''f<sub>i</sub>'' = 0 for all ''i'') and a highest weight module generated by a nonzero vector ''v''<sub>0</sub>, subject to <math>k_\lambda\cdot v_0 = q^{(\lambda,\nu)} v_0</math> for all weights ''λ'', and <math>e_i\cdot v_0 = 0</math> for all ''i''.

In the specific case where ''G'' is a finite-dimensional Lie algebra (as a special case of a Kac–Moody algebra), then the irreducible representations with dominant integral highest weights are also finite-dimensional.

In the case of a tensor product of highest weight modules, its decomposition into submodules is the same as for the tensor product of the corresponding modules of the Kac–Moody algebra (the highest weights are the same, as are their multiplicities).