The finite-dimensional simple modules over the Drinfeld double of the bosonization of the Nichols algebra ufo(7) are classified. • Standard type [A1], including Cartan type [AS1]. • Super type [AAY]. • (Super) modular type [AA]. • Unidentified type [A4]. Actually, B(V ) is the smallest Nichols algebra of unidentified type. Following the terminology introduced in [AA], we say that V is of type ufo(7). Indeed, the Nichols algebra B(V ) is finite-dimensional by [H1, Table 1, row 7]; more precisely, cf. (13),dim B(V ) = 2 4 3 2 = 144. Hence dim M (λ) = dim B(V ) = 144. Thus the PBW-basis of U + ≃ B(V ) becomes via this isomorphism a basis of M (λ).TheThe family of U -submodules of M (λ) contained in β =0 M (λ) β has a unique maximal element N (λ). We set L(λ) = M (λ)/N (λ).Since U satisfies the conditions on [RS, Section 2], [RS, Corollary 2.6] implies thatThe map λ → L(λ) provides a bijective correspondence Γ ≃ Irr U .
In this paper, we calculate the combinatorial rank of the positive part [Formula: see text] of the multiparameter version of the small Lusztig quantum group, where [Formula: see text] is a simple Lie algebra of type [Formula: see text]. Supposing that the main parameter of quantization [Formula: see text] has multiplicative order [Formula: see text], where [Formula: see text] is finite, [Formula: see text], we prove that the combinatorial rank equals 3.
In this paper, using a much simpler method than the previous existing ones, we explicitly describe the PBW-generators of the multiparameter quantum groups U + q (g), where g is a simple Lie algebra of small dimension, while the main parameter of quantization q is not a root of the unity.
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