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Polynomial elements in canonical basis $$\mathbf B$$ with two-dimensional support for type $$A_4$$. I. (English) Zbl 1242.17018
This paper continues the second author’s former work. All $$62$$ monomial elements and $$144$$ polynomial elements with one-dimensional support in the canonical basis $$\mathbb{B}$$ of the quantum group for type $$A_{4}$$ have been determined in [Y. Hu, J. Ye and X. Yue, J. Algebra 263, No. 2, 228–245 (2003; Zbl 1125.17309)] and [Y. Hu, J. Ye, Commun. Algebra 33, No. 11, 3855–3877 (2005; Zbl 1130.17007)], respectively. It is conjectured that there are other polynomial elements in $$\mathbb{B}$$ with two- or three-dimensional support. In this paper, the authors compute $$50$$ polynomial elements with two-dimensional support of different-exponent type and $$62$$ polynomial elements with two-dimensional support of same-exponent type in the canonical basis $$\mathbb{B}$$ of the quantum group for type $$A_{4}$$.

MSC:
 17B37 Quantum groups (quantized enveloping algebras) and related deformations 81R50 Quantum groups and related algebraic methods applied to problems in quantum theory 20G42 Quantum groups (quantized function algebras) and their representations
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