The alternating PBW basis for the positive part of Uq(sl^2)
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In this paper, the alternating PBW basis for Uq+ is introduced, which is related to the alternating q-shuffle algebra associated with affine sl2, and is used for the first time in the positive part of Uq(sl^2).Abstract:
The positive part Uq+ of Uq(sl^2) has a presentation with two generators A, B that satisfy the cubic q-Serre relations. We introduce a PBW basis for Uq+, said to be alternating. Each element of this PBW basis commutes with exactly one of A, B, qAB − q−1BA. This gives three types of PBW basis elements; the elements of each type mutually commute. We interpret the alternating PBW basis in terms of a q-shuffle algebra associated with affine sl2. We show how the alternating PBW basis is related to the PBW basis for Uq+ found by Damiani in 1993.The positive part Uq+ of Uq(sl^2) has a presentation with two generators A, B that satisfy the cubic q-Serre relations. We introduce a PBW basis for Uq+, said to be alternating. Each element of this PBW basis commutes with exactly one of A, B, qAB − q−1BA. This gives three types of PBW basis elements; the elements of each type mutually commute. We interpret the alternating PBW basis in terms of a q-shuffle algebra associated with affine sl2. We show how the alternating PBW basis is related to the PBW basis for Uq+ found by Damiani in 1993.read more
Citations
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The alternating presentation of Uq(gl2ˆ) from Freidel-Maillet algebras
TL;DR: In this paper, a tensor product representation of A ¯ q in terms of a Freidel-Maillet type algebra is presented, and a new tensor decomposition for U q ( s l 2 ˆ ) is given.
Journal ArticleDOI
The alternating central extension for the positive part of Uq(slˆ2)
TL;DR: In this article, the positive part U q + of the quantum group U q ( sl ˆ 2 ) has a presentation with two generators A, B that satisfy the cubic q-Serre relations.
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The alternating central extension for the positive part of $U_q(\widehat{\mathfrak{sl}}_2)$
TL;DR: In this paper, the alternating generators of the positive part of the quantum group $U_q(\widehat{\mathfrak{sl}}_2) have been used to obtain a central extension of the algebra $U^+_q.
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The alternating central extension of the $q$-Onsager algebra
TL;DR: This article showed that the alternating central extension of the O_q-Onsager algebra is related to the algebra $\mathcal A_q$ in the same way that $U^+_q+q+Q$ is related with alternating central extensions of the enveloping algebra U_q.
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A compact presentation for the alternating central extension of the positive part of $U_q(\widehat{\mathfrak{sl}}_2)$.
TL;DR: In this article, a compact presentation of the alternating central extension of the positive part of the quantum group $U+_q$ is presented, which involves a small subset of the original set of generators and a manageable set of relations.
References
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P. Podleś,E. Müller +1 more
TL;DR: In this paper, the authors give an elementary introduction to the theory of algebraic and topological quantum groups (in the spirit of S. L. Woronowicz) and recall the basic facts from Hopf (*-) algebra theory, theory of compact (matrix) quantum groups and their actions on compact quantum spaces, and provide the most important examples, including the classification of quantum SL(2)-groups, their real forms and quantum spheres.
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Vyjayanthi Chari,Andrew Pressley +1 more
TL;DR: In this article, the authors classified finite-dimensional irreducible representations of the quantum affine algebra in terms of highest weights and gave an explicit construction of all such representations by means of an evaluation homomorphism.
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Quantum groups and quantum shuffles
TL;DR: In this paper, the upper triangular part of the quantized enveloping algebra associated with a symetrizable Cartan matrix is shown to be isomorphic to the subalgebra generated by elements of degree 0 and 1 of the cotensor Hopf algebra associated by a suitable Hopf bimodule on the group algebra.
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Two relations that generalize the $q$-Serre relations and the Dolan-Grady relations
TL;DR: The Tridiagonal algebra as discussed by the authors is an algebra on two generators which is defined as follows: a field is a field, and a sequence of scalars taken from a field can be represented by two symbols A and A. The corresponding Tridiagonal algebra T is the associative K-algebra with 1 generated by A. In the first part of this paper, we survey what is known about irreducible finite di-mensional T-modules.
Journal ArticleDOI
Deformed Dolan-Grady relations in quantum integrable models
TL;DR: In this paper, a new family of quantum integrable models is proposed, which is generated by a dual pair of operators { A, A ∗ ∈ A subject to q-deformed Dolan-Grady relations.