The alternating central extension U^+_q of U^+_q is isomorphic to U^+_q tensor F[z1,z2,...] via a surjective homomorphism sending alternating generators to alternating elements, with those generators forming a PBW basis.
The Lusztig automorphism of the $q$-Onsager algebra
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abstract
Pascal Baseilhac and Stefan Kolb recently introduced the Lusztig automorphism $L$ of the $q$-Onsager algebra $\mathcal O_q$. In this paper, we express each of $L, L^{-1}$ as a formal sum involving some quantum adjoints. In addition, (i) we give a computer-free proof that $L$ exists; (ii) we establish the higher order $q$-Dolan/Grady relations previously conjectured by Baseilhac and Thao Vu; (iii) we obtain a Lusztig automorphism for the current algebra $\mathcal A_q$ associated with $\mathcal O_q$; (iv) we describe what happens when a finite-dimensional irreducible $\mathcal O_q$-module is twisted via $L$.
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The alternating central extension for the positive part of $U_q(\widehat{\mathfrak{sl}}_2)$
The alternating central extension U^+_q of U^+_q is isomorphic to U^+_q tensor F[z1,z2,...] via a surjective homomorphism sending alternating generators to alternating elements, with those generators forming a PBW basis.