  This paper is about the positive part $U^+_q$ of the quantum group
$U_q(\widehat{\mathfrak{sl}}_2)$. The algebra $U^+_q$ has a presentation with
two generators $A,B$ that satisfy the cubic $q$-Serre relations. Recently we
introduced a type of element in $U^+_q$, said to be alternating. Each
alternating element commutes with exactly one of $A$, $B$, $qBA-q^{-1}AB$,
$qAB-q^{-1}BA$; this gives four types of alternating elements. There are
infinitely many alternating elements of each type, and these mutually commute.
In the present paper we use the alternating elements to obtain a central
extension $\mathcal U^+_q$ of $U^+_q$. We define $\mathcal U^+_q$ by generators
and relations. These generators, said to be alternating, are in bijection with
the alternating elements of $U^+_q$. We display a surjective algebra
homomorphism $\mathcal U^+_q \to U^+_q$ that sends each alternating generator
of $\mathcal U^+_q$ to the corresponding alternating element in $U^+_q$. We
adjust this homomorphism to obtain an algebra isomorphism $\mathcal U_q^+ \to
U^+_q \otimes \mathbb F \lbrack z_1, z_2,\ldots\rbrack$ where $\mathbb F$ is
the ground field and $\lbrace z_n\rbrace_{n=1}^\infty$ are mutually commuting
indeterminates. We show that the alternating generators form a PBW basis for
$\mathcal U_q^+$. We discuss how $\mathcal U^+_q$ is related to the work of
Baseilhac, Koizumi, Shigechi concerning the $q$-Onsager algebra and integrable
lattice models.
