In his work on crystal bases \cite{ Kas}, Kashiwara introduced a certain degeneration of the quantized universal enveloping algebra of a semi-simple Lie algebra $\mathfrak g$, which he called a quantum boson algebra. In this paper, we construct Kashiwara operators associated to all positive roots and use them to define a variant of Kashiwara's quantum boson algebra. We show that a quasi-classical limit of the positive half of our variant is a Poisson algebra of the form $(P \simeq \mathbb C[\mathfrak n^{ \ast}], \, \{ ~~,~~\}_P)$, where $\mathfrak n$ is the positive part of $\mathfrak g$ and $\{ ~~,~~\}_P$ is a Poisson bracket that has the same rank as, but is different from, the Kirillov-Kostant bracket $\{ ~~,~~\}_{ KK}$ on $\mathfrak n^{ \ast}$. Furthermore, we prove that, in the special case of type $A$, any linear combination $a_1 \{ ~~,~~\}_P + a_2 \{ ~~,~~\}_{ KK}$, $a_1, a_2 \in \mathbb C$, is again a Poisson bracket. In the general case, we establish an isomorphism of $P$ and the Poisson algebra of regular functions on the open Bruhat cell in the flag variety. In type $A$, we also construct a Casimir function on the open Bruhat cell, together with its quantization, which may be thought of as an analog of the linear function on $\mathfrak n^{ \ast}$ defined by a root vector for the highest root.
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