Zoran Skoda Fock module and smash product

An abstract action of the Fock space type

Given a Hopf $k$-algebra $H$, a right Hopf action $\triangleleft : \mathcal{S}\otimes_k H\to\mathcal{S}$ on an algebra $\mathcal{S}$, and a homomorphism of unital algebras $\epsilon^{\mathcal{S}}:\mathcal{S}\to k$, one can, in addition to the smash product algebra $H\sharp\mathcal{S}$, define also a $k$-linear left action of the entire smash product algebra on $H$, namely $(H\sharp\mathcal{S})\otimes H \stackrel\blacktriangleright\longrightarrow H$ is the composition

$$(H\sharp\mathcal{S})\otimes H\hookrightarrow (H\sharp\mathcal{S})\otimes (H\sharp\mathcal{S}) \stackrel{m_{H\sharp\mathcal{S}}}\longrightarrow H\sharp\mathcal{S} \stackrel{H\sharp\epsilon^{\mathcal{S}}}\longrightarrow H\otimes k\cong H.$$

This action restricts along the algebra embedding $\mathcal{S}\hookrightarrow H\sharp\mathcal{S}$, $s\mapsto 1\otimes s$ to a left action $\mathcal{S}\otimes H\to H$. If the antipode $S_H: H\to H^{\mathrm{op}}$ is an isomorphism, the corresponding representation $\rho :\mathcal{S}\to\End_k(H)$ is faithful. Using the definition of the smash product algebra, we may write the restricted action $\blacktriangleright |_{\mathcal{S}\otimes H} : \mathcal{S}\otimes H\to H$ in terms of Hopf action $\triangleleft : \mathcal{S}\otimes H\to\mathcal{S}$ only:

$$s\otimes h \mapsto \sum h_{(1)} \sharp (s\triangleleft h_{(2)}) \mapsto \sum h_{(1)} \epsilon^{\mathcal{S}}(s\triangleleft h_{(2)}) = s\blacktriangleright h.$$

In particular, $s\blacktriangleright 1_{H}=\epsilon^{\mathcal{S}}(s)1_{H}$, and, if $u\in H$ is primitive and $h\in H$ then

$$\array{ s\blacktriangleright (u h) &=& \sum (u h)_{(1)} (s \triangleleft (u h)_{(2)})\blacktriangleright 1_H \\ &=& \sum u h_{(1)} (s\triangleleft h_{(2)}) + h_{(1)} ((s\triangleleft u)\triangleleft h_{(2)})\blacktriangleright 1_H\\ &=& u s\blacktriangleright h + (s\triangleleft u) \blacktriangleright h. }$$

Similar formula holds for skew-primitive elements. The symbol for the action $\blacktriangleright$ is often omitted below, unless when it is useful for clarity.

Interpretation

This construction has a spirit of Fock space construction: think of derivatives as primitive elements generating $\mathcal{S}$. The smash product $H\sharp\mathcal{S}$ has a multiplication which involves rearrangement of factors in $H$ and factors in $\mathcal{S}$. Once we put derivatives to the right we act with them on vacuum, what amounts to the map $\epsilon^{\mathcal{S}}$, while the polynomial factor in $\mathcal{S}$ stays intact.

Special case, black action

We now specialize above to the case where $\mathcal{S} := \hat{S}(\mathfrak{g}^*)$, $H := U(\mathfrak{g})$, and the Hopf action is induced by $\mathbf{\phi} : U(\mathfrak{g})\to\Der(\hat{S}(\mathfrak{g}^*),\hat{S}(\mathfrak{g}^*))$, and $\epsilon^{\mathcal{S}}$ is obtained by the application of a constant coefficient differential operator on $1$ (derivatives act in the undeformed way on $1_{U(\mathfrak{g})}$; nevertheless we view the unit $1_{U(\mathfrak{g})}$ as the deformed vacuum $|0\rangle_{\mathfrak{g}}$). Recall that in that case $H\sharp{\mathcal{S}} = \hat{A}_{\phi,k}$. The corresponding representation $\hat{A}_{\mathfrak{g},\phi}\to\End_k(U(\mathfrak{g}))$ is called $\mathbf{\phi}$-deformed Fock space. Then the restricted action $\hat{S}(\mathfrak{g}^*)\otimes U(\mathfrak{g})\to U(\mathfrak{g})$ is given by $s\blacktriangleright h = \sum h_{(1)}\epsilon^{\mathcal{S}} (\mathbf{\phi}(S_{U(\mathfrak{g})} h_{(2)})(s))$ and if $h = u \in \mathfrak{g}$ and $s = \partial \in\mathfrak{g}^*\subset \hat{S}(\mathfrak{g}^*)$, this gives

$$\partial\blacktriangleright u = \mathbf{\phi}(-u)(\partial) \blacktriangleright 1_{U(\mathfrak{g})}$$

as from $\partial 1 = 0$ the summand $u\epsilon^{\mathcal{S}}(\mathbf{\phi}(1_H)(\partial))$ vanishes. In other words, if we restrict the left action $\hat{S}(\mathfrak{g}^*)\otimes U(\mathfrak{g}) \to U(\mathfrak{g})$ to $\mathfrak{g}^*\otimes\mathfrak{g}$ it coincides with the restriction of right Hopf action $\hat{S}(\mathfrak{g}^*)\otimes U(\mathfrak{g})\to \hat{S}(\mathfrak{g}^*)$ to $\mathfrak{g}^*\otimes\mathfrak{g}$, followed by the evaluation at deformed vacuum $1_{U(\mathfrak{g})}$.