Zoran Skoda
Fock module and smash product

An abstract action of the Fock space type

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

(H𝒮)H(H𝒮)(H𝒮)m H𝒮H𝒮Hϵ 𝒮HkH. (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 𝒮H𝒮\mathcal{S}\hookrightarrow H\sharp\mathcal{S}, s1ss\mapsto 1\otimes s to a left action 𝒮HH\mathcal{S}\otimes H\to H. If the antipode S H:HH opS_H: H\to H^{\mathrm{op}} is an isomorphism, the corresponding representation ρ:𝒮End k(H)\rho :\mathcal{S}\to\End_k(H) is faithful. Using the definition of the smash product algebra, we may write the restricted action | 𝒮H:𝒮HH\blacktriangleright |_{\mathcal{S}\otimes H} : \mathcal{S}\otimes H\to H in terms of Hopf action :𝒮H𝒮\triangleleft : \mathcal{S}\otimes H\to\mathcal{S} only:

shh (1)(sh (2))h (1)ϵ 𝒮(sh (2))=sh. 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, s1 H=ϵ 𝒮(s)1 Hs\blacktriangleright 1_{H}=\epsilon^{\mathcal{S}}(s)1_{H}, and, if uHu\in H is primitive and hHh\in H then

s(uh) = (uh) (1)(s(uh) (2))1 H = uh (1)(sh (2))+h (1)((su)h (2))1 H = ush+(su)h.\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 𝒮\mathcal{S}. The smash product H𝒮H\sharp\mathcal{S} has a multiplication which involves rearrangement of factors in HH and factors in 𝒮\mathcal{S}. Once we put derivatives to the right we act with them on vacuum, what amounts to the map ϵ 𝒮\epsilon^{\mathcal{S}}, while the polynomial factor in 𝒮\mathcal{S} stays intact.

Special case, black action

We now specialize above to the case where 𝒮:=S^(𝔤 *)\mathcal{S} := \hat{S}(\mathfrak{g}^*), H:=U(𝔤)H := U(\mathfrak{g}), and the Hopf action is induced by ϕ:U(𝔤)Der(S^(𝔤 *),S^(𝔤 *))\mathbf{\phi} : U(\mathfrak{g})\to\Der(\hat{S}(\mathfrak{g}^*),\hat{S}(\mathfrak{g}^*)), and ϵ 𝒮\epsilon^{\mathcal{S}} is obtained by the application of a constant coefficient differential operator on 11 (derivatives act in the undeformed way on 1 U(𝔤)1_{U(\mathfrak{g})}; nevertheless we view the unit 1 U(𝔤)1_{U(\mathfrak{g})} as the deformed vacuum |0 𝔤|0\rangle_{\mathfrak{g}}). Recall that in that case H𝒮=A^ ϕ,kH\sharp{\mathcal{S}} = \hat{A}_{\phi,k}. The corresponding representation A^ 𝔤,ϕEnd k(U(𝔤))\hat{A}_{\mathfrak{g},\phi}\to\End_k(U(\mathfrak{g})) is called ϕ\mathbf{\phi}-deformed Fock space. Then the restricted action S^(𝔤 *)U(𝔤)U(𝔤)\hat{S}(\mathfrak{g}^*)\otimes U(\mathfrak{g})\to U(\mathfrak{g}) is given by sh=h (1)ϵ 𝒮(ϕ(S U(𝔤)h (2))(s))s\blacktriangleright h = \sum h_{(1)}\epsilon^{\mathcal{S}} (\mathbf{\phi}(S_{U(\mathfrak{g})} h_{(2)})(s)) and if h=u𝔤h = u \in \mathfrak{g} and s=𝔤 *S^(𝔤 *)s = \partial \in\mathfrak{g}^*\subset \hat{S}(\mathfrak{g}^*), this gives

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

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

Last revised on July 3, 2011 at 09:06:49. See the history of this page for a list of all contributions to it.