Lie algebra type noncommutative phase spaces are Hopf algebroids

Abstract

For a noncommutative configuration space whose coordinate algebra is the universal enveloping algebra of a finite-dimensional Lie algebra, it is known how to introduce an extension playing the role of the corresponding noncommutative phase space, namely by adding the commuting deformed derivatives in a consistent and nontrivial way; therefore, obtaining certain deformed Heisenberg algebra. This algebra has been studied in physical contexts, mainly in the case of the kappa-Minkowski space-time. Here, we equip the entire phase space algebra with a coproduct, so that it becomes an instance of a completed variant of a Hopf algebroid over a noncommutative base, where the base is the enveloping algebra.

Appendices

Appendix 1: Commutation $[\hat{x}_\alpha ,\hat{y}_\beta ] = 0$

Proposition 5

The identity $[\hat{x}_\mu ,\hat{y}_\nu ] = 0$ holds in the realization $\hat{x}_\mu = x_\sigma \phi ^\sigma _\mu = x_\sigma \left( \frac{-\mathcal {C}}{e^{-\mathcal {C}}-1}\right) ^\sigma _\mu$, $\hat{y}_\mu = x_\rho \tilde{\phi }^\rho _\mu = x_\rho \left( \frac{\mathcal {C}}{e^{\mathcal {C}}-1}\right) ^\rho _\mu$, where $\mathcal {C}^\mu _\nu = C^\mu _{\nu \gamma }\partial ^\gamma$ (cf. the Eqs.  (15,12,13)).

Proof

For any formal series $P = P(\partial )$ in $\partial$-s, $[P,\hat{x}_\mu ] = \frac{\partial P}{\partial (\partial ^\mu )} =:\delta _\mu P$. In particular (cf. [10]), from $[\hat{x}_\mu ,\hat{x}_\nu ] = C_{\mu \nu }^\lambda \hat{x}_\lambda$, one obtains a formal differential equation for $\phi ^\sigma _\mu$,

$$\begin{aligned} (\delta _\rho \phi ^\gamma _\mu )\phi ^\rho _\nu - (\delta _\rho \phi ^\gamma _\nu )\phi ^\rho _\mu = C^\sigma _{\mu \nu } \phi ^\gamma _\sigma . \end{aligned}$$

(54)

By symmetry $C^i_{jk}\mapsto -C^i_{jk}$ the same equation holds with $(-\tilde{\phi }) = \frac{-\mathcal {C}}{e^{\mathcal {C}}-1}$ in the place of $\phi$. Similarly, the equation $[\hat{x}_\mu ,\hat{y}_\nu ] = 0$, i.e., $[x_\gamma \phi ^\gamma _\mu ,x_\beta \tilde{\phi }^\beta _\nu ] = 0$, is equivalent to

$$\begin{aligned} (\delta _\rho \phi ^\gamma _\mu )\tilde{\phi }^\rho _\nu - (\delta _\rho \tilde{\phi }^\gamma _\nu )\phi ^\rho _\mu = 0 \end{aligned}$$

(55)

Recall that $\phi = \frac{-\mathcal {C}}{e^{-\mathcal {C}}-1} = \sum _{N = 0}^\infty (-1)^N \frac{B_N}{N!} (\mathcal {C}^N)^i_j$, where $B_N$ are the Bernoulli numbers, which are zero unless N is either even or $N=1$. Hence $\tilde{\phi }= \frac{\mathcal {C}}{e^{\mathcal {C}}-1} = \sum _{N = 0}^\infty \frac{B_N}{N!}\mathcal {C}^N = \frac{B_1}{2}\mathcal {C}+\sum _{N\,\text{ even }}^\infty \frac{B_N}{N!}\mathcal {C}^N$ and $\phi -\tilde{\phi }= -2\frac{B_1}{2}\mathcal {C}= \mathcal {C}$. Notice that $\frac{\partial \mathcal {C}^\alpha _\beta }{\partial (\partial ^\mu )} = C^\alpha _{\beta \mu }$. Therefore, subtracting (55) from (54) gives the condition

$$\begin{aligned} (\delta _\rho \phi ^\gamma _\mu )\mathcal {C}^\rho _\nu - C^\gamma _{\nu \rho } \phi ^\rho _\mu = C^\sigma _{\mu \nu } \phi ^\gamma _\sigma . \end{aligned}$$

$\mathcal {C}$ is homogeneous of degree 1 in $\partial ^\mu$-s, so we can split this condition into the parts of homogeneity degree N:

$$\begin{aligned}{}[\delta _\rho (\mathcal {C}^N)^\gamma _\mu ]\mathcal {C}^\rho _\nu - (\delta _\rho \mathcal {C}^\gamma _\nu )(\mathcal {C}^N)^\rho _\mu = C^\sigma _{\mu \nu }(\mathcal {C}^N)^\gamma _\sigma , \end{aligned}$$

(56)

where the overall factor of $(-1)^N B_N/N!$ has been taken out. Hence the proof is reduced to the following lemma:

Lemma 2

The identities (56) hold for $N = 0,1,2,\ldots$.

Proof

For $N = 0$, (56) reads $C^\gamma _{\nu \mu } = C^\gamma _{\mu \nu }$, which is the antisymmetry of the bracket. For $N=1$ it follows from the Jacobi identity:

$$\begin{aligned} (C^\gamma _{\mu \rho }C^\rho _{\nu \tau }-C^\gamma _{\nu \rho }C^\rho _{\mu \tau })\partial ^\tau = C^\rho _{\mu \nu }C^\gamma _{\rho \tau }\partial ^\tau . \end{aligned}$$

Suppose now (56) holds for given $N=K\ge 1$. Then

$$\begin{aligned} C^\gamma _{\mu \nu }(\mathcal {C}^K)^\rho _\sigma \mathcal {C}^\gamma _\rho = [\delta _\rho (\mathcal {C}^K)^\rho _\mu ]\mathcal {C}^\sigma _\nu \mathcal {C}^\gamma _\rho - C^\rho _{\nu \sigma }(\mathcal {C}^K)^\sigma _\mu \mathcal {C}^\gamma _\rho \end{aligned}$$

By the usual Leibniz rule for $\delta _\rho$, this yields

$$\begin{aligned} C^\gamma _{\mu \nu }(\mathcal {C}^K)^\rho _\sigma \mathcal {C}^\gamma _\rho = \delta _\rho (\mathcal {C}^{K+1})^\gamma _\rho \mathcal {C}^\sigma _\nu - (\mathcal {C}^K)^\rho _\mu C^\gamma _{\rho \sigma }\mathcal {C}^\sigma _\nu - C^\rho _{\nu \sigma }(\mathcal {C}^K)^\sigma _\mu \mathcal {C}^\gamma _\rho . \end{aligned}$$

The identity (56) follows for $N = K+1$ if the second and third summand on the right-hand side add up to $-C^\gamma _{\nu \sigma }(\mathcal {C}^{K+1})^\sigma _\mu$. After renaming the indices, one brings the sum of these two to the form

$$\begin{aligned} (\mathcal {C}^K)^\rho _\mu (- C^\sigma _{\nu \lambda }C^\gamma _{\rho \sigma } + C^\sigma _{\nu \rho }C^\gamma _{\lambda \sigma })\partial ^\lambda = -(C^K)^\rho _\mu C^\sigma _{\rho \lambda }\partial ^\lambda C^\gamma _{\nu \sigma } = -(C^{K+1})^\sigma _\mu C^\gamma _{\nu \sigma } \end{aligned}$$

as required. The Jacobi identity is used for the equality on the left. $\square$

Appendix 2: Cofiltered vector spaces and completions

We sketch the formalism treating the algebraic duals $U(\mathfrak {g})^*$ and $S(\mathfrak {g})^*$ of filtered algebras $U(\mathfrak {g}), S(\mathfrak {g})$ as cofiltered algebras. The reader can treat them alternatively as topological algebras: the basis of neighborhoods of 0 in the formal adic topology of $U(\mathfrak {g})^*$ and $S(\mathfrak {g})^*$ is given by the annihilator ideals ${\text {Ann}}\,U_i(\mathfrak {g})$ and ${\text {Ann}}\,S_i(\mathfrak {g})$, consisting of functionals vanishing on the i-th filtered component. A cofiltration on a vector space A is an inverse sequence of epimorphisms of its quotient spaces $\cdots \rightarrow A_{i+1}\rightarrow A_i\rightarrow A_{i-1}\rightarrow \cdots \rightarrow A_0$; denoting the quotient maps $\pi _i:A\rightarrow A_i$ and $\pi _{i,i+k}:A_{i+k}\rightarrow A_i$, the identities $\pi _i = \pi _{i,i+k}\circ \pi _{i+k}$, $\pi _{i,i+k+l} = \pi _{i,i+k}\circ \pi _{i+k,i+k+l}$ are required to hold. The limit $\underset{\longleftarrow }{\lim }{}_r A_r$ consists of threads, i.e., the sequences $(a_r)_{r\in \mathbb {N}_0}\in \prod _r A_r$ of compatible elements, $a_r = \pi _{r,r+k}(a_{r+k})$. The canonical map $A\rightarrow \hat{A}$ to the completion $\hat{A} := \underset{\longleftarrow }{\lim }{}_r A_r$ is 1-1 if $\forall a\in A$ $\exists r\in \mathbb {N}_0$ such that $\pi _r(a)\ne 0$. The cofiltration is complete if the canonical map $A\rightarrow \hat{A}$ is an isomorphism. Strict morphisms of cofiltered vector spaces $A\rightarrow B$ are the linear maps which induce the levelwise maps $A_r\rightarrow B_r$ on the quotients. (This makes the category of complete cofiltered vector spaces more rigid than the category of pro-vector spaces.) We say that $a= (a_r)_r$ has the cofiltered degree $\ge N$ if $a_r = 0$ for $r<N$. In our main example, $U_i(\mathfrak {g})^*:= (U(\mathfrak {g})^*)_i := U(\mathfrak {g})^*/{\text {Ann}}\,U_i(\mathfrak {g})\cong (U_i(\mathfrak {g}))^*$ and similarly for $S(\mathfrak {g})^*\cong \hat{S}(\mathfrak {g}^*)$. We use lower indices both for filtrations and for cofiltrations (but upper for gradations!). Given a family of elements in A, $a:\Lambda \rightarrow A$, $\lambda \mapsto a_\lambda$, the expression (’abstract infinite sum’) $\sum _{\lambda \in \Lambda }a_\lambda$ is called a formal sum if for each $r\ge 0$, there is only finitely many $\lambda$ such that $\pi _r(a_\lambda )\ne 0$ hence $\pi _r(\sum _{\lambda \in \Lambda }a_\lambda ) := \sum _{\lambda \in \Lambda }\pi _r(a_\lambda )\in A_r$ is well defined; and therefore there is well-defined thread $(\pi _r(\sum _{\lambda \in \Lambda }a_\lambda ))_r\in \hat{A}$, the value of the formal sum.

The usual tensor product $A\otimes B$ of cofiltered vector spaces is cofiltered with the rth cofiltered component (see [25])

$$\begin{aligned} (A\otimes B)_r = \frac{A\otimes B}{\cap _{p+q=r}{\text {ker}}\,\pi ^A_p \otimes {\text {ker}}\,\pi ^B_q}. \end{aligned}$$

(57)

$(A\otimes B)_r$ is an abelian group of finite sums of the form $\sum _\lambda a_\lambda \otimes b_\lambda \in A\otimes B$ modulo the additive relation of equivalence $\sim _r$ for which $\sum a_\mu \otimes b_\mu \sim _r 0$ iff $\pi _p(a_\mu )\otimes \pi _q(b_\mu ) = 0$ in $A_p\otimes B_q$ for all p, q such that $p+q=r$. Define the completed tensor product $A\hat{\otimes }B = \underset{\longleftarrow }{\lim }{}_r (A\otimes B)_r$, equipped with the same cofiltration, $(A\hat{\otimes }B)_r := (A\otimes B)_r$. An element in $A\hat{\otimes }B$ is thus the class of equivalence of a formal sum $\sum _\lambda a_\lambda \otimes b_\lambda$ such that for any p and q there are at most finitely many $\lambda$ such that $\pi ^A_p(a_\lambda )\otimes \pi ^B_q(b_\lambda )\ne 0$. Alternatively, we can equip $A\otimes B$ with a bicofiltration ($\mathbb {N}_0\times \mathbb {N}_0$-cofiltration), $(A\otimes B)_{r,s} = A_r\otimes B_s$. Observe the inclusions ${\text {ker}}\,\pi _{r+s}\otimes {\text {ker}}\,\pi _{r+s} \subset \cap _{p+q=r+s}{\text {ker}}\,\pi _p\otimes {\text {ker}}\,\pi _q\subset {\text {ker}}\,\pi _r\otimes {\text {ker}}\,\pi _s$, which induce projections $A_{r+s}\otimes B_{r+s}\twoheadrightarrow (A\otimes B)_{r+s}\twoheadrightarrow A_r\otimes B_s$ for all r, s; by passing to the limit we see that the completion with respect to the bicofiltration and with respect to the original cofiltration are equivalent (and alike statement for the convergence of infinite sums inside $A\hat{\otimes }B$). A linear map among cofiltered vector spaces is distributive over formal sums if it sends formal sums to formal sums summand by summand (formal version of $\sigma$-additivity). This property is weaker than being a strict morphism of complete cofiltered vector spaces. In fact ([25]), a linear map $f:C\rightarrow D$ is distributive over formal sums iff $\forall s$ $\exists r$ and a linear map $f_{sr}:C_r\rightarrow D_s$ such that $\pi _s\circ f = f_{sr}\circ \pi _r$ (in the strict case we required $s=r$). If A and B are complete, we can also consider maps $A\otimes B\rightarrow C$ distributive over formal sums in each argument separately. Unlike the strict morphisms of cofiltered spaces, such a map does not need to extend to a map $A\hat{\otimes }B\rightarrow \hat{C}$ distributive over formal sums in $A\hat{\otimes }B$ (continuity in each argument separately does not imply the joint continuity).

A (strict) cofiltered algebra A (e.g. $\hat{S}(\mathfrak {g}^*)$) is a monoid internal to the ${\varvec{k}}$-linear category of complete cofiltered vector spaces, strict morphisms and with the tensor product $\hat{\otimes }$ ([25]). The bilinear associative unital multiplication map $\hat{m}:A\hat{\otimes }A\rightarrow A$ is a strict morphism, hence inducing linear maps $m_r:(A\otimes A)_r\rightarrow A_r$ for all r. In other words, $A\hat{\otimes }A\ni \sum _\lambda a_\lambda \otimes b_\lambda \overset{\hat{m}}{\mapsto }\sum _\lambda a_\lambda \cdot b_\lambda \in A$, where $(\sum _\lambda a_\lambda \cdot b_\lambda )_r$ is an equivalence class in $A_r$ of $(\pi _r\circ \hat{m})(\sum '_\lambda a_\lambda \otimes b_\lambda )$, where $\sum '$ denotes the finite sum over all $\lambda$ such that $\exists (p,q)$ with $p+q=r$ and $\pi _p(a_\lambda )\otimes \pi _q(b_\lambda ) \ne 0$.

Any vector subspace W of a cofiltered vector space V is cofiltered by $W_p := V_p\cap W$ with a canonical linear map $\underset{\longleftarrow }{\lim }\,W_p\rightarrow \underset{\longleftarrow }{\lim }\,V_p=\hat{V}$, whose image is a cofiltered subspace $\hat{W}_{\hat{V}}\subset \hat{V}$, the completion of W in $\hat{V}$. This is compatible with many additional structures, so defining the completions of sub(bi)modules and ideals (thus $\hat{I}$, $\hat{I}'$, $\hat{I}^{(r)}$, $\hat{I}'^{(r)}$, $\hat{\bar{I}}$, $\hat{\bar{I}}'$, $\hat{\bar{I}}^{(r)}$, $\hat{\bar{I}}'^{(r)}$ in Sects. 5 and 6). If U is an associative algebra, $A_U$ a right U-module and ${}_U B$ a left U-module, where both modules are cofiltered, then define $A\hat{\otimes }_U B$ as the quotient of $A\hat{\otimes }B$ by the completion of ${\text {ker}}\,(A\otimes B\rightarrow A\otimes _U B)$ in $A\hat{\otimes }B$.

In this article, the completed tensor product $U(\mathfrak {g}^L)\hat{\otimes }\hat{S}(\mathfrak {g}^*)$ is defined by equipping the filtered ring $U(\mathfrak {g}^L)$ with the trivial cofiltration $U(\mathfrak {g}^L)$, in which every cofiltered component is the entire $U(\mathfrak {g})$ (and carries the discrete topology). The elements of $U(\mathfrak {g}^L)\hat{\otimes }\hat{S}(\mathfrak {g}^*)$ are given by the formal sums $\sum u_\lambda \otimes a_\lambda$ such that $\forall r$, $\pi _r(a_\lambda ) = 0$ for all but finitely many $\lambda$. The basis of neighborhoods of 0 in $U(\mathfrak {g}^L)\hat{\otimes }\hat{S}(\mathfrak {g}^*)$ consists of the subspaces ${\varvec{k}}\hat{f}\otimes \prod _{p>r}S^p({\mathfrak {g}^{*}})$ for all $\hat{f}\in U(\mathfrak {g})$ and $r\in \mathbb {N}$. The right Hopf action $a\otimes \hat{u}\mapsto \varvec{\phi }_+(\hat{u})(a)$ admits a completed smash product:

Theorem 6

The multiplication in $H^L = U(\mathfrak {g}^L)\sharp _{\varvec{\phi }_+}\hat{S}(\mathfrak {g}^*)$ extends to a unique multiplication $\hat{m}$ on $U(\mathfrak {g}^L)\hat{\otimes }\hat{S}(\mathfrak {g}^*)$ which distributes over formal sums in each argument, forming the completed smash product algebra ${\hat{H}^L} = U({\mathfrak {g}^{L}})\hat{\sharp }\hat{S}({\mathfrak {g}^{*}})$. Likewise, the multiplication on ${H^{R}} = \hat{S}({\mathfrak {g}^{*}})\sharp _{\tilde{\varvec{\phi }}_-}U({\mathfrak {g}^{R}})$ extends to $\hat{S}({\mathfrak {g}^{*}})\hat{\otimes }U({\mathfrak {g}^{R}})$ forming ${\hat{H}^R} = \hat{S}({\mathfrak {g}^{*}})\hat{\sharp }U({\mathfrak {g}^{R}})$. However, there are no cofiltered algebra structures on ${\hat{H}^L}$, because the multiplication does not distribute over formal sums in $H^L\hat{\otimes }{{H}^L}$.

Proof

The extended multiplication is well defined by a formal sum $\sum _{\lambda ,\mu } (u_\lambda \sharp a_\lambda )(u'_\mu \sharp a'_\mu ) = \sum _{\lambda ,\mu } u_\lambda u'_{\mu (1)}\sharp \varvec{\phi }_+(u'_{\mu (2)})(a_\lambda ) a'_\mu$ if for all $r\in \mathbb {N}_0$ the number of pairs $(\mu ,\lambda )$ such that $u_\lambda u'_{\mu (1)}\otimes \pi _r(\varvec{\phi }_+(u'_{\mu (2)})(a_\lambda ) \cdot a'_\mu )\ne 0$ (only Sweedler summation) is finite. There are only finitely many $\mu$ such that $\pi _r(a'_\mu )\ne 0$; only those contribute to the sum because $\pi _k(a)\pi _l(b)=0$ implies $\pi _{k+l}(ab)=0$ in any cofiltered ring. For each such $\mu$ fix a representation of $\Delta (u_\mu )$ as a finite sum $\sum _k u_{\mu (1)k}\otimes u_{\mu (2)k}$ and denote by $K(\mu )$ the maximal over k filtered degree of $u_{\mu (2)k}$ and by $L(\mu )$ the minimal cofiltered degree of $a'_\mu$. By Lemma 1 (iii) and induction we see that if $a_\lambda \in \hat{S}(\mathfrak {g}^*)_s$ then $\varvec{\phi }_+(u_{\mu (2)k})(a_\lambda )\in \hat{S}(\mathfrak {g}^*)_{s-K(\mu )}$. Hence for each $\mu$ there are only finitely many $\lambda$ for which $s-K(\mu )+L(\mu )\le r$. That is sufficient for the conclusion. More details will be exhibited elsewhere. $\square$