$\displaystyle\frac{1}{2}\left(S_{\text{BFBB}}[A,\Sigma]+S_{\text{BFBB}}[C,B]\right)$
$\displaystyle+\frac{1}{2}\int_{M}\langle B\wedge\chi_{A}^{*}(\wedge
C)\rangle+\langle\Sigma\wedge\eta_{C}^{*}(\wedge A)\rangle,$

which is a sum of the BF-BB theories

$S_{\text{BFBB}}[A,\Sigma]=\int_{M}\langle\Sigma\wedge
F\rangle-\langle\Sigma\wedge t\Sigma\rangle,\qquad
S_{\text{BFBB}}[C,B]=\int_{M}\langle B\wedge F^{*}\rangle-\langle B\wedge
t^{*}B\rangle$

in each individual crossed-module sector $\mathfrak{g},\mathfrak{g}^{*}[1]$
respectively, plus the interaction terms

$\langle B\wedge\chi_{A}^{*}(\wedge
C)\rangle=-\langle\operatorname{ad}_{A}^{*}(\wedge B)\wedge
C\rangle,\qquad\langle\Sigma\wedge\eta_{C}^{*}(\wedge
A)\rangle=-\langle\mathfrak{ad}_{C}^{*}(\wedge\Sigma)\wedge A\rangle.$ (5.13)

Note the invariance of $\langle\cdot,\cdot\rangle$ dualizes
$\chi^{*},\eta^{*}$ to the coadjoint
$\operatorname{ad}^{*},\mathfrak{ad}^{*}$, as opposed to the
$\Delta,\tilde{\Delta}$ maps as in Eq. (5.7). The BF-BB theory
$S_{\text{BFBB}}$ is of central interest, in particular in the trivial case
$t=\operatorname{id}$, as the 4D gravity action can be written as
$S_{\text{BFBB}}$ plus a geometric simplicity constraint [50]. We shall
explore this notion more in a following paper.

Similar to the case of the 1-gauge theory, we may consider the monster BF
theory $S_{\textbf{BF}}^{\prime\prime}$ given by a linear combination of the
alternative pairing with the canonical one

$\langle\langle\cdot,\cdot\rangle\rangle^{\prime\prime}=\alpha\langle\langle\cdot,\cdot\rangle\rangle+\beta\langle\langle\cdot,\cdot\rangle\rangle^{\prime},$

for which the real parameters $\alpha,\beta$ satisfy $\alpha^{2}+\beta^{2}\neq
0$. We would then have

$\displaystyle S_{\textbf{BF}}^{\prime\prime}[{\bf A},\boldsymbol{\Sigma}]$
$\displaystyle=$ $\displaystyle\alpha S_{\textbf{BF}}[{\bf
A},\boldsymbol{\Sigma}]+\beta S_{\textbf{BF}}^{\prime}[{\bf
A},\boldsymbol{\Sigma}]$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{M}\langle(\alpha
B+\beta\Sigma)\wedge\bar{F}\rangle+\langle(\alpha\Sigma+\beta
B)\wedge\bar{F}^{*}\rangle$ $\displaystyle-\frac{1}{2}\int_{M}\langle
B\wedge(\alpha t\Sigma+\beta t^{*}B)\rangle+\langle(\alpha t^{*}B+\beta
t\Sigma)\wedge\Sigma\rangle.$

##  6 The 2-graded classical $r$-matrix and the quadratic 2-Casimirs

##### Classical $r$-matrix and the Yang-Baxter equation.

Recall that a Lie bialgebra is characterized by a Lie algebra $\mathfrak{g}$
together with a cocycle $\psi:\mathfrak{g}\rightarrow\mathfrak{g}^{2\wedge}$
satisfying Eq. (2.2) [17, 16], which defines the dual Lie algebra
$\mathfrak{g}^{*}$. Moreover, this structure allows us to form the Manin
triple $\mathfrak{d}=\mathfrak{g}\bowtie\mathfrak{g}^{*}$, subject to the
compatibility condition Eq. (2.11).

If this cocycle is exact $\psi=dr$ for some $r\in\mathfrak{g}^{2\wedge}$, then
a particular algebra structure on the Manin triple is achieved. Such elements
$r\in\mathfrak{g}^{2\wedge}$ is called a triangular classical $r$-matrix [52].
The Jacobi identity then follows from the classical Yang-Baxter equation for
$r$:

$[X^{3\otimes},[r_{12},r_{13}]+[r_{13},r_{23}]+[r_{12},r_{23}]]=0$

for every $X^{3\otimes}=X\otimes 1\otimes 1+1\otimes X\otimes 1+1\otimes
1\otimes X\in\mathfrak{g}^{3\otimes}$. Here, the subscripts indicate the
tensor components of $\mathfrak{g}^{3\otimes}$ upon which $r$ acts.

More generally, we call $r\in\mathfrak{g}^{2\otimes}$ a quasitriangular
$r$-matrix if it satisfies

$[X^{2\otimes},r+\sigma(r)]=0,\qquad X^{2\otimes}=X\otimes 1+1\otimes X$

for each $X\in\mathfrak{g}$, in addition to the classical Yang-Baxter equation
Eq. (6), where $\sigma$ is a permutation of the tensor factors in
$\mathfrak{g}^{2\otimes}$. Now if we decompose $r=r^{\wedge}+r^{\odot}$ into
its skew-/symmetric components ($\wedge,\odot$ denote respectively the
anti/symmetrized tensors), $r^{\odot}$ must be a
$D(\operatorname{ad})$-invariant888Recall
$[X^{2\otimes},\cdot]=\operatorname{ad}_{X}\otimes
1+1\otimes\operatorname{ad}_{X}=D(\operatorname{ad}_{X})$., ie. a quadratic
Casimir [53].

By characterizing the quadratic Casimirs of $\mathfrak{g}$, we may then solve
for the skew-symmetric component $r^{\wedge}$ with the modified classical
Yang-Baxter equation [16, 53]

$\llbracket r^{\wedge},r^{\wedge}\rrbracket=\omega=-\llbracket
r^{\odot},r^{\odot}\rrbracket,$ (6.1)

where we have introduced the shorthand (the Schouten bracket) $\llbracket
r,r\rrbracket=[r_{12},r_{13}]+[r_{13},r_{23}]+[r_{12},r_{23}]$. This skew-
symmetric part $r^{\wedge}$ then generates the 1-cocycle $\psi$ on
$\mathfrak{g}$.

In the following, we shall study the 2-graded version of the above structure
by following the notions laid out in Ref. [1], in which a definition of the
"2-graded classical $r$-matrix" $R$ for a Lie bialgebra crossed-module/Lie
2-bialgebra was given. We shall pay particular attention to its symmetric
component $R^{\odot}$, and present new results characterizing these so-called
"quadratic 2-Casimirs". Some examples and applications are studied.

###  6.1 The 2-graded classical $r$-matrix

Similar to the Lie 1-algebra case, a (strict) Lie 2-bialgebra can be
classified in terms of a Lie algebra 2-cocycle $(\delta_{-1},\delta_{0})$
satisfying Eqs. (3.7), (3.9) [1, 32], which induces a dual Lie 2-algebra
$\mathfrak{g}^{*}[1]$. Moreover, the natural coadjoint representations Eq.
(3.11) gives rise to the 2-Drinfel’d double
$\mathfrak{d}=\mathfrak{g}\bowtie\mathfrak{g}^{*}[1]$.

Similar to the 1-algebra case, we begin by considering a 2-cocycle
$(\delta_{-1},\delta_{0})$ that is a "2-coboundary". We in particular focus on
the form of the 2-coboundary generated by certain elements
$r_{0}\in\mathfrak{g}_{0}\wedge\mathfrak{g}_{-1}$ and
$r_{-1}\in\mathfrak{g}_{-1}^{2\wedge}$. These elements $r_{-1},r_{0}$ form a
triangular 2-graded classical $r$-matrix [1]

$R=r_{0}-D(t)_{-1}r_{-1}=r_{0}-(t\otimes 1+1\otimes
t)r_{-1}\in\mathfrak{g}_{0}\wedge\mathfrak{g}_{-1},$

whence the 2-coboundary they form is given by

$\delta_{0}(X)=[X\otimes 1+1\otimes X,R],\qquad\delta_{-1}(Y)=[Y\otimes
1+1\otimes Y,R].$ (6.2)

Here we have used the graded Lie bracket $[\cdot,\cdot]=l_{2}$, which includes
$[\cdot,\cdot]|_{\mathfrak{g}_{0}}$ as well as the crossed-module action
$\rhd$ by Eq. (4.25).

More generally, suppose we are given
$r_{0}\in(\mathfrak{g}_{0}\otimes\mathfrak{g}_{-1})\oplus(\mathfrak{g}_{-1}\otimes\mathfrak{g}_{0}),r_{-1}\in\mathfrak{g}_{-1}^{2\otimes}$
(namely not necessarily skew-symmetric elements). It was proven that [1]

######  Theorem 6.1.

The 2-cochain $(\delta_{-1},\delta_{0})$ Eq. (6.5) makes
$(\mathfrak{g};\delta_{-1},\delta_{0})$ into a Lie 2-bialgebra iff for all
$W\in\mathfrak{g}_{0}\oplus\mathfrak{g}_{-1}$,

  * •

$[W^{2\otimes},R+\sigma(R)]=0$ where $\sigma$ is an exchange of tensor
factors, and

  * •

the 2-graded classical Yang-Baxter equations [1] are satisfied:

    1. 1.

$D(t)_{0}r_{0}=0$,

    2. 2.

$[W^{3\otimes},[R_{12},R_{13}]+[R_{13},R_{23}]+[R_{12},R_{23}]]=0$

where

$W^{3\otimes}=W\otimes 1\otimes 1+1\otimes W\otimes 1+1\otimes 1\otimes W.$

We call solutions
$R\in(\mathfrak{g}_{0}\otimes\mathfrak{g}_{-1})\oplus(\mathfrak{g}_{-1}\otimes\mathfrak{g}_{0})$
to the above criteria a quasitriangular 2-graded classical $r$-matrix.

In other words, the 2-graded classical Yang-Baxter equation implies the
2-cocycle condition [1] for 2-cochains $(\delta_{-1},\delta_{0})$ defined in
Eq. (6.2). If we write out the components

$r_{0}=\sum a\otimes b+\bar{a}\otimes\bar{b},\qquad r_{-1}=\sum c\otimes d$

for some $a,\bar{b}\in\mathfrak{g}_{0}$ and
$\bar{a},b,c,d\in\mathfrak{g}_{-1}$, then we have

$R=\sum\underbrace{(a\otimes b-tc\otimes
d)}_{\in\mathfrak{g}_{0}\otimes\mathfrak{g}_{-1}}+\underbrace{(\bar{a}\otimes\bar{b}-c\otimes
td)}_{\in\mathfrak{g}_{-1}\otimes\mathfrak{g}_{0}}\equiv\sum\rho+\bar{\rho}.$
(6.3)

By decomposing into skew-symmetric and symmetric parts
$R=R^{\wedge}+R^{\odot}$, we have $\bar{\rho}=-\sigma\rho$ in $R^{\wedge}$
while $\bar{\rho}=\sigma\rho$ in $R^{\odot}$ in terms of the components
defined in Eq. (6.3), where $\sigma$ permutes the tensor factors. In other
words, we have

$\displaystyle R^{\wedge}$ $\displaystyle=$
$\displaystyle\sum\rho-\sigma\rho=\sum a\wedge b-tc\wedge d-c\wedge td=\sum
a\wedge b-D(t)_{-1}(c\wedge d),$ $\displaystyle R^{\odot}$ $\displaystyle=$
$\displaystyle\sum\rho+\sigma\rho=\sum a\odot b-tc\odot d-c\odot d=\sum a\odot
b-D(t)_{-1}(c\odot d),$

If we write, using the graded Schouten bracket
$\llbracket\cdot,\cdot\rrbracket$ [28, 1],

$\Omega=-\llbracket
R^{\odot},R^{\odot}\rrbracket=-[R_{12}^{\odot},R_{13}^{\odot}]+[R_{13}^{\odot},R_{23}^{\odot}]+[R_{12}^{\odot},R_{23}^{\odot}],$

then the skew-symmetric part $R^{\wedge}$ satisfies the modified 2-graded
classical Yang-Baxter equation

$\llbracket R^{\wedge},R^{\wedge}\rrbracket=\Omega.$ (6.4)

This is an equivalent way of writing Eq. 2. in Theorem 6.1.

As in the 1-algebra case, the symmetric component
$R^{\odot}\in\mathfrak{g}_{0}\odot\mathfrak{g}_{-1}$ of $R$ governs the form
of Eq. (6.4), while the skew-symmetric component
$R^{\wedge}\in\mathfrak{g}_{0}\wedge\mathfrak{g}_{-1}$ contributes to the
2-coboundary Eq. (6.2). Recalling $D(t)_{-1}=t\otimes 1+1\otimes t$, the
2-coboundary Eq. (6.2) is given explicitly by

$\displaystyle\delta_{0}(X)$ $\displaystyle=$ $\displaystyle\sum[X,a]\wedge
b+a\wedge(X\rhd b)$ $\displaystyle-\leavevmode\nobreak\ \sum[X,tc]\wedge
d+tc\wedge(X\rhd d)+(c\leftrightarrow d),$ $\displaystyle\delta_{-1}(Y)$
$\displaystyle=$ $\displaystyle\sum c\wedge(td\rhd Y)+(c\leftrightarrow
d)-\sum(a\rhd Y)\wedge b,$ (6.5)

where $c\leftrightarrow d$ indicates a swap of the elements $c,d$ from the
previous term.

One particular solution for the decomposition $R=R^{\wedge}+R^{\odot}$ is if
the two quantities $r_{0},r_{-1}$ can themselves be decomposed into skew-
symmetric and symmetric parts:

$\displaystyle r_{0}=r_{0}^{\wedge}+r_{0}^{\odot},$ $\displaystyle
r_{0}^{\wedge}\in\mathfrak{g}_{0}\wedge\mathfrak{g}_{-1},\qquad
r_{-1}^{\wedge}\in\mathfrak{g}_{-1}^{2\wedge},$ $\displaystyle
r_{-1}=r_{-1}^{\wedge}+r_{-1}^{\odot},$ $\displaystyle
r_{0}^{\odot}\in\mathfrak{g}_{0}\odot\mathfrak{g}_{-1},\qquad
r_{-1}^{\odot}\in\mathfrak{g}_{-1}^{2\odot}.$

The 2-graded $r$-matrix then reads

$\displaystyle R^{\wedge}$ $\displaystyle=$ $\displaystyle
r^{\wedge}_{0}-D(t)_{-1}r_{-1}^{\wedge}=\sum a\wedge b-D(t)_{-1}(c\wedge d),$
$\displaystyle R^{\odot}$ $\displaystyle=$ $\displaystyle
r^{\odot}_{0}-D(t)_{-1}r_{-1}^{\odot}=\sum a\odot b-D(t)_{-1}(c\odot d).$
(6.6)

We stress that this may not be the most general form of the decomposition
$R=R^{\wedge}+R^{\odot}$!

Due to the first condition in Theorem 6.1, we see that the symmetric
contribution $R^{\odot}$ must be $D(\operatorname{ad})$-invariant, where
$D(\operatorname{ad})=\operatorname{ad}\otimes 1+1\otimes\operatorname{ad}$ is
defined in terms of the adjoint representation $\operatorname{ad}$ in Eq.
(3.3). In the 1-algebra case, this leads to $r^{\odot}$ being a quadratic
Casimir, and so we shall correspondingly call $R^{\odot}$ a quadratic
2-Casimir of the crossed-module $\mathfrak{g}$.

###  6.2 Quadratic 2-Casimirs

Recall that quadratic 2-Casimirs, ie. the symmetric component $R^{\odot}$ of
$R$, controls the form of the modified 2-graded classical Yang-Baxter equation
Eq. (6.4). It is therefore very important towards understanding 2-graded
classical $r$-matrices, and we shall dedicate the remainder of this section to
characterizing it.

##### Characterization of the Quadratic 2-Casimirs.

First, we note that $D(\operatorname{ad})=\operatorname{ad}\otimes
1+1\otimes\operatorname{ad}$ is the derivation on the tensor product
$\mathfrak{g}^{2\otimes}$ associated to the strict adjoint representation
$\operatorname{ad}$ Eq. (3.3). The tensor product $\mathfrak{g}^{2\otimes}$
has the structure of a three-term graded complex, as we can extend $t$ to
$D(t)_{-1,0}$ on the tensor product $\mathfrak{g}^{2\otimes}$.

$\mathfrak{g}_{-1}^{2\otimes}\xrightarrow{D(t)_{-1}}(\mathfrak{g}_{-1}\otimes\mathfrak{g}_{0})\oplus(\mathfrak{g}_{0}\otimes\mathfrak{g}_{-1})\xrightarrow{D(t)_{0}}\mathfrak{g}_{0}^{2\otimes},$
(6.7)

where

$D(t)_{-1}(Y\otimes Y^{\prime})=tY\otimes Y^{\prime}+Y\otimes
tY^{\prime},\qquad D(t)_{0}(Y\otimes X+X^{\prime}\otimes Y^{\prime})=tY\otimes
X-X^{\prime}\otimes tY^{\prime}$

for each $Y,Y^{\prime}\in\mathfrak{g}_{-1},X,X^{\prime}\in\mathfrak{g}_{0}$.

Let us thus mainly focus on the meaning of "$D(\operatorname{ad})$-invariance"
in the definition of a quadratic 2-Casimir $R^{\odot}$ by examining in detail
the action of $D(\operatorname{ad})$ on
$(\mathfrak{g}_{0}\otimes\mathfrak{g}_{-1})\oplus(\mathfrak{g}_{-1}\otimes\mathfrak{g}_{0})$.
Let $Y\otimes X+X^{\prime}\otimes Y^{\prime}$ denote an arbitrary element in
this sector, and let $X^{\prime\prime}+Y^{\prime\prime}\in\mathfrak{g}$, then

$\displaystyle
D(\operatorname{ad})_{X^{\prime\prime}+Y^{\prime\prime}}(Y\otimes X)$
$\displaystyle=$ $\displaystyle\underbrace{(X^{\prime\prime}\rhd Y)\otimes
X+Y\otimes[X^{\prime\prime},X]}_{\mathfrak{g}_{-1}\otimes\mathfrak{g}_{0}}+\underbrace{Y\otimes(X\rhd
Y^{\prime\prime})}_{\mathfrak{g}_{-1}^{2\otimes}},$ $\displaystyle
D(\operatorname{ad})_{X^{\prime\prime}+Y^{\prime\prime}}(X^{\prime}\otimes
Y^{\prime})$ $\displaystyle=$
$\displaystyle\underbrace{[X^{\prime\prime},X^{\prime}]\otimes
Y^{\prime}+X^{\prime}\otimes(X^{\prime\prime}\rhd
Y^{\prime})}_{\mathfrak{g}_{0}\otimes\mathfrak{g}_{-1}}+\underbrace{(X^{\prime}\rhd
Y^{\prime\prime})\otimes Y^{\prime}}_{\mathfrak{g}_{-1}^{2\otimes}}.$

Now if we take the symmetric tensor $Y\odot X=Y\otimes X+X\otimes Y$ and sum
the above contributions, then the $D(\operatorname{ad})$-invariance condition
$D(\operatorname{ad})_{X^{\prime\prime}+Y^{\prime\prime}}(Y\odot X)=0$ gives
rise to the following equations

$\displaystyle[X^{\prime\prime},X]\odot Y+X\odot(X^{\prime\prime}\rhd
Y)=0,\qquad(X\rhd Y^{\prime\prime})\odot Y=0$

for all $X^{\prime\prime}+Y^{\prime\prime}\in\mathfrak{g}$. The space of
solutions is the subspace

$\Theta_{\rhd}=\\{X\odot
Y\in\mathfrak{g}_{0}\odot\mathfrak{g}_{-1}\mid\operatorname{ad}X\odot
Y+X\odot\chi Y=0,\leavevmode\nobreak\ \chi_{X}=0\\},$

where we recall $\chi=\rhd$ is the crossed-module action. Now the condition
$D(t)_{0}R=D(t)_{0}r_{0}=0$ in Theorem 6.1 constrains $R^{\odot}$ to lie in
$\operatorname{ker}D(t)_{0}$, whence we assemble the elements

$a\odot
b\in\operatorname{2Cas}_{\mathfrak{g}}[0]\equiv\Theta_{\rhd}\cap\operatorname{ker}D(t)_{0}$

as the quadratic 2-Casimirs of $\mathfrak{g}$.

On the other hand, for $Y\odot Y^{\prime}\in\mathfrak{g}_{-1}^{2\odot}$ we
have

$\displaystyle
D(\operatorname{ad})_{X^{\prime\prime}+Y^{\prime\prime}}(D(t)_{-1}(Y\odot
Y^{\prime}))$ $\displaystyle=$ $\displaystyle(X^{\prime\prime}\rhd Y)\odot
tY^{\prime}+Y\odot[X^{\prime\prime},tY^{\prime}]+Y\odot(tY^{\prime}\rhd
Y^{\prime\prime})$ $\displaystyle+(X^{\prime\prime}\rhd Y^{\prime})\odot
tY+Y^{\prime}\odot[X^{\prime\prime},tY]+Y^{\prime}\odot(tY\rhd
Y^{\prime\prime})$ $\displaystyle=$ $\displaystyle(X^{\prime\prime}\rhd
Y)\odot tY^{\prime}+Y\odot t(X^{\prime\prime}\rhd
Y^{\prime})+Y\odot[Y^{\prime},Y^{\prime\prime}]$
$\displaystyle+(X^{\prime\prime}\rhd Y^{\prime})\odot tY+Y^{\prime}\odot
t(X^{\prime\prime}\rhd Y)+Y^{\prime}\odot[Y,Y^{\prime\prime}]$
$\displaystyle=$ $\displaystyle D(t)_{-1}((X^{\prime\prime}\rhd Y)\odot
Y^{\prime}+(X^{\prime\prime}\rhd Y^{\prime})\odot Y)$
$\displaystyle-([Y^{\prime\prime},Y^{\prime}]\odot
Y+Y^{\prime}\odot[Y^{\prime\prime},Y]),$

where we have used the Pfeiffer identities Eq. (3.1). Note $D(t)_{-1}=t\otimes
1+1\otimes t$ on $\mathfrak{g}_{-1}^{2\odot}$, we define the subspaces

$\displaystyle\Gamma_{t}$ $\displaystyle=$ $\displaystyle\\{Y\odot
Y^{\prime}\in\mathfrak{g}_{-1}^{2\odot}\mid\chi Y\odot Y^{\prime}+Y\odot\chi
Y^{\prime}\in\operatorname{ker}D(t)_{-1}\\},$
$\displaystyle\operatorname{Cas}_{\mathfrak{g}_{-1}}$ $\displaystyle=$
$\displaystyle\\{Y\odot
Y^{\prime}\in\mathfrak{g}_{-1}^{2\odot}\mid\operatorname{ad}Y\odot
Y^{\prime}+Y\odot\operatorname{ad}Y^{\prime}=0\\},$

we see that the space of solutions is given by the intersection

$c\odot
d\in\operatorname{2Cas}_{\mathfrak{g}}[-1]\equiv\Gamma_{t}\cap\operatorname{Cas}_{\mathfrak{g}_{-1}}.$

Recall the adjoint action $\operatorname{ad}$ on $\mathfrak{g}_{-1}$ is
defined via the second Pfeiffer identity. If each term in
$D(\operatorname{ad})_{X^{\prime\prime}+Y^{\prime\prime}}(R^{\odot})=0$
vanishes, then we obtain the following characterization of quadratic
2-Casimirs:

$R^{\odot}=\sum a\odot b+D(t)_{-1}(c\odot
d)\in\operatorname{2Cas}_{\mathfrak{g}}[0]\oplus
D(t)_{-1}(\operatorname{2Cas}_{\mathfrak{g}}[-1])\equiv\operatorname{2Cas}_{\mathfrak{g}},$

provided the decomposition Eq. (6.6) holds.

###  6.3 Explicit constructions of the quadratic 2-Casimirs for some crossed-
modules

Recall from Example 3.1 that we can obtain a Lie 2-algebra from the data of a
Lie algebra in two different ways: the canonical embedding and the suspension
embedding. Here, let us compare the characterization of the quadratic Casimirs
for the 1-algebra with that of the quadratic 2-Casimirs. We will also discuss
the quadratic 2-Casimirs in the case of the 2-Dinfel’d double.

####  6.3.1 2-Casimirs of the trivial canonical 2-algebra.

Consider the trivial 2-algebra $\operatorname{id}_{\mathfrak{l}}$ associated
to a Lie algebra $\mathfrak{l}$. We have
$(\operatorname{id}_{\mathfrak{l}})_{-1}=(\operatorname{id}_{\mathfrak{l}})_{0}=\mathfrak{l}$,
and the crossed-module map $t=\operatorname{id}$ is the identity, hence

$\displaystyle D(t)_{-1}=1,$ $\displaystyle\operatorname{ker}D(t)_{-1}=0,$
$\displaystyle D(t)_{0}=0,$
$\displaystyle\operatorname{ker}D(t)_{0}=\mathfrak{l}\odot\mathfrak{l}.$

Thus, we have $\operatorname{ad}=\chi$ and
$\Gamma_{t=1}=\operatorname{Cas}_{(\operatorname{id}_{\mathfrak{l}})_{-1}}$,
whence
$\operatorname{2Cas}_{\operatorname{id}_{\mathfrak{l}}}[-1]=\Gamma_{t=1}=\operatorname{Cas}_{\mathfrak{l}}$.

On the other hand, the condition $\chi_{a}=\operatorname{ad}_{a}=0$ implies
that $a\in\operatorname{ker}\operatorname{ad}=Z(\mathfrak{l})$ lies in the
centre of $\mathfrak{l}$. Let us take $\mathfrak{l}$ semisimple, then all
Abelian ideals such as $Z(\mathfrak{l})$ vanish, whence no contributions occur
from $\operatorname{2Cas}_{\operatorname{id}_{\mathfrak{l}}}[0]$. The only
term that survives is therefore

$R^{\odot}=-\sum D(t)_{-1}(c\odot d)=-\sum c\odot
d\in\Gamma_{t=1}=\operatorname{Cas}_{\mathfrak{g}};$

namely the quadratic 2-Casimirs
$\operatorname{2Cas}_{\operatorname{id}_{\mathfrak{l}}}$ of
$\operatorname{id}_{\mathfrak{l}}$ coincide with those
$\operatorname{Cas}_{\mathfrak{l}}$ of $\mathfrak{l}$.

In this case, it is clear that the Schouten brackets for $\mathfrak{l}$ and
$\operatorname{id}_{\mathfrak{l}}$ coincide, hence the space of solutions of
Eq. (6.1) is isomorphic to that of Eq. (6.4). The correspondence is just
$r\mapsto R$ [1].

######  Example 6.1.

Consider the simple Lie algebra $\mathfrak{l}=\mathfrak{su}(2)$ generated by
the basis $J_{1},J_{2},J_{3}$ such that

$[J_{i},J_{j}]=\varepsilon^{k}_{ij}J_{k},\qquad 1\leq i,j,k\leq 3,$

where $\varepsilon$ are the Levi-Civita symbols. The skew-symmetric component
$r^{\wedge}=J_{1}\wedge J_{2}\in\mathfrak{su}(2)^{2\wedge}$ solves the
modified classical Yang-Baxter equation Eq. (6.1) associated to the following
Casimir $r^{\odot}=-J_{i}\odot J_{i}\in\operatorname{Cas}_{\mathfrak{su}(2)}$,
and generates the cocycle

$\psi(J_{1})=J_{1}\wedge J_{3},\qquad\psi(J_{2})=J_{2}\wedge
J_{3},\qquad\psi(J_{3})=0.$

This gives a bialgebra $(\mathfrak{su}(2),\psi)$ whose Drinfel’d double
$\mathfrak{d}=\mathfrak{su}(2)\bowtie\mathfrak{su}(2)^{*}=\mathfrak{su}(2)\bowtie\mathfrak{an}_{2}$
is known as the classical double of $\mathfrak{su}(2)$ [7, 54]. It is a
deformation of the 3D Poincaré algebra
$\mathfrak{iso}(3)=\mathfrak{su}(2)\ltimes\mathbb{R}^{3}$.

Now let us consider the trivial canonical 2-algebra

$\operatorname{id}_{\mathfrak{su}(2)}=(\mathfrak{su}(2)\xrightarrow{\operatorname{id}}\mathfrak{su}(2),\operatorname{ad},[\cdot,\cdot]).$

The element $R^{\odot}_{0}=-r^{\odot}=J_{i}\odot J_{i}$ defines an admissible
2-Casimir of $\operatorname{id}_{\mathfrak{su}(2)}$, hence the choice
$R^{\wedge}=r_{0}^{\wedge}=J_{1}\wedge J_{2}$ solves the 2-graded modified
classical Yang-Baxter equation Eq. (6.4). This is consistent with the
observation made in Example 4.1, that bialgebra crossed-module
$(\operatorname{id}_{\mathfrak{su}(2)};\delta_{-1},\delta_{0})$ is then
equivalent to two copies (ie. the canonical embedding) of the bialgebra
$(\mathfrak{su}(2),\psi)$.

####  6.3.2 2-Casimirs of the skeletal suspension 2-algebra.

Consider the semidirect product $V\rtimes\mathfrak{u}$, where $V$ is Abelian.
It is known that, for $\mathfrak{u}$ semisimple and
$\operatorname{dim}\mathfrak{u}=\operatorname{dim}V$, the Casimirs
$\operatorname{Cas}_{V\rtimes\mathfrak{u}}$ span a 2-dimensional space [53].
Thus, the Killing form $K$ on $V\rtimes\mathfrak{u}$ can be decomposed into
two parts999This decomposition will have nothing to do with Proposition 6.1,
even though we use a similar notation., $K=K_{0}+K_{-1}$, given by

$K(X+Y,X^{\prime}+Y^{\prime})=K_{0}(X,X^{\prime})+K_{-1}(X,Y^{\prime})+K_{-1}(X^{\prime},Y)+K_{0}(Y,Y^{\prime})$

for each $X,X^{\prime}\in\mathfrak{u}$ and $Y,Y^{\prime}\in V$. Indeed,
$K_{-1}$ can be non-degenerate only if
$\operatorname{dim}V=\operatorname{dim}\mathfrak{u}$.

These components $K_{-1},K_{0}$ satisfy the following invariance properties

$0=\begin{cases}K_{0}(\operatorname{ad}_{X}X^{\prime},X^{\prime\prime})+K_{0}(X^{\prime},\operatorname{ad}_{X}X^{\prime\prime})\\\
K_{0}(X\rhd Y,Y^{\prime})+K_{0}(Y,X\rhd Y^{\prime})\end{cases},\qquad
0=K_{-1}(\operatorname{ad}_{X}X^{\prime},Y)+K_{-1}(X^{\prime},X\rhd Y)$

for each $X,X^{\prime},X^{\prime\prime}\in\mathfrak{u},Y,Y^{\prime}\in V$.
This gives rise to two linearly independent Casimirs [45, 53]

$X^{i}\odot X_{i}+Y^{i}\odot Y_{i},\qquad X^{i}\odot Y_{i},$

where $\\{X_{i}+Y_{i}\\}_{i}$ is a basis of $V\rtimes\mathfrak{u}$ and
$\\{X^{i}+Y^{i}\\}_{i}$ the corresponding dual basis induced by $K$.

Now let $\mathfrak{g}^{0}$ denote the skeletal 2-algebra associated to the
semidirect product $V\rtimes\mathfrak{u}$. Since $t=0$, we have
$D(t)_{-1,0}=0$ and both of these maps have full kernel. Moreover, $V$ is
Abelian, so we have
$\Gamma_{t=0}=\mathfrak{g}_{-1}^{2\odot}=\operatorname{Cas}_{\mathfrak{g}_{-1}}$.
However, since $D(t)_{-1}=0$, the only component of $R^{\odot}$ comes from

$\operatorname{2Cas}_{\mathfrak{g}^{0}}[0]=\Theta_{\rhd}.$

This space is spanned by the component $K_{-1}$ of the Killing form $K$.
Interestingly, the component $K_{0}$ is constrained to live in the space
$\Gamma_{t=0}$, but fails to contribute to $R^{\odot}$ as $D(t)_{-1}=0$.

Since the 2-algebra structure on $\mathfrak{g}^{0}$ sums to the algebra
structure on $V\rtimes\mathfrak{u}$ by Eqs. (3.4), (4.25), the Schouten
brackets in each case coincide — one is the sum of the other [1]. However, the
fact that $D(t)=0$ means that no contributions from
$\mathfrak{g}_{-1}^{2\otimes}$ arise in $R$; in other words, the solutions for
Eq. (6.4) constitute those of Eq. (6.1) lying in $(\mathfrak{u}\otimes
V)\oplus(V\otimes\mathfrak{u})$. The correspondence is $r\mapsto r_{0}$.

######  Example 6.2.

Consider the 3D Poincaré algebra
$\mathfrak{iso}(3)=\mathbb{R}^{3}\rtimes\mathfrak{so}(3)$, where the action
$\mathfrak{so}(3)\rhd\mathbb{R}^{3}$ is the vector representation. Suppose
$P_{\mu},J_{\mu},0\leq\mu\leq 2$ respectively span
$\mathbb{R}^{3},\mathfrak{so}(3)$, the following canonical Casimir

$r^{\odot}(P,J)=K_{-1}(P,J)=P_{\mu}\odot J^{\mu}$

gives rise via Eq. (6.4) to the corresponding skew-symmetric part of the
classical $r$-matrix

$r^{\wedge}=2(P_{+}\wedge J_{-}-P_{-}\wedge
J_{+}),\qquad\begin{cases}P_{\pm}=\frac{1}{2}(P_{1}\pm iP_{2})\\\
J_{\pm}=\frac{1}{2}(J_{1}\pm iJ_{2})\end{cases}$

which generates the following 1-cocycle $\psi$ on $\mathfrak{iso}(1,2)$:

$\displaystyle\psi(P_{0})=0,$ $\displaystyle\psi(P_{\pm})=iP_{\pm}\wedge
P_{0},$ $\displaystyle\psi(J_{0})=0,$
$\displaystyle\psi(J_{\pm})=i(J_{\pm}\wedge P_{0}+P_{\pm}\wedge J_{0}).$

The bialgebra $(\mathfrak{iso}(3);\psi)$ equipped with this 1-cocycle $\psi$
is known as the 3D $\kappa$-Poincaré group [55].

######  Remark 6.1.

By appending an additional Casimir $\xi K_{0}(P,P)=\frac{\xi}{2}P_{\mu}\odot
P^{\mu}$, where $\xi\in\mathbb{R}$, one obtains a one-parameter family of Hopf
algebras that deforms the $\kappa$-Poincaré algebra. However, this additional
term is not a Drinfel’d twist [53]; moreover, the 1-cocycle $\psi$ would no
longer be coisotropic
$\psi(\mathfrak{so}(3))\not\subset\mathfrak{so}(3)\wedge\mathbb{R}^{3}$.
Interestingly, this deformation is forbidden to appear in the 2-graded
classical $r$-matrix $R$.

Now let us consider the skeletal suspension 2-algebra

$\mathfrak{so}(3)^{0}=(\mathbb{R}^{3}\xrightarrow{0}\mathfrak{so}(3),\rhd,[\cdot,\cdot]),$

with $\rhd$ the vector representation. Recall, in case $t=0$, the quadratic
2-Casimir
$R^{\odot}\in\operatorname{2Cas}_{\mathfrak{so}(3)^{0}}=\Theta_{\rhd}$ is
given by the bilinear form $K_{-1}$. We recover both the Casimir
$R^{\odot}=r^{\odot}$ as well as the classical $r$-matrix
$R^{\wedge}=r^{\wedge}$ of the 3D Pincaré algebra $\mathfrak{iso}(3)$ [1]. The
1-cocycle $\psi$ is in fact coisotropic
$\psi(\mathfrak{so}(3))\subset\mathfrak{so}(3)\wedge\mathbb{R}^{3}$ [54],
hence Proposition 3.1 states that $\psi=\delta_{-1}+\delta_{0}$ defines a
2-cocycle

$\displaystyle\delta_{0}(P_{0})=0,$
$\displaystyle\delta_{0}(P_{\pm})=iP_{\pm}\wedge P_{0},$
$\displaystyle\delta_{-1}(J_{0})=0,$
$\displaystyle\delta_{-1}(J_{\pm})=i(J_{\pm}\wedge P_{0}-J_{0}\wedge P_{\pm})$

on the suspension 2-algebra $\mathfrak{so}(3)^{0}$. In other words, the
2-bialgebra $(\mathfrak{so}(3)^{0};\delta_{-1},\delta_{0})$ is equivalent to
the 3D $\kappa$-Poincaré algebra.

The above construction holds for the Poincaré algebra $\mathfrak{iso}(n)$ in
any dimension. This result was already guessed in [56]. Dimension $n=3$ is
special, as the isotropy subgroup $\mathfrak{so}(3)$ has the same dimension as
its quotient $\mathfrak{iso}(3)/\mathfrak{so}(3)\cong\mathbb{R}^{3}$. This
means that deformations of $\mathfrak{iso}(3)$ can be obtained from Drinfel’d
double structures on $\mathfrak{so}(3)$ [54], as demonstrated above.

####  6.3.3 2-Casimirs of the 2-Drinfel’d double

Once one has constructed the quadratic 2-Casimirs for $\mathfrak{g}$, we may
similarly construct the quadratic 2-Casimirs
$\operatorname{2Cas}_{\mathfrak{g}^{*}[1]}$ of the dual crossed-module
$\mathfrak{g}^{*}[1]$. One may notice that elements in
$\operatorname{2Cas}_{\mathfrak{g}}\subset\mathfrak{g}_{0}\odot\mathfrak{g}_{-1}$
and
$\operatorname{2Cas}_{\mathfrak{g}^{*}[1]}\subset\mathfrak{g}_{-1}^{*}\odot\mathfrak{g}_{0}^{*}$
satisfy invariance properties constituting the first rows of Eqs. (5.10).
Moreover, the conditions

$D(t)_{0}\operatorname{2Cas}_{\mathfrak{g}}=0,\qquad
D(t^{*})_{0}\operatorname{2Cas}_{\mathfrak{g}^{*}[1]}=0$

imply that $t,t^{*}$ are symmetric with respect to the basis diagonalizing the
quadratic 2-Casimir elements. This is nothing but the duality condition Eq.
(5.12).

On the other hand, the 2-Manin triple
$\mathfrak{d}=\mathfrak{g}\bowtie\mathfrak{g}^{*}[1]$ itself hosts a crossed-
module structure given by Eq. (4.23). This allows us to use the above
arguments to also construct the quadratic 2-Casimirs
$\operatorname{2Cas}_{\mathfrak{d}}\subset\mathfrak{d}_{0}\odot\mathfrak{d}_{-1}\cong(\mathfrak{g}_{0}\oplus\mathfrak{g}_{-1}^{*})\odot(\mathfrak{g}_{-1}\oplus\mathfrak{g}_{0}^{*})$
of the 2-Manin triple. Here, the adjoint and coadjoint actions Eq. (3.11) of
$\mathfrak{g}$ and $\mathfrak{g}^{*}[1]$ on each other both participate in the
definition of the operator $D(\operatorname{ad})$.

After a lengthy calculation, it can be explicitly shown that quadratic
2-Casimirs of $\mathfrak{d}$ satisfy the following invariance properties:

$\displaystyle(\operatorname{ad}_{X^{\prime}}X+\chi^{*}_{X^{\prime}}f)\odot(Y+g)+(X+f)\odot(X^{\prime}\rhd
Y+\operatorname{ad}_{X^{\prime}}^{*}g)$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle(\eta_{f^{\prime}}^{*}X+\mathfrak{ad}_{f^{\prime}}f)\odot(Y+g)+(X+f)\odot(\mathfrak{ad}_{f^{\prime}}^{*}Y+f^{\prime}\rhd^{*}g)$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle(X+f)\odot(\tilde{\Delta}_{g}(X^{\prime})+\Delta_{Y}(f^{\prime}))+(X^{\prime}+f^{\prime})\odot(\tilde{\Delta}_{g}(X)+\Delta_{Y}(f))$
$\displaystyle=$ $\displaystyle 0$ (6.8)

for each
$X,X^{\prime}\in\mathfrak{g}_{0},Y\in\mathfrak{g}_{-1},f,f^{\prime}\in\mathfrak{g}_{-1}^{*},g\in\mathfrak{g}_{0}^{*}$.
By expanding each row of Eq. (6.8) out, we see that these invariance
properties encompass those of both the natural evaluation pairing
$\langle\langle\cdot,\cdot\rangle\rangle$ of Eq. (3.6), as well as the
grading-inhomogeneous alternative pairing
$\langle\langle\cdot,\cdot\rangle\rangle^{\prime}$ given in Eq. (5.9). For
instance, expanding out the first equation yields

$\displaystyle 0$ $\displaystyle=$
$\displaystyle(\operatorname{ad}_{X^{\prime}}X\odot
g+X\odot\operatorname{ad}_{X^{\prime}}^{*}g)+(\chi^{*}_{X^{\prime}}f\odot
Y+f\odot X^{\prime}\rhd Y)$
$\displaystyle+(\operatorname{ad}_{X^{\prime}}X\odot Y+X\odot X^{\prime}\rhd
Y)+(\chi^{*}_{X^{\prime}}f\odot g+f\odot\operatorname{ad}_{X^{\prime}}^{*}g)$
$\displaystyle=$
$\displaystyle(g,\operatorname{ad}_{X^{\prime}}X)+(\operatorname{ad}^{*}_{X^{\prime}}g,X)+(\chi^{*}_{X^{\prime}}f,Y)+(f,\chi_{X^{\prime}}Y)$
$\displaystyle+\langle\operatorname{ad}_{X^{\prime}}X,Y\rangle+\langle
X,\chi_{X^{\prime}}Y\rangle+\langle\chi^{*}_{X^{\prime}}f,g\rangle+\langle
f,\operatorname{ad}_{X^{\prime}}^{*}g\rangle,$

where $(\cdot,\cdot)$ and $\langle\cdot,\cdot\rangle$ are the components of
the pairings Eqs. (3.6), (5.9), respectively. Similar computations can be
carried out for the other two equations. Moreover, the condition that
$D(T)_{0}\operatorname{2Cas}_{\mathfrak{d}}=0$ implies the duality condition
Eq. (5.11).

In other words, we have the following result.

######  Proposition 6.1.

Quadratic 2-Casimirs $\operatorname{2Cas}_{\mathfrak{d}}$ of the 2-Manin
triple $\mathfrak{d}$ consist of the grading-inhomogeneous pairings Eqs.
(3.6), (5.9).

Note that Eq. (6.8) follows directly from the
$D(\operatorname{ad})$-invariance of $R^{\odot}$ itself as an element of
$\mathfrak{d}_{0}\odot\mathfrak{d}_{-1}$. Assumptions about its particular
form, such as Eq. (6.6), are not necessary. In other words, Proposition 6.1 is
a general result that applies to any 2-Drinfel’d double as defined here and in
the literature [1, 32]. As we have seen previously, it is only for these
pairings that the BF theory Eq. (5.6) is non-trivial. Here, we have provided a
proof that these are in fact the only ones on $\mathfrak{d}$.

######  Example 6.3.

Consider the case $t=1$ and the canonical 2-Manin triple
$\mathfrak{1}_{\mathfrak{l}}$ associated to the bialgebra
$(\mathfrak{l};\psi)$. Recall from Example 4.1 that we have
$\mathfrak{1}_{\mathfrak{l}}=\operatorname{id}_{\mathfrak{d}}$ as crossed-
modules, where $\mathfrak{d}=\mathfrak{l}\bowtie\mathfrak{l}^{*}$ is the
Drinfel’d double.

First, as $t=1$ in the crossed-module $\operatorname{id}_{\mathfrak{d}}$,
$\operatorname{2Cas}_{\operatorname{id}_{\mathfrak{d}}}$ coincides precisely
with the usual quadratic Casimirs $\operatorname{Cas}_{\mathfrak{d}}$ (as
explained in more detail below). As such,
$\operatorname{2Cas}_{\operatorname{id}_{\mathfrak{d}}}$ consist of non-
degenerate bilinear forms $K\in\mathfrak{d}\odot\mathfrak{d}$, satisfying the
usual invariance properties.

On the other hand, the quadratic 2-Casimirs of $\mathfrak{1}_{\mathfrak{l}}$
as a 2-Manin triple consists of the two grading-inhomogeneous pairings Eqs.
(3.6), (5.9), given by
$K_{0}\in(\mathfrak{l}\odot\mathfrak{l})\oplus(\mathfrak{l}^{*}\odot\mathfrak{l}^{*})$
and $K^{0}\in\mathfrak{l}\odot\mathfrak{l}^{*}$, as shown in Proposition 6.1.
The fact $\mathfrak{1}_{\mathfrak{l}}=\operatorname{id}_{\mathfrak{d}}$ then
implies that we have a decomposition of
$\operatorname{2Cas}_{\mathfrak{1}_{\mathfrak{g}}}$ given by

$K=K_{0}+K^{0}.$

More precisely, if $Z=(X+g)\in\mathfrak{d}$ with $X\in\mathfrak{g}$ and
$g\in\mathfrak{g}^{*}$, then we have

$K(Z,Z^{\prime})=K_{0}(X,X^{\prime})+K_{0}(g,g^{\prime})+K^{0}(g,X^{\prime})+K^{0}(X,g^{\prime}).$

As a consequence, the BF theory Eq. (5.6) reads

$\displaystyle S_{\textbf{BF}}[{\bf A},\boldsymbol{\Sigma}]$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{M}K_{0}(\Sigma\wedge\bar{F})+K_{0}(B\wedge\bar{F}^{*})+K^{0}(\Sigma\wedge\bar{F}^{*})+K^{0}(B\wedge\bar{F})$
$\displaystyle+\frac{1}{2}\int_{M}K_{0}(\Sigma\wedge\Sigma)+K_{0}(B\wedge
B)+K^{0}(\Sigma\wedge B)+K^{0}(B\wedge\Sigma),$

where $\boldsymbol{\Sigma}=\Sigma+B$ is valued in
$(\operatorname{id}_{\mathfrak{d}})_{-1}=\mathfrak{l}\bowtie\mathfrak{l}^{*}$
and ${\bf A}=A+C$ is valued in
$(\operatorname{id}_{\mathfrak{d}})_{0}=\mathfrak{l}\bowtie\mathfrak{l}^{*}$.
This means that the BF theory $S_{\textbf{BF}}[{\bf A},\boldsymbol{\Sigma}]$
based on $\mathfrak{d}$ naturally hosts fake-flatness equations of motion in
each $\mathfrak{g},\mathfrak{g}^{*}[1]$-sectors that mixes the 2-connections
$\Sigma,B$. This is an artifact of the double structure of
$\mathfrak{d}=\mathfrak{g}\bowtie\mathfrak{g}^{*}[1]$; these "mixed" terms
must appear in order to preserve the invariance of the action.

As a concrete example, consider the classical double
$\mathfrak{d}=\mathfrak{su}(2)\bowtie\mathfrak{an}_{2}$. Suppose
$\\{\mathcal{J}^{\mu}\\}_{\mu}$ is a basis for $\mathfrak{d}$ diagonalizing
the non-degenerate invariant bilinear form
$K\in\operatorname{2Cas}_{\operatorname{id}_{\mathfrak{d}}}=\operatorname{Cas}_{\mathfrak{d}}$.
By writing $\mathcal{J}^{\mu}=(J^{i},\tau^{j})$ in terms of its sectors
$J^{i}\in\mathfrak{su}(2),\tau^{j}\in\mathfrak{an}_{2}$, we see that

$\delta_{\mu\nu}=K(\mathcal{J}_{\mu},\mathcal{J}_{\nu})=K_{0}(J_{i},J_{j})+K_{0}(\tau_{i},\tau_{j})+K^{0}(J_{i},\tau_{j})+K^{0}(J_{i},\tau_{j}),$

However, it is not in general possible to pick the bases
$\\{J_{i}\\}_{i},\\{\tau_{i}\\}_{i}$ that diagonalize the two bilinear forms
$K_{0},K^{0}$ simultaneously — that is, unless they commute. Generally
speaking, it is important to keep in mind that a diagonal bilinear form $K$
can decompose into components that are non-diagonal.

##  7 Conclusions

In this paper, we extracted the structures of the Drinfel’d double
$\mathfrak{d}$ of a Lie bialgebra $\mathfrak{g}$, and subsequently derived the
compatibility relations Eq. (2.11) from gauge-theoretic considerations.
Moreover, we have generalized (ie. "categorified") this method to the case of
the Lie 2-bialgebra/bialgebra crossed-module, and explicitly derived the
compatibility relations Eqs. (4.20), (4.21), (4.22) of the 2-Drinfel’d double
through 2-gauge theory.

It is important to note that our results, in particular Theorems 2.1, 4.1,
agree completely with existing mathematical literature [16, 17, 32, 1]. Though
these derived compatibility conditions are not strictly speaking new results,
the new insight is that we can in fact derive them from gauge-theoretical
considerations. This means that our approach opens the door for understanding
the structure of higher Drinfel’d doubles in a similar manner. For instance,
we could start with a 3-gauge theory [49], and apply our techniques to develop
the notion of 3-Drinfel’d double associated to 3-bialgebras (which are defined
in [57]).

We had also briefly studied topological theories based on the (2-)Drinfel’d
doubles. The BF theory Eq. (5.6) is of particular interest. Indeed, it is an
essential tool to construct quantum gravity amplitude in the spin foam
approach [34, 37]. It has been argued that it could be relevant to also
include some edge decorations to have access to the frame field, which can be
done using 2-gauge theories [58, 50, 56]. To unlock new types of symmetries
could therefore open new ways to construct spin foam models.

In general, we believe that such a procedure would lead to a systematic
understanding of the structures of a "quantum 2-group", which is very much
sought after in many areas of physics [59, 60, 61].

We would like to highlight some directions we find interesting to develop in a
near future.

##### Recovering the Crane-Yetter-Broda amplitude.

As we emphasized, the Drinfel’d double fits in a 3D Chern-Simons theory Eq.
(2.14), while the 2-Drinfel’d double fits in a 4D BF-theory Eq. (5.6). The
trivial canonical embedding
$\mathfrak{d}\mapsto\operatorname{id}_{\mathfrak{d}}$ can therefore be seen as
the most natural way of "promoting" a 3D topological gauge theory to a 4D one,
while keeping all of the algebra of the field contents identical. With this in
mind, it would interesting to see whether one can recover (more rigorously)
the Crane-Yetter-Broda amplitude (ie. a 15-$j$ symbol) defined in terms of a
quantum group [62, 36] as the amplitude associated to the $BF+BB$ action. This
would in particular bring more strength to the quantum gravity models which
implement the cosmological constant using a quantum group deformation. It
would also clarify how and why the presence of a non-zero cosmological
constant is also incorporated in the quantum gravity regime through a quantum
group structure, in 4D.

##### 2-graded integrable systems.

In this paper, we studied the 2-graded classical $r$-matrices on
$\mathfrak{d}$, as well as the quadratic 2-Casimirs. Particular examples in
the cases $t=0,1$ were worked out explicitly, and we have found that, in these
cases, the notion of duality for the 1-algebra case [17] all carry over to the
Lie 2-algebra context, including the classification of the classical
$r$-matrices.

The notion of quantum/Poisson group was initially developed in the context of
integrable systems. The use of the $r$-matrix is quite instrumental in
understanding and classifying the notion of integral system. Now that we have
Lie 2-bialgebras and the associated notion of $r$-matrix, we can explore how
such formalism can be used to develop new notions of integrable systems. This
is currently under investigation.

##### Fourier transform between dual 2-groups.

A key feature of the Drinfel’d double is that we have a duality between Lie
algebras, which can be extended to Hopf algebras [7]. In more physical terms,
this duality pertains a notion of Fourier transform, relating possibly non-
commutative structures [43, 63, 64]. This notion of Fourier transform is
different than (but related to) the standard Fourier transformation of group
elements in terms of representations. The representation theory is still
missing for most 2-groups (except the skeletal case in [65]), so that if we
intend to perform a Fourier transform for the notion of 2-groups, we need to
resort to the duality between functions over dual 2-groups. Such duality is
embedded in the definition of the 2-Drinfel’d double, hence we can expect many
of the key-notions the plane-wave need to satisfy should be recoverable from
the (quantum) 2-Drinfel’d double.

##### Weak Lie 2-algebras and their 2-double.

All along in this paper we have worked with strict Lie 2-algebras, which are
equivalent to crossed-modules. However, there is in fact an additional piece
of datum that characterizes a crossed-module $\mathfrak{g}$. If we denote by
$V=\operatorname{ker}t$ and $\operatorname{coker}t=\mathfrak{u}$ the kernel
and cokernel of the crossed-module map $t$, then this additional datum is a
Lie algebra cohomology class $\kappa\in H^{3}(\mathfrak{u},V)$ called the
Postnikov class [59], which classifies $\mathfrak{g}$ up to elementary
equivalence [66]. Working explicitly with the crossed-module structure, it
would be interesting to see how our above (2-)gauge-theoretic formalism
extends to take the Postnikov class into account.

The motivation for doing so arises from the correspondence between crossed-
modules with non-trivial Postnikov class and weak 2-algebras. In essence, the
structure of a weak Lie 2-algebra introduces a skew-trilinear homotopy map
$\eta:\mathfrak{g}_{0}^{3\wedge}\rightarrow\mathfrak{g}_{-1}$ that allows us
to relax the Jacobi identity on $\mathfrak{g}$ [32]: the 2-Jacobi identities
Eq. (3.2) are modified to

$\displaystyle[X,[X^{\prime},X^{\prime\prime}]]+[X^{\prime},[X^{\prime\prime},X]]+[X^{\prime\prime},[X,X^{\prime}]]=t\eta(X,X^{\prime},X^{\prime\prime}),$
$\displaystyle X\rhd(X^{\prime}\rhd Y)-X^{\prime}\rhd(X\rhd
Y)-[X,X^{\prime}]\rhd Y=\eta(X,X^{\prime},tY)$ (7.1)

for each
$X,X^{\prime},X^{\prime\prime}\in\mathfrak{g}_{0},Y\in\mathfrak{g}_{-1}$. Such
structures appear most naturally in string theory, where one has the string
2-algebra [46]. Notably, weak 2-algebras have also appeared in the study of
2-plectic geometry [67].

The trouble with weak 2-algebras is that they do not integrate to a 2-group.
Due to the non-trivial Jacobiator, namely the right-hand sides of Eq. (7.1),
the associator at the group level cannot be made continuous [68]. The
solution, given in Ref. [68], is precisely to pass from the description of a
weak 2-algebra to a crossed-module with non-trivial Postnikov class. As such,
based on the notion of a weak 2-bialgebra already in the literature [32], our
formalism would allow one to construct, in a manner in which the gauge content
is explicit, a "weak" classical 2-Drinfel’d double associated to a bialgebra
crossed-module with non-trivial Postnikov class.

Such a gauge-theoretic understanding of the structures of a weak 2-Drinfel’d
double is useful, as this (or more precisely the finite group version) will
allow us to pin down the 4D boundary theory of the 5D symmetry-protected
topological order $\int_{M}w_{2}\cup w_{3}$ [69]. This order hosts fermionic
point-like and string-like quasiparticles, which was also recently discovered
to be related to the new spin-$\frac{3}{2}$ $SU(2)$ anomaly [70]. This
particular order has received much attention recently from the condensed
matter theory community, thus it would be important to understand its
symmetries.

##  Appendix A Proofs of generalized (2-)covariance

In this section, we provide an explicit proof of the generalized covariance of
the combined curvature quantities Eqs. (2.13), (5.2) and (5.3) constructed in
the main text. These proofs shall demonstrate the importance of the
compatibility conditions Eq. (2.11) and Eqs. (4.20), (4.21), (4.22) in the
context of gauge and 2-gauge theory, respectively.

##### Covariance of the combined curvature.

Recall the combined curvature ${\bf F}$ constructed in Eq. (2.13). It takes
values in a Drinfel’d double
$\mathfrak{d}=\mathfrak{g}\bowtie\mathfrak{g}^{*}$ of a Lie bialgebra
$\mathfrak{g}$.

######  Theorem A.1.

Given Theorem 2.1, Eq. (2.13) is covariant

${\bf F}\rightarrow{\bf F}^{\boldsymbol{\lambda}}={\bf F}+\boldsymbol{[}{\bf
F},\boldsymbol{\lambda}\boldsymbol{]}.$

With Eq. (2.12), we may compute

$\boldsymbol{[}{\bf
F},\boldsymbol{\lambda}\boldsymbol{]}=([\bar{F},\lambda]+\mathfrak{ad}_{\tilde{\lambda}}^{*}\bar{F}-\mathfrak{ad}_{\bar{F}^{*}}^{*}\lambda)+([\bar{F}^{*},\tilde{\lambda}]+\operatorname{ad}_{\lambda}^{*}\bar{F}^{*}-\operatorname{ad}_{\bar{F}}^{*}\tilde{\lambda}).$

###### Proof.

Due to the symmetry under duality

$\mathfrak{g}\leftrightsquigarrow\mathfrak{g}^{*},\qquad\operatorname{ad}^{*}\leftrightsquigarrow\mathfrak{ad}^{*},$

we may without loss of generality (WLOG) focus on one component of ${\bf F}$,
say $\bar{F}$ that lies in the $\mathfrak{g}$-sector. This leads to the
transformations

$\displaystyle[A\wedge A]$ $\displaystyle\xrightarrow{\boldsymbol{\lambda}}$
$\displaystyle[A\wedge A]$ $\displaystyle+2[d_{A}\lambda\wedge
A]+2[\mathfrak{ad}_{\tilde{\lambda}}^{*}A\wedge A]$
$\displaystyle-2[A\wedge\mathfrak{ad}_{B}^{*}\lambda]+o(\boldsymbol{\lambda}^{2}),$
$\displaystyle\mathfrak{ad}_{B}^{*}(\wedge A)$
$\displaystyle\xrightarrow{\boldsymbol{\lambda}}$
$\displaystyle\mathfrak{ad}_{B}^{*}(\wedge A)$
$\displaystyle+\mathfrak{ad}_{B}^{*}(\wedge
d_{A}\lambda)+\mathfrak{ad}_{B}^{*}(\wedge\mathfrak{ad}_{\tilde{\lambda}}^{*}A)+\mathfrak{ad}_{d_{B}\tilde{\lambda}}^{*}(\wedge
A)+\mathfrak{ad}_{\operatorname{ad}_{\lambda}^{*}B}^{*}(\wedge A)$
$\displaystyle-\mathfrak{ad}_{\operatorname{ad}_{A}^{*}\tilde{\lambda}}^{*}A-\mathfrak{ad}_{B}^{*}(\wedge\mathfrak{ad}_{B}^{*}\lambda)+o(\boldsymbol{\lambda}^{2}),$

where we have neglected terms of quadratic and higher order in
$\boldsymbol{\lambda}=\lambda+\tilde{\lambda}$.

We first rewrite using Eq. (2.11)

$2[\mathfrak{ad}_{\tilde{\lambda}}^{*}A\wedge
A]=\mathfrak{ad}_{\tilde{\lambda}}^{*}[A\wedge
A]+2\mathfrak{ad}_{\operatorname{ad}_{A}^{*}\tilde{\lambda}}^{*}(\wedge A),$

and hence

$[\mathfrak{ad}_{\tilde{\lambda}}^{*}A\wedge
A]-\mathfrak{ad}_{\operatorname{ad}_{A}^{*}\tilde{\lambda}}^{*}A=\frac{1}{2}\mathfrak{ad}_{\tilde{\lambda}}^{*}[A\wedge
A].$

On the other hand, we have

$\mathfrak{ad}_{d_{B}\tilde{\lambda}}^{*}(\wedge
A)=\mathfrak{ad}_{\tilde{\lambda}}^{*}dA+\mathfrak{ad}_{[B,\tilde{\lambda}]}^{*}(\wedge
A),$

modulo the boundary term $d(\mathfrak{ad}_{\tilde{\lambda}}^{*}A)$. By the
identity101010This follows from the Jacobi identity for the dual bracket
$[\cdot,\cdot]_{*}$:
$(\mathfrak{ad}_{g^{\prime}}\mathfrak{ad}_{g})g^{\prime\prime}=[g^{\prime},[g,g^{\prime\prime}]_{*}]_{*}=-[g,[g^{\prime\prime},g^{\prime}]_{*}]_{*}-[g^{\prime\prime},[g^{\prime},g]_{*}]_{*}=(\mathfrak{ad}_{g}\mathfrak{ad}_{g^{\prime}})g^{\prime\prime}+\mathfrak{ad}_{[g^{\prime},g]_{*}}g^{\prime\prime}.$

$\mathfrak{ad}_{g}^{*}\mathfrak{ad}_{g^{\prime}}^{*}=\mathfrak{ad}_{g^{\prime}}^{*}\mathfrak{ad}_{g}^{*}+\mathfrak{ad}_{[g^{\prime},g]_{*}}^{*}$
(A.1)

for each $g,g^{\prime},g^{\prime\prime}\in\mathfrak{g}^{*}$, this leads to

$\mathfrak{ad}_{B}^{*}(\wedge\mathfrak{ad}_{\tilde{\lambda}}^{*}A)=\mathfrak{ad}_{\tilde{\lambda}}^{*}\mathfrak{ad}_{B}^{*}(\wedge
A)+\mathfrak{ad}_{[\tilde{\lambda},B]_{*}}^{*}A=\mathfrak{ad}_{\tilde{\lambda}}^{*}\mathfrak{ad}_{B}^{*}(\wedge
A)-\mathfrak{ad}_{[B,\tilde{\lambda}]_{*}}^{*}A,$

hence

$\mathfrak{ad}_{B}^{*}(\wedge\mathfrak{ad}_{\tilde{\lambda}}^{*}A)+\mathfrak{ad}_{d_{B}\tilde{\lambda}}^{*}(\wedge
A)=\mathfrak{ad}_{\tilde{\lambda}}^{*}dA+\mathfrak{ad}_{\tilde{\lambda}}^{*}\mathfrak{ad}_{B}^{*}(\wedge
A).$

We may then compute the combination

$\displaystyle[\mathfrak{ad}_{\tilde{\lambda}}^{*}A\wedge
A]+\mathfrak{ad}_{B}^{*}(\wedge\mathfrak{ad}_{\tilde{\lambda}}^{*}A)$
$\displaystyle\qquad+\mathfrak{ad}_{d_{B}\tilde{\lambda}}^{*}(\wedge
A)-\mathfrak{ad}_{\operatorname{ad}_{A}^{*}\tilde{\lambda}}^{*}A$
$\displaystyle=$
$\displaystyle\mathfrak{ad}_{\tilde{\lambda}}^{*}dA+\mathfrak{ad}_{\tilde{\lambda}}^{*}\mathfrak{ad}_{B}^{*}(\wedge
A)+\frac{1}{2}\mathfrak{ad}_{\tilde{\lambda}}^{*}[A\wedge A]$ (A.2)
$\displaystyle=$ $\displaystyle\mathfrak{ad}_{\tilde{\lambda}}^{*}\tilde{F}.$

Turning to the other terms, we first note the standard manipulation

$[d_{A}\lambda\wedge A]=-[\lambda\wedge dA]+[[A,\lambda]\wedge
A]=[dA,\lambda]+\frac{1}{2}[[A\wedge A],\lambda],$

where we have discarded a boundary term $d[\lambda,A]$ and used the Jacobi
identity

$[[A,\lambda]\wedge A]-[[\lambda,A]\wedge A]+[[A\wedge
A],\lambda]=2[[A,\lambda]\wedge A]+[[A\wedge A],\lambda]=0.$

Secondly, from Eq. (2.11) we have

$\mathfrak{ad}_{B}^{*}([A,\lambda])=[\mathfrak{ad}_{B}^{*}(\wedge
A),\lambda]+[A\wedge\mathfrak{ad}_{B}^{*}\lambda]-\mathfrak{ad}_{\operatorname{ad}_{\lambda}^{*}B}^{*}(\wedge
A)-\mathfrak{ad}_{\operatorname{ad}_{A}^{*}(\wedge B)}\lambda,$

which leads to

$\mathfrak{ad}_{B}^{*}(\wedge
d_{A}\lambda)=-\mathfrak{ad}_{dB}^{*}\lambda+[\mathfrak{ad}_{B}^{*}(\wedge
A),\lambda]+[A\wedge\mathfrak{ad}_{B}^{*}\lambda]-\mathfrak{ad}_{\operatorname{ad}_{\lambda}^{*}B}^{*}(\wedge
A)-\mathfrak{ad}_{\operatorname{ad}_{A}^{*}(\wedge B)}^{*}\lambda$

modulo the boundary term $d(\mathfrak{ad}_{B}^{*}\lambda)$. As such, we see
that

$\mathfrak{ad}_{B}^{*}(\wedge
d_{A}\lambda)+\mathfrak{ad}_{\operatorname{ad}_{\lambda}^{*}B}^{*}(\wedge
A)=\mathfrak{ad}_{dB}^{*}\lambda+[\mathfrak{ad}_{B}^{*}(\wedge
A),\lambda]+[A\wedge\mathfrak{ad}_{B}^{*}\lambda]+\mathfrak{ad}_{\operatorname{ad}_{A}^{*}(\wedge
B)}^{*}\lambda.$

Finally, we can rewrite

$2\mathfrak{ad}_{B}^{*}(\wedge\mathfrak{ad}_{B}^{*}\lambda)=\mathfrak{ad}_{[B\wedge
B]_{*}}^{\lambda}$

by Eq. (A.1). Hence, combining the relevant gauge-dependent terms, we have

$\displaystyle[d_{A}\lambda\wedge A]+\mathfrak{ad}_{B}^{*}(\wedge
d_{A}\lambda)+\mathfrak{ad}_{\operatorname{ad}_{\lambda}^{*}B}^{*}(\wedge A)$
$\displaystyle-[A\wedge\mathfrak{ad}_{B}^{*}\lambda]-\mathfrak{ad}_{B}^{*}(\wedge\mathfrak{ad}_{B}^{*}\lambda)$
$\displaystyle=$ $\displaystyle[dA,\lambda]+\frac{1}{2}[[A\wedge
A],\lambda]+[\mathfrak{ad}_{B}^{*}(\wedge A),\lambda],$ (A.3)
$\displaystyle-\mathfrak{ad}_{dB}^{*}\lambda-\mathfrak{ad}_{\operatorname{ad}_{A}^{*}(\wedge
B)}^{*}\lambda-\frac{1}{2}\mathfrak{ad}_{[B\wedge B]_{*}}^{*}\lambda$
$\displaystyle=$
$\displaystyle[\tilde{F},\lambda]-\mathfrak{ad}_{\tilde{F}^{*}}^{*}\lambda.$

Summing Eqs. (A.2) and (A.3) proves the lemma. ∎

##### 2-Covriance of the combined fake-flatness.

Now let us turn to the case of the 2-Deinfel’d double
$\mathfrak{d}=\mathfrak{g}\bowtie\mathfrak{g}^{*}[1]$. Recall that the
combined fake-flatness $\boldsymbol{\mathcal{F}}$ defined in Eq. 5.2, which
takes values in the degree-0 component
$\mathfrak{g}_{0}\oplus\mathfrak{g}_{-1}^{*}$ of $\mathfrak{d}$.

######  Lemma A.1.

Given Theorem 4.1, the combined fake-flatness Eq. (5.2) is 2-covariant

$\boldsymbol{\mathcal{F}}\rightarrow\begin{cases}\boldsymbol{\mathcal{F}}^{\boldsymbol{\lambda}}=\boldsymbol{\mathcal{F}}+\llbracket\boldsymbol{\mathcal{F}},\boldsymbol{\lambda}\rrbracket\\\
\boldsymbol{\mathcal{F}}^{\bf L}=\boldsymbol{\mathcal{F}}\end{cases}$

under the 2-gauge symmetry parameterized by
$\boldsymbol{\lambda}=\lambda+\tilde{\lambda}$ and ${\bf L}=L+\tilde{L}$.

###### Proof.

Let us first prove the covariance of the 1-curvature ${\bf F}$. For this, we
can copy the proof of Theorem A.1, with the modification that
$\operatorname{ad},\mathfrak{ad}$ are replaced by $\chi,\eta$, respectively.
However, we note that the argument there relies crucially on two things: the
condition Eq. (2.11) and the Jacobi identity. These conditions are analogous
in the 2-algebra context to the first conditions Eqs. (4.20) and the Jacobi
identities Eqs. (3.2), (3.8). This yields the desired covariance

${\bf F}\rightarrow{\bf F}^{\boldsymbol{\lambda}}={\bf F}+\boldsymbol{[}{\bf
F},\boldsymbol{\lambda}\boldsymbol{]}={\bf
F}+([\bar{F},\lambda]+\eta_{\tilde{\lambda}}^{*}\bar{F}-\eta_{\bar{F}^{*}}^{*}\lambda)+([\bar{F},\tilde{\lambda}]_{*}+\chi_{\lambda}^{*}\bar{F}^{*}-\chi_{\bar{F}}^{*}\tilde{\lambda}).$

Next, we examine how $T\boldsymbol{\Sigma}$ transforms under
$\boldsymbol{\lambda}$. In the $\mathfrak{g}$-valued component, we have

$t\Sigma\rightarrow
t\Sigma+t(\lambda\rhd\Sigma)+t\mathfrak{ad}_{\tilde{\lambda}}^{*}\Sigma-\eta_{\tilde{t}B}^{*}\lambda.$

By the Pfeiffer identity on $\mathfrak{g}$, we can write
$t(\lambda\rhd\Sigma)=[\lambda,t\Sigma]$. Now if moreover $\tilde{t}=t^{*}$
such that $t=\tilde{t}^{*}$, then

$t\mathfrak{ad}_{\tilde{\lambda}}^{*}\Sigma=\tilde{t}^{*}\mathfrak{ad}_{\tilde{\lambda}}^{*}\Sigma=\eta_{\tilde{\lambda}}^{*}\tilde{t}^{*}\Sigma=\eta_{\tilde{\lambda}}^{*}t\Sigma$

by Eq. (3.12). This achieves the desired covariance

$\bar{\mathcal{F}}\rightarrow\bar{\mathcal{F}}+[\bar{F},\lambda]+\eta_{\tilde{\lambda}}^{*}\bar{\mathcal{F}}-\eta_{\bar{\mathcal{F}}^{*}}^{*}\lambda=\bar{\mathcal{F}}+\boldsymbol{[}\boldsymbol{\mathcal{F}},\boldsymbol{\lambda}\boldsymbol{]}|_{\mathfrak{g}}.$

A similar argument proves the covariance of $\bar{\mathcal{F}}^{*}$ in the
dual sector $\mathfrak{g}^{*}[1]$.

Next, we compute that

$\displaystyle d{\bf A}$ $\displaystyle\rightarrow$ $\displaystyle d{\bf
A}+dT{\bf L}=d{\bf A}+Td{\bf L},$ $\displaystyle\boldsymbol{[}{\bf
A}\wedge{\bf A}\boldsymbol{]}$ $\displaystyle\rightarrow$
$\displaystyle\boldsymbol{[}{\bf A}\wedge{\bf
A}\boldsymbol{]}+2\boldsymbol{[}{\bf A}\wedge T{\bf
L}\boldsymbol{]}+\boldsymbol{[}T{\bf L}\wedge T{\bf L}\boldsymbol{]}$
$\displaystyle=\boldsymbol{[}{\bf A}\wedge{\bf A}\boldsymbol{]}+2T({\bf
A}\wedge^{\mathrlap{\rhd}{\triangleright}}{\bf L})+o({\bf L}^{2}).$

If we neglect quadratic terms in ${\bf L}^{2}$ as we have done in the main
text, then we yield the expected 2-gauge transformation

${\bf F}\rightarrow{\bf F}^{\bf L}={\bf F}+Td{\bf L}+T{\bf
A}\wedge^{\mathrlap{\rhd}{\triangleright}}{\bf
L}+\frac{1}{2}T\boldsymbol{[}{\bf L}\wedge{\bf L}\boldsymbol{]}\equiv{\bf
F}+Td_{\bf A}{\bf L}.$

As we have $\boldsymbol{\Sigma}\rightarrow\boldsymbol{\Sigma}+d_{\bf A}{\bf
L}$ from Eq. (5.4), we see that

$\boldsymbol{\mathcal{F}}\rightarrow{\bf F}^{\bf L}-T\boldsymbol{\Sigma}^{\bf
L}={\bf F}-T\boldsymbol{\Sigma}=\boldsymbol{\mathcal{F}}$

as desired.

∎

##### 2-Covariance of the combined 2-curvature.

Recall the combined 2-curvature $\boldsymbol{\mathcal{G}}$ defined in Eq. 5.3,
which takes values in the degree-(-1) component
$\mathfrak{g}_{-1}\oplus\mathfrak{g}_{0}^{*}$ of the 2-Drinfel’d double
$\mathfrak{d}$.

######  Lemma A.2.

Given Theorem 4.1, the combined 2-curvature in Eq. (5.3) is 2-covariant:

$\boldsymbol{\mathcal{G}}\rightarrow\begin{cases}\boldsymbol{\mathcal{G}}^{\boldsymbol{\lambda}}=\boldsymbol{\mathcal{G}}+\boldsymbol{\lambda}\leavevmode\nobreak\
\mathrlap{\triangleright}{\rhd}\leavevmode\nobreak\
\boldsymbol{\mathcal{G}}\\\ \boldsymbol{\mathcal{G}}^{\bf
L}=\boldsymbol{\mathcal{G}}+\boldsymbol{\mathcal{F}}\wedge^{\mathrlap{\triangleright}{\rhd}}{\bf
L}\end{cases}.$

###### Proof.

Here we make heavy use of Eqs. (5.4). We first deal with
$\boldsymbol{\lambda}$:

$\displaystyle d\boldsymbol{\Sigma}$ $\displaystyle\rightarrow$ $\displaystyle
d\boldsymbol{\Sigma}-d\boldsymbol{\lambda}\leavevmode\nobreak\
\mathrlap{\triangleright}{\rhd}\leavevmode\nobreak\
\boldsymbol{\Sigma}+\boldsymbol{\lambda}\leavevmode\nobreak\
\mathrlap{\triangleright}{\rhd}\leavevmode\nobreak\ d\boldsymbol{\Sigma},$
$\displaystyle{\bf
A}\wedge^{\mathrlap{\triangleright}{\rhd}}\boldsymbol{\Sigma}$
$\displaystyle\rightarrow$ $\displaystyle{\bf
A}\wedge^{\mathrlap{\triangleright}{\rhd}}\boldsymbol{\Sigma}+d_{\bf
A}\boldsymbol{\lambda}\wedge^{\mathrlap{\triangleright}{\rhd}}\boldsymbol{\Sigma}+{\bf
A}\wedge^{\mathrlap{\triangleright}{\rhd}}(\boldsymbol{\lambda}\leavevmode\nobreak\
\mathrlap{\triangleright}{\rhd}\leavevmode\nobreak\ \boldsymbol{\Sigma})$
$\displaystyle+o(\boldsymbol{\lambda}^{2}).$

By the 2-Jacobi identity for $\mathfrak{d}$, we have

${\bf
A}\wedge^{\mathrlap{\triangleright}{\rhd}}(\boldsymbol{\lambda}\leavevmode\nobreak\
\mathrlap{\triangleright}{\rhd}\leavevmode\nobreak\
\boldsymbol{\Sigma})=\boldsymbol{\lambda}\leavevmode\nobreak\
\mathrlap{\triangleright}{\rhd}\leavevmode\nobreak\ ({\bf
A}\wedge^{\mathrlap{\triangleright}{\rhd}}\boldsymbol{\lambda}\boldsymbol{\Sigma})-\boldsymbol{[}{\bf
A},\boldsymbol{\lambda}\boldsymbol{]}\leavevmode\nobreak\
\mathrlap{\triangleright}{\rhd}\leavevmode\nobreak\ \boldsymbol{\Sigma},$

hence

${\bf
A}\wedge^{\mathrlap{\triangleright}{\rhd}}\boldsymbol{\Sigma}\rightarrow{\bf
A}\wedge^{\mathrlap{\triangleright}{\rhd}}\boldsymbol{\Sigma}+d\boldsymbol{\lambda}\wedge^{\mathrlap{\triangleright}{\rhd}}\boldsymbol{\Sigma}+\boldsymbol{\lambda}\leavevmode\nobreak\
\mathrlap{\triangleright}{\rhd}\leavevmode\nobreak\ ({\bf
A}\wedge^{\mathrlap{\triangleright}{\rhd}}\boldsymbol{\lambda}\boldsymbol{\Sigma}).$

The sum then transforms accordingly:

$\boldsymbol{\mathcal{G}}\rightarrow\boldsymbol{\mathcal{G}}+\boldsymbol{\lambda}\leavevmode\nobreak\
\mathrlap{\triangleright}{\rhd}\leavevmode\nobreak\ \boldsymbol{\mathcal{G}}.$

Next, we deal with ${\bf L}$:

$\displaystyle d\boldsymbol{\Sigma}$ $\displaystyle\rightarrow$ $\displaystyle
d\boldsymbol{\Sigma}+d({\bf A}\wedge^{\mathrlap{\triangleright}{\rhd}}{\bf
L})+o({\bf L}^{2}),$ $\displaystyle{\bf
A}\wedge^{\mathrlap{\triangleright}{\rhd}}\boldsymbol{\Sigma}$
$\displaystyle\rightarrow$ $\displaystyle{\bf
A}\wedge^{\mathrlap{\triangleright}{\rhd}}\boldsymbol{\Sigma}+T{\bf
L}\wedge^{\mathrlap{\triangleright}{\rhd}}\boldsymbol{\Sigma}+{\bf
A}\wedge^{\mathrlap{\triangleright}{\rhd}}d_{\bf A}{\bf L}+o({\bf L}^{2}),$

where we consider terms only up to linear order in ${\bf L}$, as always. By
the Leibniz rule we have

$d({\bf A}\wedge^{\mathrlap{\triangleright}{\rhd}}{\bf L})=d{\bf
A}\wedge^{\mathrlap{\triangleright}{\rhd}}{\bf L}-{\bf
A}\wedge^{\mathrlap{\triangleright}{\rhd}}d{\bf L},$

hence the sum $\boldsymbol{\mathcal{G}}=d\boldsymbol{\Sigma}+{\bf
A}\wedge^{\mathrlap{\triangleright}{\rhd}}\boldsymbol{\Sigma}$ transforms as

$\displaystyle\boldsymbol{\mathcal{G}}$ $\displaystyle\rightarrow$
$\displaystyle\boldsymbol{\mathcal{G}}+d{\bf
A}\wedge^{\mathrlap{\triangleright}{\rhd}}{\bf L}+T{\bf
L}\wedge^{\mathrlap{\triangleright}{\rhd}}\boldsymbol{\Sigma}+{\bf
A}\wedge^{\mathrlap{\triangleright}{\rhd}}({\bf
A}\wedge^{\mathrlap{\triangleright}{\rhd}}{\bf L}).$

By the second Pfeiffer identity for $\mathfrak{d}$, we have

$\displaystyle T{\bf
L}\wedge^{\mathrlap{\triangleright}{\rhd}}\boldsymbol{\Sigma}$
$\displaystyle=$ $\displaystyle\boldsymbol{[}{\bf
L}\wedge\boldsymbol{\Sigma}\boldsymbol{]}=-\boldsymbol{[}\boldsymbol{\Sigma}\wedge{\bf
L}\boldsymbol{]}=-T\boldsymbol{\Sigma}\wedge^{\mathrlap{\triangleright}{\rhd}}{\bf
L},$

while by the 2-Jacobi identity for $\mathfrak{d}$ we have

${\bf A}\wedge^{\mathrlap{\triangleright}{\rhd}}({\bf
A}\wedge^{\mathrlap{\triangleright}{\rhd}}{\bf L})=\boldsymbol{[}{\bf
A}\wedge{\bf A}\boldsymbol{]}\wedge^{\mathrlap{\triangleright}{\rhd}}{\bf
L}-{\bf A}\wedge^{\mathrlap{\triangleright}{\rhd}}({\bf
A}\wedge^{\mathrlap{\triangleright}{\rhd}}{\bf L}).$

These terms combine to yield

$(\frac{1}{2}\boldsymbol{[}{\bf A}\wedge{\bf
A}\boldsymbol{]}-T\boldsymbol{\Sigma})\wedge^{\mathrlap{\triangleright}{\rhd}}{\bf
L}.$

The 2-curvature then transforms expectedly:

$\displaystyle\boldsymbol{\mathcal{G}}$ $\displaystyle\rightarrow$
$\displaystyle\boldsymbol{\mathcal{G}}+(d{\bf A}+\frac{1}{2}\boldsymbol{[}{\bf
A}\wedge{\bf
A}\boldsymbol{]}-T\boldsymbol{\Sigma})\wedge^{\mathrlap{\triangleright}{\rhd}}{\bf
L}$ $\displaystyle=$
$\displaystyle\boldsymbol{\mathcal{G}}+\boldsymbol{\mathcal{F}}\wedge^{\mathrlap{\triangleright}{\rhd}}{\bf
L}.$

∎

The proofs of Theorems A.1, A.1, A.2 state that the compatibility conditions
for the (2-)Manin triple are in fact crucial not only for the non-ambiguity in
the (2-)gauge transformations, but also for ensuring the covariance for the
various curvature quantities.

## References

  * [1] C. Bai, Y. Sheng and C. Zhu, _Lie 2-bialgebras_ , _Communications in Mathematical Physics_ 320 (apr, 2013) 149–172.
  * [2] E. Witten, _(2+1)-Dimensional Gravity as an Exactly Soluble System_ , _Nucl. Phys._ B311 (1988) 46.
  * [3] L. Crane and I. Frenkel, _Four-dimensional topological field theory, Hopf categories, and the canonical bases_ , _J. Math. Phys._ 35 (1994) 5136–5154, [hep-th/9405183].
  * [4] J. C. Baez and A. Lauda, _A Prehistory of n-Categorical Physics_ , 8, 2009\.  0908.2469.
  * [5] V. Turaev and A. Virelizier, _Monoidal Categories and Topological Field Theory_ , vol. 322 of _Progress in Mathematics_.  Birkhäuser, Cham, 2017, 10.1007/978-3-319-49834-8.
  * [6] V. G. Drinfeld, _Quantum groups_ , _Zap. Nauchn. Semin._ 155 (1986) 18–49.
  * [7] S. Majid, _Foundations of quantum group theory_.  Cambridge University Press, 2011.
  * [8] K. Noui, _Three Dimensional Loop Quantum Gravity: Particles and the Quantum Double_ , _J. Math. Phys._ 47 (2006) 102501, [gr-qc/0612144].
  * [9] C. Meusburger and K. Noui, _The Hilbert space of 3d gravity: quantum group symmetries and observables_ , _Adv. Theor. Math. Phys._ 14 (2010) 1651–1715, [0809.2875].
  * [10] A. Yu. Kitaev, _Fault tolerant quantum computation by anyons_ , _Annals Phys._ 303 (2003) 2–30, [quant-ph/9707021].
  * [11] O. Buerschaper, J. Martin Mombelli, M. Christandl and M. Aguado, _A hierarchy of topological tensor network states_ , _Journal of Mathematical Physics_ 54 (1, 2013) .
  * [12] F. Girelli, P. K. Osei and A. Osumanu, _Semidual Kitaev lattice model and tensor network representation_ , _JHEP_ 09 (2021) 210, [1709.00522].
  * [13] M. Dupuis, L. Freidel, F. Girelli, A. Osumanu and J. Rennert, _On the origin of the quantum group symmetry in 3d quantum gravity_ , [2006.10105].
  * [14] V. V. Fock and A. A. Roslyi, _Poisson structure on moduli of flat connections on Riemann surfaces and r matrix_ , .
  * [15] V. V. Fock and A. A. Rosly, _Poisson structure on moduli of flat connections on Riemann surfaces and r matrix_ , _Am. Math. Soc. Transl._ 191 (1999) 67–86, [math/9802054].
  * [16] V. Chari and A. Pressley, _A guide to quantum groups_.  Cambridge University Press, 1994.
  * [17] M. A. Semenov-Tyan-Shanskii, _Poisson-Lie groups. The quantum duality principle and the twisted quantum double_ , _Theoretical and Mathematical Physics_ 93 (Nov, 1992) 1292–1307.
  * [18] J. C. Baez, _Four-Dimensional BF theory with cosmological term as a topological quantum field theory_ , _Lett. Math. Phys._ 38 (1996) 129–143, [q-alg/9507006].
  * [19] J. C. Baez and J. Dolan, _Higher dimensional algebra and topological quantum field theory_ , _J. Math. Phys._ 36 (1995) 6073–6105, [q-alg/9503002].
  * [20] S. Majid, _Strict quantum 2-groups_ , [1208.6265].
  * [21] J. C. Baez and A. D. Lauda, _Higher-dimensional algebra v: 2-groups_ , [math/0307200].
  * [22] J. C. Baez, _Higher Yang-Mills theory_ , [hep-th/0206130].
  * [23] J. Baez and U. Schreiber, _Higher gauge theory: 2-connections on 2-bundles_ , [hep-th/0412325].
  * [24] J. C. Baez and U. Schreiber, _Higher gauge theory_ , [math/0511710].
  * [25] F. Girelli and H. Pfeiffer, _Higher gauge theory: Differential versus integral formulation_ , _J. Math. Phys._ 45 (2004) 3949–3971, [hep-th/0309173].
  * [26] J. C. Baez and J. Huerta, _An Invitation to Higher Gauge Theory_ , _Gen. Rel. Grav._ 43 (2011) 2335–2392, [1003.4485].
  * [27] J. C. Baez and A. S. Crans, _Higher-Dimensional Algebra VI: Lie 2-Algebras_ , _Theor. Appl. Categor._ 12 (2004) 492–528, [math/0307263].
  * [28] Z. Chen, M. Stienon and P. Xu, _Poisson 2-groups_ , _J. Diff. Geom._ 94 (2013) 209–240, [1202.0079].
  * [29] F. Girelli, H. Pfeiffer and E. M. Popescu, _Topological Higher Gauge Theory - from BF to BFCG theory_ , _J. Math. Phys._ 49 (2008) 032503, [0708.3051].
  * [30] J. F. Martins and A. Mikovic, _Lie crossed modules and gauge-invariant actions for 2-BF theories_ , _Adv. Theor. Math. Phys._ 15 (2011) 1059–1084, [1006.0903].
  * [31] A. Mikovic and M. Vojinovic, _Poincare 2-group and quantum gravity_ , _Class. Quant. Grav._ 29 (2012) 165003, [1110.4694].
  * [32] Z. Chen, M. Stiénon and P. Xu, _Weak lie 2-bialgebras_ , _Journal of Geometry and Physics_ 68 (Jun, 2013) 59–68.
  * [33] F. Girelli and P. Tsimiklis, _Discretization of 4d Poincaré BF theory: from groups to 2-groups_ , [2105.01817].
  * [34] J. W. Barrett and L. Crane, _Relativistic spin networks and quantum gravity_ , _J. Math. Phys._ 39 (1998) 3296–3302, [gr-qc/9709028].
  * [35] J. C. Baez, _Higher dimensional algebra and Planck scale physics_ , [gr-qc/9902017].
  * [36] J. C. Baez, _An Introduction to Spin Foam Models of $BF$ Theory and Quantum Gravity_, _Lect. Notes Phys._ 543 (2000) 25–93, [gr-qc/9905087].
  * [37] C. Rovelli, _Quantum Gravity_.  Cambridge Monographs on Mathematical Physics. Cambridge University Press, 2004.
  * [38] A. Yu. Alekseev, H. Grosse and V. Schomerus, _Combinatorial quantization of the Hamiltonian Chern-Simons theory_ , _Commun. Math. Phys._ 172 (1995) 317–358, [hep-th/9403066].
  * [39] A. Yu. Alekseev, H. Grosse and V. Schomerus, _Combinatorial quantization of the Hamiltonian Chern-Simons theory. 2._ , _Commun. Math. Phys._ 174 (1995) 561–604, [hep-th/9408097].
  * [40] C. Delcamp, L. Freidel and F. Girelli, _Dual loop quantizations of 3d gravity_ , [1803.03246].
  * [41] J. Roberts, _Skein theory and turaev-viro invariants_ , _Topology_ 34 (1995) 771 – 787.
  * [42] L. Freidel and D. Louapre, _Ponzano-Regge model revisited II: Equivalence with Chern-Simons_ , [gr-qc/0410141].
  * [43] S. Majid and B. J. Schroers, _q-Deformation and Semidualisation in 3d Quantum Gravity_ , _J. Phys. A_ 42 (2009) 425402, [0806.2587].
  * [44] F. Girelli, A. Osumanu and W. Wieland, _Canonical transformations generated by the boundary volume: unimodular and non-abelian teleparallel gravity_ , [2101.05347].
  * [45] P. K. Osei and B. J. Schroers, _Classical r-matrices for the generalised Chern–Simons formulation of 3d gravity_ , _Class. Quant. Grav._ 35 (2018) 075006, [1708.07650].
  * [46] H. Kim and C. Saemann, _Adjusted parallel transport for higher gauge theories_ , _J. Phys. A_ 53 (2020) 445206, [1911.06390].
  * [47] F. Wagemann, _On lie algebra crossed modules_ , _Communications in Algebra_ 34 (2006) 1699–1722, [https://doi.org/10.1080/00927870500542705].
  * [48] A. Mikovic, M. A. Oliveira and M. Vojinovic, _Hamiltonian analysis of the BFCG theory for a strict Lie 2-group_ , [1610.09621].
  * [49] T. Radenkovic and M. Vojinovic, _Higher Gauge Theories Based on 3-groups_ , _JHEP_ 10 (2019) 222, [1904.07566].
  * [50] S. K. Asante, B. Dittrich, F. Girelli, A. Riello and P. Tsimiklis, _Quantum geometry from higher gauge theory_ , _Class. Quant. Grav._ 37 (2020) 205001, [1908.05970].
  * [51] D. N. Yetter, _TQFT’s from homotopy 2 types_ , _J. Knot Theor. Ramifications_ 2 (1993) 113–123.
  * [52] C. Meusburger, _Poisson–Lie Groups and Gauge Theory_ , _Symmetry_ 13 (2021) 1324.
  * [53] N. Beisert, R. Hecht and B. Hoare, _Maximally extended $\boldsymbol{\mathfrak{sl}(2|2)}$ , q-deformed $\boldsymbol{\mathfrak{d}{(2,1;\epsilon)}}$ and 3D kappa-Poincaré_, _J. Phys. A_ 50 (2017) 314003, [1704.05093].
  * [54] I. Gutierrez-Sagredo, A. Ballesteros, G. Gubitosi and F. J. Herranz, _Quantum groups, non-commutative Lorentzian spacetimes and curved momentum spaces_ , 7, 2019.  1907.07979.
  * [55] S. Zakrzewski, _Quantum poincare group related to the kappa -poincare algebra_ , _Journal of Physics A: Mathematical and General_ 27 (mar, 1994) 2075–2082.
  * [56] F. Girelli, M. Laudonio and P. Tsimiklis, _Polyhedron phase space using 2-groups: $\kappa$-Poincaré as a Poisson 2-group_, [2105.10616].
  * [57] R. Bai, W. Guo, L. Lin and Y. Zhang, _n-lie bialgebras_ , [1607.07913].
  * [58] L. Crane and D. Yetter, _A More sensitive Lorentzian state sum_ , [gr-qc/0301017].
  * [59] A. Kapustin and R. Thorngren, _Higher symmetry and gapped phases of gauge theories_ , [1309.4721].
  * [60] C. Córdova, T. T. Dumitrescu and K. Intriligator, _Exploring 2-Group Global Symmetries_ , _JHEP_ 02 (2019) 184, [1802.04790].
  * [61] F. Benini, C. Córdova and P.-S. Hsin, _On 2-Group Global Symmetries and their Anomalies_ , _JHEP_ 03 (2019) 118, [1803.09336].
  * [62] L. Crane and D. Yetter, _On algebraic structures implicit in topological quantum field theories_ , _J. Knot Theor. Ramifications_ 8 (1999) 125–163, [hep-th/9412025].
  * [63] C. Guedes, D. Oriti and M. Raasakka, _Quantization maps, algebra representation and non-commutative Fourier transform for Lie groups_ , _J. Math. Phys._ 54 (2013) 083508, [1301.7750].
  * [64] E. Joung, J. Mourad and K. Noui, _Three Dimensional Quantum Geometry and Deformed Poincare Symmetry_ , _J. Math. Phys._ 50 (2009) 052503, [0806.4121].
  * [65] J. C. Baez, A. Baratin, L. Freidel and D. K. Wise, _Infinite-Dimensional Representations of 2-Groups_ , vol. 1032.  2012, 10.1090/S0065-9266-2012-00652-6.
  * [66] Z. Wang and X. Chen, _Twisted gauge theories in three-dimensional Walker-Wang models_ , _Phys. Rev._ B95 (2017) 115142, [1611.09334].
  * [67] J. C. Baez and C. L. Rogers, _Categorified symplectic geometry and the string Lie 2-algebra_ , _Homology, Homotopy and Applications_ 12 (2010) 221 – 236.
  * [68] J. C. Baez, D. Stevenson, A. S. Crans and U. Schreiber, _From loop groups to 2-groups_ , [math/0504123].
  * [69] R. Thorngren, _Framed wilson operators, fermionic strings, and gravitational anomaly in 4d_ , _Journal of High Energy Physics_ 2015 (2010) .
  * [70] J. Wang, X.-G. Wen and E. Witten, _A new su(2) anomaly_ , _Journal of Mathematical Physics_ 60 (May, 2019) 052301.