3.1. Weak representations of Lie–Rinehart algebras

Let ${A}$ be a commutative associative algebra over $\mathbb {K}$ . We denote by $\mathsf {Der}_{\mathbb {K}}({A})$ the set of $\mathbb {K}$ -linear derivations of ${A}$ ; that is,

$$ \begin{align*} \mathsf{Der}_{\mathbb{K}}({A})=\{ D\in\text{End}_{\mathbb{K}}(A): D(ab)=D(a)b+aD(b), \forall a,b\in{A}\}. \end{align*} $$

Definition 3.1 [Reference Rinehart39]

A Lie–Rinehart algebra over ${A}$ is a $\mathbb {K}$ -Lie algebra $(\mathcal {L},[\cdot ,\cdot ]_{\mathcal {L}})$ together with an ${A}$ -module structure on $\mathcal {L}$ and a map $\alpha :\mathcal {L}\to \mathsf {Der}_{\mathbb {K}}({A})$ (called the anchor) which is simultaneously a $\mathbb {K}$ -Lie algebra and an ${A}$ -module homomorphism such that

$$ \begin{align*} [x,ay]_{\mathcal{L}}=a[x,y]_{\mathcal{L}}+\alpha(x)(a)y,\quad \forall x,y\in \mathcal{L},~a\in {A}. \end{align*} $$

We usually denote a Lie–Rinehart algebra over ${A}$ by $({A},\mathcal {L},[\cdot ,\cdot ]_{\mathcal {L}},\alpha )$ or simply by $\mathcal {L}$ .

Example 3.5. Let ${M}$ be an ${A}$ -module. A first-order differential operator on ${M}$ is a pair $(D,\sigma )$ , where $D:{M}\,\rightarrow \,{M}$ is a $\mathbb {K}$ -linear map and $\sigma =\sigma _{D}\in \mathsf {Der}_{\mathbb {K}}({A})$ , satisfying the following compatibility condition:

(7) $$ \begin{align}D(am)=aD(m)+\sigma(a)m,\quad\forall a\in{A},m\in{M}. \end{align} $$

Denote by $\mathfrak {D}({M})$ the set of first-order differential operators on ${M}$ . It is obvious that $\mathfrak {D}({M})$ is an ${A}$ -module. Define a bracket operation $[\cdot ,\cdot ]_{C}$ on $\mathfrak {D}({M})$ by

(8) $$ \begin{align} [(D_{1},\sigma_{1}),(D_{2},\sigma_{2})]_{C}:=(D_{1}\circ D_{2}-D_{2}\circ D_{1},\sigma_{1}\circ \sigma_{2}-\sigma_{2}\circ \sigma_{1}),\quad\forall (D_{1},\sigma_{1}),(D_{2},\sigma_{2})\in\mathfrak{D}({M}), \end{align} $$

and an ${A}$ -module homomorphism $\mathrm {Pr}:\mathfrak {D}({M})\,\rightarrow \, \mathsf {Der}_{\mathbb {K}}({A})$ by $\mathrm {Pr}(D,\sigma )=\sigma $ for all $(D,\sigma )\in \mathfrak {D}({M})$ . Then $({A},\mathfrak {D}({M}),[\cdot ,\cdot ]_{C},\alpha =\mathrm {Pr})$ is a Lie–Rinehart algebra.

Remark 3.6. Let ${M}$ be an ${A}$ -module. It is straightforward to see that we have a semidirect product commutative associative algebra $A\ltimes M$ , where the multiplication is given by

$$ \begin{align*} (a,m)\cdot (b,n)=(ab,an+bm),\quad \forall a,b\in A,~m,n\in M. \end{align*} $$

Then $(D,\sigma )$ is a first-order differential operator on M if and only if $(\sigma ,D)$ is a derivation on the commutative associative algebra $A\ltimes M$ . This result is the algebraic counterpart of the fact that a first-order differential operator on a vector bundle E can be viewed as a linear vector field on the dual bundle $E^{*}$ . In fact, functions on the vector bundle $E^{*}\to N$ are generated by $C^{\infty }(N)$ and $\Gamma (E)$ , while the latter are fibre linear functions on $E^{*}$ . Since a first-order differential operator maps a fibre linear function to a fibre linear function, it is viewed as a linear vector field on $E^{*}.$

Note that zero map from $\mathcal {L}$ to $\mathcal {L}^{\prime }$ is not a Lie–Rinehart weak homomorphism if $\alpha \ne 0$ .

We denote by $\mathrm {WH}(\mathcal {L},\mathcal {L}^{\prime })$ the set of weak homomorphisms from the Lie–Rinehart algebra $({A},\mathcal {L},[\cdot ,\cdot ]_{\mathcal {L}},\alpha )$ to $({A},\mathcal {L}^{\prime },[\cdot ,\cdot ]_{\mathcal {L}^{\prime }},\alpha ^{\prime })$ . By Proposition 3.8, it is easy to see that $\mathrm {WH}(\mathcal {L},\mathcal {L})$ is a monoid.

Remark 3.10. A weak representation of a Lie–Rinehart algebra $({A},\mathcal {L},[\cdot ,\cdot ]_{\mathcal {L}},\alpha )$ on an ${A}$ -module ${M}$ means a $\mathbb {K}$ -Lie algebra homomorphism $\rho :\mathcal {L}\,\rightarrow \,\mathfrak {gl}_{\mathbb {K}}({M})$ such that

$$ \begin{align*}\rho(x)(au)=a\rho(x)(u)+\alpha(x)(a)u,\quad \forall x\in\mathcal{L}, a\in A, u\in M; \end{align*} $$

that is, $(D=\rho (x),\sigma =\alpha (x))$ is a first-order differential operator on ${M}$ .

Remark 3.11. In [Reference Huebschmann19], Huebschmann showed that there is a one-to-one correspondence between representations of a Lie–Rinehart algebra and representations of its universal enveloping algebra $\mathcal {U}(A, \mathcal {L}) :=(A\# U(\mathcal {L}))/J$ , where J is a certain ideal of the smash product $A\# U(\mathcal {L})$ . More explicitly, it is known that $U(\mathcal {L})$ is a Hopf algebra and A is a $U(\mathcal {L})$ -module algebra. Then the smash product $A\# U(\mathcal {L})$ (see [Reference Montgomery35]) is a $\mathbb {K}$ -vector space $A\otimes U(\mathcal {L})$ with elements denoted by $a\#u$ and with product defined for all $a,b\in A$ and $u,v\in U(\mathcal {L})$ by

$$ \begin{align*} (a\#u)(b\# v)=\sum a\alpha(u_{(1)})b\# u_{(2)}v, \end{align*} $$

where we use the standard Sweedler notation $\Delta (u)=\sum u_{(1)}\otimes u_{(2)}$ for the coproduct $\Delta $ . The algebra $A\# U(\mathcal {L})$ is also called the Massey–Peterson algebra in [Reference Huebschmann19]. It is not hard to see that there is a one-to-one correspondence between weak representations of a Lie–Rinehart algebra and representations of the smash product $A\# U(\mathcal {L})$ .

Example 3.12. Let $({A},\mathcal {L},[\cdot ,\cdot ]_{\mathcal {L}},\alpha )$ be a Lie–Rinehart algebra. Define $\mathsf {ad}:\mathcal {L}\,\rightarrow \,\mathfrak D(\mathcal {L})$ by

$$ \begin{align*} \mathsf{ad}_{x}y=[x,y]_{\mathcal{L}},\quad \sigma_{\mathsf{ad}_{x}}=\alpha(x),\quad \forall x,y\in\mathcal{L}. \end{align*} $$

Then $\mathsf {ad}$ is a weak representation of $\mathcal {L}$ on $\mathcal {L}$ . Note that $\mathsf {ad}$ is generally not a representation of $\mathcal {L}$ on itself.

We usually denote by $M\stackrel {\phi }{\,\rightarrow \,}M^{\prime }$ a homomorphism between the weak representations $({M};\rho )$ and $({M}^{\prime };\rho ^{\prime })$ and denote by $\mathsf {WRep}_{\mathbb {K}}(\mathcal {L})$ the category of weak representations of a Lie–Rinehart algebra $({A},\mathcal {L},[\cdot ,\cdot ]_{\mathcal {L}},\alpha )$ and $\mathsf {Rep}_{\mathbb {K}}(\mathfrak g)$ the category of representations of a $\mathbb {K}$ -Lie algebra $(\mathfrak g,[\cdot ,\cdot ]_{\mathfrak g})$ . It is obvious that the category of representations of a Lie–Rinehart algebra $({A},\mathcal {L},[\cdot ,\cdot ]_{\mathcal {L}},\alpha )$ , denoted by $\mathsf {Rep}(\mathcal {L})$ , is a full subcategory of the category $\mathsf {WRep}_{\mathbb {K}}(\mathcal {L})$ . Please note the subtle difference between the two categories $\mathsf {Rep}_{\mathbb {K}}(\mathcal {L})$ and $\mathsf {Rep}(\mathcal {L})$ .

Definition 3.15 [Reference Casas, Ladra and Pirashvili6]

Let $({A},\mathcal {L},[\cdot ,\cdot ]_{\mathcal {L}},\alpha )$ be a Lie–Rinehart algebra and $(\mathcal {G},[\cdot ,\cdot ]_{\mathcal {G}})$ be a Lie ${A}$ -algebra. We say that $\mathcal {L}$ acts on $\mathcal {G}$ if a $\mathbb {K}$ -Lie algebra homomorphism $\rho :\mathcal {L}\,\rightarrow \,\mathsf {Der}_{\mathbb {K}}({\mathcal {G}})$ is given such that

$$ \begin{align*}\rho(ax)=a\rho(x),\quad\rho(x)(au)=a\rho(x)u+ \alpha(x)(a)u,\quad \forall a\in A, x\in \mathcal{L}, u\in\mathcal{G}.\end{align*} $$

Let $({A},\mathcal {L},[\cdot ,\cdot ]_{\mathcal {L}},\alpha )$ be a Lie–Rinehart algebra and $(\mathcal {G},[\cdot ,\cdot ]_{\mathcal {G}})$ be a Lie ${A}$ -algebra on which $\mathcal {L}$ acts via $\rho :\mathcal {L}\,\rightarrow \,\mathsf {Der}_{\mathbb {K}}({\mathcal {G}})$ . On the ${A}$ -module $\mathcal {L}\oplus \mathcal {G}$ , define a bracket operation $[\cdot ,\cdot ]_{\rho }$ by

$$ \begin{align*} [(x,u),(y,v)]_{\rho}=([x,y]_{\mathcal{L}},\rho(x)v-\rho(y)u+[u,v]_{\mathcal{G}}),\quad \forall x,y\in \mathcal{L}, u,v\in\mathcal{G}, \end{align*} $$

and define an ${A}$ -module homomorphism $\tilde {\alpha }:\mathcal {L}\oplus \mathcal {G}\to \mathsf {Der}_{\mathbb {K}}({A})$ by

$$ \begin{align*} \tilde{\alpha}(x,u)=\alpha(x),\quad \forall x\in \mathcal{L}, u\in\mathcal{G}. \end{align*} $$

Then $({A},\mathcal {L}\oplus \mathcal {G},[\cdot ,\cdot ]_{\rho },\tilde {\alpha })$ is a Lie–Rinehart algebra [Reference Casas, Ladra and Pirashvili6], which is called the semi-direct product of $\mathcal {L}$ and $\mathcal {G}$ and denoted by $\mathcal {L}\ltimes _{\rho }\mathcal {G}$ .

Note that the Lie algebra $\mathcal {L}\ltimes _{\rho }\mathcal {G}$ acts on the Lie algebra $(\mathcal {G},[\cdot ,\cdot ]_{\mathcal {G}})$ by

(9) $$ \begin{align} \tilde{\rho}(x,u)v=\rho(x)v,\quad\forall x\in\mathcal{L},~u,v\in\mathcal{G}. \end{align} $$

Then using Theorem 2.7 (b), we can easily verify the following result.

3.2. Left module categories over monoidal categories

Proposition 3.17. Let $({A},\mathcal {L},[\cdot ,\cdot ]_{\mathcal {L}},\alpha )$ be a Lie–Rinehart algebra and $\rho $ an action of $\mathcal {L}$ on a Lie ${A}$ -algebra $(\mathcal {G},[\cdot ,\cdot ]_{\mathcal {G}})$ . For a crossed homomorphism ${H}:\mathcal {L}\,\rightarrow \, \mathcal {G}$ between $\mathbb {K}$ -Lie algebras, we define a $\mathbb {K}$ -linear map $\iota _{H}: \mathcal {L} \to \mathcal {L}\ltimes _{\rho }\mathcal {G}$ by

$$ \begin{align*} \iota_{H}(x)=(x,{H} x),\quad \forall x\in \mathcal{L}. \end{align*} $$

Then $\iota _{H}$ is a Lie–Rinehart injective weak homomorphism from $\mathcal {L}$ to $\mathcal {L}\ltimes _{\rho }\mathcal {G}$ .

Proof. By Theorem 2.7 (b), we know that $\iota _{H}$ is a $\mathbb {K}$ -Lie algebra monomorphism. Moreover, for all $x\in \mathcal {L}$ , we have

$$ \begin{align*} \tilde{\alpha}(\iota_{H}(x))=\tilde{\alpha}(x,{H} x)=\alpha(x), \end{align*} $$

which implies that $\alpha =\tilde {\alpha }\circ \iota _{H}$ . Thus, $\iota _{H}$ is a Lie–Rinehart injective weak homomorphism.

Let $({A},\mathcal {L},[\cdot ,\cdot ]_{\mathcal {L}},\alpha )$ be a Lie–Rinehart algebra and $(\mathfrak g,[\cdot ,\cdot ]_{\mathfrak g})$ be a $\mathbb {K}$ -Lie algebra. Then $\mathcal {G}=\mathfrak g\otimes _{\mathbb {K}} {A}$ is a Lie ${A}$ -algebra, where the ${A}$ -module structure and the Lie bracket $[\cdot ,\cdot ]_{\mathcal {G}}$ are given by

$$ \begin{align*} a(g\otimes b)=g\otimes ab,\quad [g\otimes a,h\otimes b]_{\mathcal{G}}=[g,h]_{\mathfrak g}\otimes ab,\quad\forall a,b\in {A},~g,h\in \mathfrak g. \end{align*} $$

Moreover, the Lie–Rinehart algebra $({A},\mathcal {L},[\cdot ,\cdot ]_{\mathcal {L}},\alpha )$ acts on the Lie ${A}$ -algebra $\mathfrak g\otimes _{\mathbb {K}} {A}$ by $\alpha $ as follows:

(10) $$ \begin{align} \alpha(x)(g\otimes a)=g\otimes \alpha(x)(a),\quad \forall\ x\in \mathcal{L}, a\in{A}, g\in\mathfrak g. \end{align} $$

Consequently, we have the semidirect product Lie–Rinehart algebra $({A},\mathcal {L}\ltimes _{\alpha } (\mathfrak g\otimes _{\mathbb {K}}{A}),[\cdot ,\cdot ]_{\alpha },\tilde {\alpha })$ .

Let $({A},\mathcal {L},[\cdot ,\cdot ]_{\mathcal {L}},\alpha )$ be a Lie–Rinehart algebra and $({M};\rho )$ be a Lie–Rinehart weak representation of $({A},\mathcal {L},[\cdot ,\cdot ]_{\mathcal {L}},\alpha )$ . Let $(\mathfrak g,[\cdot ,\cdot ]_{\mathfrak g})$ be a $\mathbb {K}$ -Lie algebra and $(V;\theta )$ be a representation of $\mathfrak g$ . Then ${V}\otimes _{\mathbb {K}} M$ has a natural ${A}$ -module structure:

$$ \begin{align*} a(v\otimes m)=v\otimes am,\quad \forall\ a\in{A}, v\in V, m\in{M}. \end{align*} $$

We define a $\mathbb {K}$ -linear map $ \rho \boxplus \theta :\mathcal {L}\ltimes _{\alpha } (\mathfrak g\otimes _{\mathbb {K}} {A})\,\rightarrow \,\mathfrak {gl}_{\mathbb {K}}({V}\otimes _{\mathbb {K}} M)$ by

$$ \begin{align*} (\rho\boxplus\theta)(x,g\otimes a)(v\otimes m):=v\otimes \rho(x)m+\theta(g)v\otimes am \end{align*} $$

for all $x\in \mathcal {L},~a\in {A},~g\in \mathfrak g,~m\in {M},~v\in V$ .

Proof. Since $\rho :\mathcal {L}\,\rightarrow \,\mathfrak {D}({M})$ and $\theta :\mathfrak g\,\rightarrow \,\mathfrak {gl}(V)$ are $\mathbb {K}$ -Lie algebra homomorphisms, for all $a,b\in {A},x,y\in \mathcal {L},g,h\in \mathfrak g, m\in {M},v\in V$ , we have

$$ \begin{align*} &\quad\Big([(\rho\boxplus\theta)(x,g\otimes a),(\rho\boxplus\theta)(y,h\otimes b)]_{C}-(\rho\boxplus\theta)([(x,g\otimes a),(y,h\otimes b)]_{\alpha})\Big)(v\otimes m)\\ &=(\rho\boxplus\theta)(x,g\otimes a)\Big(v\otimes \rho(y)m+\theta(h)v\otimes bm \Big)-(\rho\boxplus\theta)(y,h\otimes b)\Big(v\otimes \rho(x)m+\theta(g)v\otimes am\Big)\\ &\quad -(\rho\boxplus\theta)\Big([x,y]_{\mathcal{L}},h\otimes \alpha(x)(b)-g\otimes \alpha(y)(a)+[g,h]_{\mathfrak g}\otimes ab\Big)(v\otimes m)\\ &= v\otimes \rho(x)(\rho(y)m)+\theta(g)v\otimes a(\rho(y)m)+\theta(h)v\otimes \rho(x)(bm)+ \theta(g)(\theta(h)v)\otimes a(bm)\\ &\quad -v\otimes \rho(y)(\rho(x)m)-\theta(h)v\otimes b(\rho(x)m)-\theta(g)v\otimes \rho(y)(am)- \theta(h)(\theta(g)v)\otimes b(am)\\ &\quad -v\otimes\rho([x,y]_{\mathcal{L}})m -\theta(h)v\otimes\alpha(x)(b)m+\theta(g)v\otimes \alpha(y)(a)m-\theta([g,h]_{\mathfrak g})v\otimes (ab)m\\ &=0. \end{align*} $$

Therefore, we deduce that $\rho \boxplus \theta $ is a $\mathbb {K}$ -Lie algebra homomorphism.

Furthermore, by $\rho (x)\in \mathfrak {D}({M})$ , we have

$$ \begin{align*} (\rho\boxplus\theta)(x,g\otimes b)\Big(a(v\otimes m)\Big)&=(\rho\boxplus\theta)(x,g\otimes b)(v\otimes am)\\ &=v\otimes \rho(x)(am)+ \theta(g)v\otimes a(bm)\\ &=v\otimes \Big(a\rho(x)(m)+\alpha(x)(a)m\Big)+\theta(g)v \otimes a(bm)\\ &=a\Big( (\rho\boxplus\theta)(x,g\otimes b)(v\otimes m)\Big)+\alpha(x)(a)(v\otimes m), \end{align*} $$

which implies that $(\rho \boxplus \theta )(x,g\otimes b)\in \mathfrak {D}(V\otimes _{\mathbb {K}}{M})$ and $\tilde {\alpha }=\mathrm {Pr}\circ (\rho \boxplus \theta )$ .

Therefore, $\rho \boxplus \theta :\mathcal {L}\ltimes _{\alpha } (\mathfrak g\otimes _{\mathbb {K}} {A})\,\rightarrow \,\mathfrak {D}(V\otimes _{\mathbb {K}}{M})$ is a Lie–Rinehart weak homomorphism.

Proof. Since $\rho $ is an ${A}$ -module homomorphism, we have

$$ \begin{align*} &\quad \Big((\rho\boxplus\theta)(b(x,g\otimes a))-b(\rho\boxplus\theta)(x,g\otimes a)\Big)(v\otimes m)\\ &=v\otimes \rho(bx)m+\theta(g)v\otimes (ba)m-b\Big(v\otimes \rho(x)m+\theta(g)v\otimes am\Big)\\ &=0, \quad \forall a,b\in{A},~x\in\mathcal{L},~g\in\mathfrak g,~m\in{M},~v\in V. \end{align*} $$

Thus, $\rho \boxplus \theta $ is also an ${A}$ -module homomorphism.

Before we give the main result of the article, we recall the notions of a monoidal category and a left module category over a monoidal category.

Definition 3.21 [Reference Etingof, Gelaki, Nikshych and Ostrik10]

A monoidal category is a $6$ -tuple $({\mathcal {C}},\otimes , a,\mathbf {1},l,r)$ that consists of the following data:

• • a category ${\mathcal {C}}$ ;

• • a bifunctor $\otimes :{\mathcal {C}}\times {\mathcal {C}}\to {\mathcal {C}}$ called the monoidal product;

• • a natural isomorphism $a:\otimes \circ (\otimes \times \textrm {Id}_{\mathcal {C}})\,\rightarrow \,\otimes \circ (\textrm {Id}_{\mathcal {C}}\times \otimes )$ called the associativity isomorphism;

• • an object $\mathbf {1}\in \mathrm {Ob}({\mathcal {C}})$ called the unit object;

• • a natural isomorphism $l:\otimes \circ (\mathbf {1}\times \textrm {Id}_{\mathcal {C}})\,\rightarrow \,\textrm {Id}_{\mathcal {C}}$ called the left unit isomorphism and a natural isomorphism $r:\otimes \circ (\textrm {Id}_{\mathcal {C}}\times \mathbf {1})\,\rightarrow \,\textrm {Id}_{\mathcal {C}}$ called the right unit isomorphism.

These data satisfy the following two axioms:

$(1)$ the pentagon axiom: the pentagon diagram

commutes for all $W,X,Y,Z\in \mathrm {Ob}({\mathcal {C}})$ .

$(2)$ the triangle axiom: the triangle diagram

commutes for all $X,Y\in \mathrm {Ob}({\mathcal {C}})$ .

The monoidal category ${\mathcal {C}}$ is strict if the associativity isomorphism, left unit isomorphism and right unit isomorphism $a,l,r$ are all identities.

Example 3.23. The category of representations $\mathsf {Rep}_{\mathbb {K}}(\mathfrak g)$ of a $\mathbb {K}$ -Lie algebra $\mathfrak g$ is a monoidal category: the monoidal product of $(V_{1};\theta _{1})$ and $(V_{2};\theta _{2})$ is defined by

$$ \begin{align*} (V_{1};\theta_{1})\otimes (V_{2};\theta_{2}):=(V_{1}\otimes V_{2};\theta_{1}\otimes \text{Id}_{V_{2}}+\text{Id}_{V_{1}}\otimes \theta_{2}), \end{align*} $$

and the unit object $\mathbf {1}$ is the 1-dimensional trivial representation $(\mathbb {K};0)$ of $\mathfrak g$ . Moreover, the associativity isomorphism

$$ \begin{align*} a_{(V_{1};\theta_{1}),(V_{2};\theta_{2}),(V_{3};\theta_{3})}:((V_{1};\theta_{1})\otimes (V_{2};\theta_{2}))\otimes (V_{3};\theta_{3})\,\rightarrow\, (V_{1};\theta_{1})\otimes ((V_{2};\theta_{2})\otimes (V_{3};\theta_{3})) \end{align*} $$

is defined by

(11) $$ \begin{align} a_{(V_{1};\theta_{1}),(V_{2};\theta_{2}),(V_{3};\theta_{3})}\Big((v_{1}\otimes v_{2})\otimes v_{3}\Big):=v_{1}\otimes (v_{2}\otimes v_{3}),\,\,\,\,\forall v_{i}\in V_{i},~i=1,2,3, \end{align} $$

and the left unit isomorphism $l_{(V;\theta )}$ and the right unit isomorphism $r_{(V;\theta )}$ are defined by

(12) $$ \begin{align} l_{(V;\theta)}(k\otimes v):=kv,\,\,\,\,r_{(V;\theta)}(v\otimes k):=kv,\,\,\,\,\forall k\in\mathbb{K},~v\in V.\qquad \end{align} $$

Definition 3.24 [Reference Etingof, Gelaki, Nikshych and Ostrik10]

Let $({\mathcal {C}},\otimes , a,\mathbf {1},l,r)$ be a monoidal category. A left module category over ${\mathcal {C}}$ is a category $\mathcal {M}$ equipped with a bifunctor $\otimes ^{\mathcal {M}}:{\mathcal {C}}\times \mathcal {M}\,\rightarrow \,\mathcal {M}$ , a natural isomorphism $a^{\mathcal {M}}:\otimes ^{\mathcal {M}}\circ (\otimes \times \text {Id}_{\mathcal {M}})\,\rightarrow \,\otimes ^{\mathcal {M}}\circ (\text {Id}_{\mathcal {C}}\times \otimes ^{\mathcal {M}})$ and a natural isomorphism $l^{\mathcal {M}}:\otimes ^{\mathcal {M}}\circ (\mathbf {1}\times \textrm {Id}_{\mathcal {M}})\,\rightarrow \,\textrm {Id}_{\mathcal {M}}$ such that the pentagon diagram

and the triangle diagram

commute for all $X,Y,Z\in \mathrm {Ob}({\mathcal {C}}),~M\in \mathrm {Ob}(\mathcal {M})$ .

Let $({A},\mathcal {L},[\cdot ,\cdot ]_{\mathcal {L}},\alpha )$ be a Lie–Rinehart algebra and $\mathfrak g$ be a $\mathbb {K}$ -Lie algebra. Let ${H}$ be a crossed homomorphism from the $\mathbb {K}$ -Lie algebra $\mathcal {L}$ to $\mathfrak g\otimes _{\mathbb {K}}{A}$ with respect to the action $\alpha $ given by (10). For all $x\in \mathcal {L}$ , we set $Hx=\sum _{i} x_{i}^{\mathfrak g}\otimes x^{A}_{i}$ or $Hx=x_{i}^{\mathfrak g}\otimes x^{A}_{i}$ for simplicity.

By Corollary 3.18 and Lemma 3.19, our main theorem can be stated as follows.

Proof. By Corollary 3.18 and Lemma 3.19, $({V}\otimes _{\mathbb {K}} M;(\rho \boxplus \theta )\circ \iota _{H})$ is a weak representation of $\mathcal {L}$ . Thus, $F_{H}$ is well-defined on the set of objects. To see that $F_{H}$ is also well-defined on the set of morphisms, we need to show that the linear map $\psi \otimes \phi :V\otimes M\,\rightarrow \, V^{\prime }\otimes M^{\prime }$ is indeed a homomorphism from $({V}\otimes _{\mathbb {K}} M;(\rho \boxplus \theta )\circ \iota _{H})$ to $({V}^{\prime }\otimes _{\mathbb {K}} M^{\prime };(\rho ^{\prime }\boxplus \theta ^{\prime })\circ \iota _{H})$ . In fact, for all $a\in A,~v\in V,~m\in M$ , we have

$$ \begin{align*} (\psi\otimes \phi)\Big(a(v\otimes m)\Big)-a\Big((\psi\otimes \phi)(v\otimes m)\Big) &=(\psi\otimes \phi)(v\otimes am)-a\Big(\psi(v)\otimes \phi(m)\Big)\\ &=\psi(v)\otimes \phi(am)\otimes -\psi(v)\otimes a\phi(m)\\ &=0.\\[-15pt] \end{align*} $$

For all $x\in \mathfrak g,~v\in V$ and $m\in M$ , we have

$$ \begin{align*} &(\psi\otimes \phi)\Big(\big((\rho\boxplus\theta)\iota_{H}(x)\big)(v\otimes m)\Big)-\Big((\rho^{\prime}\boxplus\theta^{\prime})\iota_{H}(x)\Big)\Big((\psi\otimes \phi)(v\otimes m)\Big)\\ &=(\psi\otimes \phi)\Big(\big((\rho\boxplus\theta)(x,x_{i}^{\mathfrak g}\otimes x_{i}^{A})\big)(v\otimes m)\Big)-\Big((\rho^{\prime}\boxplus\theta^{\prime})(x,x_{i}^{\mathfrak g}\otimes x_{i}^{A})\Big)\Big(\psi(v)\otimes \phi(m)\Big)\\ &=(\psi\otimes \phi)\Big(v\otimes\rho(x)m+\theta(x_{i}^{\mathfrak g})v\otimes x_{i}^{A}m\Big)-\Big(\psi(v)\otimes \rho^{\prime}(x)\phi(m)+\theta^{\prime}(x_{i}^{\mathfrak g})\psi(v)\otimes x_{i}^{A}\phi(m)\Big)\\ &=\Big(\psi(v)\otimes\phi(\rho(x)m)+\psi(\theta(x_{i}^{\mathfrak g})v)\otimes \phi(x_{i}^{A}m)\Big)\\&\quad -\Big(\psi(v)\otimes \rho^{\prime}(x)\phi(m)+\theta^{\prime}(x_{i}^{\mathfrak g})\psi(v)\otimes x_{i}^{A}\phi(m)\Big)\\ &=0.\\[-15pt] \end{align*} $$

Thus, we obtain that $F_{H}(\psi ,\phi )=\psi \otimes \phi $ is a homomorphism of the weak representations. Moreover, by straightforward computations, we deduce that $F_{H}$ preserves identity morphisms and composite morphisms. Therefore, $F_{H}$ is a bifunctor.

Let $(V_{1};\theta _{1})$ and $(V_{2};\theta _{2})$ be representations of the $\mathbb {K}$ -Lie algebra $\mathfrak g$ and $({M};\rho )$ be a weak representation of the Lie–Rinehart algebra $({A},\mathcal {L},[\cdot ,\cdot ]_{\mathcal {L}},\alpha )$ . For all $b\in A,~v_{1}\in V_{1},~v_{2}\in V_{2}$ and $m\in M$ , we have

$$ \begin{align*} &\quad a_{(V_{1};\theta_{1}),(V_{2};\theta_{2}),(M;\rho)}\Big(b\big((v_{1}\otimes v_{2})\otimes m\big)\Big)-ba_{(V_{1};\theta_{1}),(V_{2};\theta_{2}),(M;\rho)}\Big((v_{1}\otimes v_{2})\otimes m\Big)\\ &=a_{(V_{1};\theta_{1}),(V_{2};\theta_{2}),(M;\rho)}\Big((v_{1}\otimes v_{2})\otimes bm\Big)-b\Big(v_{1}\otimes (v_{2}\otimes m)\Big)\\ &=v_{1}\otimes (v_{2}\otimes bm)-v_{1}\otimes b(v_{2}\otimes m)\\ &=0. \end{align*} $$

For all $x\in \mathcal {L},~v_{1}\in V_{1},~v_{2}\in V_{2}$ and $m\in M$ , we have

$$ \begin{align*} &\quad a_{(V_{1};\theta_{1}),(V_{2};\theta_{2}),(M;\rho)}\Big(\Big(\Big(\rho\boxplus\Big(\theta_{1}\otimes \text{Id}_{V_{2}}+\text{Id}_{V_{1}}\otimes \theta_{2}\Big)\Big)\iota_{H}(x)\Big)\big((v_{1}\otimes v_{2})\otimes m\big)\Big)\\ &\quad -\Big(\Big(\Big((\rho\boxplus\theta_{2})\circ\iota_{H}\Big)\boxplus\theta_{1}\Big)\iota_{H}(x)\Big)a_{(V_{1};\theta_{1}),(V_{2};\theta_{2}),(M;\rho)}((v_{1}\otimes v_{2})\otimes m)\\ &=a_{(V_{1};\theta_{1}),(V_{2};\theta_{2}),(M;\rho)}\Big(\Big(\Big(\rho\boxplus\Big(\theta_{1}\otimes \text{Id}_{V_{2}}+\text{Id}_{V_{1}}\otimes \theta_{2}\Big)\Big)(x,x_{i}^{\mathfrak g}\otimes x_{i}^{A})\Big)((v_{1}\otimes v_{2})\otimes m)\Big)\\ &\quad -\Big(\Big(\Big((\rho\boxplus\theta_{2})\circ\iota_{H}\Big)\boxplus\theta_{1}\Big)\iota_{H}(x)\Big)(v_{1}\otimes (v_{2}\otimes m))\\ &=a_{(V_{1};\theta_{1}),(V_{2};\theta_{2}),(M;\rho)}\Big((v_{1}\otimes v_{2})\otimes\rho(x)m+(\theta_{1}\otimes \text{Id}_{V_{2}}+\text{Id}_{V_{1}}\otimes \theta_{2})(x_{i}^{\mathfrak g})(v_{1}\otimes v_{2})\otimes x_{i}^{A}m\Big)\\&\quad -\Big(\Big(\Big((\rho\boxplus\theta_{2})\circ\iota_{H}\Big)\boxplus\theta_{1}\Big)(x,x_{i}^{\mathfrak g}\otimes x_{i}^{A})\Big)(v_{1}\otimes (v_{2}\otimes m))\\ &=v_{1}\otimes (v_{2}\otimes\rho(x)m)+\theta_{1}(x_{i}^{\mathfrak g})v_{1}\otimes (v_{2}\otimes x_{i}^{A}m)+v_{1}\otimes (\theta_{2}(x_{i}^{\mathfrak g})v_{2}\otimes x_{i}^{A}m)\\ &\quad -\Big(v_{1}\otimes\big((\rho\boxplus\theta_{2})\iota_{H}(x)(v_{2}\otimes m)\big)+\theta_{1}(x_{i}^{\mathfrak g})v_{1}\otimes x_{i}^{A}(v_{2}\otimes m)\Big)\\ &=v_{1}\otimes (v_{2}\otimes\rho(x)m)+\theta_{1}(x_{i}^{\mathfrak g})v_{1}\otimes (v_{2}\otimes x_{i}^{A}m)+v_{1}\otimes (\theta_{2}(x_{i}^{\mathfrak g})v_{2}\otimes x_{i}^{A}m)\\ &\quad -\Big(v_{1}\otimes\big(v_{2}\otimes\rho(x)m+\theta_{2}(x_{i}^{\mathfrak g})v_{2}\otimes x_{i}^{A}m\big)+\theta_{1}(x_{i}^{\mathfrak g})v_{1}\otimes (v_{2}\otimes x_{i}^{A}m)\Big)\\ &=0. \end{align*} $$

Thus, we obtain that $a_{(V_{1};\theta _{1}),(V_{2};\theta _{2}),(M;\rho )}$ is a homomorphism of the weak representations. Moreover, by straightforward computations, we obtain that $a_{(V_{1};\theta _{1}),(V_{2};\theta _{2}),(M;\rho )}$ is a natural isomorphism and satisfies the pentagon diagram in Definition 3.24.

Let $({M};\rho )$ be a weak representation of the Lie–Rinehart algebra $({A},\mathcal {L},[\cdot ,\cdot ]_{\mathcal {L}},\alpha )$ . We have

$$ \begin{align*} l_{({M};\rho)}(a(k\otimes m))&=l_{({M};\rho)}(k\otimes am)=k(am)=a(km)\\&=al_{({M};\rho)}(k\otimes m), \quad \forall a\in A,~k\in\mathbb{K}, ~m\in M. \end{align*} $$

For all $x\in \mathcal {L},~k\in \mathbb {K}$ and $m\in M$ , we have

$$ \begin{align*} &\quad l_{({M};\rho)}\Big(\big((\rho\boxplus 0)\iota_{H}(x)\big)(k\otimes m)\Big)-\rho(x)\Big(l_{({M};\rho)}(k\otimes m)\Big)\\ &=l_{({M};\rho)}\Big(\big((\rho\boxplus 0)(x,x_{i}^{\mathfrak g}\otimes x_{i}^{A})\big)(k\otimes m)\Big)-\rho(x)(km) =l_{({M};\rho)}\Big(k\otimes\rho(x)m\Big)-\rho(x)(km)\\ &=k(\rho(x)m)-\rho(x)(km) =0. \end{align*} $$

Thus, we deduce that $l_{({M};\rho )}$ is a homomorphism of weak representations. Moreover, by straightforward computations, we obtain that $l_{({M};\rho )}$ is a natural isomorphism and satisfies the triangle diagram in Definition 3.24. The proof is finished.

Since $({A};\alpha )$ is a representation of a Lie–Rinehart algebra $({A},\mathcal {L},[\cdot ,\cdot ]_{\mathcal {L}},\alpha )$ , which is known as the natural representation, we obtain the following result.

Corollary 3.27. Let $({A},\mathcal {L},[\cdot ,\cdot ]_{\mathcal {L}},\alpha )$ be a Lie–Rinehart algebra, $(\mathfrak g,[\cdot ,\cdot ]_{\mathfrak g})$ a $\mathbb {K}$ -Lie algebra and ${H}$ a crossed homomorphism from $\mathcal {L}$ to $\mathfrak g\otimes _{\mathbb {K}}{A}$ . Then we have a functor

$$ \begin{align*}\begin{aligned} &F_{{H}}^{A}:\mathsf{Rep}_{\mathbb{K}}(\mathfrak g)\to \mathsf{WRep}_{\mathbb{K}}(\mathcal{L}),\\ & (V;\theta)\mapsto (V\otimes_{\mathbb{K}}{A}; ({\alpha}\boxplus \theta)\circ\iota_{H}),\quad\forall(V;\theta)\in \mathsf{Rep}_{\mathbb{K}}(\mathfrak g).\end{aligned} \end{align*} $$

We can also have a very useful functor on $\mathsf {WRep}_{\mathbb {K}}(\mathcal {L})$ as follows.

Corollary 3.28. Let $({A},\mathcal {L},[\cdot ,\cdot ]_{\mathcal {L}},\alpha )$ be a Lie–Rinehart algebra, $(\mathfrak g,[\cdot ,\cdot ]_{\mathfrak g})$ a $\mathbb {K}$ -Lie algebra, ${H}$ a crossed homomorphism from $\mathcal {L}$ to $\mathfrak g\otimes _{\mathbb {K}}{A}$ and $(V;\theta )$ a given representation of $\mathfrak g$ . Then we have a functor

$$ \begin{align*}\begin{aligned} &F_{H}^{\theta} :\mathsf{WRep}_{\mathbb{K}}(\mathcal{L})\to \mathsf{WRep}_{\mathbb{K}}(\mathcal{L}),\\ & ({M};\rho)\mapsto (V\otimes_{\mathbb{K}}{M} ;({\rho}\boxplus \theta)\circ\iota_{H}),\quad\forall({M};\rho)\in\mathsf{WRep}_{\mathbb{K}}(\mathcal{L}).\end{aligned} \end{align*} $$

A special but very interesting case of the above result is that $(V;\theta )=(\mathfrak g;\mathsf {ad})$ . In the next section we will show that Corollary 3.28 is a very efficient way to construct interesting modules from easy modules.

3.3. Admissible representations of Leibniz pairs

In this subsection, we introduce the notion of an admissible representation of a Leibniz pair. In the sequel, A is always a commutative associative algebra. The notion of a Leibniz pair was originally given in [Reference Flato, Gerstenhaber and Voronov11].

We denote a Leibniz pair by $(A,\mathcal {S},[\cdot ,\cdot ]_{\mathcal {S}},\beta )$ or simply by $\mathcal {S}$ .

Definition 3.30. An admissible representation of a Leibniz pair $(A,\mathcal {S},[\cdot ,\cdot ]_{\mathcal {S}},\beta )$ consists of an A-module M and a $\mathbb {K}$ -Lie algebra homomorphism $\rho :\mathcal {S}\to \mathfrak {gl}_{\mathbb {K}}(M)$ such that

(17) $$ \begin{align}\rho(x)(am)=a\rho(x)m+\beta(x)(a)m,\quad \forall x\in \mathcal{S}, a\in A, m\in M.\end{align} $$

Admissible representations of Leibniz pairs are like weak representations of Lie–Rinehart algebras. We use $\mathsf {ARep}_{\mathbb {K}}(\mathcal {S})$ to denote the category of admissible representations of $\mathcal {S}$ .

It is straightforward to obtain the following result.

Lemma 3.32. Let $(A,\mathcal {S},[\cdot ,\cdot ]_{\mathcal {S}},\beta )$ be a Leibniz pair, M an A-module and $\rho :\mathcal {S}\to \mathfrak {gl}_{\mathbb {K}}(M)$ a $\mathbb {K}$ -linear map. Then $(M;\rho )$ is an admissible representation of $\mathcal {S}$ if and only if $(A\ltimes M, \mathcal {S}\oplus M,[\cdot ,\cdot ]_{\rho },\hat {\beta })$ is a Leibniz pair, where $A\ltimes M$ is the commutative associative algebra given in Remark 3.6, $[\cdot ,\cdot ]_{\rho }$ is the semidirect product Lie bracket and $\hat {\beta }:\mathcal {S}\oplus M\,\rightarrow \,\mathsf {Der}_{\mathbb {K}}(A\ltimes M)$ is defined by

$$ \begin{align*} \hat{\beta}(x,m)(a,n):=(\beta(x)a,\rho(x)n),\quad\forall x\in \mathcal{S}, a\in A, m,n\in M. \end{align*} $$

It is obvious that any Lie–Rinehart algebra is a Leibniz pair. A weak representation of a Lie–Rinehart algebra is naturally an admissible representation of the underlying Leibniz pair. We have the following category equivalence:

$$ \begin{align*} \mathsf{WRep}_{\mathbb{K}}(\mathcal{L})\rightleftarrows \mathsf{ARep}_{\mathbb{K}}(\mathcal{L}), \end{align*} $$

where the right-hand side $\mathcal {L}$ is considered as a Leibniz pair.

Conversely, given a Leibniz pair $(A,\mathcal {S},[\cdot ,\cdot ]_{\mathcal {S}},\beta )$ , we also have an action Lie–Rinehart algebra $(A,\mathcal {S}\otimes _{\mathbb {K}} A,[\cdot ,\cdot ],\alpha )$ , where the ${A}$ -module structure and the $\mathbb {K}$ -Lie bracket $[\cdot ,\cdot ]$ are given by

$$ \begin{align*} a(x\otimes b)=x\otimes ab,\quad [x\otimes a,y\otimes b]=[x,y]_{\mathcal{S}}\otimes ab+y\otimes (a\beta(x)b)-x\otimes (b\beta(y)a), \end{align*} $$

and an A-module homomorphism $\alpha :\mathcal {S}\otimes _{\mathbb {K}} A\,\rightarrow \,\mathsf {Der}_{\mathbb {K}}({A})$ is defined by $ \alpha (x\otimes a):=a\beta (x) $ for all $a,b\in {A},~x,y\in \mathcal {S}.$ Furthermore, we obtain the following result.

Proposition 3.33. Let $(M;\rho )$ be an admissible representation of a Leibniz pair $(A,\mathcal {S},[\cdot ,\cdot ]_{\mathcal {S}},\beta )$ . Define $\overline {\rho }:\mathcal {S}\otimes _{\mathbb {K}} A\,\rightarrow \, \mathfrak {gl}_{\mathbb {K}}(M)$ by

$$ \begin{align*} \overline{\rho}(x\otimes a):=a\rho(x),\quad \forall x\in\mathcal{S}, a\in A. \end{align*} $$

Then $(M;\overline {\rho })$ is a representation of the Lie–Rinehart algebra $(A,\mathcal {S}\otimes _{\mathbb {K}} A,[\cdot ,\cdot ],\alpha )$ .

Proof. First, it is obvious that $ \overline {\rho }$ is an A-module homomorphism from $\mathcal {S}\otimes _{\mathbb {K}} A$ to $\mathfrak {gl}_{\mathbb {K}}(M)$ . Then it is straightforward to deduce that $\overline {\rho }$ is a $\mathbb {K}$ -Lie algebra homomorphism. Finally, by (17), we deduce that

$$ \begin{align*} \overline{\rho}(x\otimes a)(bm)=a\rho(x)(bm)=a\big(b\rho(x)m+\beta(x)(b)m\big)=b\overline{\rho}(x\otimes a)m+\alpha(x\otimes a)(b)m. \end{align*} $$

Thus, $(M;\overline {\rho })$ is a representation of the Lie–Rinehart algebra $\mathcal {S}\otimes _{\mathbb {K}} A$ .

Remark 3.34. We have the following category equivalence if A is unital:

$$ \begin{align*} \mathsf{ARep}_{\mathbb{K}}(\mathcal{S})\rightleftarrows \mathsf{Rep}(\mathcal{S}\otimes_{\mathbb{K}} A). \end{align*} $$

First, the construction of Proposition 3.33 can be easily enhanced to a functor. In fact, assume that $\phi :{M}\to {M}^{\prime }$ is a homomorphism of admissible representations of a Leibniz pair $(A,\mathcal {S},[\cdot ,\cdot ]_{\mathcal {S}},\beta )$ ; then it is straightforward to deduce that

$$ \begin{align*}\phi\circ\overline{\rho}(x\otimes a)=\overline{\rho^{\prime}}(x\otimes a)\circ\phi\end{align*} $$

for all $x\in \mathcal {S}, a\in A.$ Thus, $\phi :{M}\to {M}^{\prime }$ is also a homomorphism of representations of the Lie–Rinehart algebra $\mathcal {S}\otimes _{\mathbb {K}} A.$ So we obtain a functor $P:\mathsf {ARep}_{\mathbb {K}}(\mathcal {S})\to \mathsf {Rep}(\mathcal {S}\otimes _{\mathbb {K}} A)$ , which is defined on the sets of objects and morphisms respectively by

$$ \begin{align*} P(M;\rho)&=(M;\overline{\rho}),\\ P(\phi:{M}\to{M}^{\prime})&=(\phi:{M}\to{M}^{\prime}). \end{align*} $$

Conversely, let $(M;\rho )$ be a representation of the Lie–Rinehart algebra $\mathcal {S}\otimes _{\mathbb {K}} A.$ Define $\widetilde { \rho }:\mathcal {S} \,\rightarrow \, \mathfrak {gl}_{\mathbb {K}}(M)$ by

$$ \begin{align*} \widetilde{\rho}(x ):= \rho(x\otimes 1),\quad \forall x\in\mathcal{S}. \end{align*} $$

Similar to the above discussion, this can also be enhanced to a functor and give the equivalence between the categories $\mathsf {ARep}_{\mathbb {K}}(\mathcal {S})$ and $ \mathsf {Rep}(\mathcal {S}\otimes _{\mathbb {K}} A)$ .

Let $(A,\mathcal {S},[\cdot ,\cdot ]_{\mathcal {S}},\beta )$ be a Leibniz pair and $\mathfrak h$ be a $\mathbb {K}$ -Lie algebra. Then $(A,\mathcal {S}\oplus (\mathfrak h\otimes _{\mathbb {K}} {A}),[\cdot ,\cdot ],\tilde {\beta })$ is a Leibniz pair, where the $\mathbb {K}$ -Lie algebra structure on $\mathcal {S}\oplus (\mathfrak h\otimes _{\mathbb {K}} {A})$ is given by

$$ \begin{align*} &[(x,g\otimes a),(y,h\otimes b)]\\&\quad =([x,y]_{\mathcal{S}},h\otimes\beta(x)(b)-g\otimes\beta(y)(a)+[g,h]_{\mathfrak h}\otimes ab),\quad \forall x,y\in \mathcal{S}, g\otimes a, h\otimes b\in\mathfrak h\otimes_{\mathbb{K}} A, \end{align*} $$

and $\tilde {\beta }:\mathcal {S}\oplus (\mathfrak h\otimes _{\mathbb {K}} {A})\,\rightarrow \, \mathsf {Der}_{\mathbb {K}}(A)$ is given by

$$ \begin{align*} \tilde{\beta}(x,g\otimes a)=\beta(x). \end{align*} $$

Denote this Leibniz pair by $\mathcal {S}\ltimes _{\beta } (\mathfrak h\otimes _{\mathbb {K}} {A})$ .

Let $({M};\rho )$ be an admissible representation over $\mathcal {S}$ and $(V;\theta )$ be a representation of a $\mathbb {K}$ -Lie algebra $\mathfrak h$ . Then ${V}\otimes _{\mathbb {K}} M$ has a natural ${A}$ -module structure:

$$ \begin{align*} a(v\otimes m)=v\otimes am,\quad \forall\ a\in{A}, v\in V, m\in{M}. \end{align*} $$

We define a $\mathbb {K}$ -linear map $ \rho \boxplus \theta :\mathcal {S}\ltimes _{\beta } (\mathfrak h\otimes _{\mathbb {K}} {A})\,\rightarrow \,\mathfrak {gl}_{\mathbb {K}}({V}\otimes _{\mathbb {K}} M)$ by

$$ \begin{align*} (\rho\boxplus\theta)(x,g\otimes a)(v\otimes m):=v\otimes \rho(x)m+\theta(g)v\otimes am, \quad \forall x\in\mathcal{S},~a\in{A},~g\in\mathfrak h,~m\in{M},~v\in V. \end{align*} $$

Then it is straightforward to verify the following result.

Let ${H}$ be a crossed homomorphism from the $\mathbb {K}$ -Lie algebra $\mathcal {S}$ to $ \mathfrak h\otimes _{\mathbb {K}}{A}$ . Then we have the Lie algebra homomorphism

$$ \begin{align*}\begin{aligned} &\iota_{H}: \mathcal{S} \to \mathcal{S}\ltimes_{\beta} (\mathfrak h\otimes_{\mathbb{K}} {A})\\ &\iota_{H}(x)=(x,{H} x),\quad \forall x\in \mathcal{S}.\end{aligned} \end{align*} $$

Similar to Theorem 3.26, we have the following result.

Theorem 3.36. Any crossed homomorphism $H:\mathcal {S}\,\rightarrow \,\mathfrak h\otimes _{\mathbb {K}}{A}$ induces a left module category structure of the category of admissible representations $\mathsf {ARep}_{\mathbb {K}}(\mathcal {S})$ over the monoidal category $\mathsf {Rep}_{\mathbb {K}}(\mathfrak h)$ ,

$$ \begin{align*}\begin{aligned} &\mathcal{F}_{H}:\mathsf{Rep}_{\mathbb{K}}(\mathfrak h)\times \mathsf{ARep}_{\mathbb{K}}(\mathcal{S}) \to \mathsf{ARep}_{\mathbb{K}}(\mathcal{S})\\ &\mathcal{F}_{H}\Big((V;\theta),({M};\rho)\Big)=({V}\otimes_{\mathbb{K}} M;(\rho\boxplus\theta)\circ\iota_{H}). \end{aligned}\end{align*} $$

Proof. We verify that the representation $({V}\otimes _{\mathbb {K}} M;(\rho \boxplus \theta )\circ \iota _{H})$ satisfies (17). For any $x\in \mathcal {S}, a\in A, v\in V, m\in M$ . Suppose $H(x)=Hx=\sum _{i} x_{i}^{\mathfrak h}\otimes x^{A}_{i}$ or $Hx=x_{i}^{\mathfrak h}\otimes x^{A}_{i}$ for simplicity. Then

$$ \begin{align*}\begin{aligned} ((\rho\boxplus\theta)\circ\iota_{H})(x)(a(v\otimes m)) =&(\rho\boxplus\theta) (x, x_{i}^{\mathfrak h}\otimes x^{A}_{i})(v\otimes am)\\ =&v\otimes \rho(x)(am)+ \theta(x_{i}^{\mathfrak h}) v\otimes x^{A}_{i}am\\ =&a\Big(v\otimes \rho(x)(m)+ \theta(x_{i}^{\mathfrak h}) v\otimes x^{A}_{i}m\Big)+\beta(x)(a)(v\otimes m)\\ =&a((\rho\boxplus\theta)\circ\iota_{H})(x)(v\otimes m)+\beta(x)(a)(v\otimes m).\end{aligned} \end{align*} $$

The proof is similar to Theorem 3.26, so the details will be omitted.

Since $({A};\beta )$ is an admissible representation of a Leibniz pair $(A,\mathcal {S},[\cdot ,\cdot ]_{\mathcal {S}},\beta )$ , we obtain the following result.

Corollary 3.37. Let $(A,\mathcal {S},[\cdot ,\cdot ]_{\mathcal {S}},\beta )$ be a Leibniz pair, $(\mathfrak h,[\cdot ,\cdot ]_{\mathfrak h})$ a $\mathbb {K}$ -Lie algebra and ${H}$ a crossed homomorphism from $\mathcal {S}$ to $\mathfrak h\otimes _{\mathbb {K}}{A}$ . Then we have a functor

$$ \begin{align*}\begin{aligned} &\mathcal{F}_{{H}}^{A}:\mathsf{Rep}_{\mathbb{K}}(\mathfrak h)\to \mathsf{ARep}_{\mathbb{K}}(\mathcal{S}),\\ & (V;\theta)\mapsto (V\otimes_{\mathbb{K}}{A}; ({\beta}\boxplus \theta)\circ\iota_{H}),\quad\forall(V;\theta)\in \mathsf{Rep}_{\mathbb{K}}(\mathfrak h).\end{aligned} \end{align*} $$

We can also have a very useful functor on $\mathsf {WRep}_{\mathbb {K}}(\mathcal {L})$ as follows.

Corollary 3.38. Let $(A,\mathcal {S},[\cdot ,\cdot ]_{\mathcal {S}},\beta )$ be a Leibniz pair, $(\mathfrak h,[\cdot ,\cdot ]_{\mathfrak h})$ a $\mathbb {K}$ -Lie algebra and ${H}$ a crossed homomorphism from $\mathcal {S}$ to $\mathfrak h\otimes _{\mathbb {K}}{A}$ and $(V;\theta )$ a given representation of $\mathfrak h$ . Then we have a functor

$$ \begin{align*}\begin{aligned} &\mathcal{F}_{H}^{\theta} :\mathsf{ARep}_{\mathbb{K}}(\mathcal{S})\to \mathsf{ARep}_{\mathbb{K}}(\mathcal{S}),\\ & ({M};\rho)\mapsto (V\otimes_{\mathbb{K}}{M} ;({\rho}\boxplus \theta)\circ\iota_{H}),\quad\forall({M};\rho)\in\mathsf{ARep}_{\mathbb{K}}(\mathcal{S}).\end{aligned} \end{align*} $$

A special but very interesting case of the above result is that $(V;\theta )=(\mathfrak h;\mathsf {ad})$ .

According to Corollaries 3.27 and 3.37, the bifunctors $F_{H}$ in Theorem 3.26 and $\mathcal {F}_{H}$ given in Theorem 3.36 are the actions of monoidal categories.

4.1. Shen–Larsson functors of Witt type

For $n\geq 1$ , recall the Witt algebra $\mathcal {W}_{n}=\mathsf {Der} (A_{n})$ over the Laurent polynomial algebra $A_{n}=\mathbb {C}[x_{1}^{\pm 1}, \cdots ,x_{n}^{\pm 1}]$ , which can be interpreted as the Lie algebra of (complex-valued) polynomial vector fields on an n-dimensional torus. Let $\partial _{i}=\frac {\partial }{\partial x_{i}}$ be the partial derivation with respect to the variable $x_{i}$ for $i=1,2,\dots ,n$ , and denote $d_{i}=x_{i}\partial _{i}$ and $x^{r}=x_{1}^{r_{1}}x_{2}^{r_{2}}\cdots x_{n}^{r_{n}}$ for $r=(r_{1},r_{2},\cdots , r_{n})^{T}\in \mathbb {Z}^{n}$ . Then

$$ \begin{align*}\mathcal{W}_{n}= {\text{span}}\{x^{r}d_{i}\mid r\in\mathbb{Z}^{n}, 1\leq i\leq n\}\end{align*} $$

with the Lie bracket

$$ \begin{align*} [x^{r}d_{i},x^{s}d_{j}]_{\mathcal{W}_{n}}=s_{i}x^{r+s}d_{j}-r_{j}x^{r+s}d_{i},\quad \forall\,1\leq i,j\leq n, r,s\in \mathbb {Z}^{n}. \end{align*} $$

Obviously, $(A_{n},\mathcal {W}_{n}, [\cdot ,\cdot ]_{\mathcal {W}_{n}},\textrm {Id})$ is a Lie–Rinehart algebra. Certainly, $ (A_{n}; \textrm {Id})$ is the natural representation of the Lie–Rinehart algebra $(A_{n},\mathcal {W}_{n}, [\cdot ,\cdot ]_{\mathcal {W}_{n}},\textrm {Id})$ . Let $\mathfrak g=\mathfrak {gl}_{n}$ be the Lie algebra of all $n \times n$ complex matrices. Then $\mathcal {G}=\mathfrak {gl}_{n}\otimes A_{n}$ is a Lie $A_{n}$ -algebra. For $1\leq i, j \leq n$ , we use $E_{ij}$ to denote the $n\times n$ matrix with $1$ at the $(i, j)$ entry and zeros elsewhere.

Lemma 4.1. The linear map $H:\mathcal {W}_{n}\to \mathfrak {gl}_{n}\otimes A_{n}$ defined by

$$ \begin{align*} H(x^{r}d_{j})= \sum_{i=1}^{n} r_{i} E_{ij}\otimes x^{r},\quad\forall r\in \mathbb{Z}^{n}, 1\leq j\leq n \end{align*} $$

is a crossed homomorphism from $\mathcal {W}_{n}$ to $\mathfrak {gl}_{n}\otimes A_{n}$ .

By Lemma 4.1 and Corollary 3.27, we obtain the following result.

Corollary 4.2. We have a functor $F_{{H}}^{A_{n}}:\mathsf {Rep}_{\mathbb {C}}(\mathfrak {gl}_{n})\to \mathsf {WRep}_{\mathbb {C}}(\mathcal {W}_{n})$ given by

$$ \begin{align*} F_{H}^{A_{n}}(V;\theta)=(V\otimes_{\mathbb{C}}{A_{n}}; (\mathrm{Id}\boxplus \theta)\circ\iota_{H}),\quad\forall(V;\theta)\in \mathsf{Rep}_{\mathbb{C}}(\mathfrak {gl}_{n}). \end{align*} $$

Let ${\mathcal {A}}_{n}=\mathbb {C}[x_{1}^{\pm 1},\cdots ,x_{n}^{\pm 1},\partial _{1},\cdots ,\partial _{n}]$ be the Weyl algebra, which is the universal enveloping algebra of the Lie–Rinehart algebra $(A_{n},\mathcal {W}_{n}, [\cdot ,\cdot ]_{\mathcal {W}_{n}},\textrm {Id})$ . Let $(P;\rho )$ be a representation of $\mathcal {A}_{n}$ . It is obvious that $(P;\rho |_{\mathcal {W}_{n}})$ is a $\mathcal {W}_{n}$ -module. By Lemma 4.1 and Corollary 3.28, we obtain the following result.

Corollary 4.3. We have a functor $F_{{H}}^{P}:\mathsf {Rep}_{\mathbb {C}}(\mathfrak {gl}_{n})\to \mathsf {WRep}_{\mathbb {C}}(\mathcal {W}_{n})$ given by

$$ \begin{align*} F_{H}^{P}(V;\theta)=(V\otimes_{\mathbb{C}}{P}; ({\rho|_{\mathcal{W}_{n}}}\boxplus\theta )\circ\iota_{H}),\quad\forall(V;\theta)\in \mathsf{Rep}_{\mathbb{C}}(\mathfrak {gl}_{n}). \end{align*} $$

Next we take $\mathfrak g=\mathbb {C}$ , the 1-dimensional trivial Lie algebra. Let $p=(p_{1},p_{2},\cdots ,p_{n})\in \mathbb {C}[t_{1}^{\pm 1}]\times \mathbb {C}[t_{2}^{\pm 1}]\times \cdots \times \mathbb {C}[t_{n}^{\pm 1}], q\in \mathbb {C}$ . Similar to the automorphism $\sigma _{b}$ in Section 2 of [Reference Tan and Zhao45], we can easily see that the linear map

$$ \begin{align*}\begin{aligned} &\mathcal{W}_{n}\to \mathcal{W}_{n}\ltimes_{\text{Id}} A_{n},\\ &x^{r}d_{i}\mapsto x^{r}(d_{i}+p_{i})+q r_{i}x^{r}, \end{aligned}\end{align*} $$

is a Lie algebra homomorphism. By Theorem 2.7, we see that the linear map

$$ \begin{align*}\begin{aligned} H_{p,q}: &\mathcal{W}_{n}\to \mathfrak g\otimes A_{n} \cong A_{n},\\ & x^{r}d_{i}\mapsto (p_{i}+q r_{i})x^{r},\quad\forall r\in \mathbb{Z}^{n}, 1\leq i\leq n, \end{aligned}\end{align*} $$

is a crossed homomorphism from $\mathcal {W}_{n}$ to $A_{n}$ . In fact, $H_{p,q}\in \mathsf {Der}_{\mathbb {C}}(\mathcal {W}_{n},A_{n})$ . By Lemma 4.1 and Corollary 3.28, we obtain the following result.

Corollary 4.5. We have a functor $F_{p,q} :\mathsf {WRep}_{\mathbb {C}}(\mathcal {W}_{n})\to \mathsf {WRep}_{\mathbb {C}}(\mathcal {W}_{n})$ defined by

$$ \begin{align*} F_{p,q}({M};\rho)=({M} ;{\rho}\circ\iota_{H_{p,q}}),\quad\forall({M};\rho)\in\mathsf{WRep}_{\mathbb{C}}(\mathcal{W}_{n}). \end{align*} $$

4.2. Shen–Larsson functors of divergence zero type

In this section we assume that $n\ge 2$ . Let us recall the divergence map $ \mathrm {div}: \mathcal {W}_{n}\to A_{n}$ with $ x^{r}d_{i}\mapsto r_{i}x^{r},$ for all $r\in \mathbb {Z}^{n}.$ It is well-known that

$$ \begin{align*} \mathcal{S}_{n}=\{w\in \mathcal{W}_{n}\mid \text{div}(w)=0\} \end{align*} $$

is a Lie subalgebra of $\mathcal {W}_{n}$ , called the Lie algebra of divergence zero vector fields on an n-dimensional torus. Let $d_{ij}(r)=r_{j}x^{r}d_{i}-r_{i}x^{r}d_{j}$ . Then

$$ \begin{align*} \mathcal{S}_{n}=\text{span}_{\mathbb{C}}\{d_{i}, d_{ij}(r)\mid i,j=1,2\cdots,n\} \end{align*} $$

with the Lie bracket

$$ \begin{align*} ~[d_{k},d_{ij}(r)]_{\mathcal{W}_{n}}&=r_{k}d_{ij}(r),\\ ~[d_{ij}(r),d_{pq}(s)]_{\mathcal{W}_{n}}&=r_{j}s_{p}d_{iq}(r+s)-r_{j}s_{q}d_{ip}(r+s)-r_{i}s_{p}d_{jq}(r+s)+r_{i}s_{q}d_{jp}(r+s), \end{align*} $$

for $r,s\in \mathbb {Z}^{N}, i,j,p,q=1,\cdots ,n$ .

Note that $\mathcal {S}_{n}$ is not a Lie–Rinehart subalgebra since $\mathcal {S}_{n}$ is not an $A_{n}$ -module. It is straightforward to see that $(A_{n},\mathcal {S}_{n},[\cdot ,\cdot ]_{\mathcal {W}_{n}},\textrm {Id})$ is a Leibniz pair.

Recall that $\mathfrak {sl}_{n}$ is the Lie subalgebra of $\mathfrak {gl}_{n}$ consisting of all traceless complex matrices. The restriction $H|_{\mathcal {S}_{n}}$ of the crossed homomorphism H in Lemma 4.1 is a crossed homomorphism from $\mathcal {S}_{n}$ to $\mathfrak {sl}_{n}\otimes A_{n}$ . By Corollary 3.37, we obtain the following result.

Corollary 4.7. We have a functor $\mathcal {F}_{{H}}^{A_{n}}:\mathsf {Rep}_{\mathbb {C}}(\mathfrak {sl}_{n})\to \mathsf {ARep}_{\mathbb {C}}(\mathcal {S}_{n})$ defined by

$$ \begin{align*} \mathcal{F}_{H}^{A_{n}}(V;\theta)=(V\otimes_{\mathbb{C}}{A_{n}}; (\text{Id}\boxplus \theta)\circ\iota_{H}),\quad\forall(V;\theta)\in \mathsf{Rep}_{\mathbb{C}}(\mathfrak {sl}_{n}). \end{align*} $$

Let $(P;\rho )$ be a representation of $\mathcal {A}_{n}$ . It follows that $(P;\rho |_{\mathcal {S}_{n}})$ is an admissible representation of $\mathcal {S}_{n}$ since $\mathcal {S}_{n}\subset \mathcal {A}_{n}$ . By Theorem 3.36, we obtain the following result.

Corollary 4.8. We have a functor $\mathcal {F}_{{H}}^{P}:\mathsf {Rep}_{\mathbb {C}}(\mathfrak {sl}_{n})\to \mathsf {ARep}_{\mathbb {C}}(\mathcal {S}_{n})$ defined by

$$ \begin{align*} \mathcal{F}_{H}^{P}(V;\theta)=(V\otimes_{\mathbb{C}}{P}; ({\rho|_{\mathcal{S}_{n}}}\boxplus \theta)\circ\iota_{H}),\quad\forall(V;\theta)\in \mathsf{Rep}_{\mathbb{C}}(\mathfrak {sl}_{n}). \end{align*} $$

4.3. Shen–Larsson functors of Hamiltonian type

For $r\in \mathbb {Z}^{2n}$ , let

$$ \begin{align*} h(r) = \sum_{i=1}^{n}(r_{n+i}x^{r}\partial_{i}-r_{i}x^{r}\partial_{n+i})\in \mathcal{W}_{2n}. \end{align*} $$

It is well-known that $ \mathcal {H}_{n}=\text {Span}_{\mathbb {C}}\{h(r)\mid r\in \mathbb {Z}^{2n}\} $ is a Lie subalgebra of $\mathcal {W}_{2n}$ , with

$$ \begin{align*} [h(r),h(s)]_{\mathcal{W}_{2n}}=\sum_{i=1}^{n}(r_{n+i}s_{i}-s_{n+i}r_{i})h(r+s),\quad\forall r,s\in \mathbb{Z}^{2n}.\end{align*} $$

This Lie algebra $\mathcal {H}_{n}$ is called the Lie algebra of Hamiltonian vector fields on a $2n$ -dimensional torus. Note that $\mathcal {H}_{n}$ is not a Lie–Rinehart algebra since $\mathcal {H}_{n}$ is not an $A_{2n}$ -module. It is straightforward to see that $(A_{n},\mathcal {H}_{n},[\cdot ,\cdot ]_{\mathcal {W}_{2n}},\textrm {Id})$ is a Leibniz pair.

Let $\mathfrak {sp}_{2n}$ be the Lie subalgebra of $\mathfrak {gl}_{2n}$ consisting of all symplectic matrices. The restriction $H|_{\mathcal {H}_{n}}$ of the crossed homomorphism H in Lemma 4.1 is a linear map $\mathcal {H}_{n}\to \mathfrak {sp}_{2n}\otimes A_{2n}$ given by

$$ \begin{align*} H(h(r))=\begin{pmatrix} r_{1}r_{n+1}&\cdots&r_{1}r_{2n}&-r_{1}r_{1}&\cdots&-r_{1}r_{n}\\ \vdots&\cdots&\vdots&\vdots&\cdots&\vdots\\ r_{n}r_{n+1}&\cdots&r_{n}r_{2n}&-r_{n}r_{1}&\cdots&-r_{n}r_{n}\\ r_{n+1}r_{n+1}&\cdots&r_{n+1}r_{2n}&-r_{n+1}r_{1}&\cdots&-r_{n+1}r_{n}\\ \vdots&\cdots&\vdots&\vdots&\cdots&\vdots\\ r_{2n}r_{n+1}&\cdots&r_{2n}r_{2n}&-r_{2n}r_{1}&\cdots&-r_{2n}r_{n}\\ \end{pmatrix}\otimes x^{r}\in \mathfrak{sp}_{2n}\otimes A_{2n}, \end{align*} $$

which is certainly a crossed homomorphism from $\mathcal {H}_{n}$ to $\mathfrak {sp}_{2n}\otimes A_{2n}$ . By Corollary 3.37, we obtain the following result.

Corollary 4.10. We have a functor $\mathcal {F}_{{H}}^{A_{2n}}:\mathsf {Rep}_{\mathbb {C}}(\mathfrak {sp}_{2n})\to \mathsf {ARep}_{\mathbb {C}}(\mathcal {H}_{n})$ defined by

$$ \begin{align*} \mathcal{F}_{H}^{A_{2n}}(V;\theta)=(V\otimes_{\mathbb{C}}{A_{2n}}; (\text{Id}\boxplus \theta)\circ\iota_{H}),\quad\forall(V;\theta)\in \mathsf{Rep}_{\mathbb{C}}(\mathfrak{sp}_{2n}). \end{align*} $$

4.4. Actions of monoidal categories for generalised Cartan type

Let A be a commutative associative $\mathbb {C}$ -algebra and let $\Delta $ be a nonzero $\mathbb {C}$ -vector space of commuting $\mathbb {C}$ -derivations of A. Let us first recall the construction of the generalised Witt algebras from [Reference Passman37]. The tensor product $A\Delta :=A\otimes _{\mathbb {C}}\Delta $ acts on A by

$$ \begin{align*} a\otimes\partial: x\mapsto a\partial(x),\quad a,x\in A, \partial\in \Delta. \end{align*} $$

Since A is commutative, this gives rise to a linear transformation $ \alpha : A\Delta \to \mathsf {Der}_{\mathbb {C}}(A). $ Define a bracket $[\cdot ,\cdot ]_{A\Delta }$ on $A\Delta $ by

$$ \begin{align*} [a\partial,b\delta]_{A\Delta}=a\partial(b)\delta- b\delta(a)\partial,\quad\forall a,b\in A, \partial,\delta\in\Delta, \end{align*} $$

which gives a Lie algebra structure on $A\Delta $ . Then $\alpha $ is clearly an action of $A\Delta $ on the commutative Lie algebra A. Assume that $\dim _{\mathbb {C}}\Delta <\infty $ . Then there are $\partial _{1},\cdots ,\partial _{n}\in \Delta $ such that $A\Delta $ is a free A-module with basis $\{\partial _{1},\cdots ,\partial _{n}\}$ (see [Reference Zhao49]). We denote this Lie algebra by $\mathcal {W}_{n}(A,\Delta )$ . Note that $(A,\mathcal {W}_{n}(A,\Delta ),[\cdot ,\cdot ]_{_{A\Delta }},\alpha )$ is a Lie–Rinehart algebra.

Now we have a generalisation of Lemma 4.1.

Lemma 4.12. The linear map $H: \mathcal {W}_{n}(A,\Delta )\to \mathfrak {gl}_{n} \otimes A$ defined by

$$ \begin{align*} H\left(\sum_{i=1}^{n} a_{i}\partial_{i}\right)=\sum_{i=1}^{n}\sum_{j=1}^{n} E_{ij}\otimes\partial_{i} (a_{j}),\quad a_{i}\in A \end{align*} $$

is a crossed homomorphism from $\mathcal {W}_{n}(A,\Delta )$ to $\mathfrak {gl}_{n} \otimes A$ .

Similar to Corollary 4.2, by Lemma 4.12 and Theorem 3.26 we obtain the following result.

Corollary 4.13. We have a functor $F_{{H}}^{A}:\mathsf {Rep}_{\mathbb {C}}(\mathfrak {gl}_{n})\to \mathsf {WRep}_{\mathbb {C}}(\mathcal {W}_{n}(A,\Delta ))$ defined by

$$ \begin{align*} F_{H}^{A}(V;\theta)=(V\otimes_{\mathbb{C}}{A}; ({\alpha}\boxplus \theta)\circ\iota_{H}),\quad\forall(V;\theta)\in \mathsf{Rep}_{\mathbb{C}}(\mathfrak {gl}_{n}). \end{align*} $$

Now let us define the divergence map $\mathrm {div}: \mathcal {W}_{n}(A,\Delta )\to A$ to be the $\mathbb {C}$ -linear extension of

$$ \begin{align*} \mathrm{div}(a\partial) = \partial(a),\quad \forall a\in A, \partial\in \Delta. \end{align*} $$

Let $ \mathcal {S}_{n}(A,\Delta )=\{w\in A\Delta \mid \text {div}(w)=0\}. $ Then $\mathcal {S}_{n}(A,\Delta )$ is a Lie subalgebra of $\mathcal {W}_{n}(A,\Delta )$ ; see [Reference Bergen and Passman1] for more details. If $A=\mathbb {C}[x_{1}^{\pm 1},\cdots ,x_{n}^{\pm 1}]$ and $\Delta =\text {Span}_{\mathbb {C}}\{x_{1}\frac {\partial }{\partial x_{1}},\cdots , x_{n}\frac {\partial }{\partial x_{n}}\}$ , $\mathcal {S}_{n}(A,\Delta )$ is the Lie algebra $S_{n}$ of divergence zero vector fields on an n-dimensional torus.

Note that $\mathcal {S}_{n}(A,\Delta )$ is not a Lie–Rinehart subalgebra since $\mathcal {S}_{n}(A,\Delta )$ is not an A-module. It is straightforward to see that $(A,\mathcal {S}_{n}(A,\Delta ),[\cdot ,\cdot ]_{A\Delta },\textrm {Id})$ is a Leibniz pair.

It is clear that $H|_{\mathcal {S}_{n}(A,\Delta )}$ is a crossed homomorphism from $\mathcal {S}_{n}(A,\Delta )$ to $\mathfrak {sl}_{n} \otimes A$ . Similar to Corollary 4.5, by Lemma 4.12 and Corollary 3.37 we obtain the following result.

Corollary 4.15. We have a functor $\mathcal {F}_{{H}}^{A}:\mathsf {Rep}_{\mathbb {C}}(\mathfrak {sl}_{n})\to \mathsf {ARep}_{\mathbb {C}}(\mathcal {S}_{n}(A,\Delta ))$ defined by

$$ \begin{align*} \mathcal{F}_{H}^{A}(V;\theta)=(V\otimes_{\mathbb{C}}{A}; ({\alpha}\boxplus \theta)\circ\iota_{H}),\quad\forall(V;\theta)\in \mathsf{Rep}_{\mathbb{C}}(\mathfrak {sl}_{n}). \end{align*} $$

Now let us define a map $D: A\to \mathcal {W}_{2n}(A,\Delta ) $ to be the linear extension of

$$ \begin{align*} D(a) = \sum_{i=1}^{n}(\partial_{i}(a)\partial_{n+i}-\partial_{n+i}(a)\partial_{i}),\quad \forall a\in A. \end{align*} $$

Let $ \mathcal {H}_{n}(A,\Delta )=\{D(a)\mid a\in A\}. $ Then $\mathcal {H}_{n}(A,\Delta )$ is a Lie subalgebra of $\mathcal {W}_{2n}(A,\Delta )$ , with

$$ \begin{align*} [D(a),D(b)]_{A\Delta}=D\left(\sum_{i=1}^{n}(\partial_{i}(a)\partial_{n+i}(b)-\partial_{n+i}(a)\partial_{i}(b))\right),\quad\forall a,b\in A. \end{align*} $$

If $A=\mathbb {C}[x_{1}^{\pm 1},\cdots ,x_{2n}^{\pm 1}]$ and $\Delta =\text {Span}_{\mathbb {C}}\{x_{1}\frac {\partial }{\partial x_{1}},\cdots , x_{2n}\frac {\partial }{\partial x_{2n}}\}$ , then $\mathcal {H}_{n}(A,\Delta )$ is the Lie algebra of Hamiltonian vector fields on a $2n$ -dimensional torus.

Note that $\mathcal {H}_{n}(A,\Delta )$ is not a Lie–Rinehart algebra since $\mathcal {H}_{n}(A,\Delta )$ is not an A-module. It is straightforward to see that $(A,\mathcal {H}_{n}(A,\Delta ),[\cdot ,\cdot ]_{A\Delta },\textrm {Id})$ is a Leibniz pair.

The restriction $H|_{\mathcal {H}_{n}(A,\Delta )}$ of the crossed homomorphism H in Lemma 4.12 is a crossed homomorphism from $\mathcal {H}_{n}(A,\Delta )$ to $\mathfrak {sp}_{2n}\otimes A$ . Similar to Corollary 4.10, by Lemma 4.12 and Corollary 3.37 we obtain the following result.

Corollary 4.16. We have a functor $\mathcal {F}_{{H}}^{A}:\mathsf {Rep}_{\mathbb {C}}(\mathfrak {sp}_{2n})\to \mathsf {ARep}_{\mathbb {C}}(\mathcal {H}_{n}(A,\Delta ))$ defined by

$$ \begin{align*} \mathcal{F}_{H}^{A}(V;\theta)=(V\otimes_{\mathbb{C}}{A}; ({\alpha}\boxplus \theta)\circ\iota_{H}),\quad\forall(V;\theta)\in \mathsf{Rep}_{\mathbb{C}}(\mathfrak{sp}_{2n}). \end{align*} $$

5.1. The differential graded Lie algebra controlling deformations

Definition 5.2 [Reference Loday and Vallette27]

Let $(\mathfrak g=\oplus _{k\in \mathbb Z}\mathfrak g_{k},[\cdot ,\cdot ],d)$ be a differential graded Lie algebra. A degree $1$ element $\theta \in \mathfrak g_{1}$ is called a Maurer–Cartan element of $\mathfrak g$ if it satisfies the following Maurer–Cartan equation:

(18) $$ \begin{align} d \theta+\frac{1}{2}[\theta,\theta]=0. \end{align} $$

Proposition 5.3 [Reference Loday and Vallette27]

Let $(\mathfrak g=\oplus _{k\in \mathbb Z}\mathfrak g_{k},[\cdot ,\cdot ],d)$ be a differential graded Lie algebra and let $\mu \in \mathfrak g_{1}$ be a Maurer–Cartan element. Then the map

$$ \begin{align*} d_{\mu}: \mathfrak g \longrightarrow \mathfrak g,\quad \ d_{\mu}(x):=d(x)+[\mu, x], \quad \forall x\in \mathfrak g,\end{align*} $$

is a differential on the graded Lie algebra $(\mathfrak g,[\cdot ,\cdot ])$ . For any $v\in \mathfrak g_{1}$ , the sum $\mu +v$ is a Maurer–Cartan element of the differential graded Lie algebra $(\mathfrak g, [\cdot ,\cdot ],d)$ if and only if v is a Maurer–Cartan element of the differential graded Lie algebra $(\mathfrak g, [\cdot ,\cdot ], d_{\mu })$ .

Let $(\mathfrak g,[\cdot ,\cdot ]_{\mathfrak g})$ and $(\mathfrak h,[\cdot ,\cdot ]_{\mathfrak h})$ be Lie algebras and $\rho :\mathfrak g\,\rightarrow \,\mathsf {Der}(\mathfrak h)$ be an action of $\mathfrak g$ on $\mathfrak h$ . Consider the graded vector space

$$ \begin{align*}{\mathcal{C}}^{*}(\mathfrak g,\mathfrak h):=\oplus_{k\geq 0}\mathsf{Hom}(\wedge^{k}\mathfrak g,\mathfrak h).\end{align*} $$

Define $d:\mathsf {Hom}(\wedge ^{m}\mathfrak g,\mathfrak h)\,\rightarrow \,\mathsf {Hom}(\wedge ^{m+1}\mathfrak g,\mathfrak h)$ by

(19) $$ \begin{align} (df) ( x_{1},\cdots, x_{m+1} ) &=\sum_{i=1}^{m+1}(-1)^{m+i}\rho(x_{i})f(x_{1},\cdots,\hat{x}_{i},\cdots,x_{m+1})\\ \nonumber&\quad +\sum_{1\le i< j\le m+1}(-1)^{m+i+j-1}f([x_{i},x_{j}]_{\mathfrak g},x_{1},\cdots,\hat{x}_{i},\cdots,\hat{x}_{j},\cdots,x_{m+1}), \end{align} $$

for all $f\in \mathsf {Hom}(\wedge ^{m}\mathfrak g,\mathfrak h)$ . Define a skew-symmetric bracket operation $\left [\![ \cdot ,\cdot \right ]\!] : \mathsf {Hom}(\wedge ^{m}\mathfrak g,\mathfrak h)\times \mathsf {Hom}(\wedge ^{n}\mathfrak g,\mathfrak h)\longrightarrow \mathsf {Hom}(\wedge ^{m+n}\mathfrak g,\mathfrak h)$ by

(20) $$ \begin{align} &\nonumber\left[\![ f_{1},f_{2}\right]\!] (x_{1},x_{2},\cdots,x_{m+n})\\ &=(-1)^{mn+1}\sum_{\sigma\in \mathbb S_{(m,n)}}(-1)^{\sigma}[f_{1}(x_{\sigma(1)},\cdots,x_{\sigma(m)}),f_{2}(x_{\sigma(m+1)},\cdots,x_{\sigma(m+n)})]_{\mathfrak h} \end{align} $$

for all $f_{1}\in \mathsf {Hom}(\wedge ^{m}\mathfrak g,\mathfrak h)$ and $f_{2}\in \mathsf {Hom}(\wedge ^{n}\mathfrak g,\mathfrak h)$ . Here $\mathbb S_{(m,n)}$ denotes the set of all $(m,n)$ -shuffles.

Note that for all $u,v\in \mathfrak h$ , $\left [\![ u,v\right ]\!] =-[u,v]_{\mathfrak h}$ .

Proof. In short, the graded Lie algebra $({\mathcal {C}}^{*}(\mathfrak g,\mathfrak h),\left [\![ \cdot ,\cdot \right ]\!] )$ is obtained via the derived bracket [Reference Kosmann-Schwarzbach23, Reference Voronov46]. In fact, the Nijenhuis–Richardson bracket $[\cdot ,\cdot ]_{\mathsf {NR}}$ associated to the direct sum vector space $\mathfrak g\oplus V$ gives rise to a graded Lie algebra $(\oplus _{k\geq 0}\mathsf {Hom}(\wedge ^{k}(\mathfrak g\oplus \mathfrak h),\mathfrak g\oplus \mathfrak h),[\cdot ,\cdot ]_{\mathsf {NR}})$ . Obviously, $\oplus _{k\geq 0}\mathsf {Hom}(\wedge ^{k}\mathfrak g,\mathfrak h)$ is an abelian subalgebra. We denote the Lie brackets $[\cdot ,\cdot ]_{\mathfrak g}$ and $[\cdot ,\cdot ]_{\mathfrak h}$ by $\mu _{\mathfrak g}$ and $\mu _{\mathfrak h}$ , respectively. Since $\rho $ is an action of the Lie algebra $(\mathfrak g,[\cdot ,\cdot ]_{\mathfrak g})$ , we deduce that $\mu _{\mathfrak g}+\rho $ is a semidirect product Lie algebra structure on $\mathfrak g\oplus \mathfrak h$ . Thus, $\mu _{\mathfrak g}+\rho $ and $\mu _{\mathfrak h}$ are Maurer–Cartan elements of the graded Lie algebra $({\mathcal {C}}^{*}(\mathfrak g\oplus \mathfrak h,\mathfrak g\oplus \mathfrak h),[\cdot ,\cdot ]_{\mathsf {NR}})$ . Define a differential $d_{\mu _{\mathfrak h}}$ on $({\mathcal {C}}^{*}(\mathfrak g\oplus \mathfrak h,\mathfrak g\oplus \mathfrak h),[\cdot ,\cdot ]_{\mathsf {NR}})$ via

$$ \begin{align*} d_{\mu_{\mathfrak h}}:=[\mu_{\mathfrak h},\cdot]_{\mathsf{NR}}. \end{align*} $$

Further, we define the derived bracket on the graded vector space $\oplus _{k\geq 0}\mathsf {Hom}(\wedge ^{k}\mathfrak g,\mathfrak h)$ by

$$ \begin{align*} \left[\![ f_{1},f_{2}\right]\!] :=(-1)^{m-1}[[\mu_{\mathfrak h},f_{1}]_{\mathsf{NR}},f_{2}]_{\mathsf{NR}},\quad\forall f_{1}\in\mathsf{Hom}(\wedge^{m}\mathfrak g,\mathfrak h),~f_{2}\in\mathsf{Hom}(\wedge^{n}\mathfrak g,\mathfrak h), \end{align*} $$

which is exactly the bracket given by (20). By $[\mu _{\mathfrak h},\mu _{\mathfrak h}]_{\mathsf {NR}}=0,$ we deduce that $({\mathcal {C}}^{*}(\mathfrak g,\mathfrak h),\left [\![ \cdot ,\cdot \right ]\!] )$ is a graded Lie algebra.

Moreover, by $\mathrm {Im}\rho \subset \mathsf {Der}(\mathfrak h)$ , we have $[\mu _{\mathfrak g}+\rho ,\mu _{\mathfrak h}]_{\mathsf {NR}}=0.$ We define a linear map $d=:[\mu _{\mathfrak g}+\rho ,\cdot ]_{\mathsf {NR}}$ on the graded space ${\mathcal {C}}^{*}(\mathfrak g\oplus \mathfrak h,\mathfrak g\oplus \mathfrak h)$ . We obtain that d is closed on the subspace $\oplus _{k\geq 0}\mathsf {Hom}(\wedge ^{k}\mathfrak g,\mathfrak h)$ and is given by (19).

By $[\mu _{\mathfrak g}+\rho ,\mu _{\mathfrak g}+\rho ]_{\mathsf {NR}}=0$ , we obtain that $d\circ d=0.$ Moreover, by $[\mu _{\mathfrak g}+\rho ,\mu _{\mathfrak h}]_{\mathsf {NR}}=0$ , we deduce that d is a derivation of $({\mathcal {C}}^{*}(\mathfrak g,\mathfrak h),\left [\![ \cdot ,\cdot \right ]\!] )$ . Therefore, $({\mathcal {C}}^{*}(\mathfrak g,\mathfrak h),\left [\![ \cdot ,\cdot \right ]\!] ,d)$ is a differential graded Lie algebra.

Finally, for a degree 1 element ${H}\in \mathsf {Hom}(\mathfrak g,\mathfrak h)$ , we have

$$ \begin{align*} \big(dH+\frac{1}{2}\left[\![ {H},{H}\right]\!] \big)(x,y)=\rho(x)({H} y)-\rho(y)({H} x)-{H}[x,y]_{\mathfrak g}+[{H} x,{H} y]_{\mathfrak h}. \end{align*} $$

Thus, Maurer–Cartan elements are precisely crossed homomorphisms from $(\mathfrak g,[\cdot ,\cdot ]_{\mathfrak g})$ to $(\mathfrak h,[\cdot ,\cdot ]_{\mathfrak h})$ with respect to the action $\rho $ . The proof is finished.

Let ${H}:\mathfrak g\longrightarrow \mathfrak h$ be a crossed homomorphism with respect to the action $\rho $ . Since ${H}$ is a Maurer–Cartan element of the differential graded Lie algebra $({\mathcal {C}}^{*}(\mathfrak g,\mathfrak h),\left [\![ \cdot ,\cdot \right ]\!] ,d)$ by Proposition 5.4, it follows from Proposition 5.3 that $d_{{H}}:=d+\left [\![ {H},\cdot \right ]\!] $ is a graded derivation on the graded Lie algebra $({\mathcal {C}}^{*}(\mathfrak g,\mathfrak h),\left [\![ \cdot ,\cdot \right ]\!] )$ satisfying $d^{2}_{{H}}=0$ . Therefore, $({\mathcal {C}}^{*}(\mathfrak g,\mathfrak h),\left [\![ \cdot ,\cdot \right ]\!] ,d_{{H}})$ is a differential graded Lie algebra. This differential graded Lie algebra can control deformations of crossed homomorphisms. We have obtained the following result.

5.2. Cohomologies of crossed homomorphisms

In this subsection, we define cohomologies of a crossed homomorphism, which can be used to study linear deformations in Section 5.3.

Recall that $\rho _{H}$ defined by (4) is a representation of $\mathfrak g$ on $\mathfrak h$ . Let $d_{\rho _{{H}}}: \mathsf {Hom}(\wedge ^{k}\mathfrak g,\mathfrak h)\longrightarrow \mathsf {Hom}(\wedge ^{k+1}\mathfrak g,\mathfrak h)$ be the corresponding Chevalley–Eilenberg coboundary operator. More precisely, for all $f\in \mathsf {Hom}(\wedge ^{k}\mathfrak g,\mathfrak h)$ and $x_{1},\cdots ,x_{k+1}\in \mathfrak g$ , we have

(21) $$ \begin{align} &\quad d_{\rho_{{H}}} f(x_{1},\cdots,x_{k+1})\notag\\ &=\sum_{i=1}^{k+1}(-1)^{i+1}\rho(x_{i})f(x_{1},\cdots,\hat{x}_{i},\cdots, x_{k+1})+\sum_{i=1}^{k+1}(-1)^{i+1}[{H} x_{i},f(x_{1},\cdots,\hat{x}_{i},\cdots, x_{k+1})]_{\mathfrak h}\\ &\quad +\sum_{1\le i<j\le k+1}(-1)^{i+j}f([x_{i},x_{j}]_{\mathfrak g},x_{1},\cdots,\hat{x}_{i},\cdots,\hat{x}_{j},\cdots, x_{k+1}). \notag \end{align} $$

It is obvious that $u\in \mathfrak h$ is closed if and only if $ \rho (x)u+[{H} x,u]_{\mathfrak h}=0 $ for all $x\in \mathfrak g$ , and $f\in \mathsf {Hom}(\mathfrak g,\mathfrak h)$ is closed if and only if

$$ \begin{align*} \rho(x_{1})f(x_{2})-\rho(x_{2})f(x_{1})+[{H} x_{1},f(x_{2})]_{\mathfrak h}-[{H} x_{2},f(x_{1})]_{\mathfrak h}-f([x_{1},x_{2}]_{\mathfrak g})=0,\quad\forall x_{1},x_{2}\in\mathfrak g. \end{align*} $$

Definition 5.6. Let ${H}:\mathfrak g\longrightarrow \mathfrak h$ be a crossed homomorphism with respect to the action $\rho $ . Denote by ${\mathcal {C}}^{k}(\mathfrak g,\mathfrak h)=\mathsf {Hom}(\wedge ^{k}\mathfrak g,\mathfrak h)$ and $({\mathcal {C}}^{*}(\mathfrak g,\mathfrak h)=\oplus _{k\geq 0}{\mathcal {C}}^{k}(\mathfrak g,\mathfrak h),d_{\rho _{{H}}})$ the above cochain complex. Denote the set of k-cocycles by $\mathcal {Z}^{k}(\mathfrak g,\mathfrak h)$ and the set of k-coboundaries by $\mathcal {B}^{k}(\mathfrak g,\mathfrak h)$ . Denote by

(22) $$ \begin{align} \mathcal{H}^{k}(\mathfrak g,\mathfrak h)=\mathcal{Z}^{k}(\mathfrak g,\mathfrak h)/\mathcal{B}^{k}(\mathfrak g,\mathfrak h), \quad k \geq 0, \end{align} $$

the kth cohomology group, which will be taken to be the k th cohomology group for the crossed homomorphism $ H$ .

Comparing the coboundary operators $d_{\rho _{{H}}}$ given above and the operators $d_{{H}}=d+\left [\![ {H},\cdot \right ]\!] $ defined by the Maurer–Cartan element ${H}$ , we have the following.

Proposition 5.7. Let ${H}:\mathfrak g\longrightarrow \mathfrak h$ be a crossed homomorphism. Then we have

$$ \begin{align*} d_{\rho_{{H}}}f=(-1)^{k-1}d_{{H}}f,\quad \forall f\in \mathsf{Hom}(\wedge^{k}\mathfrak g,\mathfrak h). \end{align*} $$

Proof. Indeed, for all $x_{1},x_{2},\cdots ,x_{k+1}\in \mathfrak g$ and $f\in \mathsf {Hom}(\wedge ^{k}\mathfrak g,\mathfrak h)$ , we have

$$ \begin{align*} &(-1)^{k-1}(d_{{H}}f)(x_{1},x_{2},\cdots,x_{k+1})\\ &\quad =(-1)^{k-1}(df+\left[\![ {H},f\right]\!] )(x_{1},\cdots,x_{k+1})\\ &\quad =\sum_{i=1}^{i+1}(-1)^{i+1}\rho(x_{i})f(x_{1},\cdots,\hat{x}_{i},\cdots,x_{k+1}) \nonumber \\[-14pt] \nonumber \\&\qquad +\sum_{1\le i< j\le k+1}(-1)^{i+j}f([x_{i},x_{j}]_{\mathfrak g},x_{1},\cdots,\hat{x}_{i},\cdots,\hat{x}_{j},\cdots,x_{k+1})\\ &\qquad +(-1)^{k-1}(-1)^{k+1}\sum_{\sigma\in \mathbb S_{(1,k)}}(-1)^{\sigma}[{H} x_{\sigma(1)},f(x_{\sigma(2)},\cdots,x_{\sigma(k+1)})]_{\mathfrak h}\\ &\quad =\sum_{i=1}^{i+1}(-1)^{i+1}\rho(x_{i})f(x_{1},\cdots,\hat{x}_{i},\cdots,x_{k+1})\\&\qquad +\sum_{1\le i< j\le k+1}(-1)^{i+j}f([x_{i},x_{j}]_{\mathfrak g},x_{1},\cdots,\hat{x}_{i},\cdots,\hat{x}_{j},\cdots,x_{k+1})\\ &\qquad +\sum_{i=1}^{k+1}(-1)^{i-1}[{H} x_{i},f(x_{1},\cdots,\hat{x}_{i},\cdots,x_{k+1})]_{\mathfrak h}\\ &\qquad =(d_{\rho_{{H}}}f)(x_{1},x_{2},\cdots,x_{k+1}), \end{align*} $$

which implies that $d_{\rho _{{H}}}f=(-1)^{k-1}d_{{H}}f$ .

At the end of this section, we show that certain homomorphisms between crossed homomorphisms induce homomorphisms between the corresponding cohomology groups. Let ${H}$ and $\widetilde {H}$ be two crossed homomorphisms from $\mathfrak g$ to $\mathfrak h$ with respect to the action $\rho $ and $(\phi _{\mathfrak g},\phi _{\mathfrak h})$ a homomorphism from $\widetilde {H}$ to ${H}$ in which $\phi _{\mathfrak g}$ is invertible. For all $k\geq 0$ , define

$$ \begin{align*} \Phi:&\mathsf{Hom}(\wedge^{k}\mathfrak g,\mathfrak h)\,\rightarrow\, \mathsf{Hom}(\wedge^{k}\mathfrak g,\mathfrak h)\\ &f\mapsto \phi_{\mathfrak h}\circ f\circ (\phi_{\mathfrak g}^{-1})^{\otimes k}. \end{align*} $$

Proof. By the fact that $(\phi _{\mathfrak g},\phi _{\mathfrak h})$ is a homomorphism from $\widetilde {H}$ to ${H}$ , we have

$$ \begin{align*} &\quad (\Phi (d_{\rho_{\widetilde{{H}}}}f))(x_{1},\cdots,x_{k+1})=\phi_{\mathfrak h} (d_{\rho_{\widetilde{{H}}}}f)(\phi_{\mathfrak g}^{-1} (x_{1}),\cdots,\phi_{\mathfrak g}^{-1} (x_{k+1}))\\ &=\sum_{i=1}^{i+1}(-1)^{i+1}\phi_{\mathfrak h}\rho(\phi_{\mathfrak g}^{-1}(x_{i}))f(\phi_{\mathfrak g}^{-1} (x_{1}),\cdots,\hat{x}_{i},\cdots,\phi_{\mathfrak g}^{-1} (x_{k+1}))\\ &\quad +\sum_{1\le i< j\le k+1}(-1)^{i+j}\phi_{\mathfrak h} f([\phi_{\mathfrak g}^{-1} (x_{i}),\phi_{\mathfrak g}^{-1} (x_{j})]_{\mathfrak g},\phi_{\mathfrak g} {}^{-1} (x_{1}),\cdots,\hat{x}_{i},\cdots,\hat{x}_{j},\cdots,\phi_{\mathfrak g}^{-1} (x_{k+1}))\\ &\quad +\sum_{i=1}^{k+1}(-1)^{i+1}\phi_{\mathfrak h}[\widetilde{{H}} \phi_{\mathfrak g}^{-1} (x_{i}),f(\phi_{\mathfrak g}^{-1} (x_{1}),\cdots,\hat{x}_{i},\cdots,\phi_{\mathfrak g}^{-1}( x_{k+1}))]_{\mathfrak h}\\ &=\sum_{i=1}^{i+1}(-1)^{i+1}\rho( x_{i})\phi_{\mathfrak h} f(\phi_{\mathfrak g}^{-1} (x_{1}),\cdots,\hat{x}_{i},\cdots,\phi_{\mathfrak g}^{-1} (x_{k+1})) \nonumber \\[-14pt] \nonumber \\&\quad +\sum_{1\le i< j\le k+1}(-1)^{i+j}\phi_{\mathfrak h} f(\phi_{\mathfrak g}^{-1}[ x_{i},x_{j}]_{\mathfrak g},\phi_{\mathfrak g} {}^{-1} (x_{1}),\cdots,\hat{x}_{i},\cdots,\hat{x}_{j},\cdots,\phi_{\mathfrak g}^{-1} (x_{k+1}))\\ &\quad +\sum_{i=1}^{k+1}(-1)^{i+1}[{H} (x_{i}),\phi_{\mathfrak h} f(\phi_{\mathfrak g}^{-1} (x_{1}),\cdots,\hat{x}_{i},\cdots,\phi_{\mathfrak g}^{-1}( x_{k+1}))]_{\mathfrak h}\\ &= d_{\rho_{{H}}}\Phi (f)(x_{1},\cdots,x_{k+1}), \end{align*} $$

which implies that $\Phi $ is a cochain map.

5.3. Linear deformations of crossed homomorphisms

In this subsection, we study linear deformations of crossed homomorphisms using the cohomology theory introduced in Subsection 5.2 and show that isomorphic linear deformations are identified with the same class in the second cohomology group. We give the notion of a Nijenhuis element associated to a crossed homomorphism, which gives rise to a trivial deformation.

It is direct to check that ${H}_{t}={H}+t\mathfrak H$ is a linear deformation of a crossed homomorphism ${H}$ if and only if for any $x,y\in \mathfrak g$ ,

(23) $$ \begin{align} \rho(x)\mathfrak H y-\rho(y)\mathfrak H x+[{H} x,\mathfrak H y]_{\mathfrak h}+[\mathfrak H x,{H} y]_{\mathfrak h}-\mathfrak H[x,y]_{\mathfrak g}&=0, \end{align} $$

(24) $$ \begin{align} [\mathfrak H x,\mathfrak H y]_{\mathfrak h}&=0. \end{align} $$

Note that Equation (23) means that $\mathfrak H$ is a 1-cocycle of the crossed homomorphism H.

Let $(\text {Id}_{\mathfrak g}+t\mathsf {ad}_{x},\text {Id}_{\mathfrak h}+t\rho (x))$ be a homomorphism from ${H}^{2}_{t}$ to ${H}^{1}_{t}$ . Then $\text {Id}_{\mathfrak g}+t\mathsf {ad}_{x}$ and $\text {Id}_{\mathfrak h}+t\rho (x)$ are Lie algebra endomorphisms. Thus, we have

$$ \begin{align*} (\text{Id}_{\mathfrak g}+t\mathsf{ad}_{x})[y,z]_{\mathfrak g}&=[(\text{Id}_{\mathfrak g}+t\mathsf{ad}_{x})(y),(\text{Id}_{\mathfrak g}+t\mathsf{ad}_{x})(z)]_{\mathfrak g}, \;\forall y,z\in \mathfrak g,\\ (\text{Id}_{\mathfrak h}+t\rho(x))[u,v]_{\mathfrak h}&=[(\text{Id}_{\mathfrak h}+t\rho(x))(u),(\text{Id}_{\mathfrak h}+t\rho(x))(v)]_{\mathfrak h}, \;\forall u,v\in \mathfrak h, \end{align*} $$

which implies that x satisfies

(25) $$ \begin{align} [[x,y]_{\mathfrak g},[x,z]_{\mathfrak g}]_{\mathfrak g}=0,\quad \forall y,z\in\mathfrak g, \end{align} $$

(26) $$ \begin{align} [\rho(x)u,\rho(x)v]_{\mathfrak h}=0,\quad \forall u,v\in\mathfrak h. \end{align} $$

Then by Equation (2), we get

$$ \begin{align*} ({H}+t\mathfrak H_{1})(\text{Id}_{\mathfrak g}+t\mathsf{ad}_{x})(y)=(\text{Id}_{\mathfrak h}+t\rho(x))({H}+t\mathfrak H_{2})(y),\quad\forall y\in \mathfrak g, \end{align*} $$

which implies

(27) $$ \begin{align} (\mathfrak H_2-\mathfrak H_1)(y)&=-\rho(y){H} x-[{H} y,{H} x]_{\mathfrak h}, \end{align} $$

(28) $$ \begin{align} \mathfrak H_1[x,y]_{\mathfrak g}&=\rho(x)(\mathfrak H_2y), \quad \forall y\in \mathfrak g. \end{align} $$

Finally, Equation (3) gives

$$ \begin{align*} (\text{Id}_{\mathfrak h}+t\rho(x))\rho(y)(u)=\rho((\text{Id}_{\mathfrak g}+t\mathsf{ad}_{x})(y))(\text{Id}_{\mathfrak h}+t\rho(x))(u),\quad \forall y\in\mathfrak g, u\in \mathfrak h, \end{align*} $$

which implies that x satisfies

(29) $$ \begin{align} \rho([x,y]_{\mathfrak g})\rho(x)=0,\quad\forall y\in\mathfrak g. \end{align} $$

Note that Equation (27) means that $\mathfrak H_{2}-\mathfrak H_{1}=d_{\rho _{{H}}} (-{H} x)$ . Thus, we have the following.

Definition 5.13. Let ${H}$ be a crossed homomorphism from $\mathfrak g$ to $\mathfrak h$ with respect to the action $\rho $ . An element $x\in \mathfrak g$ is called a Nijenhuis element associated to ${H}$ if x satisfies Equations (25), (26), (29) and the equation

(30) $$ \begin{align} \rho(x)\Big(\rho(y){H} x+[{H} y,{H} x]_{\mathfrak h}\Big)=0,\quad\forall y\in\mathfrak g. \end{align} $$

Denote by $\mathsf {Nij}({H})$ the set of Nijenhuis elements associated to a crossed homomorphism ${H}$ .

By Equations (25)–(29), it is obvious that a trivial linear deformation gives rise to a Nijenhuis element. The following result is in close analogy to the fact that the differential of a Nijenhuis operator on a Lie algebra generates a trivial linear deformation of the Lie algebra [Reference Dorfman7], justifying the notion of Nijenhuis elements.

We need the following lemma to prove this theorem.

The proof of Theorem 5.14

For any Nijenhuis element $x\in \mathsf {Nij}({H})$ , we define

(31) $$ \begin{align} \mathfrak H=d_{{H}} (-{H} x). \end{align} $$

By the definition of Nijenhuis elements of ${H}$ , for any t, ${H}_{t}={H}+t\mathfrak H$ satisfies

$$ \begin{align*} {H}\circ \Big(\text{Id}_{\mathfrak g}+t\mathsf{ad}_{x}\Big)&=\Big(\text{Id}_{\mathfrak h}+t\rho(x)\Big)\circ {H}_{t},\\ \Big(\text{Id}_{\mathfrak h}+t\rho(x)\Big)\circ\rho(y)&=\rho\Big((\text{Id}_{\mathfrak g}+t\mathsf{ad}_{x})(y)\Big)\circ\Big(\text{Id}_{\mathfrak h}+t\rho(x)\Big),\quad\forall y\in\mathfrak g. \end{align*} $$

For t sufficiently small, we see that $\text {Id}_{\mathfrak g}+t\mathsf {ad}_{x}$ and $\text {Id}_{\mathfrak h}+t\rho (x)$ are Lie algebra isomorphisms. Thus, we have

$$ \begin{align*}{H}_{t}=\Big(\text{Id}_{\mathfrak h}+t\rho(x)\Big)^{-1}\circ{H}\circ \Big(\text{Id}_{\mathfrak g}+t\mathsf{ad}_{x}\Big).\end{align*} $$

By Lemma 5.15, we deduce that ${H}_{t}$ is a crossed homomorphism from $\mathfrak g$ to $\mathfrak h$ , for t sufficiently small. Thus, $\mathfrak H$ given by Equation (31) satisfies the conditions (23) and (24). Therefore, ${H}_{t}$ is a crossed homomorphism for all t, which means that $\mathfrak H$ given by Equation (31) generates a deformation. It is straightforward to see that this deformation is trivial.

It is generally not easy to find Nijenhuis elements associated to a crossed homomorphism ${H}$ from a Lie algebra $\mathfrak g$ to $\mathfrak h$ . Next, we give examples on some special Lie algebras where the Nijenhuis elements can be explicitly determined.

Example 5.17. Consider the unique $2$ -dimensional nonabelian Lie algebra on $\mathbb C^{2}$ . The Lie bracket is given by $[e_{1},e_{2}]=e_{1}$ for a given basis $\{e_{1},e_{2}\}$ . For a matrix $\left (\begin {array}{@{}cc@{}}a_{11}&a_{12}\\ a_{21}&a_{22}\end {array}\right )$ , define

$$ \begin{align*} {H} e_{1}=a_{11}e_{1}+a_{21}e_{2},\quad {H} e_{2}=a_{12}e_{1}+a_{22}e_{2}. \end{align*} $$

${H}$ is a crossed homomorphism from $\mathbb C^{2}$ to $\mathbb C^{2}$ with respect to the adjoint action if and only if

$$ \begin{align*} {H}[e_{1},e_{2}]=[{H} e_{1},e_{2}]+[e_{1},{H} e_{2}]+[{H} e_{1},{H} e_{2}]. \end{align*} $$

By a straightforward computation, we conclude that ${H}$ is a crossed homomorphism if and only if $ a_{21}=0, ~(1+a_{11}) a_{22}=0. $ So we have the following two cases to consider.

(i) If $a_{22}=0$ , then we deduce that any ${H}=\left (\begin {array}{@{}cc@{}}a_{11}&a_{12}\\ 0&0\end {array}\right )$ is a crossed homomorphism. In this case, $x=t_{1}e_{1}+t_{2}e_{2}$ is a Nijenhuis element of ${H}$ if and only if $ t_{2}(t_{1}a_{11}+t_{2}a_{12})=0. $ Then for any $t_{1}\in \mathbb C, t_{1}e_{1}$ is a Nijenhuis element for the crossed homomorphism ${H}=\left (\begin {array}{@{}cc@{}}a_{11}&a_{12}\\ 0&0\end {array}\right )$ .

(ii) If $1+a_{11}=0$ , then we deduce that any ${H}=\left (\begin {array}{@{}cc@{}}-1&a_{12}\\ 0&a_{22}\end {array}\right )$ is a crossed homomorphism. In this case, $x=t_{1}e_{1}+t_{2}e_{2}$ is a Nijenhuis element of ${H}$ if and only if $ t_{2}(t_{2}a_{12}-t_{1}a_{22}-t_{1})=0. $ In particular, $e_{1}+e_{2}$ is a Nijenhuis element for the crossed homomorphism ${H}=\left (\begin {array}{@{}cc@{}}-1&2\\ 0&1\end {array}\right )$ .

Example 5.18. For any crossed homomorphism H from a finite-dimensional semisimple Lie algebra $\mathfrak g$ over $\mathbb {C}$ to another Lie algebra $\mathfrak h$ with respect to any action $\rho $ , we claim that $\mathsf {Nij}(H)=0$ .

Let $x\in \mathfrak g$ be a fixed nonzero vector and assume that $\mathfrak g_{0}=[x, \mathfrak g]$ is abelian; that is, (25) holds. We will show that this is impossible.

Denote $n=\dim \mathfrak g, \mathfrak g_{x}=\{y\in \mathfrak g:[x,y]=0\}$ . Considering the linear map $\mathsf {ad}(x):\mathfrak g\to \mathfrak g$ , we see that $\dim \mathfrak g_{0}+\dim \mathfrak g_{x}=n$ . Let $(\cdot , \cdot )$ be a nondegenerate invariant bilinear form on $\mathfrak g$ . It is easy to see that $\mathfrak g_{x}=\mathfrak g_{0}^{\perp }$ . From $0=(0, \mathfrak g)=([[x, \mathfrak g], [x, \mathfrak g]], \mathfrak g)= ([x, \mathfrak g],[[x, \mathfrak g], \mathfrak g])$ we have

$$ \begin{align*}[[x, \mathfrak g], \mathfrak g]\subset \mathfrak g_{0}^{\perp}=\mathfrak g_{x}.\end{align*} $$

We deduce that $0=[x, [[x, \mathfrak g], \mathfrak g]]=[[x, [x, \mathfrak g]], \mathfrak g]$ . Since $\mathfrak g$ is semisimple, we see that $[x, [x, \mathfrak g]]=0$ . Thus, x is nilpotent. From Jacobson–Morozov theorem, there are elements $f, h\in \mathfrak {g}$ such that

$$ \begin{align*}[h, x]=2x, \quad[h, f]=-2f,\quad [x, f]=h.\end{align*} $$

We see that $[[x, h], [x, f]]= 4x\ne 0.$ So $\mathfrak g_{0}$ is noncommutative, which is a contradiction. Therefore, $\mathsf {Nij}(H)=0$ .