Deformation and Hochschild Cohomology of Coisotropic Algebras

Coisotropic algebras consist of triples of algebras for which a reduction can be defined and unify in a very algebraic fashion coisotropic reduction in several settings. In this paper we study the theory of (formal) deformation of coisotropic algebras showing that deformations are governed by suitable coisotropic DGLAs. We define a deformation functor and prove that it commutes with reduction. Finally, we study the obstructions to existence and uniqueness of coisotropic algebras and present some geometric examples.


Introduction
Symmetry reduction plays an important role in theoretical classical mechanics and quantum physics, and its various mathematical formulations have been studied extensively during the last half century. Probably the most well-known reduction procedure of this kind is the so-called Marsden-Weinstein reduction [23] of a symplectic manifold, which can also be understood as a special case of coisotropic reduction of a Poisson manifold. This standard construction of Poisson geometry allows to construct a new Poisson manifold out of a given coisotropic submanifold of a Poisson manifold. The main motivation of such reduction scheme comes from Dirac's idea [15] of quantizing the first-class constraints, which are described by coisotropic submanifolds, and obtaining a quantized version of coisotropic reduction.
Having this motivation in mind, one can choose deformation quantization [1], see [26] for a gentle introduction, to formulate quantization of Poisson geometry. Here the idea is that a classical mechanical system which is implemented by a Poisson manifold can equivalently be described by its Poisson algebra of real-valued functions on it. The quantized system corresponds to a (formal) deformation of the commutative algebra of functions such that the Poisson bracket gets deformed into the commutator of the possibly non-commutative deformed algebra. This procedure relies on a classical principle stating that deformations of mathematical objects are governed by associated differential graded Lie algebras (DGLAs). More precisely, formal deformations of an associative algebra A in the sense of Gerstenhaber [19] are given by formal Maurer-Cartan elements of the associated Hochschild DGLA C • (A), where two such deformations are considered to be equivalent if they lie in the same orbit of the action of the canonically associated gauge group. This leads to the moduli space Def of formal deformations. An important tool to understand formal deformations of associative algebras is Hochschild cohomology: the second and third Hochschild cohomology groups contain obstructions to the existence and equivalence of formal deformations.
In the setting of deformation quantization many versions of phase space reduction are available, starting with a BRST approach in [6] and more general coisotropic reduction schemes found in e.g. [2,3,5,[9][10][11]20]. Here reduction is treated in a very algebraic fashion: the vanishing functions on the coisotropic submanifold are deformed into a left ideal of the total algebra of all functions and the reduced algebra is the quotient of the normalizer of this left ideal modulo the ideal itself.
Recently, we introduced a more algebraic approach to reduction in both the quantum and classical setting, see [13]. In particular, we defined the notion of coisotropic algebra A consisting of a unital associative algebra A tot together with a unital subalgebra A N and a two-sided ideal A 0 ⊆ A N .
Such coisotropic algebras allow for a simple reduction procedure, with the reduced algebra given by A = A N /A 0 . The eponymous example is given by a Poisson manifold M together with a coisotropic submanifold C. Then ( C ∞ (M ), B C , J C ), with J C the ideal of functions vanishing on C and B C the Poisson normalizer of J C , defines a coisotropic algebra, and its reduced algebra B C /J C is isomorphic to the algebra of functions C ∞ (M red ) on the reduced manifold M red if the reduced space is actually smooth. It turns out that one has a meaningful tensor product leading to a bicategory of bimodules over coisotropic algebras such that reduction becomes a morphism of bicategories. Moreover, reduction turns out the be compatible with classical limits in a nice and general functorial way. It is important to notice that this notion recovers other examples coming from Poisson geometry, e.g. [16] and non commutative geometry, as [24] and [12]. Motivated by the significance of coisotropic algebras and their classical limit, in this paper we develop the corresponding theory of (formal) deformations. Following the above mentioned classical principle, we introduce the notion of coisotropic DGLA and we study formal deformations of the corresponding Maurer-Cartan elements. This allows us to define a deformation functor and to prove that the deformation functor commutes with reduction, in the sense that at least an injective natural transformation exists, see Theorem 3.14. Applying these techniques to the case of the coisotropic Hochschild complex of a coisotropic algebra we prove that the existence and uniqueness of formal deformations of coisotropic algebras are obstructed by its associated coisotropic Hochschild cohomology, see Theorem 4.19,Theorem 4.20. Moreover, it is shown that the construction of the coisotropic moduli space of deformations as well as that of the associated Hochschild cohomology are compatible with reduction.
The paper is organized as follows: in Section 2 some basic coisotropic versions of classical algebraic structures, such as coisotropic modules, coistropic algebras and coisotropic complexes, are introduced. These notions lead to a definition of a coisotropic DGLA. In Section 3 coisotropic DGLAs together with their coisotropic sets of Maurer-Cartan elements, their associated coisotropic gauge groups and the formal deformation of coisotropic Maurer-Cartan elements are considered and the compatibility of these constructions with reduction is examined. In the last Section 4 we introduce coisotropic Hochschild cohomology for coisotropic algebras and apply the results of Section 3 to the case of the coisotropic Hochschild complex. Finally, some examples of formal deformations of coisotropic algebras from geometry are given.
Acknowledgements: It is a pleasure to thank Andreas Kraft for important remarks on this paper.

Preliminaries on Coisotropic Modules
In the following denotes a fixed commutative unital ring, where we adopt the convention that rings will always be associative. Let us introduce the fundamental notion of a coisotropic -module, which is crucial to all further considerations.
iii.) The category of coisotropic -modules is denoted by C 3 Mod and the set of morphisms between coisotropic -modules E and F is denoted by Hom (E, F).
If the underlying ring is clear we will often just use the term coisotropic module. We will now collect some useful categorical constructions for coisotropic modules. The following statements can be proved by straightforward checks of the categorical properties, see e.g. [21]. Let E, F be coisotropic modules and let Φ, Ψ : E → F be morphisms of coisotropic modules.
a) The morphism Φ is a monomorphism if and only if Φ tot and Φ N are injective module homomorphisms.
b) The morphism Φ is an epimorphism if and only if Φ tot and Φ N are surjective module homomorphisms.
c) The morphism Φ is a regular monomorphism if and only if it is a monomorphism with Observe that the monomorphisms (epimorphisms) in C 3 Mod do in general not agree with regular monomorphisms (epimorphisms), showing that C 3 Mod is not an abelian category, unlike the usual categories of modules. This will cause some complications later on.
e) The kernel of Φ is given by the coisotropic module f) The cokernel of Φ is given by the coisotropic module g) The coisotropic module im(Φ) := coker(ker Φ) is given by 3) It will be called the image of Φ.
h) The coisotropic module regim(Φ) := ker(coker Φ) is given by It will be called the regular image of Φ. In the case of abelian categories, there is a canonical image factorization as coker(ker Φ) ≃ ker(coker Φ) for every morphism. This is not the case in the non-abelian category C 3 Mod , leading to two different factorization systems. Using the image every morphism of coisotropic modules can be factorized into a regular epimorphism and a monomorphism while using the regular image allows for a factorization into an epimorphism and a regular monomorphism.
i) The coequalizer of Φ and Ψ is given by the coisotropic module k) The coproduct of E and F is given by with ι ⊕ = ι E + ι F . It is called the direct sum of E and F. It should be clear that also infinite direct sums can be defined this way. Finite direct sums of coisotropic modules can be shown to be biproducts for the category C 3 Mod . In particular, it is clear that also products exist. A fundamental notion in this setting is the tensor product of coisotropic modules. This is an additional piece of information and is not fixed solely from the definition of the category C 3 Mod .
i.) The triple E ⊗ F is indeed a coisotropic -module. In particular, ii.) The reason we did not insist on ι being injective in Definition 2.1 is that the injectivity of ι ⊗ may be spoiled by torsion effects. Nevertheless, in many examples this will be the case.
This definition of tensor product allows us to construct a functor ⊗ : C 3 Mod ×C 3 Mod → C 3 Mod , which together with the coisotropic module = ( , , 0) as unit object turns C 3 Mod into a (weak) monoidal category, see e.g. [18].
l) The monoidal category C 3 Mod is a symmetric monoidal category with symmetry τ : The internal Hom of E and F is given by the coisotropic module where ι : Hom (E, F) → Hom (E tot , F tot ) is the projection onto the first component. We will denote the coisotropic module of endomorphisms by C 3 End (E) := C 3 Hom (E, E). Similarly, the coisotropic automorphisms are denoted by C 3 Aut . This internal Hom is in fact right adjoint to the tensor product. More precisely, we have · ⊗ E is left adjoint to C 3 Hom(E, · ), showing that C 3 Mod is in fact closed monoidal. From this follows in particular that for every x ∈ E N and Φ : E ⊗ F → G we get a coisotropic coevaluation morphism of modules Φ(x, · ) : F → G. Let us stress that Hom (E, F) only denotes the set of coisotropic morphisms and C 3 Hom (E, F) denotes the full coisotropic module of morphisms. The definition of coisotropic modules allows us to reinterpret several (geometric) reduction procedures in a completely algebraic fashion, as stated in the following straightforward proposition. where the category Mod of -bimodules is equipped with the usual tensor product.
Remark 2.5 Since the internal Hom C 3 Hom (E, F) is a coisotropic module itself we can apply the reduction functor red to it. There is a canonical -module morphism from C 3 Hom (E, F) red to Hom (E red , F red ) given by mapping [(Φ tot , Φ N )] to the map [Φ N ] induced by Φ N on the quotient E red = E N /E 0 . Note that this morphism is injective. Therefore, we can view C 3 Hom (E, F) red as the submodule of Hom (E red , F red ) consisting of morphism that allow for a extension to the totcomponents of E and F.

Coisotropic Algebras & Derivations
Consider again the prototypical example of a coisotropic submanifold C ֒→ M of a given Poisson manifold (M, π). Then the coisotropic module ( C ∞ (M ), B C , J C ) obviously carries more structure than a mere coisotropic module. In particular, C ∞ (M ) is an associative algebra with B C ⊆ C ∞ (M ) a subalgebra and J C ⊆ B C a two-sided ideal. This is now captured by the following definition of a coisotropic algebra. ii.) A morphism Φ : A → B of coisotropic algebras is given by a pair of unital algebra homomorphisms Φ tot : A tot → B tot and Φ N : iii.) The category of coisotropic -algebras is denoted by C 3 Alg .
Coisotropic algebras can also be understood as internal algebras in the monoidal category C 3 Mod .
Here the particular definition of the tensor product of coisotropic modules, see Definition 2.2, is crucial in order to realize A 0 as a two-sided ideal in A N . Note that the definition of a coisotropic algebra as provided above generalizes the one given in [13] slightly in that we do not assume ι : A N → A tot to be injective and A 0 needs not to be a left-ideal in A tot . Nevertheless, in most of our applications these additional features (requirements in [13]) will be satisfied.
Remark 2.7 Since A 0 ⊆ A N is a two-sided ideal by definition, we can construct a reduced algebra A red = A N /A 0 similar to Proposition 2.4. This yields a functor red : C 3 Alg → Alg . ii.) Let (M, π) be a Poisson manifold together with a coisotropic submanifold C ֒→ M . Then is a coisotropic algebra and A red ∼ = B C /J C turns out to be even a Poisson algebra.
On one hand, from an algebraic point of view, representations are important in the study of algebraic structures. On the other hand, by the famous Serre-Swan theorem, vector bundles over manifolds can equivalently be understood as finitely generated projective modules over the algebra of functions on the manifold. This justifies to take a closer look at modules in our context as well. The following gives a useful notion of (bi-)module over coisotropic algebras: iii.) The category of coisotropic (B, A)-bimodules is denoted by C 3 Bimod(B, A).
Note that a coisotropic (B, A)-bimodule E can also be defined as a coisotropic -module together with morphisms λ : B ⊗ E → E and ρ : E ⊗ A → E of coisotropic modules implementing the module structure. The tensor product of coisotropic -modules as defined in Definition 2.2 can be extended to bimodules over coisotropic algebras in the following way.
Lemma 2.10 Let A, B and C be coisotropic algebras and let F ∈ C 3 Bimod( C, B) as well as E ∈ C 3 Bimod(B, A) be corresponding bimodules. Then F C B ⊗ B E B A given by the components Coisotropic -modules can be understood as bimodules for the coisotropic algebra = ( , , 0), explaining our notation for the category C 3 Mod of coisotropic -modules.

Example 2.11
Let ι : C ⊆ M be a submanifold and D ⊆ T C an integrable distribution on C. Let moreover E tot → M be vector bundle over M , E N → M a subbundle of E tot and E 0 → C a subbundle of ι # E N . Moreover, let ∇ be a flat partial D-connection on ι # E N . Then setting Note that the construction of E N strongly depends on the choice of the covariant derivative. Coisotropic modules of this form are important in a coisotropic version of the Serre-Swan theorem, see [14].
Coisotropic algebras together with coisotropic bimodules, their morphisms and their tensor product as above can be arranged in a bicategory structure. Mapping a coisotropic algebra A to its reduced algebra A red = A tot /A N and a coisotropic (A, B)-bimodule E to the (A red , B red )-bimodule E red = E N /E 0 defines a functor of bicategories, see [13].
From a geometric perspective the tangent bundle of a given manifold corresponds to the derivations of the algebra of functions on that manifold by taking sections. In order to give a definition of a derivation of a coisotropic algebra we rephrase the classical definition in an element-independent way.
holds, where ρ and λ denote the right and left A-multiplications of M, respectively. The set of derivations will be denoted by Der(A, M). If M = A we write Der(A).
We can arrange the coisotropic derivations as a coisotropic submodule of the internal homomorphism C 3 Hom (A, M) as follows.
One needs to be careful with the notation here since Der(A) has different meanings depending whether A is a coisotropic or a classical algebra. Note also that As for usual algebras the derivations turn out to be a bimodule if the algebra is commutative: Alg be a commutative coisotropic algebra. Then C 3 Der(A) is a coisotropic A-bimodule.
. The kernel of this linear map is exactly given by C 3 Der(A) 0 , thus there exists an injective module homomorphism This is simply the restriction of the canonical injective morphism from Remark 2.5 to the submodule C 3 Der(A).

Example 2.15
Our notion of a coisotropic algebra generalizes and unifies previous notions used in noncommutative geometry referring to features of the derivations: i.) A submanifold algebra in the sense of [24] and [12] can equivalently be described as a coisotropic algebra A with A tot = A N such that the canonical module morphism (2.22) is an isomorphism.
ii.) A quotient manifold algebra in the sense of [24] can equivalently be described as a coisotropic Here Z(A) denotes the coisotropic center of the coisotropic algebra A, see Proposition 4.12, i.) for the definition.
We can also define inner derivations by requiring the existence of appropriate elements in each component.

Coisotropic Homological Algebra
We collect some definitions and statements about (cochain) complexes of coisotropic modules. Most of this can be done as in every abelian category. But since C 3 Mod is not abelian we have to be careful when defining coisotropic cohomology, since we have two different notions of images, see Section 2.1, 2.1 and 2.1.

Definition 2.17 (Graded coisotropic module)
Let be a commutative unital ring.
iii.) We denote the category of graded coisotropic modules by C 3 Mod • .
We combine the indexed family of a graded coisotropic module into a single coisotropic module We will use the usual tensor product 26) and the symmetry with the usual Koszul signs.
Definition 2.18 (Coisotropic complex) Let be a commutative unital ring.
iii.) The category of coisotropic complexes is denoted by Ch(C 3 Mod ).
Since morphisms of cochain complexes commute with the differential δ, it is easy to see that we obtain a new functor by constructing the cohomology of the coisotropic complex.
Remark 2.20 ((Regular) image) Note that the coisotropic cohomology is defined by using the image of morphisms of coisotropic modules and not the regular image. However, choosing the regular image instead would not make a difference since the 0-component of the denominator is not used in the quotient of coisotropic modules, see (2.6). Moreover, note that in general we can not decide whether ker δ = im δ by computing cohomology, but we can decide if ker δ = regim δ holds.
Since graded coisotropic modules and coisotropic complexes are given by -indexed families of coisotropic modules it should be clear that applying the reduction functor in every degree yields functors red : C 3 Mod • → Mod • and red : Ch(C 3 Mod ) → Ch(Mod ). It is now natural to investigate the relation between the cohomology functor and the reduction functor. The following proposition shows that reduction and cohomology functors commute.

commutes.
Proof: Define η for every M ∈ Ch(C 3 Mod ) by showing that η is indeed a natural isomorphism.
is an isomorphism of coisotropic modules. We remark that the reduction functor red : Ch(C 3 Mod ) → Ch(Mod ) maps quasi-isomorphisms of coisotropic cochain complexes to quasi-isomorphisms of cochain complexes.

Coisotropic DGLAs
By a well-known principle of classical deformation theory, a deformation problem is controlled by a certain differential graded Lie algebra, see e.g. [22]. Thus, the first step to discuss the deformation theory of coisotropic algebras consists in introducing a suitable notion of coisotropic differential graded Lie algebra (DGLA) and a deformation functor in this realm.
Definition 3.1 (Coisotropic differential graded Lie algebra) Let be a commutative unital ring.
i.) A coisotropic DGLA g over is a pair of DGLAs ii.) For two coisotropic DGLAs g and h, a morphism Φ : The category of coisotropic DGLAs will be denoted by C 3 dgLieAlg.
Note that a morphism of coisotropic DGLAs can equivalently be understood as a morphism of coisotropic modules such that its components are DGLA morphisms. A coisotropic Lie algebra is a coisotropic DGLA with trivial differential concentrated in degree 0. Similarly a coisotropic graded Lie algebra is a coisotropic DGLA with trivial differential. Two important examples of coisotropic Lie algebras are obtained as follows: i.) Let E be a coisotropic -module. The internal endomorphisms C 3 End (E) are a coisotropic Lie algebra given by the usual commutator [ · , · ] Etot on C 3 End (E) tot and the pair ( ii.) Let A be a coisotropic algebra over . It is straightforward to see that C 3 Der(A) is a coisotropic -submodule of the coisotropic -module C 3 End (A). Moreover, C 3 Der(A) is even a coisotropic Lie subalgebra of the coisotropic Lie algebra C 3 End (A). All canonical maps like (2.22) are in fact Lie morphisms.
Since every coisotropic DGLA g is, in particular, a coisotropic cochain complex we can always construct its corresponding cohomology H(g). Moreover, every morphism Φ : g • −→ h • of coisotropic DGLAs is a morphism of coisotropic cochain complexes and therefore it induces a morphism H(Φ) : H • (g) −→ H • (h) on cohomology. Clearly, H(g) is a coisotropic graded Lie algebra and every induced morphism H(Φ) is a morphism of coisotropic graded Lie algebras. If H(Φ) is an isomorphism we call Φ a coisotropic quasi-isomorphism. From Remark 2.22 it es clear that reduction of coisotropic DGLAs preserves quasi-isomorphisms.
Following the standard way to define a deformation functor for a given DGLA, we aim to define a Maurer-Cartan functor and to introduce a notion of gauge equivalence. In order to define the Maurer-Cartan elements in the coisotropic DGLA we first need an appropriate notion of a coisotropic set: ii.) A morphism f : M → N of coisotropic sets M and N is given by a pair of maps f tot : M tot → N tot and f N : iii.) The category of coisotropic sets is denoted by C 3 Set.
Remark 3.4 Every coisotropic -module E, and hence every coisotropic algebra, coisotropic DGLA, etc., has an underlying coisotropic set in the sense that E N can be equipped with the equivalence relation induced by the submodule E 0 . In this sense coisotropic sets form the underlying structure for all the different notions of coisotropic algebraic structures.
Given a coisotropic set we can clearly define a reduced one, as for coisotropic modules, by taking the quotient M red = M/∼. This also yields a reduction functor red : C 3 Set → Set.
We can now define the coisotropic set of Maurer-Cartan elements of a coisotropic DGLA. Recall that a Maurer-Cartan element in a DGLA g • is an element ξ ∈ g 1 satisfying the Maurer-Cartan equation While up to here we did not have to make any further assumption about the ring of scalars, from now on we assume É ⊆ in order to have a well-defined Maurer-Cartan equation. We denote by MC(g) the set of all Maurer-Cartan elements of a DGLA.
Definition 3.5 (Coisotropic set of Maurer-Cartan elements) Let g be a coisotropic DGLA over a commutative unital ring . The coisotropic set MC(g) of Maurer-Cartan elements of g is given by
Note that for any coisotropic DGLA g and coisotropic algebra A the tensor product g ⊗ A is again a coisotropic DGLA by the usual construction. For this observe that Reformulating the equivalence of deformations of a given Maurer-Cartan element in terms of its twisted coisotropic DGLA requires a notion of a coisotropic gauge group. For this reason we first introduce the notion of a coisotropic group: ii.) A morphism Φ : G → H of coisotropic groups G and H is given by a pair of group homomorphisms Φ tot : G tot → H tot and Φ N : iii.) The category of coisotropic groups is denoted by C 3 Group.
Again, we obviously have a reduction functor red : C 3 Group → Group given by G red = G N /G 0 . Moreover, there is a forgetful functor C 3 Group → C 3 Set by only keeping the underlying sets and the equivalence relation induced by the normal subgroup G 0 . It can be shown that the automorphisms of a coisotropic set can be equipped with the structure of a coisotropic group. This leads to the definition of an action of a coisotropic group on a coisotropic set.  ii.) Let X = (X tot , X N , ∼) a coisotropic set. Let furthermore G be a group acting on X tot via Φ : G × X tot → X tot . Then (G, G X N , G ∼ ), with G X N the stabilizer subgroup of the subset X N and G ∼ the normal subgroup of G X N consisting of all g ∈ X X N such that Φ g (p) ∼ p for all p ∈ X N , is a coisotropic group. Clearly, (Φ, Φ G X N ) gives a coisotropic action on (X tot , X N , ∼).
To define the coisotropic gauge group we either need to assume that the DGLA we are starting with has additional properties, e.g. being nilpotent, or we can use formal power series instead. Since later on we are interested in formal deformation theory, we will choose the latter option. For this let λ = ( λ , λ , 0) denote the coisotropic ring of formal power series in .
Then the formal power series E λ of any coisotropic -module E form a coisotropic λ -module as follows: we set and use the canonical λ-linear extension ι E λ of the previous map ι E : E N −→ E tot . According to the usual convention, we denote this extension simply by ι E . Note that in general E λ is strictly larger than the tensor product E ⊗ λ : we still need to take a λ-adic completion. This is the reasons that we define E λ directly by (3.6).
It is now easy to see that g λ is a coisotropic DGLA for any coisotropic DGLA g by λ-linear extension of all structure maps. Similarly, we can extend coisotropic algebras and their modules.
Note that the gauge action will require to have É ⊆ since we need (formal) exponential series and the (formal) BCH series. Proposition 3.10 Let g be a coisotropic Lie algebra. Then G(g) = (λg tot λ , λg N λ , λg 0 λ ) with multiplication • given by the Baker-Campbell-Hausdorff formula is a coisotropic group.
Proof: The additional prefactor λ makes all the BCH series λ-adically convergent. The well-known group structures on g tot λ and g N λ are given by the BCH formula and we clearly have a group morphism g N λ → g tot λ . Finally, we need to show that λg 0 λ is a normal subgroup of λg N λ . For this let λg ∈ λg N λ and λh ∈ λg 0 λ be given. Since by the BCH formula λg • λh • (λg) −1 = λg 0 + λh 0 − λg 0 + λ 2 (· · · ), where all higher order terms are given by Lie brackets and g 0 is a Lie ideal in g N , we see that λg • λh • (λg) −1 ∈ λg 0 λ .
By abuse of notation we will write G(g) = G(g 0 ) for every coisotropic DGLA g. With the composition • on G(g) defined by the Baker-Campbell-Hausdorff formula it is immediately clear that every morphism Φ : g → h of coisotropic DGLAs induces a morphism G(Φ) : G(g) → G(h) of the corresponding gauge groups, given by the λ-linear extension of Φ. In other words, we obtain a functor G : The usual gauge action of the formal group on the (formal) Maurer-Cartan elements can now be extended to a coisotropic DGLA as follows: for λg ∈ G(g) tot and ξ ∈ MC(λg λ ) tot as well as for λg ∈ G(g) N and ξ ∈ MC(λg λ ) N .

Deformation functor and reduction
Maurer-Cartan elements are said to be equivalent if they lie in the same orbit of the gauge action. Hence the object of interest for deformation theory is not the set of Maurer-Cartan elements itself but its set of equivalence classes. More precisely let us denote by Def(g) the pair given by with an equivalence relation on Def(g) N defined by Proposition 3.12 Let g be a coisotropic DGLA. Then Def(g) is a coisotropic set.
We have seen in Lemma 3.6 that morphisms of coisotropic DGLAs induce morphisms between the corresponding coisotropic sets of Maurer-Cartan elements. This is still true after taking the quotient by the coisotropic gauge group.
and similar for the tot-component, showing that MC(Φ) is equivariant along G(Φ) and hence inducing a morphism Def(Φ) as needed.
The question arises if the above constructions of the coisotropic set of Maurer-Cartan elements, the coisotropic gauge group and the deformation functor commute with reduction. The next theorem shows that this is partially true, in the sense that at least an injective natural transformation exists.  ii.) There exists a natural isomorphism η : red • G =⇒ G • red, i.e.
showing that η is natural.
ii.) Then η g : G(g) red → G(g red ) given by [λg] G → λ[g] g , where [g] g denotes the equivalence class of g in g red , is well-defined. Indeed, η g is just the λ-linear extension of the obvious identity g N /g 0 = g red . Moreover, η g is a group morphism, since [ · ] g : g N → g red is a morphism of DGLAs and • is given by sums of iterated brackets. Naturality follows directly.

Deformations of Coisotropic Algebras
In formal deformation quantization one is interested in algebras of formal power series over a ring  = ( λ , λ , 0). Given a coisotropic -algebra A ∈ C 3 Alg , we can define a formal deformation to be a coisotropic λ -algebra B together with an isomorphism α : cl(B) −→ A.
Here cl(B) denotes the classical limit as introduced in [13]: The classical limit of a coisotropic λalgebra B is the coisotropic -algebra defined as cl(B) = B/λB = (B tot /λB tot , B N /λB N , B 0 /λB N ).
It is easy to see that this definition agrees with the one from deformation via Artin rings, see e.g. [22]. Usually, one is interested in more specific deformations, namely those that are e.g. free -modules. This leads us to the following definition: Definition 4.1 (Deformation of coisotropic algebra) Let A ∈ C 3 Alg be a coisotropic algebra. A (formal associative) deformation of A is given by an associative multiplication µ : A λ ⊗ A λ −→ A λ on A λ turning it into a coisotropic λ -algebra such that cl(A λ , µ) ≃ A.
Let us comment on this definition. First recall that we have with the structure map ι A λ = ι A being just the λ-linear extension of the previous map according to (3.6). Then we have two formal associative deformations µ tot and µ N for A tot λ and A N λ of the , respectively, such that the undeformed map ι A is an algebra homomorphism and such that A 0 λ is a two-sided ideal in A N λ with respect to µ N . Note that we insist on the A N and A 0 being the same up to taking formal series.
Also the algebra morphism ι A is not deformed.
One particular scenario we will be interested in the context of deformation quantization of phase space reduction is the following: Example 4.2 For convenience, we will assume that is actually a field and not just a ring. Let A = (A tot ⊇ A N ⊇ A 0 ) be a coisotropic triple such that A 0 ⊆ A tot is a left ideal and A N ⊆ N(A 0 ) is a unital subalgebra of the normalizer of this left ideal. In particular, ι A is just the inclusion. Consider now a formal associative deformation µ tot of A tot with the additional property that the formal series A 0 λ are still a left ideal inside A tot λ with respect to µ tot . Moreover, assume that the normalizer A A A N = N µtot (J λ ) ⊆ A tot λ with respect to µ tot satisfies It is now easy to check that A A A N ⊆ A tot λ is a closed subspace with respect to the λ-adic topology.  where q = ι D + ∞ r=1 λ r q r with ι D being the canonical inclusion of the subspace. By our assumption, D ⊆ A N but the inclusion could be proper. Moreover, since by our assumption A 0 λ ⊆ N(A 0 λ ) = A A A N , we have A 0 ⊆ D.
Since we work over a field, we can find a complement C ⊆ D such that A 0 ⊕ C = D. This allows to redefine the maps q r to q ′ r C = q r C and q ′ r A 0 = 0.

(4.4)
The resulting map q ′ then satisfies q ′ (D λ ) = A A A N and q ′ A 0 = id A 0 . We can then use q ′ to pass to a new deformation µ ′ tot of A tot with the property that A 0 λ is still a left ideal in A tot λ with respect to µ ′ tot and the normalizer of this left ideal is now given by D λ ⊆ A tot λ . It follows that µ ′ tot provides a deformation of the coisotropic triple (A tot , D, A 0 ) in the sense of Definition 4.1.
Of course, it might happen that D = A N and hence this construction will not provide a deformation of the original coisotropic triple, in general. It turns out that this can be controlled as follows: we assume in addition that the deformed normalizer A A A N is large enough in the sense that the classical limit cl : between the reduced algebras is surjective. As is a field, this gives us a split Q : A red −→ A A A red which we can extend λ-linearly to It is then easy to see that this is in fact a λ -linear isomorphism. It follows, that in this case we necessarily have (4.7) Thus the previous construction gives indeed a deformation µ ′ tot of the original coisotropic triple. This seemingly very special situation will turn out to be responsible for one of the main examples from deformation quantization.

Coisotropic Hochschild Cohomology
From now on we assume that É ⊆ . Let M, N ∈ C 3 Mod be coisotropic modules. We define for any n ∈ AE In other words, f N maps to N 0 if at least one tensor factor comes from M 0 . This clearly defines a graded coisotropic -module C • (M, N). Then we need to show that [ · , · ] M N preserves the 0-components. This follows directly from the usual formula for the Gerstenhaber bracket, see [19]. [ · , · ] forms a graded coisotropic Lie algebra.

Remark 4.4
The coisotropic Gerstenhaber bracket can also be derived from a coisotropic pre-Lie algebra structure on C • (M), which in turn results from a sort of partial composition. These partial compositions can be interpreted as the usual endomorphism operad structure of M in C 3 Mod . The theory of operads in the (non-abelian) category C 3 Mod will be the subject of a future project.
As in the standard theory of deformation of associative algebras, we can characterize associative multiplications by using the Gerstenhaber bracket. Note that (4.10) only involves the N-component of the coisotropic Gerstenhaber bracket [ · , · ]. Using the coisotropic structure of C 2 (M) we get ι 2 (µ) = µ tot ∈ C 2 (M) tot , from which directly [µ tot , µ tot ] tot = 0 follows.
Let us now move from a module M to an algebra (A, µ). Then we can use the multiplication to construct a differential on C • (A). is a coisotropic chain map of degree 1 and δ 2 = 0.
Proof: Since µ tot is an associative multiplication on A tot we know that δ tot : is a differential. Moreover, it is clear that δ N : C • (A) N −→ C • (A) N is also a differential and it preserves the 0-component by the definition of [ · , · ] N . Finally, we have for ) holds, and hence (δ tot , δ N ) is a coisotropic morphism.
The coisotropic Hochschild differential can be interpreted as twisting the coisotropic DGLA (C • (A), [ · , · ], 0) with the Maurer-Cartan element µ ∈ C 2 (A) N , but with signs chosen in such a way that it corresponds to the usual Hochschild differential. More explicitly we have the following result.   Assigning the Hochschild complex to a given (coisotropic) algebra is not functorial on all of C 3 Alg . But if we restrict ourselves to the subcategory C 3 Alg × of coisotropic algebras with invertible morphisms we get a functor C • : C 3 Alg × → C 3 dgLieAlg by mapping each coisotropic algebra to its Hochschild complex and every algebra isomorphism φ : A similar construction clearly also works for usual algebras. We can now show that this functor commutes with reduction up to an injective natural transformation. Proposition 4.9 (Hochschild complex vs. reduction) There exists an injective natural transformation η : red • C • =⇒ C • • red, i.e.

Proof: For every coisotropic algebra
red → A red is well defined since if a i ∈ A 0 for any i = 1, . . . , n we have f N (a 1 , . . . , a n ) ∈ A 0 and hence [f N (a 1 , . . . , a n )] = 0. Moreover, η A is well-defined since for f ∈ C n (A) 0 we have f N (a 1 , . . . , a n ) ∈ A 0 and thus η([f ]) = 0. To see that η is indeed a natural transformation we need to show that for every isomorphism φ :   and With this we can compute the zeroth and first Hochschild cohomology of a given coisotropic algebra. The following also shows that in low degrees the interpretation of the coisotropic Hochschild cohomology is analogous to that for usual algebras.
Proposition 4.12 Let A ∈ C 3 Alg be a coisotropic algebra.
i.) We have  ii.) We have

Formal Deformations
Throughout this section we will assume that the scalars satisfy É ⊆ in order to make use of the description of deformations by Maurer-Cartan elements. Let (A, µ 0 ) ∈ C 3 Alg be a coisotropic associative -algebra. By Definition 4.1 a formal associative deformation (A λ , µ) is given by an associative multiplication µ : A λ ⊗ A λ −→ A λ making A λ a coisotropic λ -algebra such that cl(A, µ) is given by (A, µ 0 ), or in other words where we used the associativity of µ 0tot and the graded skew-symmetry of Gerstenhaber bracket. The very same holds for the N-component.
Let us now consider two formal associative deformations µ and µ ′ of (A, µ 0 ). We say that they are equivalent if there exists T = id +λ(. for a, b ∈ A tot/N . Thus, as in the case of associative algebras, there exists a unique D = ∞ k=0 λ k D k ∈ C 3 Hom (A λ , A λ ) such that T = exp(λD). This allows us to conclude the following claim. where ad(D) = [D, · ] using the coisotropic Gerstenhaber bracket.
Note that (4.29) is equivalent to Summing up the above lemmas we can state the relation between formal deformations and the deformation functor. Finally, we can reformulate the classical theorem about the extension of a deformation up to a given order for coisotropic algebras.
Proof: By the classical deformation theory of associative algebras it is clear that (4.32) is closed can be extended via (µ k+1 ) tot and (µ k+1 ) N , respectively. Thus µ k+1 yields an extension of µ (k) . On the other hand, if µ (k) can be extended, we know that Thus the obstructions for deformations of an associative structure on a coisotropic module are given by HH 3 (A) N . The coisotropic module HH 3 (A) carries more information than just the obstructions to deformations of the coisotropic algebra A. Since HH 3 (A) tot = HH 3 (A tot ) it also encodes the obstructions of deformations of the classical algebra A tot . Moreover, HH 3 (A) 0 is important for the reduction of HH 3 (A) and hence controls which obstructions on A descend to obstructions on A red .
In particular, we have seen at the end of Section 4.2 that HH 3 (A) red ⊆ HH 3 (A red ).

Example I: BRST Reduction
The above definition of a deformation of a coisotropic algebra recovers the following two interesting examples from deformation quantization. The first comes from BRST reduction of star products.
We recall the situation of [6,20]. Consider a Poisson manifold M with a strongly Hamiltonian action of a connected Lie group G and momentum map J : M −→ g * , where g is the Lie algebra of G. One assumes that the classical level surface C = J −1 ({0}) ⊆ M is a non-empty (necessarily coisotropic) submanifold by requiring 0 to be a regular value of J. Moreover, we assume that the action of C is proper. Then we have the classical coisotropic triple where J C = ker ι * ⊆ C ∞ (M ) is the vanishing ideal of the constraint surface C ⊆ M and B C its Poisson normalizer. Next, we assume to have a star product ⋆ invariant under the action of G which admits a deformation J of J into a quantum momentum map. In the symplectic case such star products always exist since we assume the action of G to be proper, see [25] for a complete classification and further references. In the general Poisson case the situation is less clear.
Out of this a coisotropic λ -algebra A A A := ( C ∞ (M ) λ , B B B C ,J J J C ) is then constructed, where J J J C = ker ι * ⊆ C ∞ (M ) λ is the quantum vanishing ideal given by the kernel of the deformed restriction ι * = ι * • S. Here S = id + ∞ k=1 λ k S k is a formal power series of differential operators guaranteeing that J J J C is indeed a left ideal with respect to ⋆. In fact, S can be chosen to be G-

invariant.
We now want to construct a coisotropic algebra ( C ∞ (M ) λ , B C λ ,J C λ ) which is isomorphic to ( C ∞ (M ) λ , B B B C ,J J J C ). For this, note that S : C ∞ (M ) λ −→ C ∞ (M ) λ is invertible, hence we get a star product on C ∞ (M ) λ . From ι * = ι * • S directly follows, that S mapsJ J J C to J C λ . It is slightly less evident, but follows from the characterization of the normalizer B B B C as those functions whose restriction to C are G-invariant, that S maps the normalizer B B B C to the normalizer B B B S C of J C λ with respect to ⋆ S . Finally, we know that f ∈ B B B C if and only if for all ξ ∈ g it holds that 0 = L ξ C ι * f = L ξ C ι * Sf . Hence f ∈ B B B C if and only if Sf ∈ B C λ . Thus S is an isomorphism of coisotropic triples In particular, we have a deformation A A A S = (( C ∞ (M ) λ , ⋆ S ), B C λ ,J C λ ) of the classical coisotropic triple in this case, and the coisotropic triple A A A is isomorphic to it.

Example II: Coisotropic Reduction in the Symplectic Case
While the previous example makes use of a Lie group symmetry, the following relies on a coisotropic submanifold only. However, at the present state, we have to restrict ourselves to a symplectic manifold (M, ω). Thus let ι : C −→ M be a coisotropic submanifold. We assume that the classical reduced phase space M red = C ∼ is smooth with the projection map π : C −→ M red being a surjective submersion. It follows that there is a unique symplectic form ω red on M red with π * ω red = ι * ω. We follow closely the construction of Bordemann in [2,3] to construct a deformation of the classical coisotropic triple To this end, one considers the product M ×M − red with the symplectic structure pr * M ω −pr * M red ω red . Then I : C ∋ p → (ι(p), π(p)) ∈ M × M red (4.36) is an embedding of C as a Lagrangian submanifold. By Weinstein's Lagrangian neighbourhood theorem one has a tubular neighbourhood U ⊆ M × M red and an open neighbourhood V ⊆ T * C of the zero section ι C : C −→ T * C in the cotangent bundle π C : T * C −→ C with a symplectomorphism Ψ : U −→ V , where T * C is equipped with its canonical symplectic structure, such that Ψ • I = ι C . In the symplectic case, star products ⋆ are classified by their characteristic or Fedosov class c(⋆) The assumption of having a smooth reduced phase space allows us now to choose star products ⋆ on M and ⋆ red on M red in such a way that ι * c(⋆ U ) = π * c(⋆ red ). Note that this is a non-trivial condition on the relation between ⋆ and ⋆ red which, nevertheless, always has solutions. Given such a matching pair we have a star product ⋆⊗ ⋆ opp red on M × M − red by taking the tensor product of the individual ones. Note that we need to take the opposite star product on the second factor as we also took the negative of ω red needed to have a Lagrangian embedding in (4.36). It follows that the characteristic class c (⋆ ⊗ ⋆ opp red ) U = 0 is trivial. On the cotangent bundle T * C the choice of a covariant derivative induces a standard-ordered star product ⋆ std together with a left module structure on C ∞ (C) λ via the corresponding symbol calculus, see [8]. The characteristic class of ⋆ std is known to be trivial, c(⋆ std ) = 0, see [7]. Hence the pull-back star product Ψ * (⋆ std V ) is equivalent to (⋆ ⊗ ⋆ opp red ) U . Hence we find an equivalence transformation between Ψ * (⋆ std ) and ⋆ ⊗ ⋆ red on the tubular neighbourhood U . Using this, we can also pull-back the left module structure to obtain a left module structure on C ∞ (C) λ for the algebra C ∞ (M × M red ) λ . Note that here we even get an extension to all functions since the left module structure with respect to ⋆ std coming from the symbol calculus is by differential operators and Ψ • I = ι C . Hence the module structure with respect to ⋆ ⊗ ⋆ opp red is by differential operators as well.
This ultimately induces a left module structure ⊲ on C ∞ (C) λ with respect to ⋆ and a right module structure ⊳ with respect to ⋆ red such that the two module structures commute: we have a bimodule structure. Moreover, it is easy to see that the module endomorphisms of the left ⋆-module are given by the right multiplications with functions from C ∞ (M red ) λ , i.e.
End ( C ∞ (M ) λ ,⋆) ( C ∞ (C) λ ) opp ∼ = C ∞ (M red ) λ . (4.37) Moreover, one can construct from the above equivalences a formal series S = id + ∞ r=1 λ r S r of differential operators S r on M such that the left module structure is given by f ⊲ ψ = ι * (S(f ) ⋆ prol(ψ)), Thanks to the explicit formula for ⊲ we can use the series S to pass to a new equivalent star product ⋆ ′ such that J J J ′ C = J C λ . We see that this brings us precisely in the situation of Example 4.2: The coisotropic algebra A A A = (A A A tot , A A A N , A A A 0 ) with A A A tot = ( C ∞ (M ) λ , ⋆) and A A A N = B B B C as well as A A A 0 =J J J C is isomorphic to a deformation of the classical coisotropic algebra A we started with. Note that it might not be directly a deformation of A as we still might have to untwist first J J J C using S and then B B B C as in Example 4.2. This way we can give a re-interpretation of Bordemann's construction in the language of deformations of coisotropic algebras.

Outlook
When working with coisotropic algebras and related structures it is a recurring theme to investigate the compatibility of a given construction with the reduction functor. We have seen in Theorem 3.14 and Proposition 4.9 that the compatibility with reduction might only be given up to an injective natural transformation, and in general it seems that one can not expect much more. Nevertheless, it would be rewarding to find special situations in which the deformation functor Def or the construction of the Hochschild complex commute with reduction up to a natural isomorphism.
Given a bimodule over a coisotropic algebra it should be clear that one can define the coisotropic Hochschild complex and its cohomology also with coefficients in the bimodule. This can then be used to formulate also the deformation problem for (bi-)modules.
Having established coisotropic Hochschild cohomology and its importance in deformation theory of coisotropic algebras one would like to be able to actually compute it in certain cases. A first important example known from classical differential geometry is the Hochschild-Kostant-Rosenberg theorem, implementing a bijection between the Hochschild cohomology of the algebra of functions on a manifold and its multivector fields. A coisotropic version of this result for coisotropic algebras of the form ( C ∞ (M ), C ∞ (M ) F , J C ), with M a smooth manifold, J C the vanishing ideal of a submanifold and C ∞ (M ) F the functions on M which are constant along a foliation F on C, would be desirable. To achieve this it will be necessary to carry over other notions of differential geometry, like multivector fields etc., to the coisotropic setting. It will be important to consider also geometrically motivated bimodules for the coefficients in such scenarios. The cohomologies computed in [4] should be related to the coisotropic Hochschild cohomology, at least for particular and simple cases of submanifolds and foliations.