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Bialgebra actions, twists, and universal deformation formulas

We introduce a general theory of twisting algebraic structures based on actions of a bialgebra. These twists are closely related to algebraic deformations and also to the theory of quasi-triangular bialgebras. In particular, a deformation produced from a universal deformation formula (UDF) is a special case of a twist. The most familiar example of a deformation produced from a UDF is perhaps the"Moyal product"which (locally) is the canonical quantization of the algebra of functions on a symplectic manifold in the direction of the Poisson bracket. In this case, the derivations comprising the Poisson bracket mutually commute and so this quantization is essentially obtained by exponentiating this bracket. For more general Poisson manifolds, this formula is not applicable since the associated derivations may no longer commute. We provide here generalizations of the Moyal formula which (locally) give canonical quantizations of various Poisson manifolds. Specifically, whenever a certain central extension of a Heisenberg Lie group acts on a manifold, we obtain a quantization of its algebra of functions in the direction of a suitable Poisson bracket obtained from noncommuting derivations.


Introduction
The purpose of this paper is to introduce a general notion of "twisting" algebraic structures based on actions of a bialgebra B and to relate these twists to deformation theory. We show that elements of B ⊗ B can be used to twist the multiplication of any B-module algebra or B-module coalgebra. Moreover, we focus on a certain subcategory of left A-modules which naturally twists to a subcategory of modules over the twisted algebra and we show that these two categories are equivalent. Similar statements can be made for certain categories of comodules. When the bialgebra is (kG) * with G a finite group or monoid, the algebras to which our construction applies are the G-graded algebras A = g∈G A g (these being the (kG) * -module algebras) and the modules which twist are the graded modules.
For algebras graded by an infinite group or monoid it is more convenient to use the "dual" of our theory which will produce twists of A as a kG-comodule algebra. In these cases, the twists we obtain are precisely the "cocycle" twists of [AST].
Another important case is when the bialgebra is the universal enveloping algebra of a Lie algebra. Here, the twists which naturally arise are related to deformation theory and, in particular, "universal deformation formulas" (cf [GGS,Section 6] or [CGS]). Our theory of twisting provides alternative proofs to the basic properties of these formulas. In particular, we bypass the cohomological arguments previously used to establish that the deformations obtained from these formulas are in fact associative. The supply of such formulas is scarce.
Indeed, apart from those based on commutative, bialgebras only a few isolated examples are known, even though some results of Drinfel'd ([D]) imply that there is such a formula associated to every (constant) unitary solution to the classical Yang-Baxter equation. We broaden the supply of universal deformation formulas here by providing the first family based on non-commutative bialgebras. More specifically, for every n ≥ 3 we present a formula based on the universal enveloping algebra U H where H ⊂ sl(n) is a central extension of a Heisenberg Lie algebra. These formulas are generalizations of the "quasi-exponential" formula based on the two-dimensional solvable Lie algebra, see [CGG]. A natural use of a universal deformation formula based on an enveloping algebra U g is to deform coordinate rings of various algebraic varieties. In particular, if an algebraic group with Lie algebra g acts as automorphisms of a variety V , then the formula induces a deformation of the coordinate ring O(V ). That is, the quantization takes place solely because of the action of the group. These quantizations are generalizations of the classical " Moyal product" ([M]) which, in our language is a deformation of O (R 2n ) obtained from a universal deformation formula associated to an abelian Lie algebra. This quantization is in the "direction" of the standard Poisson bracket on O (R 2n ). In a similar way, a universal deformation formula provides deformations of coordinate rings in the direction of a suitable Poisson bracket. In the analytic case, if G is a Lie group which acts as diffeomorphisms of a manifold M , then a formula based on U g will deform the ring C ∞ (M ). For actions of R d , Rieffel has shown more -namely that this action induces a strict deformation quantization of C ∞ (M ) as a C * -algebra. It is natural to ask whether similar constructions exist for actions of non-abelian Lie groups.
Finally, we give here a skew form of the "quasi-exponential" formula mentioned above.
A skew form of a universal deformation formula has certain parity properties that make it more convenient in the study of * -products on Poisson manifolds. We do not, as of yet, have a skew form for our formula based on U H although, in theory, every formula based on an enveloping algebra can be skew-symmetrized.

Twisting and Deformations
Let k be a fixed commutative ring. Throughout this paper, all algebras, coalgebras, and their respective modules and comodules will be symmetric k-modules and their tensor product over k will be denoted ⊗. We will generally follow [Mon] for basic definitions and notation about bialgebras and their actions.
we similarly have the right multiplication map b r : N → N . In this situation, we can make M ⊗ M and N ⊗ N into, respectively, left and right B ⊗ B-modules via the maps Condition (1.1.2) is equivalent to commutativity of the diagram It is also easy to see that if A is a left B-module algebra then the primitive elements of B act as derivations of A and the grouplike elements act as automorphisms. In a similar way, we can define the notion of a right B-module algebra. The foregoing may be easily dualized to obtain the notion of a left or right B-module coalgebra.
Note that (1.2.2) is an expression which must hold in B ⊗ B ⊗ B. As we shall see, such F can be used to "twist" the multiplication of any left B-module algebra A and the 3 comultiplication of any right B-module coalgebra C. More precisely, a twisting element provides a new multiplication on the underlying k-module of a B-module algebra and a new comultiplication on the underlying k-module of a B-module coalgebra. For the algebra case, the twisted multiplication is defined to be the composite µ A • F l : A ⊗ A → A and for the coalgebra case the twisted comultiplication is F r • ∆ C : C → C ⊗ C.
is an associative k-algebra.
(2) If C is a right B-module coalgebra, then Proof. We only prove (1) since its dual establishes (2). The associativity of µ A • F l can be determined by considering the following diagram: Each of the four inner squares commutes: the lower right square represents the associativity of µ A , the top left square commutes by (1.2.2) since we are assuming that F is a twisting element, and the commutativity of the two "off-diagonal" squares follows from (1.
Similarly, 1 A also serves as the right unit under µ A • F .

Remarks 1.4.
(1) In the same manner, if P ∈ B ⊗ B satisfies then P will twist any right B-module algebra and any left B-module coalgebra.
(2) If A = g∈G A G is a graded algebra with G a finite group or monoid, (i.e. A is a left (kG) * -module algebra) and F is a twisting element based on (kG) * , then A F is a cocycle twist in the sense of [AST]. When G is infinite, we instead can use a dual construction and twist A as a kG-comodule algebra. The twists of graded algebras appearing in [Z] are, in general, not obtainable by twisting elements.
A natural question is whether a twisting element can be used to twist bialgebras. So far, we are unaware of any general method of twisting both the multipliction and comultiplication of a bialgebra. There is, however, a notable exception: it is possible twist the comultiplication of B itself in such a way that it remains compatible with its original multiplication, unit, and counit. Moreover, this twisted bialgebra acts in a natural on the twisted algebras A F and coalgebras C F formed using Theorem 1.3. To establish this we use the fact that the right multiplication map b r : B → B sending x to xb gives B the structure of a right B-module coalgebra. Similarly, B is also a left B-module coalgebra via left multiplication Theorem 1.5. Let F be an invertible twisting element based on a bialgebra B. ( twisting element. This comultiplication, however, is generally not compatible with µ B , but it further twists to one which is. To see this, first note that inverting both sides of F only involves right multiplication by F . In light of (1.6) and Remark 1.4, it follows that ∆ ′ and so this comultiplication is compatible with µ B . Moreover, since the algebra structure remains unchanged, the original counit ε B : B → k is an algebra map and consequently, B(µ B , ∆ ′ B , 1 B , ε B ) is a bialgebra. To prove (1.5.2), we first need to establish that A F is actually a left B F module. To do so, we define the action of b ∈ B F on a ∈ A F to be just b · a, the action given by the original B-module structure of A. This action is well-defined since A and B coincide as k-modules with A F and B F and the multiplications of B and B F are identical. With this module structure, (1.1.1) is automatically satisfied since the counits of B and B F are also identical.
To show that condition (1.1.2) holds we need to use the twisted multiplication in A F and If B is commutative then the twist B F is just B itself and so in this case A F and C F are still, respectively, a left B-module algebra and right C-module coalgebra.
We now turn to the topic of twisting modules and comodules along with algebras and coalgebras. Only the details for the module/algebra case are given here as those for the comodule/coalgebra case are easily obtainable by dualization. If M is a left A-module and A F is a twist of A, it is natural to ask whether M twists to a left A F -module. This is, in general, not possible as we must take into account the action of the bialgebra B. We will be concerned with the following class of modules.
Definition 1.7. Suppose that A is a left B-module and let A-Mod be the category of all left A-modules. Define (B, A)-Mod to be the subcategory of A-Mod whose objects are those for all b ∈ B, a ∈ A, and m ∈ M . Morphisms in (B, A)-Mod are k-linear maps which are simultaneously A-module and B-module maps.
This equation is equivalent to having Now the latter equation is satisfied since inverting both sides yields (1.2.2) which holds since F is a twisting element based on B. We may thus form ( are clearly just the original A and B and, in the same way as above, we get a functor Mod which is clearly a two-sided inverse to F and so these two categories are equivalent.
There are remarkably few techniques to explicitly produce twisting elements. In some cases though, it is possible to multiply two twisting elements to produce a new one. Proof. Since elements of B ′ commute with those of B ′′ we have that Because F and F ′ are UDF's, this may be expressed as For any bialgebra B, the element 1 B ⊗1 B is obvioulsy a twisting element; when used as in Theorems 1.3 or 1.5, it is the "identity twist", that is, it effects no change. Apart from this trivial example and the cocycle twists previously mentioned, the class of quasi-triangular bialgebras gives other twisting elements.
Definition 1.10 [D2]. A bialgebra B is quasi-triangular if there is an invertible element A basic fact about quasi-triangular bialgebras is that the associated R satisfies the quantum Yang-Baxter equation (1) (Majid [Ma]) Let G = Z/(nZ) be the cyclic group of order n with generator g. If n is invertible in k and q ∈ k is a primitive nth root of unity then then the group bialgebra K[G] then has a non-trivial quasi-triangular structure in which (2) (Drinfel'd [D2]) If B is a finite dimensional bialgebra then its "double" D(B) is a quasi-triangular bialgebra. As a k-module, D(B) = B * ⊗ B; its bialgebra structure is intricate and not necessary for this paper and we refer the reader to [D2] or [Mon] for a careful discussion of this topic.
(3) The classic example of a quasi-triangular bialgebra is the quantized enveloping algebra U q g, see [D2]. To obtain its quasi-triangularity, it is necessary to consider U q g as a k[[t]]-module with q = e t and take its completion with respect to the t-adic topology. The corresponding R is then expressible as an infinite series in t.
Our main interest in the remainder of this paper will be twisting elements which can be used to obtain deformations of various algebraic structures and so we will consider those based on k[[t]]-bialgebras. We now briefly recall the main aspect of formal algebraic deformations.
Definition 1.13. Let A be an associative k-algebra with multiplication µ A : A ⊗ A → A.
(1) Note that µ ′ A is completely determined by its effect on A ⊗ A. Even though this a formal construction it turns out that in many cases µ ′ A (a⊗b) is actually a polynomial in t and so, by specialization, we can view the deformation as a k-algebra. We can obtain the same conclusion in many cases by convergence considerations (when the base field is R or C.) (2) In a more general context it is possible to consider deformations over other rings in the following sense: If R is a commutative ring with epimorphism R → k then a deformation of A over R is an R-algebra A ′ which is a flat R-module together with an isomorphism A ′ ⊗ R k ∼ = A. The case R = k[q, q −1 ] is of particular interest in the study of quantum groups.

Returning now to formal deformations, we say that deformations
( 1.15) for all n ≥ 0 and a, b, c ∈ A. In particular, when n = 1 we have that the infinitesimal, µ 1 , must satisfy Given an element of Z 2 (A, A), it is natural to ask whether there is a deformation of A with that infinitesimal. In general, this is not possible since there may be obstructions, all lying in H 3 (A, A), to "integrating" an infinitesimal to a full deformation. Namely, if µ + tµ 1 + t 2 µ 2 + · · · + t n−1 µ n−1 defines an associative multiplication on A[t]/t n then i+j=n i,j>0 is automatically a Hochschild 3-cocycle and must be a coboundary if the multiplication is extendible to an associative product on A[t]/t n+1 . Even when an infinitesimal µ 1 is known to be integrable to a full deformation, e.g. whenever H 3 (A, A) = 0, very little is known on how to find µ 2 , µ 3 , . . . such that µ t = i≥0 t i µ i satisfies (1.15).
It is straightforward to dualize the deformation theory of algebras to that of coalgebras.
If C is a coalgebra with comultiplication ∆ C : C → C ⊗ C, then a formal deformation of

C is a k[[t]]-coalgebra structure on C[[t]] with comultiplication of the form ∆
and so we must consider its completion, (C ⊗ C) [[t]] with respect to the t-adic topology. Now the coassociativity imposes restrictions on the maps ∆ i which may be interpreted in terms of the coalgebra cohomology of C. As in the algebra case, it is generally difficult to explicitly produce deformations of a coalgebra.
One way to produce deformations for a wide class of algebras and coalgebras is by use of certain twisting elements. ] is strictly unchanged, this will be a "preferred deformation" of B in the sense of [GGS].

Definition 1.16. A universal deformation formula (UDF) based on a bialgebra B is a twisting element F based on B[[t]] of the form
A fundamental question is to determine "all" UDF's based on a bialgebra B. This must be understood under a natural notion of equivalence. Namely, if F is a UDF and is also a UDF and we say F and F are equivalent. When used as in Theorem 1.3 or Theorem 1.5, a UDF produces equivalent deformations. Little is known about the equivalence classes of UDF's except when B = U g. This case is of greatest interest to us in part due to its extensive connections to the theory of quantum groups. For simplicity, we denote the comultiplication of U g simply by ∆ instead of ∆ Ug .
Up to equivalence, the existence problem for UDF's based on U g [[t]] has been completely settled by in [D1], (see [MV] for a more detailed discussion of the ideas in [D1]). Drinfel'd also gives an important connection between UDF's and solutions of the Yang-Baxter equations.
Recall that r ∈ g ⊗ g is a solution to the classical Yang-Baxter equation if Solutions to these equations are unitary if r 12 + r 21 = 0 in the classical case (i.e. r is skew-symmetric) and R 12 R 21 = 1 in the quantum case.
(1) r is a unitary solution to the classical Yang-Baxter equation. It turns out that there is generally a rich supply of U g-module algebras. First, it is easy to check that any algebra A which admits an action of g as derivations is a U gmodule algebra. A natural source of this phenomenon comes from group actions. Suppose

Explicit Formulas
The first UDF's we will consider are built from an abelian bialgebra B. In this case, the UDF's have a remarkably simple form. While widely used, a formal proof of the following has not appeared in the literature and since it has extensive applications we include one for completeness.
Theorem 2.1. Suppose that k ⊃ Q. If B is a commutative bialgebra and P is its space of primitive elements, then for any r ∈ P ⊗ P exp (tr) = ∞ i=0 t i i! r i = 1 ⊗ 1 + t r + t 2 2! r 2 + · · · + t n n! r n + · · · is a UDF. which clearly holds whenever r ∈ P ⊗ P .
Up to equivalence, we only need to consider those r ∈ P ⊗ P which are skew-symmetric.
In this case, exp(tr) satisfies (1.17.3) and thus produces * -products when used to deform coordinate rings of algebraic varieties.
(1) The classic use of the exponential deformation formula is to quantize the coordinate ring O(R 2n ) = R[x 1 , . . . , x n , y 1 , . . . y n ] of polynomial functions, (or C ∞ (R 2n ) in the analytic case), with respect to the canonical Poisson bracket. For f, g ∈ O (R 2n ), this bracket is This Poisson bracket is induced from a translation action of R 2n on itself. Since the derivations ∂/∂x i and ∂/∂y i mutually commute, we can expontiate the Poisson bracket to deform O (R 2n ). If we let f * g denote the deformed product, then it is easy to verify that the quantized algebra has the Heisenberg relations Note that for any f, g ∈ O (R 2n ) the product f * g is a polynomial in t and so it is meaningful to specialize t to any real number. This construction also provides a deformation of k[x 1 , . . . , x n , y 1 , . . . y n ] over k[t] whenever k ⊃ Q.
(2) Consider the commuting derivations x i (∂/∂x i ) of k[x 1 , . . . , x n ]. If k ⊃ Q then for any scalars p ij we can exponentiate i<j p ij x i (∂/∂x i ) ∧ x j (∂/∂x j ) to deform the polynomial ring k[x 1 , . . . x n ]. The new relations are x i * x j = P ij x j * x i where P ij = exp(tp ij ). This algebra is the quantum n-space associated with the standard multi-parameter quantization of O(SL(n)). If k = R or C then for any f and g in k[x 1 , . . . , x n ], the deformed product f * g converges for all t ∈ k.
(3) Our final application of Theorem 2.2 concerns the enveloping algebra U gl 2 . Since the matrix units E 11 and E 22 commute, Theorem 1.5 provides a preferred bialgebra It is easy to check that both E 11 and E 22 remain primitive while ∆ ′ (E 12 ) = E 12 ⊗ L −1 + L ⊗ E 12 and ∆ ′ (E 21 ) = E 21 ⊗ L + L −1 ⊗ E 21 14 with L = 1 2 exp(t(E 11 − E 22 )). As this is a preferred deformation, ∆ ′ is compatible with the original multiplication of gl 2 . It is interesting to compare this with the standard deformation U q gl 2 in which the multiplication must also be changed.
In contrast to the commutative case, very little is known about UDF's based on noncommutative bialgebras. Even for the case of enveloping algebras of non-abelian Lie algebras, only one family of UDF's is known. Most of the remainder of this paper will be devoted to this family. It is based on an abelian extension of a Heisenberg Lie algebra. Specifically, let k ⊃ Q and let H ⊂ sl(n) be the k-Lie-algebra generated by the diagonal matrices for 1 ≤ i < n and elements of the form E 1p or E pn for p = 2, . . . , n − 1.
The element of H ⊗ H which will serve as the infinitesimal of the UDF is where c 1 , . . . c n−1 are scalars with c 1 + c n−1 = 1. The infinitesimal for the special case (along with an incorrect UDF) where all c i = 1/2 appears both in [GGS] and [CGS]. Before presenting the correct UDF for the general case, we need some notation which will be used for its definition and proof. Let Also set c 1 = c and c n−1 = d and so c + d = 1. In this notation, the infinitesimal (

15
Then the series For n = 2 we have X = 0 and F 1 = H ⊗ E = (1/2)(E 11 − E 22 ) ⊗ E 12 ; in this case, the corresponding UDF is the "quasi-exponential" formula ∞ m=0 (t m /m!)H m ⊗ E m of [CGG]. Before proving Theorem 2.6, we first need to establish some elementary formulas.
Lemma 2.7. In the above notation, the following identities hold for all non-negative integers r and s and all scalars a and b: and (1 ⊗ E)X = X (1 ⊗ E).
(5) X 13 X 23 − X 23 X 13 = 0 and X 13 X 12 − X 12 X 13 = 0. Proof. All of these are straightforward to establish. Properties (2.7.1) and (2.7.2) are immediate and (2.7.3) through (2.7.6) follow from the relations among the generators of the Lie algebra H. Finally, (2.7.7) through (2.7.9) can each be proved using induction on r.
Proof. We use induction on i. For i = 0, the assertion is trivial and so assume it holds for i. For i + 1, the top line of (2.9) becomes, by (2.7.8), k[x 1 , . . . , x n ] where E ij acts as x i (∂/∂x j ). The UDF thus deforms k[x 1 , . . . , x n ] and it is not hard to see that the new relations become x 1 * x n − x n * x 1 = (c 1 /2)x 1 * x 1 while x p * x n − x n * x p = λ p x 1 * x p with λ p = (1/2)(c p − c p−1 ) + 1 and all other We conclude this paper with some questions about universal deformation formulas.
(1) Is there a "conceptual" way to determine whether a given series is a UDF? That is, can the direct computational method of proof be avoided?
(2) Do the UDF's of Theorem 2.6 and Theorem 2.21 globalize in an analogous way that the formula of Example 2.4.1 does? The global version of this Example 2.4.1 is the DeWilde-Lecomte Theorem which says that if M is any symplectic manifold then there is a canonical * -product on C ∞ (M ), see [DL] or, for a geometric proof see [F]. Locally, any such manifold looks like R 2n with the symplectic structure given in (2.4.1) and so the exponential formula gives the local * -product. The difficult part is to show that these local deformations are compatible. Now if M is just a Poisson manifold and locally a UDF gives a deformation of C ∞ (M ), does there exist a global deformation?
(3) Are there analogs of UDF's which give strict deformation quantizations of C *algebras in the sense of Rieffel (see [R])?
(4) Is there a procedure which produces a UDF based on U g from a constant unitary solution to the classical Yang-Baxter equation based on a Lie algebra g? As stated earlier, there always is such a formula but its explicit form remains a mystery.