On the Associative Analog of Lie Bialgebras

Journal of Algebra 244, 492᎐532Ž. 2001 doi:10.1006rjabr.2001.8877, available online at http:rrwww.idealibrary.com on

Marcelo Aguiar1

Centre de Recherches Mathematiques,´´´ Uni¨ersite de Montreal, C.P. 6128, View metadata, citation and similarSucc. Centre-Ville,papers at core.ac.uk Montreal,´´ Quebec H3C 3J7, Canada brought to you by CORE E-mail: [email protected] provided by Elsevier - Publisher Connector Communicated by Corrado de Concini

Received July 20, 2000

An infinitesimal bialgebra is at the same time an associative algebra and coalgebra in such a way that the comultiplication is a derivation. This paper continues the basic study of these objects, with emphasis on the connections with the theory of Lie bialgebras. It is shown that non-degenerate antisymmetric solutions of the associative Yang᎐Baxter equation are in one to one correspondence with non-degenerate cyclic 2-cocycles. The associative and classical Yang᎐Baxter equations are compared: it is studied when a solution to the first is also a solution to the second. Necessary and sufficient conditions for obtaining a Lie bialgebra from an infinitesimal one are found, in terms of a canonical map that behaves simultaneously as a commutator and a cocommutator. The class of balanced infinitesimal bialgebras is introduced; they have an associated Lie bialgebra. Several well known Lie bialgebras are shown to arise in this way. The construction of Drinfeld’s double from earlier work by the authorŽ in press, in Contemp. Math., Amer. Math. Soc., Providence. for arbitrary infinitesimal bialgebras is complemented with the construction of the balanced double, for balanced ones. This construction commutes with the passage from balanced infinitesimal bialgebras to Lie bialgebras. ᮊ 2001 Academic Press Key Words: Hopf algebras; Lie bialgebras; derivations; Yang᎐Baxter equation; Drinfeld double.

1. INTRODUCTION

An infinitesimal bialgebraŽ.Ž. abbreviated ⑀-bialgebra is a triple A, m, ⌬ where Ž.A, m is an associative algebra Ž possibly without unit .Ž. , A, ⌬ is a coassociative coalgebraŽ. possibly without counit , and, for each a, b g A, ⌬ s m q m Ž.1.1 Ž.ab ab1212b a ab.

1 Research supported by a Postdoctoral Fellowship at the Centre de Recherches Mathema-´ tiques and Institut des Sciences Mathematiques,´´´ Montreal, Quebec.

492 0021-8693r01 $35.00 Copyright ᮊ 2001 by Academic Press All rights of reproduction in any form reserved. ANALOG OF LIE BIALGEBRAS 493

In other words, ⌬ is required to be a derivation of the associative algebra A with values on the A-bimodule A m A. The notion of a ⑀-bialgebra is manifestly analogous to that of a Lie bialgebra. Gian-Carlo Rota argued in favor of the study of ⑀-bialgebras on several occasions, starting perhaps withwx J-R, Sect. XII , which precedes the introduction of Lie bialgebras by Drinfeld in the early 1980s. Apparently, such a program remained uncontested until recently, when new motiva- tions were found, coming on one hand from interesting applications to combinatoricswx A2, E-R and on the other from the close relation to the theory of Lie bialgebras. Inwx A1 , the basic theory of ⑀-bialgebras was presented. This includes the introduction of several examples and the notions of antipode, Drinfeld’s double, and the associative Yang᎐Baxter equation. The study of ⑀-bialgebras is continued in the present paper, with special emphasis on the analogy and connection with Lie bialgebras. The results obtained are summarized below, after the introduction of the necessary notation. The contents are described in more detail at the beginning of each section.

Notation. All vector spaces and algebras are over a fixed field k. MknŽ. denotes the algebra of matrices. Sum symbols are often omitted from ⌬ s m ⌬ Sweedler’s notation: we write Ž.a a12a when is a coassociative comultiplication. Let A be an associative algebra, possibly non-unital. The space A m A is viewed as an A-bimodule via a и Ž.u m ¨ s au m ¨ and Ž.u m ¨ и a s u m ¨a. g m ⌬ ª Let r A A. The principal derivation defined by r is the map r : A A m A defined by ⌬ s и y и r Ž.a a r r a. The symmetric and antisymmetric parts are denoted by 11 rqs Ž.r q ␶ Ž.r g SA2 Ž . and rys Ž.r y ␶ Ž.r g ⌳2 ŽA ., 22 where ␶ : A m A ª A m A is the switch ␶ Ž.u m ¨ s ¨ m u. Consider the elements s y q AŽ.r rr13 12rr 12 23rr 23 13 , s wxwxwxq q CŽ.r r13, r 12r 23, r 12r 23, r 13 . These belong to A m A m A and are well defined even if A is non-unital. The associative Yang᎐Baxter equation iswx A1, Sect. 5 Ž.AYB AŽ.r s 0. 494 MARCELO AGUIAR

It is analogous to the classical Yang᎐Baxter equation

Ž.CYB CŽ.r s 0. wx ⌬ According to A1, Proposition 5.1 , a principal derivation r is coassociative if and only if AŽ.r is an A-invariant element of the A-bimodule A m A m A. In this case one says that Ž.A, r is a coboundary ⑀-bialgebra. If the stronger conditionŽ. AYB is satisfied, one says that Ž.A, r is a quasitriangular ⑀-bialgebra. Contents. In Section 2, antisymmetric solutions of the associative Yang᎐Baxter equation are studied. The main result, Proposition 2.7, establishes a one to one correspondence between such solutions and those subalgebras of A carrying a non-degenerate cyclic 2-cocycle. As an illus- ރ tration, all antisymmetric solutions ofŽ. AYB in M2 Ž.are found. An algebra carrying a non-degenerate cyclic 2-cocycle is called antisymmetric, in this paper. Examples and basic properties of such algebras are given throughout the section. In Section 3, the associative and classical Yang᎐Baxter equations are compared, beyond the obvious formal analogy. It is shown that a solution r ofŽ. AYB for which the symmetric part rq is invariant is also a solution of Ž.ŽCYB Theorem 3.5 . . This allows us to obtain solutions of Ž. CYB from the solutions ofŽ. AYB found in Section 2. It is shown that some solutions of Ž.CYB by Belavin and Drinfeld arise in this way. There is an obvious way of attempting to construct a Lie bialgebra from an ⑀-bialgebra, namely by endowing it with the commutator bracket and cocommutator cobracket. The necessary compatibility condition between the two is, however, not always satisfied. In Section 4, this situation is studied in detail. A canonical map B : A m A ª A m A, defined on any ⑀-bialgebra A, is introduced. It is called the balanceator, for its basic properties are analogous to those of the commutator of an associative algebra, or the cocommutator of a coassociative coalgebra, simultaneously. These are given in Proposition 4.3. An ⑀-bialgebra is called balanced if its balanceator is zero. This guarantees that passing to the commutator and cocommutator results in a Lie bialgebra. Two constructions of balanced ⑀-bialgebras from arbitrary ones are givenŽ. Proposition 4.6 . These are dual to each other and analogous to the construction of the center of an associative algebra. An important class of balanced ⑀-bialgebras is provided by quasitriangular ⑀-bialgebras Ž.A, r for which rq is A-invariant Ž.Proposition 4.7 . This ‘‘explains’’ the result of Corollary 3.7, about the passage from coboundary or quasitriangular ⑀-bialgebras to the respective classes of Lie bialgebras. In Section 5, the constructions of Drinfeld’s doubles for ⑀-bialgebras and Lie bialgebras are compared. The Drinfeld double DAŽ.of a finite ANALOG OF LIE BIALGEBRAS 495 dimensional ⑀-bialgebra A carries a canonical associative form, but unlike the case of Lie bialgebras, this fails to be non-degenerate. Theorem 5.5 establishes a connection between the radical of this form and the fact that A is balanced. When A is balanced, the radical of the form admits a particularly simple description, and the quotient of DAŽ.by this ideal turns out to be again a balanced ⑀-bialgebra, called the balanced Drinfeld double and denoted DAbŽ.. This is perhaps one of the main constructions s [ of the paper. As a vector space, DAbŽ. A A*, and the multiplication in DAbŽ.is Ž.Ž.Ža, f и b, g s ab q a ¤ g q f ª b, a ª g q f ¤ b q fg ., where the arrows denote the natural actions of A and A* on each other. While DAŽ.is a quasitriangular ⑀-bialgebra that is universally attached to ⑀ ⑀ any -bialgebra A, DAbŽ.is a balanced quasitriangular -bialgebra that is universally attached to any balanced ⑀-bialgebra A, as spelled out in Theorem 5.9. The balanced double is thus strictly analogous to the double of a Lie bialgebra, but is defined only for balanced ⑀-bialgebras. This analogy is made more precise by Proposition 5.12, where it is shown that the doubles’ constructions commute with the functor from balanced ⑀-bialgebras to Lie bialgebras.

2. ANTISYMMETRIC SOLUTIONS OF THE ASSOCIATIVE YANG᎐BAXTER EQUATION

There is a one to one correspondence between non-degenerate antisymmetric solutions of the classical Yang᎐Baxter equation in a Lie algebra ᒄ and non-degenerate 2-cocycles ᒄ m ᒄ ª k wxC-P, Corollary 2.2.4 . This can be used to classify all antisymmetric solutions in ᒄ wxC-P, Proposition 2.2.6 . In this section we obtain the analogous results for the associative Yang᎐Baxter equation. Non-degenerate antisymmetric solutions of the associative Yang᎐Baxter equation in A are in one to one correspondence with non-degenerate cyclic 2-cocycles A m A ª k in the sense of Connes Ž.also called 1-multitraces . Let A be an associative, perhaps non-unital, algebra. A cyclic 2-cocycle in the sense of Connes is an antisymmetric form ␻ : A m A ª k such that Ž.Ž.Ž.Ž.2.1 ␻ ab m c y ␻ a m bc q ␻ ca m b s 0 ᭙a, b, c g A. This corresponds to the original definition of cyclic cohomology by Connes wxCon, pp. 310᎐318 . It agrees with what is today called periodic cyclic cohomology, when k = ޑ. Seewx Ros, 6.1.34 and 6.1.35 . An example of a ␻ ª 2-cocycle is the coboundary f of a 1-form f : A k, defined by ␻ m s y f Ž.Ža b fab ba .. 496 MARCELO AGUIAR

s m ¨ g m We say that an antisymmetric tensor r ÝuiiA A is non- degenerate if the map ª ¬ ¨ sy ¨ Ž.2.2 ˜r : A* A, f ÝÝfuŽ.iiuf i Ž. i is bijective. Similarly, an antisymmetric form ␻ : A m A ª k is non-degenerate if Ž.2.3 ␻˜˜: A ª A*, ␻ Ž.Ž.abs ␻ Ža m b .sy␻ Žb m a . is bijective. Obviously, there is a one to one correspondence between such non-degenerate tensors and forms via Ž.2.4 ␻˜˜s ry1 .

PROPOSITION 2.1. Let A be a finite dimensional algebra. There is a one to one correspondence between non-degenerate antisymmetric solutions r g A m A of the associati¨e Yang᎐Baxter equation and non-degenerate Connes 2- cocycles ␻ : A m A ª k. Proof. Let r g A m A be an antisymmetric non-degenerate tensor. We will show that AŽ.r s 0 m ˜˜ry1 s ␻ : A ª A* is a derivation m ␻ is a Connes 2-cocycle. Here, A* is viewed as an A-bimodule in the usual way: Ž.Ž.Ž.a ª fbs fba and Ž.Ž.Ž.f ¤ abs fab. s m ¨ g Write r Ýuii. For f, g A*, we have Ž.Ž.f m id m g Ž.A r s ¨ ¨ y ¨ ¨ q ¨¨ ÝÝÝfuuŽ.Ž.ij jg i fu Ž.Ž.i ijug j fu Ž.Ž.j ug i ij

Ž.2.2 sy ¨ q q ¨ ÝÝfrguŽ.˜˜˜˜Ž.jj rfrg Ž.Ž. ugiŽ. irfŽ. sy ¤ ¨ q q ª ¨ ÝÝŽ.f ˜˜˜˜rgŽ.Ž ujj .rfrg Ž.Ž.urfiŽ. Ž.g Ži .

Ž.2.2 sy˜˜rfŽ.¤ rgŽ.q ˜˜˜˜rfrg Ž.Ž.y rrfŽ Ž Ž.ª g .. Since ˜r : A* ª A is bijective we conclude that AŽ.r s 0 m ᭙f , g g A*, ˜˜rfrgŽ.Ž.s ˜˜rrfŽ Ž.Ž. Ž.ª g q ˜rf¤ ˜rgŽ. m ᭙a, b g A, ˜˜˜ry1 Ž.ab s a ª ry1 Ž.b q ry1 Ž.a ¤ b m ˜ry1 : A ª A* is a derivation. ANALOG OF LIE BIALGEBRAS 497

Evaluating the last equality on c g A, and usingŽ. 2.4 , we obtain the additional equivalent condition: ᭙a, b, c g A, ␻˜˜˜Ž.Ž.ab c s ␻ Ž.Ž.bcaq ␻ Ž.Ž.abc

Ž.2.3 m ᭙a, b, c g A, ␻ Ž.ab m c sy␻ Ž.Ž.ca m b q ␻ a m bc which says that ␻ is a Connes 2-cocycle as needed. ª g Let cu : A A denote conjugation by an invertible element u A, s y1 g m cauŽ. uau .If r A A is aŽ. non-degenerate, antisymmetric solution m ofŽ. AYB , then so is Žcuucr .. Two such solutions are said to be conjugate.

PROPOSITION 2.2. If two non-degenerate antisymmetric solutions of the associati¨e Yang᎐Baxter equation are conjugate, then the corresponding cocycles are cohomologous. s m Proof. Suppose r and s are two conjugate solutions, s Ž.cuucr.It ␻ s ␻ follows readily that the corresponding cocycles are related by rsuŽc m ␻ ␻ cur.. Hence, the cohomology classes of and sare related by the conjugation action on periodic cyclic cohomology. Now, according tow Lod, x ␻ ␻ Proposition 4.1.3 , this action is trivial. Hence rsand are cohomologous. Often, an algebra A is called symmetric if it possesses a non-degenerate symmetric form²: , : A m A ª k that is associative, in the sense that ²:²:ab, c s a, bc ᭙a, b, c g A. We will say that an algebra A is antisymmetric if it possesses a non-degenerate Connes 2-cocycle.

EXAMPLE 2.3.Ž. 1 Let A be the two dimensional algebra with basis Ä4x, y and multiplication x 2 s 0, xy s 0, yx s x, and y 2 s y. A is then an associative algebra; in fact, it can be realized as a subalgebra Ž. s wx01 s wx10 of Mk2 via x 00 and y 00. Let ␻ : A m A ª k be the form ␻ Ž.Ž.Ž.ax q by m cx q dy s ad y bc.

␻ ; wx0 y1 10is non-degenerate, and it is a Connes 2-cocycle; in fact, it is the coboundary of f : A ª k, fxŽ.sy1, fy Ž.s 0. Thus A is an antisymmetric algebra. The corresponding non-degenerate solution ofŽ. AYB is r s y m x y x m y. Ž.A, r is the quasitriangular ⑀-bialgebra ofwx A1, Example 5.4.5e . 498 MARCELO AGUIAR

Ž.2 There are, up to isomorphism, only three antisymmetric algebras of dimension 2. These are the algebra of Example 2.3.1, its opposite, and the trivial algebra with zero multiplication. We sketch a proof next. Consider a 2-dimensional algebra carrying a non-degenderate antisymmetric solution r ofŽ. AYB . Choose a basis Ä4x, y such that r s y m x y x m y Žor equivalently, ␻ ; wx0 y1 . ᎐ 10with respect to this basis . The associative Yang Baxter equation for r Ž.or the cocycle condition for ␻ imposes some conditions on the structure constants of the algebra, from which it follows that the multiplication table must have the form x 2 s Ž.a q bx, xy s dx q ay y 2 s Ž.c q dy, yx s cx q by for some scalars a, b, c, d. Associativity leads to the equations Ž.) ab s bc s cd s da s 0. Let us denote such an algebra by Aawx, b, c, d . The case a s b s c s d s 0 yields the trivial algebra with zero multiplication. Otherwise, we may assume that either c / 0ord / 0, since, in general, Aawx, b, c, d ( Acwx, d, a, b via the map that interchanges x with y. If c / 0 then Ž.) implies b s d s 0, and Aawxwx,0,c,0 ( A 0, 0, 1, 0 via xЈ s cx y ay and yЈ s y . Note that Awx0, 0, 1, 0 is precisely the antisym- c metric algebra of Example 2.3.1. If d / 0 then Ž.) implies a s c s 0, and Awxwx0, b,0,d ( A 0, 0, 0, 1 via xЈ s cx y ay and yЈ s y . Note that Awx0, 0, 0, 1 is the opposite algebra of d Awx0, 0, 1, 0 .Ž. They are not isomorphic. Ž.3 A higher dimensional antisymmetric algebra, which is a general-

ization of Example 1, can be obtained as follows. Let ZkmŽ.denote the Ž.non-unital subalgebra of Mkm Ž.consisting of those matrices whose last ª ␣ s my1 ␣ row is equal to zero, and f : ZkmiŽ. k the functional fŽ. Ý s1 i, iq1; i.e., fŽ.␣ is the sum of the entries directly above the diagonal. The coboundary of f is then my1 ␻␣m ␤ s ␣␤ y ␤␣ fiŽ.Ý , jj, iq1 i, jj, iq1 . i, js1 ␻ It is easy to see that f is non-degenerate and that the corresponding solution ofŽ. AYB is my1 maxŽ.i, j s n r ÝÝei, iqjykq1 ej, k , i, js1 ks1 n s m y m where the ei, j denote the elementary matrices and x y x y y x. ANALOG OF LIE BIALGEBRAS 499

␻ Zk2Ž.is the algebra of Example 1 Žf, and r have been multiplied by y1.. There is a simple way of constructing new Connes cocycles from old, which is reminiscent of the construction of a new Lie algebra as the tensor product of a Lie algebra and a commutative algebra.

PROPOSITION 2.4. Let A be an antisymmetric algebra with Connes cocycle ␻ m A and B a symmetric algebra with form ²:,.B Then A B is an antisymmetric algebra with Connes cocycle ␻ m Ј m Ј s ␻ Ј Ј Ž.Ž.a b, a b A a, a ²:b, b B . Proof. Non-degeneracy is obvious. We check the cocycle condition Ž.2.1 . Since ²: ,B is associative and symmetric, we have

²bЉ , b, bЈ :²:²:²:BBBBs bЉ , bbЈ s bbЈ, bЉ s b, bЈbЉ . Therefore ␻ Ž.a m b и aЈ m bЈ, aЉ m bЉ y ␻ Ž.a m b, aЈ m bЈиaЉ m bЉ q ␻ Ž.aЉ m bЉиa m b, aЈ m bЈ s ␻ Ј Љ Ј Љ y ␻ Ј Љ Ј Љ AAŽ.aa , a ²:bb , b BB Ž.a, a a ²:b, b b q ␻ Љ Ј Љ Ј AŽ.a a, a ²:b b, b B s ␻ Ј Љ y ␻ Ј Љ q ␻ Љ Ј Ј Љ s Ž.AAAŽ.Ž.Ž.aa , a a, a a a a, a ²:bb , b B 0.

EXAMPLE 2.5. Let B be any symmetric algebra and ZBmŽ.the subalgebra of MBmŽ.consisting of those matrices with entries in B whose last row is zero. Then ZBmŽ.is an antisymmetric algebra, since it can be identified with the tensor product of B with the antisymmetric algebra

ZkmŽ.of Example 2.3.3. s In particular we may take B MknmŽ., with the trace form. Then ZBŽ. can be described as the subalgebra of MkmnŽ.consisting of those matrices whose last n rows are zero. We recall an important result about solutions ofŽ. AYB .

PROPOSITION 2.6. Let A be a finite dimensional algebra and r g A m A an arbitrary solution of the associati¨e Yang᎐Baxter equation. When A* is equipped with an appropriate dual structure, the map ª ¬ ¨ ˜r : A* A, f Ý fuŽ.ii is a morphism of ⑀-bialgebras. 500 MARCELO AGUIAR

Proof. Seewx A1, Proposition 5.6 . From this we conclude that antisymmetric solutions ofŽ. AYB correspond to antisymmetric subalgebras.

PROPOSITION 2.7. Let A be a finite dimensional algebra. There is a one to one correspondence between antisymmetric solutions in A of the associati¨e Yang᎐Baxter equation and pairsŽ. B, ␻ where B is an Ž antisymmetric . subalgebra of A and ␻ a non-degenerate Connes 2-cocycle on B. s m ¨ g m Proof. Let r ÝuiiA A be an antisymmetric solution of Ž.AYB . By Proposition 2.6, ImŽ.˜˜r is an ⑀-subalgebra of A. ByŽ 2.2 . , Im Ž.r ¨ is the subspace of A linearly spanned by either set Ä4uiior Ä4. Hence, r is a non-degenerate antisymmetric solution ofŽ. AYB in Im Ž.˜r . By Proposi- tion 2.1, ImŽ.˜r is an antisymmetric subalgebra of A. Conversely, starting from an antisymmetric subalgebra B of A, Proposi- tion 2.1 yields an antisymmetric solution ofŽ. AYB in B, and hence also in A.

EXAMPLE 2.8. As an application, we find all antisymmetric solutions of ރ Ž.AYB in the algebra M2 Ž.. First of all, there are the antisymmetric subalgebras

ab Z Ž.ރ s a, b g ރ 2 ½500 and its transpose

a 0 Z t Ž.ރ s a, b g ރ 2 ½5b 0 of Examples 2.3. These yield the solution

10mym 01 01 10 00 00 00 00 and its transpose. Suppose that r is another anitsymmetric solution and let B s ImŽ.˜r be the corresponding antisymmetric algebra. According to the remarks below, dim B must be even, and moreover B cannot be unital. Hence dim B is either 0 or 2. The first case corresponds to the trivial solution r s 0. In the second case, according to the result of Example 2.3.2, B must be isomor- ރ phic to either Z2Ž., its opposite algebra, or the 2-dimensional algebra ރ with zero multiplication. The opposite algebra of Z2Ž.is obviously t ރ isomorphic to Z2Ž.. The 2-dimensional algebra with zero multiplication is ރ not a subalgebra of M2Ž.Žbeing commutative, its elements could be ANALOG OF LIE BIALGEBRAS 501 simultaneously conjugated to nilpotent Jordan forms, but there is only one ރ such form in M2Ž.., so this case is excluded. ރ t ރ Thus B is isomorphic to either Z22Ž.or Z Ž.. We claim that in each case the isomorphism is given by conjugation by an invertible element of ރ M2Ž.. Consider for instance the first case. Let Ä4x, y be a basis of B Äwxwx01 104 Ž.ރ corresponding to the basis00 , 00 of Z2 . Then x and y satisfy the 2 s g ރ relations of Example 2.3.1. Since x 0, there is an invertible u M2Ž. y1 s wx01 s s such that uxu 00. The relations xy 0 and yx x imply that y1 s wx1 b g ރ ¨ s wx1 b uyu 00 for some b . Further conjugating by 01 one ob- ¨ y1 ¨y1 s wx01 ¨ y1 ¨y1 s wx10 tains uxu 00 and uyu 00. Thus the isomorphism ރ ¨ between B and Z2Ž.is realized as conjugation by u. We concluce that, up to conjugation and transposes, the only antisym- Ž. Ž.ރ s smywx10 wx 01 metric solutions of AYB in M2 are r 0 and r 00 00 wx01m wx 10 00 00.

We finish this section with some general remarks on antisymmetric algebras and solutions ofŽ. AYB . Any vector space carrying a non-degenerate antisymmetric form must obviously be even dimensional, so the same is true for antisymmetric s algebrasŽŽ.Ž for instance, for the algebra of Example 2.5, dim ZBm mm y 1.. dim B is an even number . More interestingly, antisymmetric algebras cannot be unital. In fact, if A is a unital algebra, then any Connes cocycle ␻ on A must be degenerate, since otherwise the bijective map ␻˜ : A ª A*, being a derivation by Proposition 2.1, would vanish at 1. Regarding solutions ofŽ. AYB for unital algebras, a stronger conclusion may in fact be derived:

PROPOSITION 2.9. Let A / 0 be a unital algebra. Then any solution r g A m A of the associati¨e Yang᎐Baxter equation, antisymmetric or not, is necessarily degenerate.

Proof. Suppose there is a non-degenerate solution r. In particular A is finite dimensional. By Proposition 2.6, ˜r : A* ª A is a morphism of ⑀-bialgebras. Hence A is self dual as ⑀-bialgebra, in particular it is both unital and counital. But this forces A s 0 bywx A1, Remark 2.2 .

Remark 2.10. It is known that any antisymmetric solution ofŽ. CYB in a semisimple Lie algebra must be degeneratewx C-P, Proposition 2.2.5 . There is an analogous result for semisimple ⑀-bialgebras. However, this becomes trivial in view of Proposition 2.9, since it is well known that any semisimple associative algebra is necessarily unital. 502 MARCELO AGUIAR

3. FROM THE ASSOCIATIVE TO THE CLASSICAL YANG᎐BAXTER EQUATION

The main result we will prove in this section, Theorem 3.5, states: Suppose rq is A-inäriant. Ž.1 If A Ž.r s 0 then C Ž.r s 0. Ž.2 If A Ž.risA-inäriant then C Ž.risAlie-inäriant. Here Alie denotes the Lie algebra obtained by endowing A with the commutator bracket. These are formal but not completely straightforward consequences of the definitions. Several results of independent interest will be derived in the course of the proof. The significance of Theorem 3.5 is that a coboundary ⑀-bialgebra Ž.A, r for which rq is A-invariant yields a coboundary Lie bialgebra by passing to the commutator and cocommutator bracketsŽ. Corollary 3.7 . g m s y q g m LEMMA 3.1. For s, t A A, letÄ4 t, s ts13 12ts 12 23ts 23 13 A A m A. Ž.1 Let s g ⌳2 Ž.A be an antisymmetric tensor and t g S2 Ž. A a symmetric one. Suppose that t is A-inäriant. ThenÄ4 t, s sy Ä4s, t . Ž.2 Let r g A m A be such that rq is A-inäriant. Then AŽ.r s A Ž.rq q A Ž.ry . s m ¨ s ¨ m s m sy Proof. Ž.1 Write t ÝuiiÝ iu iand s Ý x jjy Ý y j m x j. We compute s m m ¨ s ¨ m m s ¨ m m ts13 12 ÝÝÝuxijy j i ijx y ju i iy jxu ji s m m ¨ sy m m ¨ sy ÝÝuijjiy x u ijjix y st23 13 by invariance, symmetry, and antisymmetry. Invariance alone implies s m ¨ m s m ¨ m s ts12 23 ÝÝuiijjx y xu jiijy st13 12 s m m ¨ s m m ¨ s ts23 13 ÝÝx jiijjjiiu y x yu st12 23 . Thus s y q sy y q sy Ä4t, s ts13 12ts 12 23ts 23 13st 23 13st 13 12st 12 23 Ä4s, t . Ž.2 Note that AŽ.r s Ä4r, r . By partŽ. 1 applied to t s rqyand s s r , Ärqy, r 4Äsyryq, r 4. Hence AŽ.r s Ärqyqyq r , r q r 4Äs rqq, r 4 q Ä4Ä4Ä4rqy, r q r yq, r q r yy, r s AŽ.r qq A Ž.r y. ANALOG OF LIE BIALGEBRAS 503

LEMMA 3.2. Let t g A m A be symmetric and A-in¨ariant. Then t is also Aop-in¨ariant. Proof. The A-bimodule and Aop-bimodule structures on the space A m A s Aop m Aop are respectively a и Ž.u m ¨ и b s au m ¨b and aop и Ž.u m ¨ и b op s ua m b¨ . s m ¨ s ¨ m If t ÝuiiÝ iu iis symmetric then и и s m ¨ s ¨ m s ␶ m ¨ a t b ÝÝauiib a iub iŽ. Ýub ia i s ␶ Ž.b op и t и aop . In particular, a и t y t и a s ␶ Ž.Ž.t и aop y ␶ aop и t sy␶ Žaop и t y t и aop .. Hence t is A-invariant if and only if t is Aop-invariant.

LEMMA 3.3. Let r g A m AbeA-in¨ariant. Then s s s AŽ.r rr13 12rr 12 23rr 23 13 and this element of A m A m AisA-in¨ariant. s m ¨ Proof. Write r Ýuii. We compute s m ¨ m ¨ s m ¨ m ¨ s rr13 12 ÝÝuuij j iu j jiu i rr12 23 and s m m ¨¨ s m ¨ m ¨ s rr23 13 ÝÝujiijjjiiu u u rr12 23 from which s s s rr13 12rr 12 23rr 23 13 AŽ.r .

The A-invariance of, say, rr13 12 , follows immediately from that of r. Suppose r g ⌳2 Ž.A is antisymmetric. It is known that in this case CŽ.r is antisymmetric too, i.e., CŽ.r g ⌳3Ž.A . Next, we consider to what extent AŽ.r fails to be antisymmetric. To this end, consider the permutations ␴ , ␳ : A m A m A ª A m A m A, ␴ Ž.x m y m z s z m y m x and ␳ Ž.x m y m z s y m z m x. We are interested in the more general situation where it is only assumed that the symmetric part of r is A-invariantŽ. not necessarily zero . Consider the element Ј s y q A Ž.r rr12 13rr 23 12rr 13 23 . 504 MARCELO AGUIAR

Note that Ž.3.1 CŽ.r s A Ž.r y AЈ Ž.r .

LEMMA 3.4.Ž. 1 Let s g A m A be antisymmetric and t g A m Abe symmetric. Then

␴ Ž.AŽ.s s AЈ Ž.s , ␴ Ž.A Ž.t s AЈ Ž.t , and ␳ Ž.AŽ.s s A Ž.s .

Ž.2 Let r g A m A be such that rq is A-in¨ariant. Then

␴ Ž.AŽ.r s AЈ Ž.r and ␳ Ž.AŽ.r s A Ž.r . ␴ s m Proof. Ž.1 We first prove the assertions involving . Write s Ýui ¨ sy ¨ m iiiÝ u . We have s m ¨ m ¨ sy ¨ m m ¨ s ¨¨ m m ss13 12 ÝÝÝuuij j iu iju j i iju ju i « ␴ s m m ¨¨ s Ž.ss13 12 Ý uijiju ss13 23 .

Also,

s m ¨ m ¨ sy ¨ m m ¨ s ¨ m ¨ m ss12 23 ÝÝÝujjiiu jjiiuu jjiiu u « ␴ s m ¨ m ¨ s Ž.ss12 23 Ý uijiju ss23 13 ; and

s m m ¨¨ sy ¨ m m ¨ s ¨ m ¨ m ss23 13 ÝÝÝujiiju jiijjiiju u uu « ␴ s m ¨ m ¨ s Ž.ss23 13 Ý uuij i j ss12 13 .

Thus ␴ s ␴ y q s y q s Ј Ž.AŽ.s Žss13 12ss 12 23ss 23 13. ss 13 23ss 23 12ss 12 13 A Ž.s .

The same argument shows that ␴ ŽŽ..A t s AЈ Ž.t , since there were two sign changes involved at each step. Finally,

␳ s ␳ m ¨ m ¨ y m ¨ m ¨ q m m ¨¨ Ž.AŽ.s Ž.ÝÝÝuuij j iu j jiu iu ju i ij s ¨ m ¨ m y ¨ m ¨ m q m ¨¨ m ÝÝÝjiuu ij jiiu u ju iijju s m m ¨¨ q m ¨ m ¨ y m ¨ m ¨ ÝÝÝujiijjiiju uu u iijju s AŽ.s . ANALOG OF LIE BIALGEBRAS 505

Ž.2 For the permutation ␴ , we apply the results just proved to t s rq and s s ry, together with Lemma 3.1.2, which applies since rq is A-invariant. We obtain

␴ Ž.Ž.Ž.AŽ.r s ␴ A Ž.rq q ␴ A Ž.ry s AЈ Ž.rq q AЈ Ž.ry s AЈ Ž.r . The last equality holds by Lemma 3.1.2 applied to the algebra Aop: first, notice that rq is also Aop-invariant by Lemma 3.2, so the lemma indeed s X applies; second, observe that A AAopŽ.r A Ž.r . For the permutation ␳, we need a special argument for the term AŽrq.. Since rq is A-invariant, we can apply Lemma 3.3: q s q q s q q AŽ.r r13r 12r 23r 13. qs m Writing r Ý xiiy , we have q s q q s m m AŽ.r r13r 12 Ý xxijy jy i « ␳ q s m m s m m s q q s q Ž.AŽ.r ÝÝyjiijy xx x jiijx yy r23r 13 AŽ.r .

Using this and the result of the previous partŽ for s s ry. we conclude ␳ Ž.Ž.Ž.AŽ.r s ␳ A Ž.rq q ␳ A Ž.ry s A Ž.rq q A Ž.ry s A Ž.r .

Let Alie denote the Lie algebra obtained from the associative algebra A via the commutator wxa, b s ab y ba. Given an A-bimodule M, let M lie denote the same space M but viewed as an Alie-module via a ª m s a и m y m и a.

n The space Mm can then be seen as an A-bimodule via и m m иии m и s m m иии m a Ž.m12m mn b am12m mbn ,

n as the Alie-module Ž Mm .lie ª m m иии m a Ž.m12m mn s m m иии m y m m иии m am12m mn m12m man ,

n or as the Alie-module Ž M lie.m ¸ m m иии m a Ž.m12m mn s ª m m иии m q m ª m иии m Ž.a m12m mn m12Ž.a m mn q иии q m m иии m ª m12m Ž.a mn . 506 MARCELO AGUIAR

It is easy to see that these structures are related by means of the n-cycle ␳ mn ª mn n: M M , ␳ m m иии m s m иии m m nŽ.m12m mn m2 mn m1

n as follows: for a g A and u g Mm ,

¸ s ª q ␳y1 ª ␳ Ž.3.2 a u a u nnŽ.a Ž.u q иии q␳yŽ ny1. ª ␳ Žny1. nnŽ.a Ž.u .

n We will consider Alie-invariant elements of Mm always with respect to the n Lie structure ¸ . Thus, an element m g Mm is A-invariant if a ª m s 0 ᭙a g A, while it is Alie-invariant if a ¸ m s 0 ᭙a g A. We are mostly interested in the case M s A and n s 2 or 3. Notice that ␳ s ␶ ␳ s ␳ the permutations 23and have been considered already. We are now in position to prove the main result of this section. q THEOREM 3.5. Let r g A m A be a tensor for which its symmetric part r is A-inäriant. Ž.1 rq is then also Alie-inäriant. Ž.2 If A Ž.r s 0 then C Ž.r s 0. Ž.3 If A Ž.risA-inäriant then C Ž.risAlie-inäriant. Proof. Ž.1 Since ␶ Žrq. s rq, Eq.Ž. 3.2 gives

a ¸ rqs a ª rqq ␶ Ž.a ª rq s 0 for any a g A, as needed. Ž.2 By Lemma 3.4.2, AЈŽ.r s ␴ ŽA Ž..r s 0. Hence

Ž.3.1 CŽ.r s A Ž.r y AЈ Ž.r s 0.

Ž.3 By Lemma 3.4.2, AŽ.r s ␳ ŽA Ž..r s ␳ 2 ŽŽ..A r . By hypothesis this element is A-invariant. It follows from Eq.Ž. 3.2 that A Ž.r is Alie-invariant. Now, according to Eq.Ž. 3.1 and Lemma 3.4.2,

CŽ.r s A Ž.r y AЈ Ž.r s A Ž.r y ␴ Ž.A Ž.r .

Clearly, ␴ : Alie m Alie m Alie ª Alie m Alie m Alie is a morphism of Alie- modulesŽ.Ž. though not of A-bimodules . It follows that C r is Alie- invariant.

EXAMPLES 3.6.Ž. 1 consider the solution of Ž AYB . of Example 2.3.3, in the algebra ZkmŽ.of matrices with last row equal to zero. Since it is ANALOG OF LIE BIALGEBRAS 507 antisymmetric, it is also a solution ofŽ. CYB by Theorem 3.5. Similarly, we may obtain a solution ofŽ. CYB in the Lie algebra of matrices of size mn whose last n rows are zero, according to Example 2.5. This solution of Ž.CYB was constructed by Belavin and Drinfeldwx B-D, Example, Sect. 7 . The non-degenerate Lie cocycle to which it corresponds already appears in wxOom, p. 497 . The constructions of the present paper show that these solutions and cocycles in the Lie setting actually come from the associative setting. Ž.2 Let A be a symmetric algebra Ž Section 2 . and t g A m A the symmetric invariant tensor corresponding to the given form; such an element is often called a Casimir element. Then

t r s x y y is an antisymmetric solution ofŽ. AYB , belonging to a certain completed tensor product containing Ž.A m A wxwx, y A1, Example 5.4.3x . In a sense, this may be seen as constructed from the solution1 in kwxx and the x y y symmetric form on A, as in Proposition 2.4. Allowing ourselves to apply Theorem 3.5 in this situation, partŽ. 2 says that r is also a solution ofŽ. CYB in the Loop Lie algebra Aliewxx . Notice that by partŽ. 1 of the theorem, t is also a Casimir element for the Lie algebra Alie; therefore, this solution is the usual ‘‘rational’’ solution of Drinfeldwx Dri, Example 3.3, C-P, Example 2.1.9 . We will now interpret the results of Theorem 3.5 in terms of infinitesi- s m ¨ g m ⌬ ª m mal and Lie bialgebras. Given r ÝuiiA A, let r: A A A ⌬lie lie ª lie m lie and r : A A A be

⌬ s ª s m ¨ y m ¨ riŽ.a a r ÝÝau iuiia and

⌬lie s ¸ s wxm ¨ q m wx¨ riŽ.a a r ÝÝa, u iuia, i.

⑀ ⌬ Recall that the pair Ž.A, r is called a coboundary -bialgebra if Ž.A, r is an ⑀-bialgebra. According towx A1, Proposition 5.1 , this is the case if and only if AŽ.r is A-invariant. Similarly, ŽAlie, r. is called a coboundary Lie lie⌬ lie bialgebra if Ž A , r . is a Lie bialgebra. It is well known that this is the case if and only if CŽ.r and rq are Alie-invariantwx C-P, Proposition 2.1.2 . Also, Ž.A, r is called a quasitriangular ⑀-bialgebra if A Ž.r s 0 and ŽAlie, r. is called a quasitriangular Lie bialgebra if CŽ.r s 0 and rq is Alie-invariant. 508 MARCELO AGUIAR

COROLLARY 3.7. LetŽ. A, r be a coboundary ⑀-bialgebra for which the symmetric part rq is A-in¨ariant. Ž.1 Then Ž Alie, r. is a coboundary Lie bialgebra. Moreo¨er,

⌬lie s ⌬ y ␶⌬ rr r. Ž.2 If in additionŽ. A, r is a quasitriangular ⑀-bialgebra thenŽ Alie, ris. a quasitriangular Lie bialgebra. Proof. If Ž.A, r is a coboundary ⑀-bialgebra then A Ž.r is A-invariant. Hence, by Theorem 3.5.3, CŽ.r is Alie-invariant. Also, by Theorem 3.5.1, rq is Alie-invariant. Thus Ž Alie, r. is a Lie bialgebra. Moreover,

Ž.3.2 ⌬lie s ¸ s ª q ␶ ª ␶ s ⌬ q ␶⌬ rrŽ.a a r a r Ž.a Ž.r Ž.a ␶ Ž r. Ž.a , and since rq is A-invariant,

⌬ s ⌬ y ⌬ sy⌬ y rr and ␶ Žr. r . Hence

lie ⌬ s ⌬ yy ␶⌬ ys ⌬ y ␶⌬ rr rrr. The second assertion is just a restatement of Theorem 3.5.2. Remark 3.8. For arbitrary ⑀-bialgebras Ž.A, m, ⌬ , it is not true that Ž.Ž.A, m y m␶ , ⌬ y ␶⌬ is a Lie bialgebra see Example 4.12.1 . However, ⌬ s ⌬ the corollary shows that this does hold in the special case when r and rq is A-invariant. More generally, we will show that if Ž.A, m, ⌬ is a balanced ⑀-bialgebra, then Ž.A, m y m␶ , ⌬ y ␶⌬ is indeed a Lie bialgebra ⌬ s ⌬ q ⌬ Ž.Proposition 4.10 . When rrand r is A-invariant, Ž.A, m, is balancedŽ. Proposition 4.7 .

4. FROM INFINITESIMAL TO LIE BIALGEBRAS: BALANCED INFINITESIMAL BIALGEBRAS

Let Ž.A, m, ⌬ be an ⑀-bialgebra. The commutator bracket m y m␶ and the cocommutator cobracket ⌬ y ␶⌬ endow A with structures of Lie algebra and coalgebra, but these are not always compatible: Ž A, m y m␶ , ⌬ y ␶⌬. need not be a Lie bialgebra. In this section we introduce the class of balanced ⑀-bialgebras, for which this conclusion does hold. This class also arises naturally in the context of Drinfeld’s double, as will be seen in Section 5. Any ⑀-bialgebra that is both commutative and cocommutative is balancedŽ. in this case the corresponding Lie bialgebra is trivial . A ANALOG OF LIE BIALGEBRAS 509 different family of balanced ⑀-bialgebras is provided by those coboundary ⑀-bialgebras Ž.A, r for which rq is A-invariant. 4.1. The Balanceator. Balanced ⑀-bialgebras are defined below, in terms of a natural map B : A m A ª A m A that plays a role somewhat analogous to that of the commutator of an associative algebra and the cocommutator of a coassociative coalgebra, simultaneously. The basic properties of B, as well the construction of a balanced ⑀-bialgebra from an arbitrary oneŽ. analogous to the construction of the center of an associative algebra , are also presented in this section.

DEFINITION 4.1. Let Ž.A, m, ⌬ be an ⑀-bialgebra. The map B : A m A ª A m A defined by

BŽ.a, b s a ª ␶⌬ Ž.b q ␶ Ž.b ª ␶⌬ Ž.a s m y m q m y m ab2121b b ba a 1ba 21ab a 2 is called the balanceator of A. The ⑀-bialgebra A is called balanced if B ' 0, or equivalently if for every a, b g A, m q m s m q m Ž.B ab1a 221b ba ab 211b a ba 2.

The relevance of these notions will hopefully become clear from the results of this section. Remark 4.2. One may easily check that conditionŽ. B is equivalent to the commutativity of any of the diagrams below:

⌬mid 6 idm␶ 6 A m A A m A m A A m A m A

m m id y id m m 6 m ␶ A 6 A

mop m id y id m m op

A m6 AA6 m A m AA6 m A m A ⌬copmid idm␶

idm␶ 6 mmid 6 A m6 A m AAm A m AAm A

⌬ m id y id m ⌬ A m A ␶

⌬cop m id y id m ⌬cop 6

A m A m A 6 A m A m A 6 A m6 A idm␶ m opmid 510 MARCELO AGUIAR

Since these diagrams are clearly dual to each other, conditionŽ. B is self dual. Recall fromwx A1, Sect. 2 that if A [ Ž.A, m, ⌬ is an ⑀-bialgebra then so are Aop, cop [ Ž A, mop, ⌬ cop., Ay, q[ Ž.A, ym, ⌬ , Aq, y[ Ž.A, m, y⌬ and, if A is finite dimensional, A* [ Ž.A*, ⌬*, m*.

PROPOSITION 4.3. The balanceator B enjoys the following properties. Ž.1 B is natural: for any morphism f : A ª Bof⑀-bialgebras, the diagram

B 6 A m AAA m A

fmffmf

6 6 B m BB6 m B BB

commutes. Ž.2 The diagram

B 6 A m AAm A

␶ ␶

6 6 A m AA6 m A B

commutes. Ž.3 For any ⑀-bialgebra A,

sy␶ s y qsy BAAAAAAop, cop B , B Ž.B *, and B , B . Hence, if A is balanced, so are the other associated ⑀-bialgebras. Ž.4 For any a, b, c g A, Ž.4.1 B Žab, c .s Ža m 1 .Ž.B b, c q B Ža, c .Ž.1 m b , Ž.4.2 B Ža, bc .s B Ža, bc .Ž.m 1 q Ž1 m b .Ž.B a, c .

Ž.5 For any a, b g A, ⌬ m s m q m ␶ m Ž.Ž4.3 id .Ž.B a, b a12B Ža , b . Žid .ŽŽ.B a, b12 .b , m ⌬ s m q ␶ m m Ž.Ž4.4 id .Ž.B a, b B Ža12, b .a Žid .Ž. b 1B Ža, b 2 ..

Proof. PartsŽ. 1 and Ž. 2 are immediate from the definition. PartŽ. 3 can be checked by direct computation or by expressing B in terms of the ANALOG OF LIE BIALGEBRAS 511 diagrams of Remark 4.2, and noting that they are dual to each other, and that they remain unchanged after replacing m for mop and ⌬ for ⌬cop. ⌬ s m q m To verifyŽ. 4.1 , recall that by Ž.Ž. 1.1 , ab ab1212b a ab, hence s m y m q m q m BŽ.ab, c abc2121c c cab ab 1cb 21a ca 2 b y m y m ab1222 c b ac ab s m q m y m abc21c ab 1cb 2ab 1 c b 2 qy m q m Ž.ac21cb ac 21cb y m q m y m c21cab a 1ca 2 b ac 1ab 2 s Ž.Ž.Ž.Ž.a m 1 B b, c q B a, c 1 m b .

EquationŽ. 4.2 follows from Ž. 4.1 and part Ž. 2 . Equations Ž. 4.3 and Ž. 4.4 are their formal duals, so they must also hold according to Remark 4.2; a direct verification is also possible. Remark 4.4. EquationsŽ. 4.1 and Ž. 4.2 involve a unit element 1, but it is of course not necessary to assume that it belongs to A, since we may always embed A in a unital algebra. If A does have a unit element 1, then it follows from these equations that

BŽ.1, a s B Ž.a,1 s 0 ᭙a g A.

EquationsŽ. 4.1 and Ž. 4.2 are reminiscent of the familiar properties of the commutator bracket of an associative algebra, e.g., wxwxab, c s ab, c q wxa, cb. Also,Ž. 4.3 and Ž. 4.4 are similar to the dual properties of cocommu- tators. This suggests that conditionŽ. B may be seen as a simultaneous weakening of commutativity and cocommutativity. This is further supported by the results of Propositions 4.5 and 4.6 below.

PROPOSITION 4.5. LetŽ. A, m, ⌬ be an ⑀-bialgebra that is both commuta- ti¨e and cocommutati¨e. Then A is balanced. Proof. We verify that conditionŽ. B holds, as

Ž. m q m s m q m s1.1 ⌬ s ⌬ ab12211212a b ba ba a b ba Ž.ba Ž.ab Ž. s1.1 m q m s m q m ab1212b a ab ab 211b a ba 2.

The analogy between the balanceator of an ⑀-bialgebra and the commutator of an algebra suggests the possibility of defining for ⑀-bialgebras a notion analogous to that of the center of an algebra. It is pleasant that this 512 MARCELO AGUIAR indeed results in a balanced ⑀-subbialgebra of the given ⑀-bialgebra. Moreover, since the balanceator is self dual, a construction of a balanced quotient of an arbitrary ⑀-bialgebra, dual to the previous one, is also possible.

PROPOSITION 4.6. Let A be an arbitrary ⑀-bialgebra. Ž.1 Define s g r s ᭙ g s g r s ᭙ g ZAb Ž.Ä4a A B Ž.a, b 0 b A Ä4b A BŽ.a, b 0 a A . ⑀ Then ZbŽ. A is an -subbialgebra of A, and as such, it is balanced. Ž.2 Define

s m s m IAb Ž.ÝÝImŽ. Ž.id f B Im Ž. Ž.f id B . fgA* fgA*

r Then IbbŽ. A is a biideal of A, and the quotient A IŽ. A is a balanced ⑀-bialgebra.

Proof. First notice that the two definitions of ZAbŽ.indeed agree, by symmetry of B Ž.ŽProposition 4.3.2 . Equation 4.1. immediately implies that s F y ZAbbŽ.is a subalgebra. Notice that ZAŽ. bg A KerŽŽB , b ..; therefore, to show that it is a subcoalgebra we must show that

⌬ : m s m l m Ž.ZAbbbbŽ.ZA Ž.ZA Ž.ŽZA Ž. A .ŽA ZAbŽ. . s F KerŽ.Ž.BŽ.y, b m id l Ker id m BŽ.y, b . bgA g Take a ZAbŽ.. Then Ž.BŽ.y, b m id ⌬ Ž.a Ž. s m s4.4 m ⌬ y ␶ m m BŽ.a12, b a Žid .Ž.Ž.B a, b idŽ. b1B Ž.a, b 2 s 0 Ž.id m BŽ.Ž.y, b ⌬ a Ž. s m s4.3 ⌬ m y m ␶ m a12BŽ.Ža , b id .Ž.Ž.Ž.B a, b id Ž.B a, b12b s 0 as needed. Thus, ZAbŽ.is a subbialgebra of A. It is obviously balanced since, by naturally of B, B Ž.a, b s B Ž.a, b s 0 ᭙a, b g ZAŽ.. Z bŽ A. Ab PartŽ. 2 is formally dual to partŽ. 1 ; nevertheless, we provide a direct verification. ANALOG OF LIE BIALGEBRAS 513

m The elements of the form Ž.Ž.id f B b, c are linear generators of IAb Ž.. IdentityŽ. 4.1 implies that a и Ž.Ž.id m f B b, c s m y m ª g Ž.Ž.id f B ab, c Ž.id Ž.Ž.Ž.b f B a, c IAb , ª where b f is the usual left action of A on A*; seeŽ. 5.10 . Thus, IAb Ž.is a left ideal. Similarly, one deduces fromŽ. 4.2 that it is a right ideal. Also, Ž.4.3 implies that ⌬ m s m m q m m Ž.Ž.Ž.Ž.Ž.Ž.id f B a, b a12id f B a , b id f B a, b 1b2 g m q m A IAbbŽ.IA Ž. A r which shows that IAbbŽ.is a coideal. Finally, to show that A IAŽ.is balanced one may use the following general fact: if S is a subspace of m s m s m : A A, S1 Ý f g A*2Ž.Ž.f id S , and S Ý f g A*2 Ž.Ž.id fS, then S S m s : m S1. Applied to S ImŽ.B this gives ImŽ.B IAbb Ž.IA Ž., which cer- tainly implies that B r ' 0. A IbŽ A. Examples of balanced ⑀-bialgebras and of the constructions of Proposi- tion 4.6 will be given belowŽ. Examples 4.8 and 4.12 . In Proposition 5.10, the balanceator of the Drinfeld double will be explicitly calculated. Balanced Quasitriangular ⑀-Bialgebras. Below we find a sufficient condition for a quasitriangularŽ. or more generally, coboundary ⑀-bialgebra to be balanced. This provides an important source of examples of balanced ⑀-bialgebras.

PROPOSITION 4.7. LetŽ. A, r be a coboundary ⑀-bialgebra. Then

s и ª ␶⌬ q ᭙ g BŽ.a, b 2 Ž.a r Ž.b a, b A. In particular, if rq is A-in¨ariant then A is balanced. s m ¨ ⌬ s m ¨ y m ¨ Proof. Let r Ýuii,so rŽ.a Ýau iiu iia. We compute ª ␶⌬ s ª ¨ m y ¨ m a riŽ.b a Ý Ž buiib ui. s a ª Ž.Ž1 m b .Ž.␶ r y ␶ Ž.Žrbm 1 . s Ž.Ž.Ž.Ž.Ž.a m b ␶ r y a m 1 ␶ rbm 1 yŽ1 m b .Ž.Ž␶ r 1 m a .q ␶ Ž.Žrbm a .. Hence ª ␶⌬ s m ␶ y m ␶ m b r Ž.a Žb a .Ž.r Žb 1 .Ž.Žra 1 . yŽ.Ž.Ž.Ž.Ž.1 m a ␶ r 1 m b q ␶ ram b 514 MARCELO AGUIAR and ␶ ª ␶⌬ s m y m m Ž.b r Ž.a Ža br . Ž1 br .Ž1 a . yŽ.Ž.Ž.a m 1 rbm 1 q rbm a s ª ␶⌬ a ␶ Žr.Ž.b . Hence s ª ␶⌬ q ␶ ª ␶⌬ s ª ␶⌬ BŽ.a, b a rrr Ž.b Ž.b Ž.a a q␶ Ž r .Ž.b

s и ª ␶⌬ q 2 Ž.a r Ž.b .

EXAMPLES 4.8.Ž. 1 The ⑀-bialgebra kwx␧ rŽ␧ 2 . of dual numbers is quasitriangular with r s 1 m ␧ wxA1, Example 5.4.1 . Since rqs 1 m ␧ q ␧ m 1is A-invariant, this ⑀-bialgebra is balanced.Ž Actually, it is commutative and cocommutative.. Ž.2 The condition of Proposition 4.7 is not necessary: a coboundary ⑀-bialgebra Ž.A, r may be balanced even if rq is not invariant. Consider s wx␧ ␧ the three dimensional algebra A k 12, where

␧ 2 s ␧ 2 s ␧␧ s ␧␧s 1212210.

A can be realized as the subalgebra of Mk3Ž.of matrices of the form

ab0 0 a 0 . 0 ca s m ␧ wx A is quasitriuangular with r 1 1 A1, Example 5.4.1 . We have

⌬ q s ⌬ q ␧ s rrŽ.1 0, Ž.1 0, but 1 ⌬ q Ž.␧ s Ž␧ m ␧ y ␧ m ␧ ./ 0. r 22 2112

In particular rq is not A-invariant. However,

ª ⌬ q ␧ s ␧ ª ⌬ q ␧ s ␧ ª ⌬ q ␧ s 1 r Ž.210, r Ž.220, and r Ž.2 0.

s и ª ␶⌬ q s Therefore, according to Proposition 4.7, BŽ.a, b 2 Ža r Ž..b 0 ᭙a, b g A,so A is balanced. ANALOG OF LIE BIALGEBRAS 515

From Balanced ⑀-Bialgebras to Lie Bialgebras. Below we will show that balanced ⑀-bialgebras give rise to Lie bialgebras by passing to the commutator and cocommutator brackets. In fact, the necessary and sufficient condition for obtaining a Lie bialgebra is that the balanceator satisfies BŽ.a, b s B Ž.b, a . Several examples will be given.

LEMMA 4.9. Let M be an A-bimodule and M lie the associated Alie-module: a ª m s a и m y m и a. Then DerŽ.A, M : Der ŽAlie, M lie.. Proof. If D g DerŽ.A, M then

DaŽ.wx, b s DabŽ.y Dba Ž. s a и DbŽ.y Da Ž.и b y Ž.b и Da Ž.y Db Ž.и a s a ª DbŽ.y b ª Da Ž. so D g DerŽ.Alie, M lie .

PROPOSITION 4.10. LetŽ. A, m, ⌬ be an ⑀-bialgebra. Then Ž A, m y m␶ , ⌬ y ␶⌬. is a Lie bialgebra if and only if B s B␶. In particular, if Ž.A, m, ⌬ is balanced Ž.ŽB s 0 then A, m y m␶ , ⌬ y ␶⌬ .is a Lie bialgebra. Proof. By hypothesis and Lemma 4.9,

⌬ g DerŽ.A, A m A : DerŽ.Alie, Ž.A m A lie .

Thus,

⌬Ž.wxa, b s a ª ⌬Ž.b y b ª ⌬ Ž.a and, letting ⌬lie s ⌬ y ␶⌬,

Ž.) ⌬lieŽ.wxa, b s aª⌬Ž.b y bª⌬ Ž.a y␶ Ž.a ª ⌬Ž.b y b ª ⌬ Ž.a .

On the other hand, we have

a ª ⌬lieŽ.b s a ¸ ⌬ Ž.b y a ¸ ␶⌬ Ž.b

Ž.3.2 s a ª ⌬Ž.b q ␶ Ž.a ª ␶⌬Ž.b ya ª ␶⌬Ž.b y ␶ Ž.a ª ⌬ Ž.b . 516 MARCELO AGUIAR

Therefore

a ¸ ⌬lieŽ.b y b ¸ ⌬lie Ž.a s a ª ⌬Ž.b y b ª ⌬ Ž.a q ␶ Ž.a ª ␶⌬Ž.b y b ª ␶⌬ Ž.a ya ª ␶⌬Ž.b q b ª ␶⌬ Ž.a y ␶ Ž.a ª ⌬ Ž.b y b ª ⌬ Ž.a Ž.) s ⌬lieŽ.wxa, b q ␶ Ž.a ª ␶⌬Ž.b y b ª ␶⌬ Ž.a ya ª ␶⌬Ž.b q b ª ␶⌬ Ž.a s ⌬lieŽ.wxa, b q BŽ.b, a y B Ž.a, b . Thus, ⌬lie g DerŽ Alie, A liem A lie.Žif and only if B b, a.Žs B a, b.for every a, b g A, which is the desired conclusion. Remarks 4.11.Ž. 1 Recall that B␶ s ␶ B always holds, by Proposition 4.3. Thus, when the condition B s B␶ holds, the balanceator induces a map SA2 Ž.ª SA2 Ž., and conversely. We will say in this case that the balanceator is symmetric. Ž.2 It is easy to see the balanceator is symmetric if and only if a ª ␶⌬Ž.b y b ª ␶⌬ Ž.a is a symmetric element of A m A, for every a, b g A.

EXAMPLES 4.12.Ž. 1 Not every ⑀-bialgebra yields a Lie bialgebra. To see this, we must produce an ⑀-bialgebra for which B is not symmetric.

Consider the algebra of matrices Mk2Ž.equipped with the comultiplication xy 0 x 01 01 zw ⌬ smym. zw 0 z 00 00 00 ⑀ wx Mk2Ž.is then an -bialgebra A1, Example 2.3.7 . One calculates xy pq B , ž/zw r s 0 x 0 p 01 s 2 и myxr xs m ž 0 z 0 r zr zs 00 01 pz pw rs zw ym q m . 00rz rw 00 00/ Thus B is not symmetric, so there is no associated Lie bialgebra. It is easy to see that the constructions of Proposition 4.6 produce the wxxy following results: ZAbŽ.is the subalgebra of matrices of the form 0 x r s Ž.where the comultiplication is zero , while A IAbŽ.0. ANALOG OF LIE BIALGEBRAS 517

Ž.2 This is an example of a non-balanced ⑀-bialgebra for which the balanceator is symmetricŽ and therefore there is an associated Lie bialgebra. . Consider the algebra A s kwxx rŽx 4 ., equipped with the comultiplication ⌬Ž.x i s x i m x 2 y 1 m x iq2 .

A is then an ⑀-bialgebra. In fact, if r s 1 m x 2 then Ž.A, r is quasitriangu- ⌬ s ⌬ wx lar and r A1, Example 5.4.1 . In this case

BxŽ.i , x j s x iq2 m x j y x iqjq2 m 1 y x 2 m x iqj q x jq2 m x i q x j m x iq2 y 1 m x iqjq2 y x iqj m x 2 q x i m x jq2 , which is clearly symmetricŽŽ. this had to be the case since A, m is commutative, so Ž.A, m y m␶ , ⌬ y ␶⌬ is a Lie bialgebra trivially . . How- ever, B / 0so A is not balanced.

The constructions of Proposition 4.6 are as follows: ZAbŽ.is the subalgebra spanned by 1 and x 2, which is isomorphic to the ⑀-bialgebra of dual numbersŽ.Ž. Example 4.8.1 ; IAb is the ideal generated by x,so r ( A IAbŽ. k with zero comultiplication. Ž.3 Let ᒄ be the two dimensional non-abelian Lie algebra: ᒄ has a basis Ä4x, y such that wxy, x s x. There are, up to isomorphism and scalar multiplies, only two Lie bialgebra structures on ᒄ wC-P, Examples 1.3.7 and 2.1.5x : ␦ s ␦ s m y m 11Ž.x 0, Ž.y y x x y and ␦ s m y m ␦ s 22Ž.x x y y x, Ž.y 0.

Let us discuss these examples in connection with balanced ⑀-bialgebras. Let A be the two dimensional algebra with basis Ä4x, y and structure

x 2 s 0, xy s 0, yx s x, and y 2 s y, ⌬Ž.x s x m x and ⌬Ž.y s y m x.

A is then an ⑀-bialgebra; in fact, it is the quasitriangular ⑀-bialgebra of Example 2.3.1, with r s y m x y x m y. In this case, rqs 0, so by Proposi- tions 4.7 and 4.10, A is balanced and Alie is a Lie bialgebra. This is precisely the first Lie bialgebra structure on ᒄ above. Direct calculations show that the second Lie bialgebra structure does not come from any ⑀-bialgebra structure. 518 MARCELO AGUIAR

5. DRINFELD’S DOUBLE OF LIE AND INFINITESIMAL BIALGEBRAS

Recall that the Drinfeld double DŽ.ᒄ of a Lie bialgebra ᒄ carries a non-degnerate symmetric form that is associative, in the sense that

²:²:wx␣ , ␤ , ␥ s ␣ , wx␤ , ␥ ᭙␣ , ␤ , ␥ g DŽ.ᒄ . Inwx A1 , we have defined the Drinfeld double DAŽ.of an ⑀-bialgebra A. Since this is defined for arbitrary ⑀-bialgebras A Ž.balanced or not , there may be no Lie bialgebras associated to A or DAŽ.. DA Ž.always carries a canonical symmetric associative form, but this may be degenerate. The main result of this section describes a necessary and sufficient condition for A to be balanced in terms of the radical of this formŽ. Theorem 5.5 . In this case, the quotient DAbŽ.of DA Ž.by the radical of the form is again a balanced ⑀-bialgebraŽ. Proposition 5.10 , and the corresponding Lie bialgebra is, up to a sign, the Drinfeld double of the Lie bialgebra corresponding to A Ž.Proposition 5.12 . Another important result in this section is Theo- rem 5.9, which contains the universal property of DAbŽ.. We begin by recalling the construction of the double DAŽ.of a finite dimensional ⑀-bialgebra Ž.A, m, ⌬ . Consider the following version of the dual of A AЈ [ Ž.A*, ⌬*op, ym*cop . Explicitly, the structure of AЈ is и s ᭙ g g Ј Ž.5.1 Žf ga .Ž.Ž.Ž.ga12 fa a A, f , g A and ⌬ s m m sy ᭙ g Ј g Ž.5.2 Ž.f f12f fabŽ.fafb2 Ž.Ž. 1 f A , a, b A. Below we always refer to this structure when dealing with multiplications or comultiplications of elements of AЈ. Consider also the actions of AЈ on A and A on AЈ defined by ª s ¤ sy Ž.5.3 f a faŽ.12 a and f a faf2Ž. 1 or equivalently Ž.Ž.Ž.Ž.5.4 gfª a s gf a and Ž.Ž.Ž.f ¤ abs fab.

PROPOSITION 5.1. Let A be a finite dimensional ⑀-bialgebra, consider the ¨ector space DAŽ.[ ŽA m AЈ .[ A [ AЈ, ANALOG OF LIE BIALGEBRAS 519 and denote the element a m f g A m AЈ : DŽ. A by a j f. Then D Ž. A admits a unique ⑀-bialgebra structure such that: Ž.a A and AЈ are subalgebras, a и f s a j f, f и a s f ª a q f ¤ a, and Ž.b A and AЈ are subcoalgebras. Explicitly, the multiplication and comultiplication are gi¨en in the remaining cases by a и Ž.b j f s ab j f , Ž.a j f и g s a j fg, Ž.a j f и b s af Žª b .q a j Žf ¤ b ., f и Ž.Ž.Ž.a j g s f ª a j g q f ¤ ag, Ž.Ž.Ž.Ž.5.5 a j f и b j g s afª b j g q a j Ž.f ¤ bg, ⌬ j s j m q m j Ž.5.6 Ža f . Ža f1212 .f a Ža f .. Proof. Seewx A1, Theorem 7.3 . If ᒄ is a finite dimensional Lie bialgebra, its double DŽ.ᒄ s ᒄ [ ᒄ* carries a non-degenerate symmetric associative form, defined by ²:a q f , b q g s fbŽ.q ga Ž.. For ⑀-bialgebra, the analogous form is still symmetric and associative, in the sense that ²:²:␣␤, ␥ s ␣ , ␤␥ ᭙␣ , ␤ , ␥ g DAŽ., but this form is usually degenerate.

PROPOSITION 5.2. Let A be a finite dimensional ⑀-bialgebra. There is a symmetric associati¨e form on DŽ. A uniquely determined by Ž.5.7 ²:a q f , b q g s fbŽ.Ž.q ga. Proof. Define a bilinear form on DAŽ.by Ž.5.8 ²:a q f q b j g, aЈ q fЈ q bЈ j gЈ s fЈŽ.a q fa ŽЈ .q gЈ ŽabЈ .q ga ŽЈb .q ŽgЈfb .Ž.Ј q Ј q X Ј X q Ј X Žgf .Ž.b gb Ž.Ž12g bb .g Ž.Žbgb 12b .. This form is clearly symmetric and satisfies Eq.Ž. 5.7 . Conversely, symmetry, associativity, andŽ. 5.7 force us to define the form as above: for instance,

Ž.5.7 ²:²:²:²:b j g, aЈ s aЈ, b j g s aЈ, b и g s aЈиb, g s gaŽ.Јb . 520 MARCELO AGUIAR

Thus, uniqueness holds. To show that this form is indeed associative one must check several cases. We do it only for one of the most relevant, involving elements of A m AЈ : DAŽ.. In the computation, we will use that

Ž. ) ⌬ ª s5.3 ⌬ Ž.Ž.af Žb .fb Ž12 .Žab . Ž. s1.1 m q m fbŽ.12 ab b 3fb Ž. 1122 a ab. We calculate

²:Ž.Ž.a j f и b j g , c j h

Ž.5.5 s ²:afŽ.ª b j g q a j Ž.f ¤ bg, c j h

Ž.Ž.) 5.8s , ª q gcŽ.121 hafŽ.Ž. Žbc .hfb Ž. abgcb2 Ž3 . q q ¤ hfbŽ.Ž.11 a gcab Ž 22 .Ž. Žf bg . Ž.Ž c1 hac 2 . q ¤ haŽ.Ž12Ž. f bg . Ž ca .

Ž.Ž.5.3 , 5.4 s gc fb habc q fb habgcb ^`_^`_Ž.Ž.Ž11 221 . Ž.Ž 2 .Ž 3 . 12

q fb ha gca, b q gc fbchac ^`_^`_Ž.Ž.Ž11 22 . Ž.Ž.Ž 1 2 3 . 34

q ha gcafba q ha gc fbca . ^`_^`Ž.Ž123 .Ž . Ž.Ž.Ž 1122 _. 56 Since the form is symmetric, we can also conclude that

²:a j f , Ž.Ž.b j g и c j h s²:Ž.Ž.b j g и c j h , a j f s ha gc fbca q gc fbchac ^`_^`_Ž.Ž.Ž11 221 . Ž.Ž 2 .Ž 3 . 64

q gc fb habc q ha gcafba ^`_^`_Ž.Ž.Ž11 221 . Ž.Ž 2 .Ž 3 . 15

q fb habgcb q fb ha gcab . ^`_^`Ž.Ž123 .Ž . Ž.Ž.Ž 1122 _. 23 ANALOG OF LIE BIALGEBRAS 521

Thus ²:Ž.Ž.a j f и b j g , c j h s²:a j f , Ž.Ž.b j g и c j h as needed. We will refer to the form of Proposition 5.2 as the canonical form on the double. We now turn towards the proof of the main theorem of this section, which relates the radical of the canonical form with a certain coideal of the double, to be defined next. Let A be a finite dimensional ⑀-bialgebra. There are four natural actions of A and AЈ on each other. Our definition of DAŽ.involves only two of them, namely the actionsŽ. 5.3 and Ž. 5.4 . It is time now to consider the other two actions, defined by ¤ s ª sy Ž.5.9 a f faŽ.21 a and a f faf1Ž. 2 or equivalently Ž.Ž.Ž.Ž.5.10 ga¤ f s fg a and Ž.Ž.Ž.a ª fbs fba. Let I denote the subspace of DAŽ.linearly spanned by the set Ž.5.11 Ä4a j f y a ¤ f y a ª fra g A, f g AЈ . Recall that in DAŽ.we have Ž by Proposition 5.1. f и a s f ª a q f ¤ a and a и f s a j f ; therefore, modulo I, the multiplication in DAŽ.acquires the more symmetric form f и a ' f ª a q f ¤ a and a и f ' a ¤ f q a ª f. We will return to this after discussing the basic properties of I.

LEMMA 5.3. Let A be a finite dimensional ⑀-bialgebra. Ž.1 For any a g A and f g AЈ, ¤ m q m ª s Ž.5.12 Ža f1212 .f a Ža f .0.

Ž.2 I is a coideal of DŽ. A . Proof. Evaluating the left hand side on b g A one obtains ¤ q ª fba2121Ž.Žf . Ža fba .Ž. Ž.Ž. Ž. 5.9s , 5.10 q s5.1 fbfaa2Ž. 121 Ž .fbaa Ž 21 . 0 which proves the first part. 522 MARCELO AGUIAR

For the second, we calculate

⌬Ž.a j f y a ¤ f y a ª f Ž. s5.9 ⌬ j y ⌬ q ⌬ Ž.Ž.Ž.Ž.Ž.a f fa211a fa f 2 Ž. s5.6 j m q m j y m q m Ž.a f1212f a Ža f .Ž.fa 312123 a a faf Ž.f Ž. s5.9 j m q m j y m ¤ Ž.a f1212f a Ža f .a 1 Ža 2f . y ª m Ž.a f12f Ž. 5.12s j y ª y ¤ m Ž.a f1112a f a f f q m j y ¤ y ª a12Ž.a f a 2f a 2f . Thus, ⌬Ž.I : I m DA Ž.q DA Ž.m I as needed.

LEMMA 5.4. As ¨ector spaces, DAŽ.( I [ A [ AЈ. Proof. Consider the linear map ␲ : DAŽ.ª A [ AЈ given by

␲ Ž.a s a, ␲ Ž.f s f , and ␲ Ž.a j f s a ¤ f q a ª f. : ␲ ␣ s j q q g ␲ By definition, I KerŽ.. Conversely, if Ýaiif a f KerŽ. ¤ q ª q q s ␣ s j y ¤ y then ÝŽ.aiiiif a f a f 0, hence Ýaiif Ýa iif ª g s ␲ Ýaiif I. Thus I KerŽ.. Let i : A [ AЈ ª DAŽ.be the inclusion. Then ␲ i s id, hence DAŽ.( KerŽ.␲ [ Im Ž.i s I [ A [ AЈ. We are now in position to derive the main result of this section, which characterizes balanced ⑀-bialgebras in terms of the radical of the canonical form on the double and the coideal I.

THEOREM 5.5. Let A be a finite dimensional ⑀-bialgebra and I the subspace linearly spanned by

Ä4a j f y a ¤ f y a ª fra g A, f g AЈ

Ž.as abo¨e . The following statements are equi¨alent: Ž.1 A is balanced; Ž.2 I is the radical of the canonical form on D Ž A .; Ž.3 I is an ideal of D Ž A .. ANALOG OF LIE BIALGEBRAS 523

Proof. Ž.1 « Ž.2 . We first show that I : rad ² , : . For this we consider three cases:

Ž.5.8 Ž.i ²:a j f y a ¤ f y a ª f , b s fbaŽ .y Ža ª fb .Ž .

Ž.5.9 s fbaŽ.y fba Ž.s 0;

Ž.5.8 Ž.ii ²:a j f y a ¤ f y a ª f , g s Žfg .Ž. a y ga Ž¤ f .

Ž.5.10 s Ž.Ž.Ž.Ž.fg a y fg a s 0; Ž.iii ²:a j f y a ¤ f y a ª f , b j g

Ž. s5.8 q fbŽ.Ž12 gab .ga Ž.Ž 12 fba . y gaŽ.Ž.Ž.¤ fby ga Ž.Ž.ª fb Ž.Ž. 5.9s , 5.10 q y fbŽ.Ž12 gab .ga Ž.Ž 12 fba .fa Ž.Ž 21 gab . y s gbŽ.Ž.21 fba 0; the last equality holds precisely by conditionŽ.Ž B Definition 4.1. . Since I : rad²: , , the form ²: , descends to a bilinear form on DAŽ.rI with radical rad² , :rI. By Lemma 5.4, DAŽ.rI ( A [ AЈ, and the resulting form is simply ²:Ž.Ž.a q f, b q g s fbq ga, which is clearly non-degenerate. Hence rad²: , s I. Ž.2 « Ž.3 . This is clear since, by Proposition 5.2, the canonical form is associative. Ž.3 « Ž.1 . Since I is an ideal, we have in particular that Ž.a j f y a ¤ f y a ª fb' 0 mod I ᭙a, b g A, f g AЈ. Now, according to Proposition 5.1, Ž.a j f y a ¤ f y a ª fb s afŽ.ª b q a j Ž.Ž.Ž.f ¤ b y a ¤ fby a ª f ª b y Ž.a ª f ¤ b s q j ¤ y y y ª ¤ fbŽ.12 ab a Ž.Ž.Ž.f b fa21 ab fbab 12a Ž.f b , by Eqs.Ž.Ž.Ž. 5.3 , 5.4 , 5.9 , and Ž 5.10 . . Hence, modulo I, we have ' q ¤ ¤ y y 0 fbŽ.12 ab a Žf b .fa Ž.21 ab fbab Ž 12 . mod I ' q y y 0 fbŽ.1 ab 2fbaa Ž 21 .fa Ž. 21 ab fbab Ž 1 . 2 mod I. 524 MARCELO AGUIAR

Since I l A s 0Ž. by Lemma 5.4 , we deduce that s q y y 0 fbŽ.1 ab 2fbaa Ž 21 .fa Ž. 21 ab fbab Ž 1 . 2 and since this holds ᭙f g A*, we conclude that s m q m y m y m 0 b12212112ab ba a a ab ba b , which is conditionŽ. B . Let A be a finite dimensional balanced ⑀-bialgebra. It follows from Theorem 5.5 and Lemma 5.3 that DAŽ.rI is an ⑀-bialgebra. According to Lemma 5.4, the inclusion A [ AЈ ¨ DAŽ.induces an isomorphism of r ( [ Ј vector spaces DAŽ.I A A . We denote by DAb Ž.the resulting ⑀-bialgebra structure on A [ AЈ, and call it the balanced Drinfeld double of A. From the definition of I, this structure is as follows. The algebra structure is determined by Ž.a A and AЈ are subalgebras, Ž.b a и f s a ¤ f q a ª f, and Ž.c f и a s f ª a q f ¤ a. Equivalently, and writing Ž.a, f instead of a q f, Ž.Ž.Ža, f и b, g s ab q a ¤ g q f ª b, a ª g q f ¤ b q fg .. Ј As a coalgebra, DAbŽ.is simply the direct sum of A and A . Theorem 5.5 also shows that DAbŽ.is a symmetric algebra. The non- degenerate symmetric associative form is simply ²:a q f , b q g s fbŽ.q ga Ž.. EXAMPLE 5.6. The ⑀-bialgebra kwx␧ rŽ␧ 2 .Žof dual numbers Example 4.8.1. is the simplest instance of a balanced Drinfeld double. In fact, if we view the one dimensional algebra k as a balanced ⑀-bialgebra with ⌬ s ␧ comultiplication Ž.1 0, then Dkb Ž .is two dimensional, with basisÄ4 1, , whereÄ4 1 and Ä␧ 4are dual bases of k and kЈ. The algebra structure on the balanced double is precisely that of the algebra of dual numbers, while the coalgebra structure has been multiplied by y1 with respect to that of Example 4.8.1.

We will derive next a universal property DAbŽ.as a quasitriangular ⑀-bialgebra. First, we recall the corresponding result for the full double DAŽ.and state a lemma needed for the proof. ⑀ LEMMA 5.7. Let A be a finite dimensional -bialgebra, Ä4ei be a linear Ј ᭙ g g Ј basis of A, andÄ4 fi the dual basis of A . Then a A and f A , s s Ž.5.13 ÝÝfaeii Ž. a and fŽ. eii f f ii m s m ¤ Ž.5.14 ÝÝaeiif e iŽ.f ia ii ANALOG OF LIE BIALGEBRAS 525

ª m s m Ž.5.15 ÝÝŽ.f eiif e iiff ii m s m ª Ž.5.16 ÝÝeaiif e iŽ.a f i ii ¤ m s m Ž.5.17 ÝÝŽ.eiiiif f e ff ii m ª s ¤ m Ž.5.18 ÝÝeiiŽ.Ž.f a a fiie ii m ¤ s ª m Ž.5.19 ÝÝfiiiiŽ.Ž.f e e f f . ii Proof. These are all straightforward. A proof ofŽ.Ž. 5.13 to 5.15 can be found inwx A1, Lemma 7.2 ; the others are similar.

PROPOSITION 5.8. Let A be a finite dimensional ⑀-bialgebra and DŽ. A its Ј Drinfeld double. LetÄ4 eii and Ä4 f be dual bases of A and A and sy m g m Ј : m R Ý eiif A A DAŽ.DA Ž.. i Ž.1 ŽDA Ž ., R . is a quasitriangular ⑀-bialgebra. s m ¨ m ␲ r ª Ž.2 Let r Ý jju jbe an element of A A and : DAŽ. A the map defined by ␲ r s ␲ r sy ¨ Ž.a a, Ž.f Ý fu Žjj . , and j ␲ r j sy ¨ Ž.a f Ý fu Ž.jj a. j

ThenŽ. A, r is quasitriangular if and only if ␲ r is a morphism of ⑀-bialgebras. Ž.3 A is quasitriangular if and only if the canonical inclusion A ¨ DA Ž . splits as a morphism of ⑀-bialgebras. Proof. Seewx A1, Theorem 7.3 and Proposition 7.5 The balanced Drinfeld double satisfies a similar universal property, but among quasitriangular ⑀-bialgebras Ž.A, r for which the symmetric part of r is A-invariant.

THEOREM 5.9. Let A be a finite dimensional balanced ⑀-bialgebra and sy m g m Ј : m RbiibbÝ e f A A DAŽ.DA Ž.. i ⑀ q Ž.1 ŽDAbb Ž ., R . is a quasitriangular -bialgebra and Rbb is DŽ. A - in¨ariant. 526 MARCELO AGUIAR

s m ¨ m ␲ r ª Ž.2 Let r Ý jju jbe an element of A A and b: DA bŽ. A the map defined by

␲ r s ␲ r sy ¨ bbjŽ.a a and Ž.Ž.f Ý fu j. j

␲ r ⑀ Then b is a morphism of -bialgebras if and only ifŽ. A, r is quasitriangular and rq is A-inäriant. ¨ ⑀ Ž.3 The canonical inclusion A Db Ž A . splits as a morphism of -bialgebras if and only if there is r g A m A such thatŽ. A, r is quasitriangular and rq is A-inäriant. Proof. Ž.1 Since ŽDA Ž ., R .is a quasitriangular ⑀-bialgebra, so is its q quotient ŽŽ..DAbb, R . To show that Rbbis DAŽ.-invariant we must check и qи и qs qи ᭙ g g Ј that a Rbba and f R Rbf a A, f A . Using the description of DAbŽ.following Theorem 5.5, we compute и qs и q и ␶ sy и m y и m a 2 Rbba R a Ž.R bÝÝ Ža eii .f Ža f ii .e sy ae m f y Ž.a ¤ f m e y Ž.a ª f m e . ^`_^ÝÝii ìi _^ Ý ìi _ AB C On the other hand,

qи s и q ␶ и sy m и y m и 2 Rbba R a Ž.R ba ÝÝe iiŽ.Ž.f a f iie a sy e m Ž.f ª a y e m Ž.f ¤ a y f m ea. ^`_^`_^`_ÝÝÝii ii ii BЈ AЈ CЈ Note that A s AЈ byŽ. 5.14 , B s BЈ by Ž. 5.18 , and C s CЈ byŽ. 5.16 . Thus и qs qи a RbbR a. Similarly,

и qs и q и ␶ sy и m y и m f 2 Rbbf R f Ž.R bÝÝ Žf eii .f Žf f ii .e sy Ž.f ª e m f y Ž.f ¤ e m f y ff m e ^`_^`_^`_ÝÝÝii ii ii ABC and

qи s и q ␶ и sy m и y m и 2 Rbbf R f Ž.R bf ÝÝe iiŽ.Ž.f f f iie f sy e m ffy f m Ž.e ¤ f y f m Ž.e ª f . ^`_^ÝÝii i ` i _^ Ýi ` i _ AЈ CЈ BЈ ANALOG OF LIE BIALGEBRAS 527

We have that A s AЈ byŽ. 5.15 , B s BЈ by Ž. 5.19 , and C s CЈ by Ž. 5.17 . и qqs и Thus f RbbR f. This completes the proof of partŽ. 1 . ␲ b Ž.2 The map r is surjective. Moreover, ␲ b m ␲ b sy ␲ b m ␲ b Ž.rrbŽ.R Ý riri Ž.e Ž.f i

Ž.5.13 s m ¨ s m ¨ s ÝÝfueijiŽ. ju j j r. i, jj ␲ b ⑀ Therefore, if r is a morphism of -bialgebras, Ž.A, r is a quotient ⑀ q -bialgebra of ŽŽ..DAbb, R , so it is quasitriangular and r is A-invariant. ␲ ª Conversely, if Ž.A, r is quasitriangular then the map r : DAŽ.A of Proposition 5.8 is a morphism of ⑀-bialgebras. We claim that ␲ s m q r Ž.I 0 r is A-invariant. ␲ This will complete the proof of partŽ. 2 , since the map induced by r on r s ␲ b the quotient DAŽ.I DAbr Ž.is clearly . To verify the claim, we calculate ␲ b j y ¤ y ª r Ž.a f a f a f sy ¨ y ¤ y ª ¨ ÝÝfuŽ.jj a a f Ža fu .Ž.jj jj sy ¨ y q ¨ ÝÝfuŽ.jj a fa Ž.21 a fua Žjj .. jj Since Ž.A, r is quasitriangular, m s ⌬ s m ¨ y m ¨ a12a rjjjjŽ.a ÝÝau u a jj « s ¨ y ¨ faŽ.21 a ÝÝf Ž.jjau f Ž jjau .. jj Therefore, ␲ s m y ¨ y ¨ q ¨ rŽ.I 0 ÝÝÝfuŽ.Ž.Ž.jjjjjj a f au f au jjj q ¨ s ᭙ g g Ј Ý fuaŽ.jj0 a A, f A j m ¨ m q m ¨ s m ¨ q ¨ m ÝÝÝÝa ju jau jju jja jua j jjj j ᭙a g A m a и rqs rqи a ᭙a g A m rq is A-invariant. 528 MARCELO AGUIAR

Ž.3If ŽA, r .is quasitriangular and rq is A-invariant then the inclusion ¨ ␲ r A DAbbŽ.is split by the morphism , according to partŽ. 2 . ¨ ⑀ Conversely, if the inclusion A DAbŽ.splits as a morphism of -bial- [ ␲ m ␲ g m gebras, then by partŽ. 1 , r Ž .ŽRb . A A is such that Ž.A, r is q ␲ ª quasitriangular and r is invariant, where : DAbŽ. A is a splitting.

The properties of DAbŽ.thus obtained are strictly analogous to those of the double DŽ.ᒄ of a Lie bialgebra ᒄ. This analogy can be made more precise: we will show below that the functor from balanced ⑀-bialgebras to Lie bialgebras commutes with the double constructions. First we must ⑀ verify that DAbŽ.is indeed a balanced -bialgebra. This follows at once from Theorem 5.9 and Proposition 4.7. However, we find of independent interest to deduce this result from the explicit expression of the balanceator of the full double DAŽ., as follows.

PROPOSITION 5.10. Let A be a finite dimensional ⑀-bialgebra, a g A, and f g AЈ. Then the balanceator of the full double DŽ. A is gi¨en by s j y ª y ¤ m Ž.5.20 BDŽ A. Ž.Ža, f a f2221a f a f .f q ª q ¤ y j m Ž.a111f a f a f a 2 s m j y ª y ¤ Ž.5.21 BDŽ A. Ž.f , a f1 Ža f 222a f a f . q m ª q ¤ y j a21Ž.a f a 1f a 1f .

In particular, if A is balanced, then so is DbŽ. A . Proof. We have

BŽ A.Ž.a, f s m y m q m y m af2121f f fa a 1fa 2af 1a 2 s5.1 j m y m ª y m ¤ q m ª Ž.a f2121f f Ž.f a f 21 Ž.f a a 1 Ž.f a 2 q m ¤ y j m a1212Ž.Ž.f a a f a Ž. s5.3 j m y m q m q m Ž.a f2f 11122231faf Ž.a faf Ž.Ž.f fa 213 a a y m y j m faa221Ž.f 1 Ža 1f . a 2 s j q y m y m Ž.a f22fafŽ. 322111122faa Ž . f fafŽ. a q m y j m faŽ.21 a a 3 Ža 1f . a 2 Ž. s5.9 j y ª y ¤ m Ž.a f2221a f a f f q ª q ¤ y j m Ž.a111f a f a f a 2. ANALOG OF LIE BIALGEBRAS 529

Hence also

Ž. 4.3.2s ␶ s m j y ª y ¤ BDŽ A.Ž.f , a BDŽ A. Ž.a, f f1 Ža f 222a f a f . q m ª q ¤ y j a21Ž.a f a 1f a 1f .

Now suppose that A is balanced. Then so is AЈ by Proposition 4.3.3. Hence, since A and AЈ are ⑀-subbialgebras of DAŽ., and by naturality of B, s s ᭙ g g Ј BDŽ A.Ž.a, b BDŽ A. Ž.f , g 0 a, b A, f , g A .

On the other hand, by definition of I Ž.5.11 , Eqs. Ž. 5.20 and Ž. 5.21 imply that : m BDŽ A.Ž.a, f I DA Ž. and : m ᭙ g g Ј BDŽ A.Ž.Ž.f , a A I a A, f A .

Since A is balanced, I is an ideal of DAŽ., by Theorem 5.5. Together with the multiplicativity properties of B ŽŽ.Ž..Eqs. 4.1 and 4.2 , and since DA Ž.is generated as an algebra by A and AЈ, these facts ensure that ␣ ␤ g m q m ᭙␣ ␤ g BDŽ A.Ž, .I DA Ž.DA Ž. I , DAŽ., which in turn says that B ' 0, D bŽ A. i.e., DAbŽ.is balanced. Remark 5.11. In Proposition 4.6 we gave a canonical construction of a r ⑀ balanced quotient A IAbŽ.of an arbitrary -bialgebra A. The expression for the balanceator of the full double suggests a connection between r DAŽ.IDAbb ŽŽ..and DA Ž..

CLAIM. Let A be a balanced finite dimensional ⑀-bialgebra. If A is unital or counital, there is a surjection ¸ r DAbbŽ.DA Ž.IDAŽ. Ž..

Proof. Suppose that A is unital. Let F g DAŽ.* be given by FaŽ.s 0 sy ᭙ g g Ј sy and FfŽ.f Ž.1 a A, f A . Since f12Ž.1 f f Žby Ž 5.2 .. , we have that

Ž. m 5.20s j y ª y ¤ ᭙ g g Ј Ž.Ž.id F BDŽ A. a, f a f a f a f a A, f A . 530 MARCELO AGUIAR

: s r ¸ This shows that I IDAbbŽŽ..and hence DA Ž.DA Ž.I r DAŽ.IDAb ŽŽ... When A is counital, the same conclusion can be obtained similarly. r In general, there is no connection between DAŽ.IDAbb ŽŽ..and DA Ž.. Consider for instance the trivial case when both the multiplication and comultiplication of A are zero. FromŽ. 5.11 we see that I s A m AЈ : ' DAŽ.. On the other hand, it follows from Ž. 5.20 and Ž. 5.21 that BDŽ A. 0, s so IDAbŽŽ.. 0. For our final result, we must recall the construction of the Drinfeld double of a finite dimensional Lie bialgebra Žᒄ,,,wx␦ .. First, the dual Lie bialgebra ᒄ* has Lie bracket and cobracket defined by wxf , gxŽ.[ Žf m g .Ž.␦ x and ␦ s m wxs ᭙ g ᒄ Ž.f f12f iff fx, y fxfy1Ž. 2 Ž. x, y . The right coadjoint actions of ᒄ and ᒄ* on each other are defined by Ž.Ž.f · xy[ fxwx, y and gxŽ.· f [ wxf , gx Ž.. The Drinfeld double of ᒄ is the vector space DŽ.ᒄ s ᒄ [ ᒄ* with the following Lie bracket and cobracket: Ž.Ž.Ž.5.22 x, f , y, g [ Ž.wxx, y q x · g y y · f , yg · x q f · y q wxf , g and Ž.5.23 ␦ Ž.x, f [ ␦ Ž.Ž.x y ␦ f . ᒄ ᒄ Let Ä4eiiand Ä4f be dual bases of and *. It is well known that ᒄ m wx ŽŽ.D , Ýeiif .is a quasitriangular Lie bialgebra E-S, Sects. 4.1 and 4.2 . According to Proposition 4.10, if A [ Ž.A, m, ⌬ is a balanced ⑀-bialge- lie [ y ␶ ⌬ y ␶⌬ bra, then A Ž.A, m m , is a Lie bialgebra. Since DAb Ž.is also balanced, it has an associated Lie bialgebra. As our last result, we verify that this is, up to isomorphism, the Drinfeld double of the Lie bialgebra associated to A.

PROPOSITION 5.12. Let A be a finite dimensional ⑀-bialgebra. Then the map

lie ª lie ª y DAŽ.DAb Ž., Ž.Ža, f a, f . is an isomorphism of quasitriangular Lie bialgebras. Proof. We only need to check that the map preserves brackets and cobrackets. From the description of DAbŽ.following Theorem 5.5 it ANALOG OF LIE BIALGEBRAS 531

lie follows that theŽ. commutator bracket on DAb Ž.is Ž.Ž.a, f , b, g s Ž ab y ba q a ¤ g y g ª a q f ª b y b ¤ f , a ª g y g ¤ a q f ¤ b y b ª f q fg y gf . . Comparing definitions we find that ab y ba s wxa, b ;

Ž. y s5.1 y Žfg gf .Ž. a ga Ž.Ž.12 fa fa Ž.Ž. 1 ga 2 sy m ␦ sywx Žf g .Alie Ž.a f , gaAlie Ž. « y sywx fg gf f , g Alie ; Ž.Ž. ¤ y ª 5.10s , 5.4 y sywx faŽ g g a. Žgf fg .Ž. a g, faAlie Ž. « a ¤ g y g ª a sya · g and hence also f ª b y b ¤ f s b · f ;

Ž.Ž.5.10 , 5.4 Ž.a ª g y g ¤ abŽ.Žs gba.Žy gab.sygawx, b « a ª g y g ¤ a syg · a and hence also f ¤ b y b ª f s f · b.

lie Thus, the Lie bracket on DAbŽ. is Ž.Ž.a, f , b, g s Ž.wxa, b y a · g q b · f , yg · a q f · b y wxf , g . Under the map Ž.Ža, f ¬ a, yf ., this does correspond to the Lie bracket on DAŽ lie.Ž.given by Eq. 5.22 .

On the other hand, the comultiplication on DAbŽ.is ⌬ s ⌬ q ⌬ s ⌬ y ␶⌬ Ža, f .AA Ž.a Ј Ž.f AA Ž.a * Ž.f , lie hence the cobracket on DAbŽ. is ⌬lieŽ.a, f s ⌬ Ž.a, f y ␶⌬ Ž.a, f s ⌬ y ␶⌬ y ␶⌬ q ⌬ AAŽ.a * Ž.f AA Ž.a * Ž.f s ␦ q ␦ Alie Ž.a Ž A*.lie Ž.f . Under the map Ž.Ža, f ¬ a, yf ., this does correspond to the cobracket on DAŽ lie., Ž. ␦ 5.23s ␦ y ␦ Ž.a, f Alie Ž.a Ž A*.lie Ž.f . 532 MARCELO AGUIAR

lie y m Finally, recall that by Corollary 3.7, ŽŽ.DAbi, Ýe fi. is a quasitriangular Lie bialgebra. This obviously corresponds to the quasitriangular structure on DAŽ.lie under the isomorphism above. One may summarize the result of Proposition 5.12 by means of the diagram below, which commjutes up to isomorphismŽ. or up to sign

D 6 Ä4balanced ⑀-bialgebrasb Ä4 balanced ⑀-bialgebras

Ž.lie Ž.lie 6 6 Ä4Lie bialgebras 6 Ä4Lie bialgebras D

The change of sign is not really essential; it may be avoided by making a different choice of signs in the definition of DAŽ.Žand hence of DAb Ž...

ACKNOWLEDGMENTS

I thank Steve Chase and Andre´ Joyal for fruitful conversations.

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