Homotopy relative Rota-Baxter Lie algebras, triangular L ∞ -bialgebras and higher derived brackets
aa r X i v : . [ m a t h . QA ] J u l HOMOTOPY RELATIVE ROTA-BAXTER LIE ALGEBRAS, TRIANGULAR L ∞ -BIALGEBRAS AND HIGHER DERIVED BRACKETS ANDREY LAZAREV, YUNHE SHENG, AND RONG TANG
Abstract.
We describe L ∞ -algebras governing homotopy relative Rota-Baxter Lie algebrasand triangular L ∞ -bialgebras, and establish a map between them. Our formulas are based on afunctorial approach to Voronov’s higher derived brackets construction which is of independentinterest. Contents
1. Introduction 11.1. Higher derived brackets 11.2. Homotopy relative Rota-Baxter Lie algebras 21.3. Triangular L ∞ -bialgebras 21.4. Notation and conventions 22. L ∞ -algebras and their extensions 33. Voronov’s higher derived brackets and MC elements 84. Homotopy relative Rota-Baxter Lie algebras 115. Shifted Poisson algebras, r ∞ -matrices and triangular L ∞ -bialgebras 145.1. Doubling construction for shifted Poisson algebras 145.2. r ∞ -matrices and triangular L ∞ -bialgebras 146. From triangular L ∞ -bialgebras to homotopy relative Rota-Baxter Lie algebras 17References 191. Introduction
The subject of this paper is the study of two important algebraic structures: homotopyrelative Rota-Baxter (RB) Lie algebras and triangular L ∞ -bialgebras and their relationship.1.1. Higher derived brackets.
Given an inclusion of differential graded Lie algebras (dglas) i : g → L with a direct complement h (so that L ∼ = h ⊕ g ), a homotopy fiber of i is quasi-isomorphic to h [ − h . Under the additional assumption that the Liebracket on g restricted to h vanishes, T. Voronov constructed L ∞ -structures on the cocylinderand homotopy fiber of i in [35]. The higher products of these L ∞ -algebras are called higherderived brackets . This construction, subsequently generalized in [2, 7] proved to be extremelyuseful and showed up in a variety of situations such as the study of simultaneous deformationsof two compatible structures in [15, 16], quantization of coisotropic submanifolds of Poissonmanifolds [8] and many others. We present a functorial approach to higher derived bracketsand identify explicitly Maurer-Cartan (MC) elements in the corresponding L ∞ -algebras. This isour first collection of results that are subsequently applied to the two concrete cases of interest(homotopy relative Rota-Baxter Lie algebras and triangular L ∞ -bialgebras); it is clear thatthere are many other situations where they are relevant. Mathematics Subject Classification.
Key words and phrases.
Triangular L ∞ -bialgebras, homotopy relative Rota-Baxter Lie algebras, Maurer-Cartan elements, higher derived brackets. .2. Homotopy relative Rota-Baxter Lie algebras.
The concept of a Rota-Baxter (RB)operator was introduced by G. E. Baxter [5] with motivation from probability theory; it was laterput in an abstract context as an operator on an associative algebra, satisfying a certain identity[33]. Recently, it played an important role in the Connes-Kreimer’s study of renormalization inquantum field theory [12]. In the context of Lie algebras, RB operators are closely related withthe classical Yang-Baxter equation [21] and thus, with the study of integrable systems, see thebook [18] for more details.It is well-known that many homotopy invariant algebraic structures are themselves MC ele-ments in certain dglas; this point of view underlies the modern approach to algebraic deformationtheory in characteristic zero (cf. for example the survey [17] explaining this). In such a case wesay that an algebraic structure is governed by the corresponding dgla. In the previous work [25],the authors introduced the notion of a strong homotopy version of a RB Lie algebra, a so-calledhomotopy relative RB Lie algebra as an L ∞ -algebra together with an appropriate generalizationof a RB operator. In the present paper we explicitly find an L ∞ -algebra governing this algebraicstructure and, using our functorial approach to higher derived brackets, express the homotopyRB identities in a compact, ‘synthetic’ way.1.3. Triangular L ∞ -bialgebras. A triangular Lie bialgebra is a Lie bialgebra g whose co-bracket g → Λ g is the coboundary of an r -matrix, i.e. a skew-symmetric quadratic elementsatisfying the classical Yang-Baxter equation [ r, r ] = 0 in the Schouten Lie algebra Λ ∗ g of g .Thus, a triangular Lie bialgebra can be viewed as a pair ( g , r ) where g is a Lie algebra and r ∈ Λ g is an r -matrix. Triangular Lie bialgebras play an important role in deformation quan-tization [11, 31, 32]; in particular, it is known that they can be quantized [14], in the sense thattheir universal enveloping algebras admit formal deformations as triangular Hopf algebras, animportant notion that goes beyond the scope of this paper and will not be discussed here.Just as a Lie algebra has a strong homotopy generalization, called an L ∞ -algebra, Lie bial-gebras have a strong homotopy version called L ∞ -bialgebras, cf. [3, 20]. More recently, a stronghomotopy generalization of a triangular Lie bialgebra was introduced in [4], together with an ap-propriate infinity analogue of an r -matrix, the so-called r ∞ -matrix. A triangular L ∞ -bialgebracan then be defined as a pair ( g , r ) where g is an L ∞ -algebra and r is a (suitably defined) r ∞ -matrix. One of the motivations behind studying triangular L ∞ -bialgebras is the problem ofquantizing them and, in particular, making sense of the notion of a quantum r ∞ -matrix (or, inother words, A ∞ quantum Yang-Baxter equation).It is well-known that L ∞ -bialgebras with a fixed underlying graded vector space are governedby a certain dgla. This point of view, for example, allows interpreting deformation quantizationof Lie bialgebras as an L ∞ -quasi-isomorphism between dglas (or L ∞ -algebras) governing L ∞ -bialgebras and strong homotopy associative bialgebras, cf. [31] regarding this approach. Bycontrast, triangular L ∞ -bialgebras are not governed by any dgla but rather, by a certain L ∞ -algebra that we explicitly identify. This suggests that the perspective of [31] may be applicableto quantization of triangular L ∞ -bialgebras.Finally, we show that a triangular L ∞ -bialgebra gives rise to an associated homotopy relativeRB Lie algebra; this correspondence comes from a map between the L ∞ -algebras governingthe corresponding structures. This substantially strengthens a result in the authors’ previouspaper [25] where it was proved in the ungraded case and only on the level of cohomology. It isinteresting to observe that, while the construction of homotopy RB Lie algebras carries over eas-ily to the associative (or A ∞ ) context, the corresponding analogue of triangular L ∞ -bialgebras(that we can putatively call triangular A ∞ -Hopf algebras ) is rather less straightforward to con-struct. We hope to return to this problem in a future work.1.4. Notation and conventions.
We work in the category DGVect of differential graded (dg)vector spaces over a field k of characteristic zero; the grading is always cohomological. The n-fold suspension of a graded vector space g is defined by the convention g [ n ] i = g i + n ; thedifferential is therefore a map d : g → g [1]. Given an element x ∈ g k , the corresponding elementin g [ n ] k − n will be denoted by x [ n ]. There is an isomorphism ( g [ n ]) ∗ ∼ = g ∗ [ − n ]. he category DGVect is symmetric monoidal, and monoids (respectively commutative monoids)in it are called dg algebras (dgas) and commutative dg algebras (cdgas). We also need to workwith pseudocompact dg vector spaces, or projective limits of finite-dimensional vector spaces;thus a pseudocompact dg vector space V can be written as V = lim ←− α V α for a projective system { V α } of finite dimensional dg vector spaces. The category of pseudocompact dg vector spaces isequivalent to the opposite category of DGVect with anti-equivalence established by the k -linearduality functor. This category also admits a symmetric monoidal structure which we will denotesimply by ⊗ and monoids (respectively commutative monoids) in it are called pseudocompactdgas (respectively pseudocompact cdgas). We will call local pseudocompact dgas (or cdgas) complete , another name for a complete (c)dga is a local pro-Artinian (c)dga. Occasionally weneed to consider the tensor product of a pseudocompact dg vector space V = lim ←− α V α and a dis-crete one U ; in this situation we will always write V ⊗ U for lim ←− α V α ⊗ U ; such a tensor product isin general neither discrete nor pseudocompact. The category of complete cdgas and continuousmultiplicative maps will be denoted by CDGA ∧ k . The maximal ideal of a complete cdga A willbe denoted by A ≥ ; thus A = k ⊕ A ≥ ; there is a filtration A =: A ≥ ⊃ A ≥ ⊃ . . . ⊃ A ≥ n ⊃ . . . where A ≥ n stands for the n -th power of the maximal ideal.We will also need the notion of a dgla; this is a dg vector space ( g , d ) with an anti-symmetricproduct [ − , − ] : g ⊗ g → g satisfying the graded Jacobi identity. A Maurer-Cartan (MC) elementin a dgla g is an element x ∈ g such that dx + [ x, x ] = 0; the set of MC elements in a dgla g will be denoted by MC( g ).For a graded Lie algebra g and x ∈ g we denote by ad x the right adjoint action of x definedby ad x y = [ y, x ] for y ∈ g .2. L ∞ -algebras and their extensions Let g be a graded vector space. We denote by Der ˆ S g ∗ [ −
1] the graded Lie algebra consistingof continuous derivations of the complete symmetric algebra ˆ S g ∗ [ −
1] and by Der ˆ S g ∗ [ −
1] itsgraded Lie subalgebra of derivations vanishing at zero. We will briefly recall the definition ofan L ∞ -algebra following e.g. [24, 26]. See also [22, 23] for more details and a different point ofview. Definition 2.1. An L ∞ -algebra structure on g is a continuous degree 1 derivation m of thecomplete cdga ˆ S g ∗ [ − m ◦ m = 0 and m has no constant term. The pair ( g , m ) iscalled an L ∞ -algebra, and ( ˆ S g ∗ [ − , m ) is called its representing complete cdga. Sometimes wewill refer to g as an L ∞ -algebra leaving m understood. Remark 2.2.
Thus, an L ∞ -algebra structure is an MC element in the graded Lie algebraDer ˆ S g ∗ [ − S g ∗ [ −
1] is taken in place of Der ˆ S g ∗ [ −
1] we getthe definition of a curved L ∞ -algebra. Many of our results hold, with appropriate modifications,for curved L ∞ -algebras. Remark 2.3.
Let ( g , m ) be an L ∞ -algebra. The element m can be written as a sum m = m + · · · + m n + · · · where m n is the order n part of m so we can write m n : g ∗ [ − → ˆ S n g ∗ [ − m n , thus we have the degree 1 map ˇ m n : S n g [1] → g [1] for n = 1 , , · · · . Namely an L ∞ -algebra on a graded vector space g is a sequence of linear maps of degree 1:ˇ m n : S n g [1] → g [1] , n ≥ x , · · · , x n ∈ g [1]: n X i =1 X σ ∈ S ( i,n − i ) ε ( σ ; x , · · · , x n ) ˇ m n − i +1 ( ˇ m i ( x σ (1) , · · · , x σ ( i ) ) , x σ ( i +1) , · · · , x σ ( n ) ) = 0 . (2.1) efinition 2.4. Let ( g , m ) and ( h , m ′ ) be two L ∞ -algebras. An L ∞ -map f from g to h is,by definition, a continuous map of degree 0 between the corresponding representing completecdgas so that f : ˆ S h ∗ [ − → ˆ S g ∗ [ − Remark 2.5. An L ∞ -map f : g → h can be represented as f = f + · · · + f n + · · · where f n is the order n part of f so that f n : h ∗ [ − → ˆ S n g ∗ [ − f n = 0 for n = 1 then f is called a strict L ∞ -map . Furthermore, dualizing, an L ∞ -map f : g → h is a sequence of linear maps ofdegree 0: ˇ f n : S n g [1] → h [1] , n ≥ x , · · · , x n ∈ g [1]: n X i =1 X σ ∈ S ( i,n − i ) ε ( σ ; x , · · · , x n ) ˇ f n − i +1 ( ˇ m i ( x σ (1) , · · · , x σ ( i ) ) , x σ ( i +1) , · · · , x σ ( n ) )= X k + ··· + k j = n X σ ∈ S ( k , ··· ,kj ) ε ( σ ; x , · · · , x n )1 j ! ˇ m ′ j ( ˇ f k ( x σ (1) , · · · , x σ ( k ) ) , · · · , ˇ f k j ( x σ ( k + ··· + k j − +1) , · · · , x σ ( n ) )) . Definition 2.6.
Let ( g , m ) be an L ∞ -algebra. Then an element ξ ∈ ( g [1]) is an MC element if it satisfies the MC equation ∞ X i =1 i ! ˇ m i ( ξ, · · · , ξ ) = 0 . (2.2)The set of MC elements in an L ∞ -algebra ( g , m ) will be denoted by MC( g ). Remark 2.7.
The definition of an MC element in an L ∞ -algebra assumes that the left handside of (2.2) converges. If this is the case for all ξ ∈ g , we will say that g contains MC elements.For example, if m n = 0 for n > g is essentially a dgla) then g contains MC elements.We now formulate some other sufficient conditions for an L ∞ -algebra to contain MC elements. Definition 2.8.
Let g be an L ∞ -algebra and F • g be a descending filtration of the graded vectorspace g such that g = F g ⊃ · · · ⊃ F n g ⊃ · · · and g is complete with respect to this filtration,i.e. there is an isomorphism of graded vector spaces g ∼ = lim ←− g / F n g .(1) If for all k, n , . . . , n k ≥ m k ( F n g [1] , · · · , F n k g [1]) ⊆ F n + ··· + n k g [1] , we say that the pair ( g , F • g ) is a filtered L ∞ -algebra.(2) If there exists l ≥ k > l it holds thatˇ m k ( g [1] , · · · , g [1]) ⊆ F k g [1] , we say that the pair ( g , F • g ) is a weakly filtered L ∞ -algebra. Remark 2.9.
The definition of a filtered L ∞ -algebra belongs to Dolgushev and Rogers [13].It is the weak notion that is relevant to our immediate purposes while Dolgushev-Rogers’snotion is given for comparison. Taking in the definition of a filtered L ∞ -algebra l = 0 and n = n = · · · = n k = 1, we obtain the condition of being weakly filtered, i.e. one notionis indeed stronger than the other. Furthermore, when g is finite-dimensional, it is easy to seethat the condition of being weakly filtered is equivalent to requiring that the differential m onthe representing complete cdga ˆ S g ∗ [ −
1] of g restricts to the (uncompleted) symmetric algebra S g ∗ [ −
1] whereas the stronger Dolgushev-Rogers condition means that the cdga ( S g ∗ [ − , m ) iscofibrant in the model category of cdgas.Furthermore, it is clear that a weakly filtered L ∞ -algebra contains MC elements. iven a complete cdga A and an L ∞ -algebra ( g , m ), consider the tensor product A ≥ ⊗ g andextend the L ∞ -structure maps ˇ m n : g [1] ⊗ n → g [1] by A -linearity to mapsˇ m An : ( A ≥ ⊗ g [1]) ⊗ n → A ≥ ⊗ g [1]so thatˇ m An ( a ⊗ x , · · · , a n ⊗ x n ) = ( d A ( a ) ⊗ x + ( − | a | a ⊗ ˇ m ( x ) , n = 1 , ( − P ni =1 | a i | ( | x | + ··· + | x i − | +1) ( a · · · a n ) ⊗ ˇ m n ( x , · · · , x n ) , n ≥ , where a , · · · , a n ∈ A ≥ and x , · · · , x n ∈ g [1]. With this, A ≥ ⊗ g becomes an L ∞ -algebra. Proposition 2.10.
Given an L ∞ -algebra g and a complete cdga A , the L ∞ -algebra A ≥ ⊗ g isfiltered.Proof. We define a filtration on A ≥ ⊗ g as follows: F n ( A ≥ ⊗ g ) = A ≥ n ⊗ g . The corresponding conditions of Definition 2.8 are trivial to check. (cid:3)
Given a complete cdga A and an L ∞ -map f : g → h with components ˇ f n , there is an L ∞ -map f A : A ≥ ⊗ g → A ≥ ⊗ h with components ˇ f An defined by the formulas:ˇ f An ( a ⊗ x , · · · , a n ⊗ x n ) = ( − P ni =1 | a i | ( | x | + ··· + | x i − | ) ( a · · · a n ) ⊗ ˇ f n ( x , · · · , x n ) , where a , · · · , a n ∈ A ≥ and x , · · · , x n ∈ g [1]. Proposition 2.11.
Let ( g , m ) be an L ∞ -algebra and f : A → B be a morphism of completecdgas. Then f ⊗ Id g : A ≥ ⊗ g → B ≥ ⊗ g is a strict L ∞ -map.Proof. Since f is a morphism of complete cdgas, it follows that f ( A ≥ ) ⊂ B ≥ . For any a ⊗ x , · · · , a n ⊗ x n ∈ A ≥ ⊗ g [1], we have( f ⊗ Id g )( ˇ m An ( a ⊗ x , · · · , a n ⊗ x n ))= ( f ( d A ( a )) ⊗ x + ( − | a | f ( a ) ⊗ ˇ m ( x ) , n = 1 , ( − P ni =1 | a i | ( | x | + ··· + | x i − | +1) f ( a · · · a n ) ⊗ ˇ m n ( x , · · · , x n ) , n ≥ , = ˇ m Bn (cid:0) ( f ⊗ Id g )( a ⊗ x ) , · · · , ( f ⊗ Id g )( a n ⊗ x n ) (cid:1) . Thus, we obtain that f ⊗ Id g is a strict L ∞ -map. (cid:3) Moreover, given an L ∞ -algebra ( g , m ), there is a set-valued functor MC g on the categoryCDGA ∧ k , which is defined on the set of objects and on the set of morphisms respectively by:MC g ( A ) = MC( g , A ) , (2.3) MC g ( A f → B ) = MC( g , A ) f ⊗ Id g → MC( g , B ) , (2.4)for A, B ∈ CDGA ∧ k and f ∈ Hom
CDGA ∧ k ( A, B ). Since f ⊗ Id g is a strict L ∞ -map, we concludethat f ⊗ Id g takes elements in MC( g , A ) to elements in MC( g , B ). So the functor MC g iswell-defined. Moreover, it is representable. Theorem 2.12.
Let ( g , m ) be an L ∞ -algebra. Then the functor MC g is represented by thecomplete cdga ( ˆ S g ∗ [ − , m ) . In other words, for any complete cdga A there is an isomorphism MC g ( A ) ∼ = Hom CDGA ∧ (cid:16) ( ˆ S g ∗ [ − , m ) , A (cid:17) , functorial in A .Proof. See, e.g. [9, Proposition 2.2 (1)] where this result is proved in the Z / Z -graded case. (cid:3) y the Yoneda embedding theorem, the functor MC g determines the L ∞ -algebra g up toa canonical L ∞ -isomorphism. Conversely, given a functor F on CDGA ∧ k , we will often beinterested in whether it is isomorphic to MC g for a suitable L ∞ -algebra ( g , m ).Furthermore, for a fixed complete cdga A , the correspondence g MC( g , A ) is functorialwith respect to L ∞ -maps. Namely, the following result holds. Proposition 2.13.
Let f : g → h with components ( ˇ f , · · · , ˇ f n , · · · ) . Then for any completecdga A it induces a map f ∗ : MC( g , A ) → MC( h , A ) according to the formula (2.5) ξ f ∗ ( ξ ) = ∞ X k =1 k ! ˇ f Ak ( ξ, · · · , ξ ) , ∀ ξ ∈ MC( g , A ) . Proof.
Since f : ˆ S h ∗ [ − → ˆ S g ∗ [ −
1] is a map of complete cdgas, it clearly induces a map of setsMC( g , A ) ∼ = Hom CDGA ∧ ( ˆ S g ∗ [ − , A ) → Hom
CDGA ∧ ( ˆ S h ∗ [ − , A ) ∼ = MC( h , A ) , and a straightforward inspection shows that it is given in components by the stated formula. (cid:3) Remark 2.14.
Proposition 2.13 is well-known in formal deformation theory. It was formulatedexplicitly in [19, Section 4.2] for dglas and in [30, Section 2.5.5] in general.
Remark 2.15.
All told, the set MC( − , − ) can be viewed as a functor of two arguments. It isnatural with respect to L ∞ -maps in the first argument and maps of complete cdgas in the secondargument. In order to determine an L ∞ -map g → h it suffices to specify, for any complete cdga A , a map MC( g , A ) → MC( h , A ), functorial in A (by Yoneda’s lemma). Moreover, it is clearthat if f = ( ˇ f , · · · , ˇ f n , · · · ) is such that for any ξ ∈ MC( g , A ) the element f ∗ ( ξ ) ∈ MC( h , A )given by formula (2.5) is an MC element, then f is an L ∞ -map. This can sometimes be usedfor explicit constructions of L ∞ -maps out of MC elements.Recall that given a Lie algebra g , its representation in a vector space V is a Lie algebra mapfrom g to gl ( V ). This generalizes in a straightforward way to the L ∞ -case. Definition 2.16.
Let ( g , m ) be an L ∞ -algebra with the representing cdga ( ˆ S g ∗ [ − , m ) and V be a graded vector space. Then a representation of g in V is an L ∞ -map f from g to gl ( V ),where gl ( V ) is the graded Lie algebra of endomorphisms of V . Remark 2.17.
Let ρ be a representation of an L ∞ -algebra ( g , m ) in a graded vector space V .By Definition 2.4, we deduce that ρ ∈ Hom
CDGA ∧ k ( ˆ S gl ( V ) ∗ [ − , ˆ S g ∗ [ − ρ can be viewed as an MC element of the dgla ˆ S ≥ g ∗ [ − ⊗ gl ( V ).Given a complete cdga A , we will need the notion of an A -linear L ∞ -algebra; this notion,with a slight modification, was used in [9]. Definition 2.18.
Let A be a complete cdga and g be a graded vector space. Then an A -linear L ∞ -algebra structure on A ⊗ g is an MC element of the dgla A ≥ ⊗ Der ˆ S g ∗ [ − B is anothercomplete cdga and A → B is a map, then an A -linear L ∞ -algebra on A ⊗ g obviously determinesa B -linear L ∞ -algebra on B ⊗ g that will be referred to as obtained from A ⊗ g by change ofscalars. Remark 2.19.
Note that an A -linear L ∞ -algebra structure on A ⊗ g is a deformation of thetrivial L ∞ -algebra structure on g with a dg base A . Alternatively, we could have called an A -linear L ∞ -algebra structure on A ⊗ g an MC element of A ⊗ Der ˆ S g ∗ [ − not a generalization of an ordinary L ∞ -algebra over k (because k ≥ = 0).We now have the following result. Proposition 2.20.
Let g be a graded vector space and F HL be the functor associating to acomplete cdga A the set of A -linear L ∞ -algebra structures on A ⊗ g . Then F HL is representedby the complete cdga ˆ S (cid:0) Der ˆ S g ∗ [ − (cid:1) ∗ [ − (cid:1) . roof. By Definition 2.18, we deduce that F HL ( A ) = MC(Der ˆ S g ∗ [ − , A ). Moreover, by Theo-rem 2.12, it follows that F HL ( A ) ∼ = Hom CDGA ∧ k ( ˆ S (cid:0)(cid:0) Der ˆ S g ∗ [ − (cid:1) ∗ [ − (cid:1) , A ) and we are done. (cid:3) Corollary 2.21.
Given a graded vector space g , there exists a ‘universal’ ˆ S (cid:0)(cid:0) Der ˆ S g ∗ [ − (cid:1) ∗ [ − (cid:1) -linear L ∞ -algebra structure on ˆ S (cid:0)(cid:0) Der ˆ S g ∗ [ − (cid:1) ∗ [ − (cid:1) ⊗ g such that any other A -linear L ∞ -structure on A ⊗ g is obtained by change of scalars from a unique map of complete cdgas ˆ S (cid:0)(cid:0) Der ˆ S g ∗ [ − (cid:1) ∗ [ − (cid:1) → A .Proof. By Proposition 2.20, there is a natural isomorphism α as following: α : Hom CDGA ∧ k ( ˆ S (cid:0)(cid:0) Der ˆ S g ∗ [ − (cid:1) ∗ [ − (cid:1) , · ) → F HL . By the universality of the representing cdga, we deduce that for any complete cdga A and any y ∈ F HL ( A ), there exists a unique map of complete cdgas f : ˆ S (cid:0)(cid:0) Der ˆ S g ∗ [ − (cid:1) ∗ [ − (cid:1) → A suchthat y = F HL ( f )( X ). Here X ∈ F HL ( ˆ S (cid:0)(cid:0) Der ˆ S g ∗ [ − (cid:1) ∗ [ − (cid:1) ) is given by X = α (cid:0) ˆ S (cid:0)(cid:0) Der ˆ S g ∗ [ − (cid:1) ∗ [ − (cid:1)(cid:1) (Id) . This completes the proof. (cid:3)
Remark 2.22.
The universal L ∞ -algebra ˆ S (cid:0)(cid:0) Der ˆ S g ∗ [ − (cid:1) ∗ [ − (cid:1) ⊗ g can be viewed as a uni-versal deformation of the trivial L ∞ -algebra on g ; its universal properties persist upon passingto the homotopy category of complete cdgas cf. [17, 26] regarding this approach to deformationtheory. In the present paper we will not be concerned with this aspect of the theory.Suppose that we have an A -linear L ∞ -algebra structure m A on A ⊗ g , where A = ( ˆ SU ∗ [ − , m U )is itself the representing complete cdga of an L ∞ -algebra U . The element m A is an A -linearderivation of A ⊗ ˆ S g ∗ [ −
1] = ˆ SU ∗ [ − ⊗ ˆ S g ∗ [ − ∼ = ˆ S ( U ⊕ g ) ∗ [ − . Forgetting that m A is A -linear, we can view it as an MC element in Der ˆ S ( U ⊕ g ) ∗ [ − L ∞ -structure on U ⊕ g with the representing cdga ( ˆ S ( U ⊕ g ) ∗ [ − , m A ). Moreover, A =( ˆ SU ∗ [ − , m U ) is a sub-cdga of ( ˆ S ( U ⊕ g ) ∗ [ − , m A ) and we can form an L ∞ -structure on g with the representing cdga ˆ S g ∗ [ − L ∞ -algebras and strict L ∞ -maps:(2.6) U → U ⊕ g → g . This leads naturally to the notion of an extension of L ∞ -algebras, cf. [10, 26, 28]. Definition 2.23.
The sequence of L ∞ -algebras and strict L ∞ -maps of the form (2.6) is calledan extension of g by U . Example 2.24.
The universal ˆ S (cid:0)(cid:0) Der ˆ S g ∗ [ − (cid:1) ∗ [ − (cid:1) -linear L ∞ -algebraˆ S (cid:0)(cid:0) Der ˆ S g ∗ [ − (cid:1) ∗ [ − (cid:1) ⊗ g gives rise to an L ∞ -extension g → Der ˆ S g ∗ [ − ⊕ g → Der ˆ S g ∗ [ − . Here g is given the trivial L ∞ -structure and the L ∞ -structure on Der ˆ S g ∗ [ − ⊕ g can be readoff the ˆ S (cid:0)(cid:0) Der ˆ S g ∗ [ − (cid:1) ∗ [ − (cid:1) -linear L ∞ -structure on ˆ S (cid:0)(cid:0) Der ˆ S g ∗ [ − (cid:1) ∗ [ − (cid:1) ⊗ g . Specifically(cf. [10, Example 3.8]), for φ [1] ∈ Hom( g ⊗ n , g ) ⊂ Der ˆ S g ∗ [ −
1] and v , · · · , v n ∈ g we haveˇ m n ( φ [1] , v , · · · , v n ) = φ ( v , · · · , v n ) . Note that the L ∞ -algebra Der ˆ S g ∗ [ − ⊕ g represents the functor associating to a completecdga A an A -linear L ∞ -algebra on A ⊗ g together with an MC element in it. Later on, we willconsider a higher version of this construction with an MC element replaced with the so-called r ∞ -matrix , cf. Definition 5.7 below associated to an L ∞ -algebra and see how that leads totriangular L ∞ -bialgebras. ur next task is to describe a dgla controlling the pair of an L ∞ -algebra and its representationin a graded vector space. We arrange the set of such pairs as a functor on complete cdgas. Definition 2.25. An L ∞ Rep pair consists of an L ∞ -algebra ( g , m ) and a representation ρ : g −→ gl ( V ) of g in a graded vector space V .There is a natural action of the graded Lie algebra Der ˆ S g ∗ [ −
1] on ˆ S ≥ g ∗ [ − ⊗ gl ( V ) givenby [ φ, x ⊗ y ] = φ ( x ) ⊗ y, ∀ φ ∈ Der ˆ S g ∗ [ − , x ⊗ y ∈ ˆ S ≥ g ∗ [ − ⊗ gl ( V ) . Let L L ∞ Rep ( g , V ) = Der ˆ S g ∗ [ − ⋉ (cid:0) ˆ S ≥ g ∗ [ − ⊗ gl ( V ) (cid:1) be the corresponding semidirect productgraded Lie algebra. Note that an L ∞ Rep pair (( g , m ) , ρ ) is nothing but an MC element in thegraded Lie algebra L L ∞ Rep ( g , V ). Definition 2.26.
Let A be a complete cdga. Then an A -linear L ∞ Rep pair with the underlyinggraded vector spaces g and V is an element in MC( L L ∞ Rep ( g , V ) , A ).Let F L ∞ Rep be the functor associating to a complete cdga A the set of A -linear L ∞ Rep pairswith the underlying graded vector spaces g and V . Then we have the following result. Proposition 2.27.
The functor F L ∞ Rep is represented by the complete cdga ˆ S L ∗ L ∞ Rep ( g , V )[ − .Proof. Let ( m, x ⊗ y ) be a degree 1 element in L L ∞ Rep ( g , V ). We have[( m, x ⊗ y ) , ( m, x ⊗ y )] = ([ m, m ] , m ( x ) ⊗ y + x ⊗ [ y, y ]) . Thus, ( m, x ⊗ y ) is an MC element of L L ∞ Rep ( g , V ) if and only if[ m, m ] = 0 , m ( x ) ⊗ y + 12 x ⊗ [ y, y ] = 0 . By Remark 2.2 and Remark 2.17, we deduce that ( m, x ⊗ y ) is an MC element of L L ∞ Rep ( g , V )if and only if ( g , m ) is an L ∞ -algebra and x ⊗ y is a representation of the L ∞ -algebra ( g , m ) ina graded vector space V . Thus, we obtain that F L ∞ Rep ( A ) = MC( L L ∞ Rep ( g , V ) , A ). Moreover,by Theorem 2.12, F L ∞ Rep is represented by the complete cdga ˆ S L ∗ L ∞ Rep ( g , V )[ − (cid:3) Remark 2.28.
There is an inclusion of graded Lie algebras i : L L ∞ Rep ( g , V ) ⊂ Der ˆ S ( g ⊕ V ) ∗ [ −
1] where Der ˆ S g ∗ [ − ⊂ Der ˆ S ( g ⊕ V ) ∗ [ −
1] in an obvious way and ˆ S ≥ g ∗ [ − ⊗ gl ( V ) ⊂ Der ˆ S ( g ⊕ V ) ∗ [ −
1] via the isomorphism ˆ S n g ∗ [ − ⊗ gl ( V ) ∼ = Hom( V ∗ [ − , ˆ S n g ∗ [ − ⊗ V ∗ [ − L ∞ -algebra on g together with a representation of g ina graded vector space V is equivalent to an MC element ( m, x ⊗ y ) ∈ L L ∞ Rep ( g , V ) and thus, i ( m, x ⊗ y ) ∈ MC (cid:0) Der ˆ S ( g ⊕ V ) ∗ [ − (cid:1) . The graded Lie algebra Der ˆ S ( g ⊕ V ) ∗ [ −
1] supplied withthe differential d = [ i ( m, x ⊗ y ) , · ] can be identified with the Chevalley-Eilenberg complex of the L ∞ -algebra ( g , m ) with coefficients in the representation V .3. Voronov’s higher derived brackets and MC elements
In this section we review Voronov’s constructions [35] of higher derived brackets from thepoint of view of MC elements and L ∞ -extensions. Related results are contained in [15]. Definition 3.1.
Let L be a dgla, x ∈ MC( L ) and h ∈ L . The right gauge transformation by h on x is given by the following formula: x x ∗ h := x + ∞ X n =1 n ! (ad nh ( x ) + ad n − h ( d ( h ))) . Remark 3.2.
In the above definition it is assumed that the e ad h := P ∞ n =0 (ad h ) n n ! is a well-defined operator on L . This is the case, e.g. when L is a pronilpotent dgla. In that case it iseasy to see that this is a well-defined action (i.e. x ∗ h ∈ MC( L )); moreover x ∗ h agrees withthe ordinary (left) gauge action on x by the element − h . efinition 3.3. We say that a dgla L is supplied with a V -structure if there is given an operator P : L → L with P = P (so that P is a projector) such that(1) The subspace ker P is a sub-dgla of L ,(2) The image of P (denoted hereafter by h ) is an abelian graded Lie subalgebra of L .From now on, we will denote a V -structure by a pair ( L, P ). If, for a given V -structure, thereis the following filtration on L : L ⊃ P [ L, h ] ⊃ P [[ L, h ] , h ] ⊃ · · · which is complete (e.g. if the adjoint action of h on L is pronilpotent), the corresponding V -structure is called admissible . Remark 3.4.
Note that for an admissible V -structure on L , the operator e ad h : L → L makessense for any h ∈ h .Associated to an admissible V -structure on a dgla L is the notion of a VMC functor. Definition 3.5.
Let (
L, P ) be an admissible V -structure. A VMC element associated to it isa pair ( x, h ) where x ∈ MC( L ) and h ∈ h such that P ( x ∗ h ) = 0. The set of VMC elementsassociated to ( L, P ) will be denoted by VMC( L ) (leaving P understood).Just as the ordinary MC set in a dgla, the VMC set can be made into a functor of twoarguments. Let L be a dgla with a V -structure and A be a complete cdga. Then A ≥ ⊗ L ispronilpotent and has an induced admissible V -structure given by the projector id ⊗ P . Definition 3.6.
The VMC set of L with values in A is defined as VMC( A ≥ ⊗ L ) and denotedby VMC( L, A ).It is clear that VMC( − , − ) is a functor in the second variable. We will show that it isrepresented by a complete cdga that is the representing cdga of a certain L ∞ -algebra. Recallthe following result by Voronov [35]. Theorem 3.7.
Let ( L, P ) be a V -structure. Then the graded vector space L ⊕ h [ − is an L ∞ -algebra where ˇ m ( x [1] , h ) = ( − d ( x )[1] , P ( x + d ( h ))) , ˇ m ( x [1] , y [1]) = ( − | x | [ x, y ][1] , ˇ m k ( x [1] , h , h , · · · , h k − ) = P [ · · · [[ x, h ] , h ] · · · , h k − ] , k ≥ , ˇ m k ( h , h , · · · , h k ) = P [ · · · [ d ( h ) , h ] · · · , h k ] , k ≥ . Here h, h , · · · , h k are homogeneous elements of h and x, y are homogeneous elements of L . Allthe other L ∞ -algebra products that are not obtained from the ones written above by permutationsof arguments, will vanish. (cid:3) Remark 3.8.
The L ∞ -products on L ⊕ h [ −
1] restrict to h [ −
1] making the latter into an L ∞ -algebra given byˇ m k ( h , · · · , h k ) = P [ · · · [ d ( h ) , h ] · · · , h k ] , for homogeneous h , · · · , h k ∈ h . (3.1)It is included into an L ∞ -extension(3.2) h [ − → h [ − ⊕ L → L, where the second arrow is the natural projection and L is viewed as a dgla (hence an L ∞ -algebra).For later use we record the following obvious observation. Remark 3.9.
Let L ′ be a graded Lie subalgebra of L that satisfies d ( L ′ ) ⊂ L ′ . Then L ′ ⊕ h [ − L ∞ -subalgebra of the above L ∞ -algebra ( L ⊕ h [ − , { ˇ m k } ∞ k =1 ).The following key lemma interprets a VMC element of an admissible V-structure as an MCelement. emma 3.10. Let L be a dgla with an admissible V-structure. Then (1) The L ∞ -algebra L ⊕ h [ − is weakly filtered (so it contains MC elements). (2) The following isomorphism of sets holds:
MC( L ⊕ h [ − ∼ = VMC( L ) . Proof.
For (1) consider the following filtration on L ⊕ h [ − F := L ⊕ h [ − ⊃ F := P ([ L, h ])[ − ⊃ F := P ([[ L, h ] , h ])[ − ⊃ · · · . By the definition of an admissible V -structure, the above filtration is complete; moreover itclearly satisfies condition (2) of Definition (2.8) with l = 3.For (2), let ( x [1] , h ) ∈ MC( L ⊕ h [ − x ∈ L and h ∈ h . We have ∞ X n =1 n ! ˇ m n (cid:16) ( x [1] , h ) , . . . , ( x [1] , h ) (cid:17) = ˇ m ( x [1] , h ) + 12 ˇ m (cid:16) ( x [1] , h ) , ( x [1] , h ) (cid:17) + ∞ X n =3 n ! ˇ m n (cid:16) ( x [1] , h ) , . . . , ( x [1] , h ) (cid:17) = (cid:16) − d ( x )[1] , P ( x + d ( h )) (cid:17) + (cid:16) −
12 [ x, x ][1] , P ad h ( x ) + 12 P ad h ( d ( h )) (cid:17) + (cid:16) , P ∞ X n =3 n − n − h ( x ) (cid:17) + (cid:16) , P ∞ X n =3 n ! ad n − h ( d ( h )) (cid:17) from which it follows that d ( x ) + 12 [ x, x ] = 0 ,P ( x ) + ∞ X n =1 n ! P (cid:16) (ad h ) n ( x ) + ad n − h ( dh ) (cid:17) = 0 . Therefore, x ∈ MC( L ) and P ( x ∗ h ) = 0. All told, we obtain that ( x, h ) ∈ VMC( L ). The samecalculation performed in the reverse order, shows that, conversely, if ( x, h ) ∈ VMC( L ), then( x [1] , h ) ∈ MC( L ⊕ h [ − (cid:3) Furthermore, the following result holds.
Proposition 3.11.
Let L be a dgla with a V-structure. Then the functor VMC( L, − ) is repre-sentable. The complete cdga representing it is the representing cdga of the L ∞ -algebra L ⊕ h [ − constructed in Theorem 3.7.Proof. Let A be a complete cdga. Then ( A ≥ ⊗ L, id ⊗ P ) is a pronilpotent dgla with a V-structure (which is then admissible) and by Lemma 3.10 we have:VMC( L, A ) ∼ = VMC( A ≥ ⊗ L ) ∼ = MC( A ≥ ⊗ ( L ⊕ h [ − ∼ = MC( L ⊕ h [ − , A )as required. (cid:3) It is clear that the L ∞ -algebra L ⊕ h [ −
1] defined above, is quasi-isomorphic to the dgla ker P and thus, there is an L ∞ -map j : L ⊕ h [ − → ker P that is homotopy inverse to the inclusionker P ֒ → L ֒ → L ⊕ h [ − L ∞ -algebras and maps:(3.3) h [ − → ker P → L. Denote by i : h [ − → ker P the corresponding L ∞ -map in the above homotopy fibre sequence. Ifthe given V -structure is admissible, there is an induced map MC( i ) : MC( h [ − → MC(ker P ).We will find this map explicitly. Proposition 3.12.
Let L be a pronilpotent dgla with a V-structure. Then: The L ∞ -map j : L ⊕ h [ − → ker P induces the map MC( j ) : MC( L ⊕ h [ − → MC(ker P ) , so that for ( x [1] , h ) ∈ MC( L ⊕ h [ − it holds that MC( j )( x [1] , h ) = x ∗ h. (2) The L ∞ -map i : h [ − → ker P induces the map MC( i ) : MC( h [ − → MC(ker P ) , so that for h ∈ MC( h [ − it holds that MC( i )( h ) = 0 ∗ h := ∞ X n =1 n ! ad n − h d ( h ) . Proof.
It is clear that (2) follows from (1). For (1) let ( x [1] , h ) ∈ MC( L ⊕ h [ − x ∈ L and h ∈ h . Then the element x ∗ h is an MC element in L (gaugeequivalent to x ) and it belongs to ker P by definition of a VMC element. To finish the proof itsuffices to observe that this morphism splits the canonical map MC(ker P ) → MC( L ⊕ h [ − P ֒ → L ⊕ h [ − (cid:3) Corollary 3.13.
Let L be as in Proposition 3.12. Then: (1) The L ∞ -map j : L ⊕ h [ − → ker P has the form: ˇ j ( x [1] , h ) =(id − P )( x )[1] , ˇ j k ( x [1] , h , · · · , h k − ) =(id − P )[ · · · [[ x, h ] , h ] · · · , h k − ][1] , ˇ j k ( h , . . . , h k ) =(id − P )[ · · · [ d ( h ) , h ] · · · , h k ][1] , k ≥ . (2) The L ∞ -map i : h [ − → ker P from (3.3) has the following form: ˇ i k ( h , . . . , h k ) =(id − P )[ · · · [ d ( h ) , h ] · · · , h k ][1] , k ≥ . Proof.
It suffices to show that the map f := ( f , . . . , f k . . . ) as defined above induces the correctmap on MC elements with values in arbitrary complete cdgas (i.e. it agrees with the formula ofProposition 3.12 (2)). This is a straightforward calculation similar to that of Lemma 3.10. (cid:3) Example 3.14.
Here is one of the simplest yet nontrivial examples of the above construc-tion. Let (
V, m ) be an L ∞ -algebra. Set L := Der ˆ SV ∗ [ −
1] and P be the natural projectionDer ˆ SV ∗ [ − → V [1] and ∆ := m ∈ Der ˆ SV ∗ [ − L, [∆ , − ]) is nothing but theChevalley-Eilenberg complex of the L ∞ -algebra V whereas the dg Lie subalgebra Der ˆ SV ∗ [ − h ∼ = V [1] and the L ∞ -structure on V coming from higher derived brack-ets is just the original L ∞ -structure. This result, obtained by a different method, as well as anexplicit L ∞ -map V → Der ˆ SV ∗ [ −
1] is contained in [10].
Remark 3.15.
The L ∞ -algebra h [ −
1] is a model for a homotopy fiber of the inclusion of dglasker P → L . Our methods can be extended to the case when h = Im( P ) is not abelian, but wewill not pursue this route as our applications are only concerned with the abelian case. Thenon-abelian case was treated in [2] using different methods.The functor VMC constructed above is analogous to the functor Def χ associated to an arbi-trary morphism (not necessarily an inclusion) of dglas χ : M → L introduced in [27] in the sensethat both can be interpreted as MC functors of homotopy fibers of the corresponding maps.4. Homotopy relative Rota-Baxter Lie algebras
Let ( g , m ) be an L ∞ -algebra and V be a representation of g , i.e. a graded vector spacetogether with an L ∞ -map ρ : g → gl ( V ). Recall that this data can be represented as an MCelement Φ := ( m, ρ ) ∈ L L ∞ Rep ( g , V ) := Der ˆ S g ∗ [ − ⋉ (cid:0) ˆ S ≥ g ∗ [ − ⊗ gl ( V ) (cid:1) . ia the natural inclusion L L ∞ Rep ( g , V ) ⊂ Der ˆ S ( g ⊕ V ) ∗ [ − S ( g ⊕ V ) ∗ [ − h = Hom( g ∗ [ − , ˆ S ≥ V ∗ [ − . Then h is the abelian Lie subalgebra in Der ˆ S ( g ⊕ V ) ∗ [ −
1] consisting of continuous derivationsvanishing on V ∗ [ −
1] and mapping g ∗ [ −
1] to ˆ S ≥ V ∗ [ − h ⊥ of h in Der ˆ S ( g ⊕ V ) ∗ [ −
1] is closed with respect to the commutator bracket andcontains Φ. We will define P to be the projector in Der ˆ S ( g ⊕ V ) ∗ [ −
1] onto h . Thus, we obtaina V -structure as following: Proposition 4.1.
Let ( g , m ) be an L ∞ -algebra and ( V, ρ ) be a representation of g . Then theprojection onto the subspace h determines an admissible V -structure on the dgla Der ˆ S ( g ⊕ V ) ∗ [ − supplied with the commutator bracket and the differential d = [Φ , · ] .Consequently, ( h [ − , { ˇ m k } ∞ k =1 ) is a weakly filtered L ∞ -algebra, where higher products ˇ m k aregiven by formulas (3.1) .Proof. We only need to show that the given V -structure is admissible. To that end let T ∈ h ∼ =Hom( g ∗ [ − , ˆ S ≥ V ∗ [ − T can be written as a sum(4.1) T = T + T + · · · , where T n is the order n part of T so we can write T n : g ∗ [ − → ˆ S n V ∗ [ − T is aderivation of ˆ S ( g ⊕ V ) ∗ [ −
1] but we will view it merely as an endomorphism of ˆ S ( g ⊕ V ) ∗ [ − T is a nilpotent endomorphism (in fact T = 0) and it follows that the adjointaction of T is pronilpotent. (cid:3) Remark 4.2.
In fact, the L ∞ -algebra h is even filtered , cf. [25, Remark 5.9]We can now give a compact definition of a homotopy relative RB operator. Definition 4.3. A homotopy relative RB operator on an L ∞ -algebra ( g , m ) supplied with arepresentation ( V, ρ ) is an element T ∈ h such that P ( e ad T ( m + ρ )) = 0 (i.e. T is an MCelement of the L ∞ -algebra ( h [ − , { ˇ m k } ∞ k =1 )). Remark 4.4.
Let T be a homotopy relative RB operator compatible with an L ∞ -algebra ( g , m )and its representation ( V, ρ ) and T n , n = 1 , , · · · be its components as in (4.1). Consider thedual map of T n , thus we have the degree 0 map ˇ T n : ˆ S n V [1] → g [1] for n = 1 , , · · · . Moreover,ˇ T n is graded symmetric and satisfy the homotopy relative RB relation. See [25] for more detailsabout homotopy relative RB operators. Remark 4.5.
By the well-known formula for the exponential of the adjoint representation, thehomotopy relative RB condition P ( e ad T ( m + ρ )) = 0 can be rewritten as P ( e − T ( m + ρ ) e T ) = 0. Remark 4.6.
In the classical case for ( g , [ − , − ] g ) is an ordinary Lie algebra and ρ : g → gl ( V )is a representation of g on V . For any T ∈ Hom( V, g ), we have e ad T ( m + ρ ) = e − T ◦ ( m + ρ ) ◦ ( e T ⊗ e T ) . For all u, v ∈ V , we have (cid:0) e − T ◦ ( m + ρ ) ◦ ( e T ⊗ e T ) (cid:1) ( u, v ) = (1 − T )( m + ρ )( u + T u, v + T v )= (1 − T )([ T u, T v ] g + ρ ( T u ) v − ρ ( T v ) u )= ρ ( T u ) v − ρ ( T v ) u + [ T u, T v ] g − T ( ρ ( T u ) v − ρ ( T v ) u ) . Therefore, P ( e ad T ( m + ρ )) = 0 will give us[ T u, T v ] g = T ( ρ ( T u ) v − ρ ( T v ) u ) , which implies that T is a relative RB operator (also called an O -operator) on the Lie algebra( g , m ) with respect to the representation ( V, ρ ). See [1] for more details. efinition 4.7. Let g and V be graded vector spaces. A homotopy relative RB Lie algebra on g and V is a triple ( m, ρ, T ), where m is an L ∞ -algebra structure on g , ρ is a representation of( g , m ) on V and T is a homotopy relative RB operator compatible the L ∞ -algebra ( g , m ) andits representation ( V, ρ ).Let g and V be graded vector spaces. Since L L ∞ Rep ( g , V ) is a graded Lie subalgebra ofDer ˆ S ( g ⊕ V ) ∗ [ − Corollary 4.8.
With the above notation, ( L L ∞ Rep ( g , V ) ⊕ h [ − , { ˇ m i } ∞ i =1 ) is an L ∞ -algebra,where ˇ m i are given by ˇ m ( Q [1] , Q ′ [1]) = ( − | Q | [ Q, Q ′ ][1] , ˇ m k ( Q [1] , θ , · · · , θ k − ) = P [ · · · [ Q, θ ] , · · · , θ k − ] , for homogeneous elements θ , · · · , θ k − ∈ h , homogeneous elements Q, Q ′ ∈ L , and all the otherhigher products vanish, unless they can be obtained from the ones listed by permutations ofarguments. We denote the L ∞ -algebra ( L L ∞ Rep ( g , V ) ⊕ h [ − , { ˇ m i } ∞ i =1 ) by L HRB ( g , V ) .Proof. It follows from Theorem 3.7, Remark 3.9 and Proposition 4.1. (cid:3)
Now we show that the L ∞ -algebra L HRB ( g , V ) governs homotopy relative RB Lie algebras. Theorem 4.9.
Let g and V be graded vector spaces, m ∈ Der ˆ S g ∗ [ − , ρ ∈ (cid:0) ˆ S ≥ g ∗ [ − ⊗ gl ( V ) (cid:1) and T ∈ Hom ( g ∗ [ − , ˆ S ≥ V ∗ [ − . Then the triple ( m, ρ, T ) is a homotopy relative RB Lie algebra structure on g and V if and only if (Φ[1] , T ) is an MC element of the L ∞ -algebra L HRB ( g , V ) given in Corollary 4.8, where Φ = ( m, ρ ) .Proof. Let (Φ[1] , T ) be an MC element of the L ∞ -algebra L HRB ( g , V ). Using the inclusion of L ∞ -algebras( L L ∞ Rep ( g , V ) ⊕ h [ − , { ˇ m i } ∞ i =1 ) ⊂ Der ˆ S ( g ⊕ V ) ∗ [ − ⊕ h [ − , { ˇ m k } ∞ k =1 ) , the pair (Φ[1] , T ) can be viewed as an MC element in Der ˆ S ( g ⊕ V ) ∗ [ − ⊕ h [ − , { ˇ m k } ∞ k =1 ).This implies, by Lemma 3.10, that P ( e − T ( m + ρ ) e T ) = 0, i.e. T is a homotopy relative RBoperator on g with respect to the representation ρ . Conversely, given a homotopy relative RBoperator T , the same argument traced in the reverse order, shows that the pair (Φ[1] , T ) is anMC element of the L ∞ -algebra L HRB ( g , V ). (cid:3) Remark 4.10.
It is clear that the L ∞ -algebra L HRB is included in an L ∞ -extension h [ − → L HRB ( g , V ) → L L ∞ Rep ( g , V )where h [ −
1] has the trivial L ∞ -structure. Definition 4.11.
Let A be a complete cdga, g and V be graded vector spaces. Then an A -linearhomotopy relative RB Lie algebra structure on A ⊗ g and A ⊗ V is a pair (Φ A [1] , T A ), which isan MC element of the L ∞ -algebra A ≥ ⊗ L HRB ( g , V ), where Φ A = ( m A , ρ A ).Let F HRB be the functor associating to a complete cdga A the set of A -linear homotopyrelative RB Lie algebra structures on A ⊗ g and A ⊗ V . Then we have the following result. Theorem 4.12.
The functor F HRB is represented by the complete cdga ˆ S L ∗ HRB ( g , V )[ − .Proof. By the definition of an A -linear homotopy relative RB Lie algebra and Theorem 4.9, wededuce that F HRB ( A ) = MC( L HRB ( g , V ) , A ). Therefore, by Theorem 2.12, we obtain that thefunctor F HRB is represented by the complete cdga ˆ S L ∗ HRB ( g , V )[ − (cid:3) . Shifted Poisson algebras, r ∞ -matrices and triangular L ∞ -bialgebras In this section, we describe a certain doubling construction for shifted Poisson algebras. Ourexposition is a straightforward modification of the corresponding Z / r ∞ -matrices. Thenwe show that r ∞ -matrices give rise to triangular L ∞ -bialgebras.5.1. Doubling construction for shifted Poisson algebras.
Let g be a graded vector space,here and later on assumed to be finite-dimensional. Then the n -shifted cotangent bundle T ∗ [ n ] g [1] is a graded symplectic manifold equipped with a degree n symplectic structure. Con-sequently, the pairing of degree − n g ∗ [ − ⊗ g [1 − n ] → k determines an n -shifted Poisson algebra structure on the complete pseudocompact algebraˆ S ( g ∗ [ − ⊕ g [1 − n ]). The corresponding Poisson bracket {− , −} can be viewed as a gradedLie algebra structure on ( ˆ S ( g ∗ [ − ⊕ g [1 − n ]))[ n ], in other words, it is a degree zero map {− , −} : ( ˆ S ( g ∗ [ − ⊕ g [1 − n ]))[ n ] ⊗ ( ˆ S g ∗ [ − ⊕ g [1 − n ]))[ n ] → ( ˆ S ( g ∗ [ − ⊕ g [1 − n ]))[ n ]satisfying the graded Jacobi identity.Later on we will also need to work with the graded vector spaceˆ S ′ ( g ∗ [ − ⊕ g [1 − n ]) := ˆ S ≥ g ∗ [ − ⊗ ˆ S ≥ ( g [1 − n ]) , which is clearly a (shifted) Poisson subalgebra of ˆ S g ∗ [ − ⊗ ˆ S ( g [1 − n ]).Consider the graded Lie algebra Der ˆ S g ∗ [ −
1] with respect to the commutator bracket. Sinceany derivation is determined by its value on g ∗ [ − S g ∗ [ −
1] with the gradedvector space Hom( g ∗ [ − , ˆ S g ∗ [ − ∼ = ˆ S g ∗ [ − ⊗ g [1] . Definition 5.1.
The n -shifted double is the map D n : Der ˆ S g ∗ [ − ∼ = ˆ S g ∗ [ − ⊗ g [1] → ( ˆ S g ∗ [ − n ] ⊗ ˆ S ≥ ( g [1 − n ]) ⊂ ( ˆ S ≥ ( g ∗ [ − ⊕ g [1 − n ]))[ n ]given by the formulaˆ S g ∗ [ − ⊗ g [1] ⊃ f ⊗ w ( − n | f | f [ n ] ⊗ ( w [ − n ]) ∈ ( ˆ S g ∗ [ − n ] ⊗ ˆ S ≥ ( g [1 − n ]) . Then we have the following result.
Proposition 5.2.
The map D n is a map of graded Lie algebras.Proof. The proof of [6, Theorem 3.2] carries over with only notational modifications. The caseswhen n is odd or even are slightly different. (cid:3) r ∞ -matrices and triangular L ∞ -bialgebras. Let ( g , m ) be an L ∞ -algebra. In thissection we define, using the doubling construction and Voronov’s higher derived brackets, thehigher shifted Schouten Lie algebra and an n -shifted r ∞ -matrix for g . This leads naturally tothe notion of an n -shifted triangular L ∞ -bialgebra. Ordinary (or 0-shifted in our terminology) L ∞ -bialgebras were introduced in [20] and its shifted version was considered in [3, 29].Consider the graded Lie algebra ( ˆ S ( g ∗ [ − ⊕ g [1 − n ]))[ n ]where the graded Lie bracket was defined in the previous subsection. The L ∞ -algebra structure m is an MC element in Der ˆ S g ∗ [ −
1] and so, D n ( m ) is an MC element in the graded Lie algebra( ˆ S ( g ∗ [ − ⊕ g [1 − n ]))[ n ] making it a dgla. Note that ( ˆ S g [1 − n ])[ n ] ⊂ ( ˆ S ( g ∗ [ − ⊕ g [1 − n ]))[ n ]is an abelian Lie subalgebra whose direct complement is a dg Lie subalgebra. Then, Voronov’sderived brackets construction implies the following result. heorem 5.3. Let ( g , m ) be an L ∞ -algebra. Then the differential graded Lie algebra (cid:16) L := ( ˆ S ( g ∗ [ − ⊕ g [1 − n ]))[ n ] , {− , −} , d = { D n ( m ) , ·} (cid:17) is endowed with an admissible V-structure P : L → L , which is the projection to h := ( ˆ S ≥ g [1 − n ])[ n ] . Consequently, ( h [ − , { ˇ m k } ∞ k =1 ) is an L ∞ -algebra, where ˇ m k is given by (3.1) .Moreover, there is an L ∞ -algebra structure on L ⊕ h [ − given by (5.1) ˇ m ( x [1] , h ) = ( − d ( x )[1] , P ( x + d ( h ))) , ˇ m ( x [1] , y [1]) = ( − | x | { x, y } [1] , ˇ m k ( x [1] , h , · · · , h k − ) = P {· · · {{ x, h } , h } , · · · , h k − } , k ≥ , ˇ m k ( h , · · · , h k − , h k ) = P {· · · { d ( h ) , h } , · · · , h k } , k ≥ , for all x, y ∈ L and h, h , . . . , h k ∈ h . The remaining L ∞ -products vanish, unless they areobtained from those above by permutations of arguments. Moreover there exists the following L ∞ -extension: (5.2)( ˆ S ≥ g [1 − n ])[ n − → ( ˆ S ( g ∗ [ − ⊕ g [1 − n ]))[ n ] ⊕ ( ˆ S ≥ g [1 − n ])[ n − → ( ˆ S ( g ∗ [ − ⊕ g [1 − n ]))[ n ] . Proof.
Arguing as in the proof of Proposition 4.1, let h ∈ h ; then h = h + h + · · · where h n ∈ ( S ≥ n +1 g [1 − n ])[ n − h ∈ ( S ≥ g [1 − n ])[ n −
1] viewed as anendomorphism of L is nilpotent (even has square zero) and we conclude that the specified V -structure is admissible. The stated formulas for the L ∞ -products follow from Theorem 3.7. (cid:3) Remark 5.4.
If we choose ( ˆ S g [1 − n ])[ n ] as the abelian subalgebra in L , then the correspondingV-structure will not be admissible. Nevertheless, the derived brackets construction is applicableand ( ˆ S g [1 − n ])[ n −
1] becomes an L ∞ -algebra where formulas (5.1) still hold. Clearly, h =( ˆ S ≥ g [1 − n ])[ n −
1] is an L ∞ -subalgebra in ( ˆ S g [1 − n ])[ n − Definition 5.5.
The L ∞ -algebra ( ˆ S g [1 − n ])[ n − , { ˇ m k } ∞ k =1 ) is called the higher Schouten Liealgebra of the L ∞ -algebra ( g , m ). Remark 5.6.
It is well known that for an ordinary Lie algebra g , the exterior algebra Λ( g ) := ⊕ ∞ k =0 ∧ k +1 g is a graded Lie algebra ( ∧ k +1 g is of degree k ), which is also called the SchoutenLie algebra. Now the L ∞ -algebra structure on ( ˆ S g [1 − n ])[ n −
1] given in Theorem 5.3 onlycontains m . If, furthermore, n = 2, it reduces to the Schouten Lie algebra, i.e.( ˆ S g [ − g ) . If, on the other hand, n = 1, we obtain a graded Lie algebra structure on ˆ S g ; this is thewell-known Poisson bracket on the completion of gr U g , the associated graded to the universalenveloping algebra of g (which is isomorphic, by the Poincare-Birkhoff-Witt theorem, to S g ).We can now give the definition of an n -shifted r ∞ -matrix. Definition 5.7.
Let ( g , m ) be an L ∞ -algebra.(1) An n -shifted r ∞ -matrix for g is a degree 0 element r ∈ ( ˆ S ≥ g [1 − n ])[ n ] such that P ( e ad r D n ( m )) = 0, i.e. r is an MC element in the L ∞ -algebra ( ˆ S ≥ g [1 − n ])[ n − A be a complete cdga. Then an n -shifted r ∞ -matrix for g with values in A is adegree 0 element r A ∈ A ≥ ⊗ ( ˆ S ≥ g [1 − n ])[ n ] such that P ( e ad rA D n ( m )) = 0, i.e. r A isan MC element in the L ∞ -algebra ( ˆ S ≥ g [1 − n ])[ n −
1] with values in A . Remark 5.8.
It is possible to define an r ∞ -matrix using the L ∞ -algebra ( ˆ S g [1 − n ])[ n − L ∞ -algebra ( ˆ S ≥ g [1 − n ])[ n − k [[ λ ]] as a complete cdga, was given in [4] using adifferent method. As far as we know, n -shifted r ∞ -matrices for n odd have not been considered,even in the case of ordinary (graded) Lie algebras. Our definition is closer to the classical otion of an r -matrix, [34] and specializes to this notion in the case when the formal powerseries r ∈ ( ˆ S ≥ g [1 − n ])[ n −
1] contains only the first, quadratic, term.
Definition 5.9.
The structure of an n -shifted L ∞ -bialgebra on a graded vector space g is anMC element in ˆ S ′ ( g ∗ [ − ⊕ g [1 − n ])[ n ], i.e. an element h ∈ ˆ S ′ ( g ∗ [ − ⊕ g [1 − n ])[ n ] of degree 1such that { h, h } = 0. Remark 5.10.
The projectionsˆ S ′ ( g ∗ [ − ⊕ g [1 − n ])[ n ] → g [1] ⊗ ˆ S g ∗ [ − ∼ = Der ˆ S g ∗ [ − S ′ ( g ∗ [ − ⊕ g [1 − n ])[ n ] → ( ˆ S g [1 − n ])[ n ] ⊗ g ∗ [ − ∼ = Der ˆ S ( g [1 − n ]) ∼ = Der ˆ S ( g ∗ [ n − ∗ [ − n -shifted L ∞ -bialgebra structure on g determines • an L ∞ -algebra structure on g and • an L ∞ -algebra structure on g ∗ [ n −
2] (equivalently, an L ∞ - coalgebra structure on g [2 − n ]).Additionally, there are appropriate compatibility conditions between these structures. Thisexplains the terminology ‘shifted L ∞ -bialgebra’.Note also that the Poisson algebra ˆ S ≥ ( g ∗ [ − ⊕ g [1 − n ])[ n ] (as opposed to its Poissonsubalgebra ˆ S ′ ( g ∗ [ − ⊕ g [1 − n ])[ n ]) leads to the notion of an L ∞ -quasi-bialgebra.Classically, an r -matrix in a Lie algebra g gives rise to a Lie bialgebra structure on g , called triangular . We will now formulate a higher version of this result. Theorem 5.11.
Let ( g , m ) be an L ∞ -algebra and r ∈ ( ˆ S ≥ g [1 − n ])[ n ] be an n -shifted r ∞ -matrix.Define r ( m ) ∈ ( ˆ S ( g ∗ [ − ⊕ g [1 − n ]))[ n ] by (5.3) r ( m ) := e ad r D n ( m ) . Then r ( m ) is an MC element in ( ˆ S ′ ( g ∗ [ − ⊕ g [1 − n ]))[ n ] and determines the structure of an n -shifted L ∞ -bialgebra on g .Proof. Note that D n ( m ) is an MC element in the graded Lie algebra ( ˆ S ( g ∗ [ − ⊕ g [1 − n ]))[ n ]as the image of the MC element m under the gla map D n , cf. Proposition 5.2. It followsthat e ad r D n ( m ) is likewise an MC element as obtained from D n ( m ) by a gauge transformation.Next, since r is an r ∞ -matrix, we have P ( e ad r D n ( m )) = 0 and it follows that e ad r D n ( m ) ∈ ˆ S ′ ( g ∗ [ − ⊕ g [1 − n ])[ n ] and we are done. (cid:3) Definition 5.12.
Given an L ∞ -algebra ( g , m ) and an n -shifted r ∞ -matrix r , the n -shifted L ∞ -bialgebra r ( m ) defined by formula (5.3) will be called a triangular n -shifted L ∞ -bialgebra .By abuse of terminology, we will also refer to the triple ( g , m, r ) as a triangular n -shifted L ∞ -bialgebra . Remark 5.13. (1) The statement of Theorem 5.11 for n = 2 and in the context of k [[ λ ]]-linear r ∞ -matrices is essentially the main result of [4], cf. Theorem 2 and the discussionfollowing it in op. cit.(2) For an ordinary Lie algebra, Theorem 5.11 reduces to the standard construction of atriangular Lie bialgebra.Let g be a vector space;recall that h ∼ = ( ˆ S ≥ g [1 − n ])[ n ]. Since Der ˆ S g ∗ [ −
1] is a graded Liesubalgebra of ( ˆ S ( g ∗ [ − ⊕ g [1 − n ]))[ n ], we have the following corollary: Corollary 5.14.
With the above notation, (Der ˆ S g ∗ [ − ⊕ h [ − , { ˇ m i } ∞ i =1 ) is an L ∞ -algebra,where l i are given by ˇ m ( Q [1] , Q ′ [1]) = ( − | Q | [ Q, Q ′ ][1] , ˇ m k ( Q [1] , θ , · · · , θ k − ) = P [ · · · [ Q, θ ] , · · · , θ k − ] , for homogeneous elements θ , · · · , θ k − ∈ h , homogeneous elements Q, Q ′ ∈ Der ˆ S g ∗ [ − , andall the other possible combinations vanish. roof. It follows from Remark 3.9 and Theorem 5.3. (cid:3)
We denote the above L ∞ -algebra (Der ˆ S g ∗ [ − ⊕ h [ − , { ˇ m i } ∞ i =1 ) by L T L ∞ Bi ( g ). We will seethat it governs triangular L ∞ -bialgebras; the proof is formally analogous to that of Theorem4.9. Theorem 5.15.
Let g be a graded vector space, m ∈ Der ˆ S g ∗ [ − and r ∈ ( ˆ S ≥ g [1 − n ])[ n ] .Then the pair ( m [1] , r ) is a triangular L ∞ -bialgebra structure on g if and only if ( m [1] , r ) is an MC element of the L ∞ -algebra L T L ∞ Bi ( g ) given in Corollary 5.14.Proof. Let ( m [1] , r ) be an MC element of the L ∞ -algebra L T L ∞ Bi ( g ). Using the inclusion of L ∞ -algebras Der ˆ S g ∗ [ − ⊕ h [ − ⊂ ( ˆ S ( g ∗ [ − ⊕ g [1 − n ]))[ n ] ⊕ h [ − m [1] , r ) can be viewed as an MC element in ( ˆ S ( g ∗ [ − ⊕ g [1 − n ]))[ n ] ⊕ h [ − P ( e ad r D n ( m )) = 0, i.e. r is an r ∞ -matrix for g . Conversely, givenan r ∞ -matrix r , the same argument traced in the reverse order, shows that the pair ( m [1] , r ) isan MC element of the L ∞ -algebra L T L ∞ Bi ( g ). (cid:3) Remark 5.16.
It follows from the above that there is a commutative diagram where horizontalarrows are L ∞ -extensions and vertical arrows are strict L ∞ -maps or graded Lie algebra maps:( ˆ S g [1 − n ])[ n − / / L T L ∞ Bi ( g ) (cid:15) (cid:15) / / Der ˆ S g ∗ [ − D n (cid:15) (cid:15) ( ˆ S g [1 − n ])[ n − / / K / / ( ˆ S ( g ∗ [ − ⊕ g [1 − n ]))[ n ] . and K = ( ˆ S ( g ∗ [ − ⊕ g [1 − n ]))[ n ] ⊕ ( ˆ S g [1 − n ])[ n − Definition 5.17.
Let A be a complete cdga and g be a graded vector space. Then an A -lineartriangular n -shifted L ∞ -bialgebra structure on A ⊗ g is a pair ( m A [1] , r A ), which is an MCelement in the L ∞ -algebra A ≥ ⊗ L T L ∞ Bi ( g ).Let g be a graded vector space and F T L ∞ Bi be the functor associating to a complete cdga A the set of A -linear triangular n -shifted L ∞ -bialgebra structures on A ⊗ g . Then we have thefollowing result. Theorem 5.18.
The functor F T L ∞ Bi is represented by the complete cdga ˆ S L ∗ T L ∞ Bi ( g )[ − .Proof. By the definition of an A -linear triangular n -shifted L ∞ -bialgebra and Theorem 5.15, wededuce that F T L ∞ Bi ( A ) = MC( L T L ∞ Bi ( g ) , A ). Therefore, by Theorem 2.12, we obtain that thefunctor F T L ∞ Bi is represented by the complete cdga ˆ S L ∗ T L ∞ Bi ( g )[ − (cid:3) From triangular L ∞ -bialgebras to homotopy relative Rota-Baxter Liealgebras In this section, we establish the relation between the L ∞ -algebra governing triangular L ∞ -bialgebras and the L ∞ -algebra governing homotopy relative RB Lie algebras.Consider again the graded Lie algebraˆ S g ∗ [ − ⊗ ( ˆ S g [1 − n ])[ n ] ∼ = ˆ S ( g ∗ [ − ⊕ g [1 − n ])[ n ]together with its Poisson bracket {− , −} . An element in it (a hamiltonian) can be viewed asa hamiltonian (formal) vector field on g ∗ [ − ⊕ g [1 − n ], and Poisson brackets correspond tocommutators of vector fields. Thus, we have an inclusion of graded Lie algebras H : ˆ S ≥ ( g ∗ [ − ⊕ g [1 − n ])[ n ] ֒ → Der ˆ S ( g ∗ [ − ⊕ g [1 − n ]) ∼ = Der ˆ S (cid:0) ( g ⊕ g ∗ [ n − ∗ [ − (cid:1) . MC elements in ˆ S ≥ ( g ∗ [ − ⊕ g [1 − n ])[ n ] correspond, under this map, to elements in the gradedLie algebra Der ˆ S (cid:0) ( g ⊕ g ∗ [ n − ∗ [ − (cid:1) ⊂ Der ˆ S (cid:0) ( g ⊕ g ∗ [ n − ∗ [ − (cid:1) hat are cyclic L ∞ -algebra structures on g ⊕ g ∗ [ n − S ≥ g [1 − n ])[ n ] ⊂ ˆ S ≥ ( g ∗ [ − ⊕ g [1 − n ])[ n ]under the map H is inside the abelian Lie subalgebra in Der ˆ S (cid:0) ( g ⊕ g ∗ [ n − ∗ [ − (cid:1) having theform Hom( g ∗ [ − , ˆ S ( g ∗ [ n − ∗ ) ∼ = Hom( g ∗ [ − , ˆ S g [1 − n ]) ∼ = Hom( ˆ S g ∗ [ n − , g [1]) . This is precisely the abelian subalgebra that was used to construct (using derived brackets) the L ∞ -algebra controlling homotopy relative RB Lie algebras given in Theorem 4.9.Taking into account Remark 5.16, we obtain the following commutative diagram whose rowsare L ∞ -extensions:( ˆ S ≥ g [1 − n ])[ n − / / L T L ∞ Bi ( g ) (cid:15) (cid:15) / / Der ˆ S g ∗ [ − D n (cid:15) (cid:15) ( ˆ S ≥ g [1 − n ])[ n − / / H (cid:15) (cid:15) K / / (cid:15) (cid:15) ˆ S ≥ ( g ∗ [ − ⊕ g [1 − n ])[ n ] H (cid:15) (cid:15) Hom( g ∗ [ − , ˆ S ≥ g [1 − n ]) / / X / / Der ˆ S ( g ∗ [ − ⊕ g [1 − n ])where K = ˆ S ≥ ( g ∗ [ − ⊕ g [1 − n ])[ n ] ⊕ ( ˆ S g [1 − n ])[ n −
1] and X is the L ∞ -extension obtained fromthe graded Lie algebra Der ˆ S ( g ∗ [ − ⊕ g [1 − n ]) and its abelian subalgebra Hom( g ∗ [ − , ˆ S ≥ g [1 − n ]) by the derived brackets construction. The vertical maps in the above diagram are dgla mapsor strict L ∞ -maps.Now recall that there is an L ∞ -extensionHom( g ∗ [ − , ˆ S ≥ g [1 − n ]) → L HRB ( g , g ∗ [ n − → Der ˆ S g ∗ [ − ⋉ ˆ S ≥ g ∗ [ − ⊗ gl ( g ∗ )where the L ∞ -algebra L HRB ( g , g ∗ [ n − g with a representation on g ∗ [ n − gl ( g ∗ [ n − gl ( g ∗ )).Noting that the image of Der ˆ S g ∗ [ −
1] inside Der ˆ S (cid:0) ( g ⊕ g ∗ [ n − ∗ [ − (cid:1) under the map H ◦ D n is contained in the Lie subalgebra L L ∞ Rep ( g , g ∗ [ n − S g ∗ [ − ⋉ ˆ S ≥ g ∗ [ − ⊗ gl ( g ∗ ),we obtain the following commutative diagram where, as above, the rows are L ∞ -extensions andvertical arrows are graded Lie algebra maps or strict L ∞ -maps:(6.1) ( ˆ S g [1 − n ])[ n − / / H (cid:15) (cid:15) L T L ∞ Bi ( g ) (cid:15) (cid:15) / / Der ˆ S g ∗ [ − H ◦ D n (cid:15) (cid:15) Hom( g ∗ [ − , ˆ S g [1 − n ]) / / L HRB ( g , g ∗ [ n − / / L L ∞ Rep ( g , g ∗ [ n − Theorem 6.1.
Let r be an n -shifted r ∞ -matrix in an L ∞ -algebra ( g , m ) , i.e. the pair ( g , m, r ) is an n -shifted triangular L ∞ -bialgebra. Then H ( r ) is a homotopy relative RB operator on g with respect to the coadjoint representation ad ∗ of g on g ∗ [ n − . The resulting correspondence ( m, r ) ( m, ad ∗ , H ( r )) between triangular L ∞ -bialgebras and homotopy relative RB Lie algebras is realized as a strict L ∞ -map L T L ∞ Bi ( g ) → L HRB ( g , g ∗ [ n − between L ∞ -algebras governing the corresponding structures. (cid:3) emark 6.2. When we consider the ungraded case (i.e. when g is an ordinary Lie algebra),the strict L ∞ -map L T L ∞ Bi ( g ) → L HRB ( g , g ∗ [ n − i givenin [25, Proposition 4.19]; note that i was only constructed in op. cit. as a chain map ratherthan a strict L ∞ -map. Remark 6.3.
It is easy to see that the image of the map L T L ∞ Bi ( g ) → L HRB ( g , g ∗ [ n − L ∞ -algebra whose underlying graded vector space isHom( g ∗ [ − , ˆ S g [1 − n ]) ⊕ ( H ◦ D n )(Der ˆ S g ∗ [ − ∼ = Hom( g ∗ [ − , ˆ S g [1 − n ]) ⊕ Der ˆ S g ∗ [ − g with the fixed coadjoint representation. Weomit the details. Acknowledgements.
This research was partially supported by NSFC (11922110). This work wascompleted in part while the first author was visiting Max Planck Institute for Mathematics inBonn and he wishes to thank this institution for excellent working conditions.
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Department of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF, UK
E-mail address : [email protected] Department of Mathematics, Jilin University, Changchun 130012, Jilin, China
E-mail address : [email protected] Department of Mathematics, Jilin University, Changchun 130012, Jilin, China
E-mail address : [email protected]@jlu.edu.cn