We define a homotopy algebra associated to classical open-closed strings. We
call it an open-closed homotopy algebra (OCHA). It is inspired by Zwiebach's
open-closed string field theory and also is related to the situation of
Kontsevich's deformation quantization. We show that it is actually a homotopy
invariant notion; for instance, the minimal model theorem holds. Also, we show
that our open-closed homotopy algebra gives us a general scheme for deformation
of open string structures (A(infinity)-algebras) by closed strings
(L(infinity)-algebras).Comment: 30 pages, 14 figures; v2: added an appendix by M.Markl, ambiguous
  terminology fixed, minor corrections; v3: published versio

Kajiura, Hiroshige

Stasheff, Jim

English

arXiv

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summary:We recall the definition of strong homotopy derivations of $A_\infty $ algebras and introduce the corresponding definition for $L_\infty $ algebras. We define strong homotopy inner derivations for both algebras and exhibit explicit examples of both

Lada, Tom

Tolley, Melissa

Czech digital mathematics library

Derivations of homotopy algebras

Tom Lada

Melissa Tolley

Archivum Mathematicum

Institute of Mathematics AS CR, v. v. i.

Archivum MathematicumTom Lada; Melissa TolleyDerivations of homotopy algebrasArchivum Mathematicum, Vol. 49 (2013), No. 5, 309--315Persistent URL: http://dml.cz/dmlcz/143555Terms of use:© Masaryk University, 2013Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czARCHIVUM MATHEMATICUM (BRNO)Tomus 49 (2013), 309–315DERIVATIONS OF HOMOTOPY ALGEBRASTom Lada and Melissa TolleyAbstract. We recall the definition of strong homotopy derivations of A∞algebras and introduce the corresponding definition for L∞ algebras. Wedefine strong homotopy inner derivations for both algebras and exhibit explicitexamples of both.1. IntroductionThe concept of a strong homotopy derivation of an A∞ algebra was introducedby Kajiura and Stasheff in [2]. In this note we will introduce the correspondingconcept for L∞ algebras. We will discuss several concrete examples of such algebrasand strong homotopy inner derivations on them.In Section 2, we recall the definitions of A∞ and L∞ algebras and discuss anexplicit example of each. We will use these examples to exhibit examples of stronghomotopy derivations.In Section 3, we review the definition of a strong homotopy derivation of an A∞algebra. We introduce the concept of an inner such derivation of these algebrasand present an explicit example of this concept by using the A∞ algebra in theprevious section.The next section contains our definition of a strong homotopy derivation of anL∞ algebra. We discuss the concept of inner derivation and present a concreteexample of such a derivation on the L∞ algebra in Section 2.In the final section we will discuss the relationship between the A∞ data andthe L∞ data using symmetrization.We work in the setting of Z graded vector spaces and will occasionally use thenotation |x| to denote the degree of an element x.2. A∞ and L∞ algebrasDefinition 1. An A∞ algebra [6] structure on a Z graded vector space V is acollection of degree one linear maps mn : V ⊗n → V that satisfy the relations(2.1)∑k+l=n+1k∑i=1(−1)αmk(v1, . . . , vi−1,ml(vi, . . . , vi+l−1), vi+l, . . . , vn)= 02010 Mathematics Subject Classification: primary 18G55.Key words and phrases: L∞ algebra, A∞ algebra, strong homotopy derivation.DOI: 10.5817/AM2013-5-309310 T. LADA AND M. TOLLEYfor n ≥ 1 and α is the sum of the degrees of the elements v1, . . . , vi−1.We remark that this definition differs from but is equivalent to the originaldefinition [6] in which the maps mn have degree n− 2 and the signs are adjustedaccordingly. It is well known [6] that the structure maps mn’s may be extendedto a degree +1 coderivation m on the tensor coalgebra T c(V ) of V , and that therelations are equivalent to the equation m2 = 0.Example 2 ([1]). Consider the graded vector space in which V−1 has basis 〈x1, x2〉,V0 has basis 〈y〉, and Vn = 0 otherwise. Define degree one mapsm1(x1) = m1(x2) = ymn(x1 ⊗ y⊗k ⊗ x1 ⊗ y⊗n−2−k) = x1 , 0 ≤ k ≤ n− 2mn(x1 ⊗ y⊗n−2 ⊗ x2) = x1mn(x1 ⊗ y⊗n−1) = yand mn = 0 on the remaining elements of V . This determines an A∞ algebrastructure on V .We next recall the definition of L∞ algebras.Definition 3 ([5]). An L∞ algebra structure on a Z graded vector space V is acollection of degree one graded symmetric linear maps ln : V ⊗n → V , n ≥ 1, thatsatisfy the relations (higher order Jacobi relations)∑j+k=n∑σ(−1)e(σ)l1+j(lk(vσ(1), . . . , vσ(k)), vσ(k+1), . . . , vσ(n))= 0where σ runs over all (k, n− k) unshuffle permutations. The exponent e(σ) is thesum of the products of the degrees of the elements that are permuted, sometimesknown as the Koszul sign.Again, we remark that this definition differs from but is equivalent to the originaldefinition [5] in which the maps ln have degree n−2 and are graded skew symmetricwith the signs adjusted. Also, the structure maps may be extended to a degree+1 coderivation l on the symmetric coalgebra Sc(V ) on V , and the relations areequivalent to l2 = 0, [4],[5].It is well known that skew symmetrization of an A∞ algebra structure yields anL∞ algebra structure [4] when one utilizes the original definitions. However, withthe definitions that we use here, we symmetrize the data in the example above toobtainExample 4. Consider the graded vector space in which V−1 has basis 〈x1, x2〉, V0has basis 〈y〉, and Vn = 0 otherwise. Define degree one symmetric mapsl1(x1) = l1(x2) = yln(x1 ⊗ y⊗n−1) = (n− 1)! yln(x1 ⊗ y⊗n−2 ⊗ x2) = (n− 2)! x1DERIVATIONS OF HOMOTOPY ALGEBRAS 311and extend the maps using symmetry. This yields an L∞ algebra structure on V .We will use these examples above to illustrate examples of strong homotopyderivations which we will define in the next two sections.3. Strong homotopy derivations of A∞ algebrasKajiura and Stasheff [2] have formulated the following definition:Definition 5. A strong homotopy derivation of degree one of an A∞ algebra(V, {mn}) is a collection of degree one linear maps θq : V ⊗q → V , q ≥ 1, thatsatisfy the relations0 =∑r+s=q+1r−1∑i=0(−1)β(s,i)θr(v1, . . . , vi,ms(vi+1, . . . , vi+s), . . . , vq)+ (−1)β(s,i)mr(v1, . . . , vi, θs(vi+1, . . . , vi+s), . . . , vq).(3.1)The exponent β(s, i) results from moving the degree one maps ms and θs past(v1, . . . , vi). The θq’s may be extended to a degree +1 coderivation θ on T c(V ) andthe relations then can be described by the equation [m,θ] = 0.As an example of such a structure, we can define a strong homotopy innerderivation of an A∞ algebra.Proposition 6. Let (V, {mn}) be an A∞ algebra and let a ∈ V have the propertythat m1(a) = 0 and the degree of a is even. Then the mapsθn(v1, . . . , vn) = mn+1(a, v1, . . . , vn) + · · ·+mn+1(v1, . . . , vi, a, vi+1, . . . , vn)+ · · ·+mn+1(v1, . . . , vn, a)(3.2)define a strong homotopy derivation of V . We call such a derivation inner.Proof. It can be calculated that the defining relations for a strong homotopyderivation in this case result in n + 1 copies of the defining relations for an A∞algebra except for terms that involve m1(a). Because of our requirement thatm1(a) = 0, we may add in the missing terms and utilize the A∞ algebra relationsn+ 1 times to obtain the result. Recall the example of an A∞ algebra in Section 2. There we had the followingdata. Consider the graded vector space in which V−1 has basis 〈x1, x2〉, V0 has basis〈y〉, and Vn = 0 otherwise. We may construct a strong homotopy inner derivationon V by letting a = y. One may then calculate resulting θn’s to beθ1(x1) = yθn(x1 ⊗ y⊗k ⊗ x1 ⊗ y⊗n−2−k) = nx1θn(x1 ⊗ y⊗n−1) = nyθn(x1 ⊗ y⊗n−2 ⊗ x2) = (n− 1)x1and θn = 0 on the terms not mentioned.312 T. LADA AND M. TOLLEY4. Strong homotopy derivations of L∞ algebrasWe now turn our attention to L∞ algebras.Definition 7. A strong homotopy derivation of degree one of an L∞ algebra(V, {ln}) is a collection of degree one graded symmetric linear maps θq : V ⊗q → V ,q ≥ 1, that satisfy the relationsn∑j=1∑σ(−1)e(σ)θn−j+1(lj(vσ(1), . . . , vσ(j)), vσ(j+1), . . . , vσ(n))+(−1)e(σ)ln−j+1(θj(vσ(1), . . . , vσ(j)), vσ(j+1), . . . , vσ(n)) = 0(4.1)where σ runs over all (j, n− j) unshuffle permutations.The exponent e(σ) is the sum of the products of the degrees of the permutedelements. As we saw for A∞ derivations, we may express the defining relations forstrong homotopy derivations on L∞ algebras by the equation [l,θ] = 0 where θ isthe degree +1 coderivation on Sc(V ) induced by the θn’s. See [7] for details.As an example, we define a strong homotopy inner derivation of an L∞ algebra.Proposition 8. Let (V, {ln}) be an L∞ algebra and let a ∈ V have the propertythat l1(a) = 0 and the degree of a is even. Then the maps(4.2) θn(v1, . . . , vn) = ln+1(v1, . . . , vn, a)define a strong homotopy derivation of V .Proof. We computen∑j=1σ(−1)e(σ)θn−j+1(lj(vσ(1), . . . , vσ(j)), vσ(j+1), . . . , vσ(n))+ (−1)e(σ)ln−j+1(θj(vσ(1), . . . , vσ(j)), vσ(j+1), . . . , vσ(n))=n∑j=1σ(−1)e(σ)ln−j+2(lj(vσ(1), . . . , vσ(j)), vσ(j+1), . . . , vσ(n), a)+ (−1)e(σ)ln−j+1(lj+1(vσ(1), . . . , vσ(j), a), vσ(j+1), . . . , vσ(n))=n∑j=1σ(−1)e(σ)ln−j+2(lj(vσ(1), . . . , vσ(j)), vσ(j+1), . . . , vσ(n), a)+ (−1)e(σ)(−1)αln−j+1(lj+1(vσ(1) . . . , vσ(j), a), vσ(j+1), . . . , vσ(n))+ (−1)βln+1(l1(a), v1, . . . , vn)= 0because these are precisely the L∞ algebra relations on (v1, . . . , vn, a). Note that itis necessary to add the last line, where β = |a|∑ni=1 |vi|, to the homotopy derivationrelations to obtain the L∞ algebra relations; this term, however, is zero because ofDERIVATIONS OF HOMOTOPY ALGEBRAS 313our assumptions on the element a. The sign in the next to last line reduces to therequired (−1)e(σ) because α = |a|∑ni=j+1 |vσ(i)| is even. Recall the example of an L∞ algebra in Section 2. There, V = V−1 ⊕ V0 withbasis for V−1 = 〈x1, x2〉 and basis for V0 = 〈y〉 and the degree one graded symmetricmaps given byl1(x1) = l1(x2) = yln(x1 ⊗ y⊗n−1) = (n− 1)! yln(x1 ⊗ y⊗n−2 ⊗ x2) = (n− 2)! x1 .We construct a strong homotopy derivation of V by letting a = y and thencalculate the resulting θn’s to beθ1(x1) = yθn(x1 ⊗ y⊗n−1) = n! yθn(x1 ⊗ y⊗n−2 ⊗ x2) = (n− 1)! x1and θn is zero on the elements not listed.For example,θn(x1 ⊗ y⊗n−2 ⊗ x2) := ln+1(x1 ⊗ y⊗n−2 ⊗ x2 ⊗ y)= ln+1(x1 ⊗ y⊗n−1 ⊗ x2) = (n− 1)! x1 .5. Symmetrization of A∞ derivationsWe recall that there is a well known injective coalgebra map χ : Sc(V ) −→ T c(V )given byχ(v1, . . . , vn) =∑σ∈Sn(−1)e(σ)vσ(1) ⊗ · · · ⊗ vσ(n)where (−1)e(σ) is the Koszul sign.Suppose that f : T c(V ) −→ V is a linear map which extends to the coderi-vation f : T c(V ) −→ T c(V ) such that π1 ◦ f = f , where π1 : T c(V ) −→ V isprojection. Then the linear map f ◦ χ : Sc(V ) −→ V extends to the coderivationf ◦ χ : Sc(V ) −→ Sc(V ) and the following diagram commutes ([3, Prop. 5])Sc(V ) χ // T c(V )π1""DDDDDDDDDSc(V )f◦χOOχ // T c(V )fOOf // VThe symmetrization of an A∞ algebra structure that was mentioned in Section 2may then be described by the commutative diagram314 T. LADA AND M. TOLLEYSc(V ) χ // T c(V )π1""DDDDDDDDDSc(V )lOOχ // T c(V )mOOm // Vwhere m =∑mn : T c(V ) −→ V is the collection of the A∞ algebra structuremaps, m is the lift of m to a coderivation on T c(V ) with m2 = 0, and the L∞algebra structure l is the lift of the map m ◦ χ : Sc(V ) −→ V to a coderivation onSc(V ).We now address the issue of symmetrization of strong homotopy derivations ofA∞ algebras.Proposition 9. Let θ = {θn} denote the the collection of maps giving a stronghomotopy derivation on the A∞ algebra (V,m). Regard θ as a map T c(V ) −→V and lift it to the coderivation θ on T c(V ). Then the extension of the mapθ◦χ : Sc(V ) −→ V to the coderivation θ′ on Sc(V ) is a strong homotopy derivationon the L∞ algebra V with algebra structure given by m ◦ χ.Proof. We claim that [l,θ′] = 0. We have the commutative diagramSc(V ) χ // T c(V )π1""DDDDDDDDDSc(V )θ′OOχ // T c(V )θOOθ // Vand we calculateχ[l,θ′] = χ(lθ′ + θ′l)= (χl)θ′ + (χθ′)l= m(χθ′) + θ(χl)= mθχ+ θmχ= [m,θ]χ = 0because χ ◦ l = m ◦ χ from the commutative diagram and [m,θ] = 0 because θ isa strong homotopy derivation of an A∞ algebra. Because χ is injective, it followsthat [l,θ′] = 0. Acknowledgement. We thank the referee for the careful reading of this articleand for the helpful corrections.References[1] Allocca, M., Lada, T., A finite dimensional A∞ algebra example, Georgian Math. J. 12 (10)(2010), 1–12.[2] Kajiura, H., Stasheff, J., Homotopy algebras inspired by classical open–closed string fieldtheory, Comm. Math. Phys. 263 (3) (2006), 553–581.DERIVATIONS OF HOMOTOPY ALGEBRAS 315[3] Lada, T., Commutators of A∞ structures, Contemporary Mathematics, 1999, pp. 227–233.[4] Lada, T., Markl, M., Strongly homotopy Lie algebras, Comm. Algebra 23 (6) (1995), 2147–2161.[5] Lada, T., Stasheff, J., Introduction to SH Lie algebras for physicists, Internat. J. Theoret.Phys. 32 (7) (1993), 1087–1103.[6] Stasheff, J., Homotopy associativity of H-spaces II, Trans. Amer. Math. Soc. 108 (1963),293–312.[7] Tolley, M., The connections between A∞ and L∞ algebras, Ph.D. thesis, NCSU, 2013.Department of Mathematics,North Carolina State University,Raleigh, NC 27695E-mail: lada@math.ncsu.edu mmtolley@ncsu.edu

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Homotopy algebras inspired by classical open-closed string field theory

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