Stasheffs A -algebras and Homotopy Gerstenhaber Algebras. Tornike Kadeishvili A. Razmadze Mathematical Institute of Tbilisi State University

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1 1 Stasheffs A -algebras and Homotopy Gerstenhaber Algebras Tornike Kadeishvili A. Razmadze Mathematical Institute of Tbilisi State University 1

2 Twisting Elements 2 A dg algebra (A, d : A A +1, : A A A ) Twisting element (E. Brown) a A 1, da = a a. For a dg A-module (M, d M, A M M) a twisting element a A determines on (M, d M ) new, perturbed differential d a (m) = d M (m) + a m, da = a a <=> d m d m = 0.

3 Perturbation of Twisting Elements (N. Berikashvili) 3 Let g A 0 be an invertible element. Then g a = g a g 1 + dg g 1 is a twisting element too and g : (M, d a ) (M, d g a ) is an isomorphism of dg modules. tw(a) = {a A 1, da = a a} D(A) = tw(a)/

4 4 Products in Hochschild Complex Let A be an algebra and M be A bimodule. The Hochschild cochain complex with differential C n (A; M) = Hom(A n, M) df(a 1... a n ) = a 1 f(a 2... a n )+ n 1 k=1 f(a 1... a k 1 a k a k+1.. a n )+ f(a 1... a n 1 ) a n.

5 We focus on the case M = A 5 Gerstenhaber s product on C (A, A) f g(a 1... a n+m ) = f(a 1... a n ) g(a n+1... a n+m ) satisfies Leibnitz rule d(f g) = df g + f dg

6 Gerstenhaber s Circle Product 6 : C n (A; A) C m (A; A) C n+m 1 (A; A) f g(a 1... a n+m 1 ) = k f(a 1... a k g(a k+1... a k+m )... a m+n 1 )

7 Gerstenhaber s Circle Product 7 In terms of this circle product the Hochschild differential df(a 1... a n ) = a 1 f(a 2... a n )+ n 1 k=1 f(a 1... a k 1 a k a k+1.. a n )+ f(a 1... a n 1 ) a n. can be rewritten as df = µ f + f µ where µ : A A A is the multiplication of A.

8 Stasheff s A -algebra 8 is a graded module M = {M k } k Z with a sequence of operations {m i : M...(i times)... M M, i = 1, 2, 3,...} satisfying the conditions m i (( i M)q) M q+i 2 and i 1 i k k=0 j=1 ± m i j+1 (a 1... a k m j (a k+1... a k+j ) a k+j+1... a i ) = 0.

9 Particularly: 9 m 1 : M p M p 1, m 2 : (M M) p M p, m 3 : (M M M) p M p+1, m 1 m 1 = 0 (so m 1 is a differential); m 1 m 2 (a 1 a 2 ) + m 2 (m 1 (a 1 ) a 2 ) + m 2 (a 1 m 1 (a 2 )) = 0 (Leibnitz rule); m 1 m 3 (a 1 a 2 a 3 )+m 3 (m 1 a 1 a 2 a 3 )+ m 3 (a 1 m 1 a 2 a 3 ) + m 3 (a 1 a 2 m 1 a 3 )+ m 2 (m 2 (a 1 a 2 ) a 3 ) + m 2 (a 1 m 2 (a 2 a 3 )) = 0 (m 2 is associative just up to homotopy m 3 )

10 A algebra (M, {m i } is minimal if m 1 = In this case m 2 (m 2 (a 1 a 2 ) a 3 ) + m 2 (a 1 m 2 (a 2 a 3 )) = m 1 m 3 (a 1 a 2 a 3 )+m 3 (m 1 a 1 a 2 a 3 )+ m 3 (a 1 m 1 a 2 a 3 ) + m 3 (a 1 a 2 m 1 a 3 )

11 A algebra (M, {m i } is minimal if m 1 = In this case Stasheff s defining condition gives m 2 (m 2 (a 1 a 2 ) a 3 ) + m 2 (a 1 m 2 (a 2 a 3 ) = 0 the operation m 2 is strictly associative, so (M, m 2 ) is an associative graded algebra.

12 Stasheff s next condition 12 m 2 (m 3 (a 1 a 2 a 3 ) a 4 ) + m 2 (a 1 m 3 (a 2 a 3 a 4 ))+ m 3 (m 2 (a 1 a 2 ) a 3 a 4 ) + m 3 (a 1 m 2 (a 2 a 3 ) a 4 )+ m 3 (a 1 a 2 m 2 (a 3 a 4 )) = 0 i.e. m 2 m 3 + m 3 m 2 = 0 is exactly dm 3 = 0 i.e. m 3 is a 3-cocycle in the Hochschild complex C (M, M) of our strictly associative algebra (M, m 2 ).

13 Next condition looks as 13 m 2 m 4 + m 3 m 3 + m 4 m 2 = 0 that is dm 4 = m 3 m 3

14 Generally, for a minimal A -algebra 14 (M, {m 1 = 0, m 2, m 3,... }) Stasheff s defining conditions look as dm = m m where m = m 3 + m m i +... C (M, M) that is m is a twisting cochain in the Hochschild complex of the associative algebra (M, m 2 ) with respect to Gerstenhabers circle product.

15 Equivalence of minimal A -algebras 15 Suppose we have two minimal A -algebra structures (M, {m 1 = 0, m 2 = µ, m 3, m 4,...}) and (M, {m 1 = 0, m 2 = µ, m 3, m 4,...}) both extending given associative algebra (M, µ).

16 Let us call them equivalent if there exists an (iso)morphism of A -algebras 16 {f i } : (M, {m i }) (M, {m i }), f 1 = id M which, by definition is a collection of homomorphisms {f i : i M M, i = 1, 2, 3,... } satisfying certain coherence conditions.

17 Both our minimal A algebras are represented by Hochschild twisting cochains from C (M, M) 17 m = m 3 + m 4 + m , dm = m m m = m 3 + m 4 + m , dm = m m So we need a notion of equivalence of Hochschild twisting cochains which describe isomorphic extended A -algebras.

18 Both our minimal A algebras are represented by Hochschild twisting cochains from C (M, M) 18 m = m 3 + m 4 + m , dm = m m m = m 3 + m 4 + m , dm = m m So we need a notion of equivalence of Hochschild twisting cochains which describes isomorphic extended A -algebras. For this only Gerstenhaber s circle product is not enough. Certain higher operations are needed, which form on the Hochschild complex a structure of homotopy Gerstenhaber algebra (hga).

19 Some further properties of circle product 20 This Gerstenhaber s circle product satisfies the conditions similar to Steenrod s 1 product Steenrod formula: d(f 1 g) = df 1 g + f 1 dg + f g + g f, Left Hirsh formula: (f g) 1 h = f (g 1 h) + (f 1 h) g

20 Left Hirsh formula 21 (f g) 1 h + f (g 1 h) + (f 1 h) g = 0 As for right Hirsh formula f 1 (g h) + g (f 1 h) + (f 1 g) h 0

21 Left Hirsh formula 22 (f g) 1 h + f (g 1 h) + (f 1 h) g = 0 As for right Hirsh formula f 1 (g h) + g (f 1 h) + (f 1 g) h = de 1,2 (f; g, h) + E 1,2 (f; dg, h) + E 1,2 (f; g, dh) where the homotopy E 1,2 : C p (A; A) C q (A; A) C r (A; A) C p+q+r 2 (A; A) is certain three argument operation which gives reise to hga structure on Hochschild complex.

22 Although the 1 = E 1,1 is not associative, it satisfies the pre- Jacobi identity 22 a 1 (b 1 c) (a 1 b) 1 c = a 1 (c 1 b) (a 1 c) 1 b which guarantees that the commutator [a, b] = a 1 b b 1 a satisfies the Jacobi identity, i.e. turns the Hochschild complex C (A, A) into a dg Lie algebra

23 Although the 1 = E 1,1 is not associative, it satisfies the pre- Jacobi identity 23 a 1 (b 1 c) (a 1 b) 1 c = a 1 (c 1 b) (a 1 c) 1 b which guarantees that the commutator [a, b] = a 1 b b 1 a satisfies the Jacobi identity, i.e. turns the Hochschild complex C (A, A) into a dg Lie algebra and induces on Hochschild cohomologies Hoch (A, A) = H (C (A, A) a structure of Gerstenhaber algebra

24 hga structure on Hochschild complex 24 The circle prduct a b = a 1 b = E 1,1 (a; b) and the mentioned operation E 1,2 (a; b, c) are parts of higher structure, a hga on the Hochshild complex C (A, A).

25 Homotopy Gerstenhaber Algebras (hga) (Gerstenhaber- Voronov) 25 A hga (C, d,, {E 1,1, E 1,2, E 1,3,...}) is defined as a differential graded algebra (dga) (C, d, ) together with sequence of operations E 1,k : C C k C, k = 1, 2, 3,... subject of some coherency conditions. Notation E 1,k (a; b 1,..., b k ) = a 1 {b 1,..., b k } = a{b 1,..., b k }

26 26 de 1,1 (a; b) + E 1,1 (da; b) + E 1,1 (a; db) = a b b a, E 1,1 = 1 d(a 1 b) + da 1 b + a 1 db = a b b a (a b) 1 c + a (b 1 c) + (a 1 c) b = 0, a 1 (b c) + b (a 1 c) + (a 1 b) c = de 1,2 (a; b, c) + E 1,2 (da; b, c) + E 1,2 (a; db, c) + E 1,2 (a; b, dc), (a 1 b) 1 c a 1 (b 1 c) = E 1,2 (a; b, c) + E 1,2 (a; c, b), (a 1 b) 1 c a 1 (b 1 c) = (a 1 c) 1 b a 1 (c 1 b)

27 hga as operad 27 The operations E 1,k forming hga have nice description in terms of the surjection operad χ. Namely, to the dot product corresponds (1, 2) χ 0 (2); to E 1,1 = 1 corresponds (1, 2, 1) χ 1 (2); and generally to the operation E 1,k corresponds E 1,k = (1, 2, 1, 3,..., 1, k, 1, k + 1, 1) χ k (k + 1).

28 Three Aspects of hga 28 A hga is a 1. Homotopy Gerstenhaber algebra; 2. B algebra; 3. Strong homotopy commutative algebra.

29 Three Examples of hga Cochain complex of a topological space (C (X), d,, {E 1,1 = 1, E 1,2, E 1,3,... }) The starting operation E 1,1 is the classical Steenrod s 1 product. The existence of higher operations E 1,k>1 follows from the diagonal constructed on the cobar construction ΩC (X) by Baues.

30 2. The cobar construction of a dg bialgebra 29 For a dg coalgebra (C, d : C C, : C C C) its cobar construction ΩC of a is a dg aalgebra. If C is additionally equipped with a multiplication µ : C C C turning it into a dg bialgebra, how this structure reflects on the cobar construction ΩC? In [Kade] it is shown that µ gives rise to a hga structure on ΩC. And again the starting operation E 1,1 is classical: it is Adams s 1 -product defined for ΩC using the multiplication of C.

31 3. Hochschild cochain complex C (A, A) of an associative algebra A 30 (C (A, A), d,, {E 1,1 =, E 1,2, E 1,3,... }) The starting operation E 1,1 is the classical Gerstenhaber s circle product which is sort of 1 -product. The higher operations E 1,k (now called brace operations) were defined with the purpose to describe A( )-algebras in terms of Hochschild cochains.

32 hga operations in Hochschild complex 31 E 1,1 (f; g)(a 1... a n+m 1 ) = f g(a 1... a n+m 1 ) = f 1 g(a 1... a n+m 1 ) = k f(a 1... a k g(a k+1... a k+m ) a k+m+1... a n+m 1 )

33 31 E 1,2 (f; g, h)(a 1... a p+q+r 2 ) = f 1 (g, h)(a 1... a p+q+r 2 ) = k,l f(a 1... a k g(a k+1... a k+q ) a k+m+1... a l h(a l+1... a l+r ) a l+r+1... a p+q+r 2 ).

34 32 E 1,i (f; g 1,..., g i ) = f 1 (g 1,..., g i )(a 1... a n ) = k1,...,k i f(a 1... a k1 g 1 (a k1 +1..a k1 +n 1 )... a ki g i (a ki +1...a ki +n i )... a n ).

35 So, the Hochschild comlex is a hga 33 (C (A, A), d,, {E 1,1 = = 1, E 1,2, E 1,3,... }). This structure is involved in perturbation of Hochschild twisting cochains:

36 Let 34 m = m 3 + m m p +... ; m p C p (A, A), dm = m 1 m and let g = g 2 + g g p +... ; g p C p. Then m = m 3 + m m p +... ; m p C p (A, A) given by m = m + dg + g g + E 1,1 (g; m) + satisfies dm = m 1 m k=1 E 1,k (m ; g,..., g)

37 Particularly 36 m 3 = m 3 + dg 2 ; m 4 = m 4 + dg 3 + g 2 g 2 + g 2 1 m 3 + m 3 1 g 2 ; m 5 = m 5 + dg 4 + g 2 g 3 + g 3 g 2 + g 2 1 m 4 + g 3 1 m 3 + m 3 1 g 3 + m 4 1 g 2 + E 1,2 (m 3 ; g 2, g 2 ). (this is inductive formula)

38 This action allows us to perturb twisting elements in the following sense: 37 Let g n C n be an arbitrary element, then for g = g n the twisting element m looks as m = m m n + (m n+1 + dg n ) + m n+2 + m n so the components m 3,..., m n remain unchanged and m n+1 = m n+1 + dg n

39 We use this perturbation to prove the following 38 Theorem. If for an associative algebra (M, µ) all Hochshild homology modules Hoch 3 (M, M) = H 3 (C (M, M)) = 0 then each Hochschild twisting element m = m 3 + m m p +... ; m p C p (M, M) is equivalent to m = 0.

40 Proof. 38 m = m 3 + m 4 + m

41 Proof. 38 m = m 3 + m 4 + m dg 2

42 Proof. 38 m = 0 + m 4 + m

43 Proof. 38 m = 0 + m 4 + m dg 3

44 Proof. 38 m = m

45 Proof. 38 m = m dg 4

46 39 Corollary. homology modules If for an associative algebra (M, µ) all Hochshild Hoch 3 (M, M) = H 3 (C (M, M)) = 0 then each minimal A -algebra structure (M, {m 1 = 0, m 2 = µ, m 3, m 4,...}) which extends µ, is trivial, i.e. is isomorphic as A -algebra to (M, {0, µ, 0, 0,...})

47 Minimality theorem [Kade] 40 For a dga (A, d, ) in homology H(A) there exists a structure of minimal A -algebra (H(A), {m i }) such that (A, {m 1 = d, m 2 =, m 3 = 0, m 4 = 0,...) (H(A), {m i }) in the category of A -algebras. The A -algebra (H(A), {m i }) is called a minimal model of a dga (A, d, ). Main property of minimal model: A A iff (H(A), {m i }) (H(A ), {m i })

48 41 Generalizations: Minimal model of an A algebra: (M, {m i }) (H(M, m 1 ), {m i }) [Kade] Minimal model of an A category: A category minimal A category [Fukaya] Minimal model of a commutative dg algebra: (A, d, ) minimal C algebra [Kade], [Smirnov], [Markl] Minimal model of a dg Lie algebra: (L, d, [, ]) minimal L algebra [Huebschman-Stasheff], [Kontsevich] Minimal model of a dg Hopf algebra: (A, d,, ) minimal H algebra [Saneblidze-Umble]

49 Some applications 42 Cohomology A -algebra (H (X), {m i }) determines H (ΩX), when only cohomology algebra (H (X), m 2 ) does not. When it does? Obstruction for formality are in Hochschild cohomologies Hoch (H (X), H (X)).

50 43 Rational cohomology C -algebra (H (X, Q), {m i }) determines the rational homotopy type of X, when only rational cohomology algebra (H (X, Q), m 2 ) does not. When it does? Obstruction to formality are in Harrison cohomologies Harr (H (X, Q), H (X, Q)) (Tanre).

51 Deformation of an algebra (A, ) 44 B i : A A A, i = 0, 1, 2,...; B 0 (a b) = a b i+j=n B i (a B j (b c)) = i+j=n B i (B j (a b) c) B = B 1 + b , db = B 1 B Equivalence: {B i } {B i } if {g i : A A; i = 0, 1, 2,...; g 0 = id} s.t. r+s=n g r (B s (a b)) = i+j+k=n B i (g j(a) g k (b)) B = B + δg + g g + g 1 B + E 1,1 (B ; g) + E 1,2 (B ; g, g) B = B + δg + g g + g 1 B + k=1 E 1,k (B ; g,..., g))

52 C 2, 3 b 3 C 2,3 C 3,3 C 4,3 C 5,3 C 2, 3 b 2 C 2,2 C 3,2 C 4,2 C 5,2 C 2, 3 b 1 C 2,1 C 3,1 C 4,1 C 5,1 C 2, 3 C 2,0 C 3,0 C 4,0 C 5,0 C 2, 3 C 2, 1 m 3 C 3, 1 C 4, 1 C 5, 1 C 2, 3 C 2, 2 C 3, 2 m 4 C 4, 2 C 5, 2 C 2, 3 C 2, 3 C 3, 3 C 4, 3 m 5 C 5, Twisting cochains reprezenting: Stasheff s min. A algebras : m = m 3 + m dm = m 1 m, Gerstenhabers s deformations : b = b 1 + b db = b 1 b

53 Congratulations!!! 45