Lie Algebra Structure on Hochschild Cohomology Tolulope Oke Texas A&M University Early Commutative Algebra Researchers (eCARs) Conference June 27-28, 2020 1
This talk is organized in the following way ❼ MOTIVATION ❼ HOCHSCHILD COHOMOLOGY ❼ QUIVER & KOSZUL ALGEBRAS ❼ HOMOTOPY LIFTING MAPS ❼ EXAMPLES & APPLICATIONS 2
Motivation
Let k be a field of characteristic 0. Defnition: A differential graded Lie algebra (DGLA) over k is a i ∈ I L i with a bilinear map graded vector space L = � [ · , · ] : L i ⊗ L j → L i + j and a differential d : L i → L i +1 such that ❼ bracket is anticommutative i.e. [ x , y ] = − ( − 1) | x || y | [ y , x ] ❼ bracket satisfies the Jacobi identity i.e. ( − 1) | x || z | [ x , [ y , z ]]+( − 1) | y || x | [ y , [ z , x ]]+( − 1) | z || y | [ z , [ x , y ]] = 0 ❼ bracket satisfies the Liebniz rule i.e. d [ x , y ] = [ d ( x ) , y ] + ( − 1) | x | [ x , d ( y )] 3
Examples 1 Every Lie algebra is a DGLA concentrated in degree 0. i A i be an associative graded-commutative 2 Let A = � k -algebra i.e. ab = ( − 1) | a || b | ba for a , b homogeneous and i L i a DGLA. Then L ⊗ k A has a natural structure of L = � DGLA by setting: ( L ⊗ k A ) n = ( L i ⊗ k A n − i ) , d ( x ⊗ a ) = d ( x ) ⊗ a , � i [ x ⊗ a , y ⊗ b ] = ( − 1) | a || y | [ x , y ] ⊗ ab . 3 Space of Hochschild cochains C ∗ (Λ , M ) of an algebra Λ is a DGLA where [ · , · ] is the Gerstenhaber bracket, and M a Λ-bimodule. 4
Deformation philosophy Over a field of characteristic 0, it is well known that every deformation problem is governed by a differential graded Lie algebra (DGLA) via solutions of the Maurer-Cartan equation modulo gauge action.[6] { Deformation problem } � { DGLA } � { Deformation functor } The first arrow is saying that the DGLA you obtain depends on the data from the deformation problem and the second arrow is saying for DGLAs that are quasi-isomorphic, we obtain an isomorphism of deformation functor. Definition: An element x of a DGLA is said to satisfy the Maurer-Cartan equation if d ( x ) + 1 2[ x , x ] = 0 . 5
Hochschild cohomology
Hochschild cohomology Let B = B • (Λ) denote the bar resolution of Λ. Λ e = Λ ⊗ Λ op the enveloping algebra of Λ. → Λ ⊗ ( n +1) → · · · B : · · · → Λ ⊗ ( n +2) δ n → Λ ⊗ 3 δ 1 δ 2 → Λ ⊗ 2 ( π → Λ) The differentials δ n ’s are given by n � ( − 1) i a 0 ⊗ · · · ⊗ a i a i +1 ⊗ · · · ⊗ a n +1 δ n ( a 0 ⊗ a 1 ⊗ · · · ⊗ a n +1 ) = i =0 for each elements a i ∈ Λ (0 ≤ i ≤ n + 1) and π , the multplication map. 6
Hochschild cohomology Let B = B • (Λ) denote the bar resolution of Λ. Λ e = Λ ⊗ Λ op the enveloping algebra of Λ. → Λ ⊗ ( n +1) → · · · B : · · · → Λ ⊗ ( n +2) δ n → Λ ⊗ 3 δ 1 δ 2 → Λ ⊗ 2 ( π → Λ) The differentials δ n ’s are given by n � ( − 1) i a 0 ⊗ · · · ⊗ a i a i +1 ⊗ · · · ⊗ a n +1 δ n ( a 0 ⊗ a 1 ⊗ · · · ⊗ a n +1 ) = i =0 for each elements a i ∈ Λ (0 ≤ i ≤ n + 1) and π , the multplication map.Let M be a left Λ e -module. The Hochschild cohomology of Λ with coefficients in M is defined as � HH ∗ (Λ , M ) = C ∗ (Λ , M ) = H n ( Hom Λ e ( B • (Λ) , M )) n ≥ 0 If M = Λ, we write HH ∗ (Λ). 6
Multiplicative structures on HH ∗ (Λ) ❼ Cup product � : HH m (Λ) × HH n (Λ) → HH m + n (Λ) α � β ( a 1 ⊗· · ·⊗ a m + n ) = ( − 1) mn α ( a 1 ⊗· · ·⊗ a m ) β ( a m +1 ⊗· · ·⊗ a m + n ) ❼ Gerstenhaber bracket of degree − 1. [ · , · ] : HH m (Λ) × HH n (Λ) → HH m + n − 1 (Λ) defined originally on the bar resolution by [ α, β ] = α ◦ β − ( − 1) ( m − 1)( n − 1) β ◦ α where where α ◦ β = � m j =1 ( − 1) ( n − 1)( j − 1) α ◦ j β with ( α ◦ j β )( a 1 ⊗ · · · ⊗ a m + n − 1 ) = α ( a 1 ⊗ · · · ⊗ a j − 1 ⊗ β ( a j ⊗ · · · ⊗ a j + n − 1 ) ⊗ a j + n ⊗ · · · ⊗ a m + n − 1 ) . (1) 7
Make sense of Equation (1) without using B ❼ Hochschild cohomology as the Lie algebra of the derived Picard group (B. Keller) - 2004 ❼ Brackets via contracting homotopy using certain resolutions (C. Negron and S. Witherspoon) - 2014 [ α, β ] = α ◦ φ β − ( − 1) ( m − 1)( n − 1) β ◦ φ α ❼ Completely determine [ HH 1 ( A ) , HH m ( A )] using derivation arez-´ operators on any resolution P . (M. Su´ Alvarez) - 2016 [ α 1 , β ] = α 1 β − β ˜ α m where ˜ α m : P m → P m . ❼ Completely determine [ HH ∗ ( A ) , HH ∗ ( A )] using homotopy lifting on any resolution. (Y. Volkov) - 2016 [ α, β ] = αψ β − ( − 1) ( m − 1)( n − 1) βψ α 8
Quiver algebras and Koszul algebras
Quiver algebras A quiver is a directed graph where loops and multiple arrows between vertices are allowed. It is often denoted by Q = ( Q 0 , Q 1 , o , t ), where Q 0 is the set of vertices, Q 1 set of arrows and o , t : Q 1 → Q 0 taking every path a ∈ Q to its origin vertex o ( a ) and terminal vertex t ( a ). Define kQ to be the vector k -vector space having the set of all paths as its basis. If p and q are two paths, we say pq is possible if t ( p ) = o ( q ) otherwise, pq = 0. By this, kQ becomes an associative algebra. Let kQ i be a vector subspace spanned by all paths of length i , then kQ is graded. � kQ = kQ n n ≥ 0 9
Examples of quiver algebras ❼ Let Q be the quiver with a vertex 1 (with a trivial path e 1 of length 0). Then kQ ∼ = k . ❼ Let Q be the quiver with two vertices and a path: 1 α → 2. There are two trivial paths e 1 and e 2 associated with the vertices 1 , 2. There is a relation e 1 α = e 1 α e 2 = α e 2 . Define a � � � � 0 0 1 0 map kQ → M 2 ( k ), by e 1 �→ , e 2 �→ and 0 1 0 0 � � 0 0 . Then kQ ∼ α �→ = { A ∈ M 2 ( k ) : A 12 = 0 } . 1 0 ❼ Let Q be the quiver with a vertex and 3 paths x , y , z . x z Then kQ ∼ 1 = k � x , y , z � . y 10
Koszul algebras A relation on Q is a k -linear combination of paths of length n ≥ 2 having same origin and terminal vertex. Let I be the subspace spanned by some relations, we denote by ( Q , I ) a quiver with relations and kQ / I the quiver algebra associated to ( Q , I ). We are interested in quiver algebras that are Koszul. Let Λ = kQ / I be Koszul: ❼ Λ is quadratic. This means that I is a homogenous admissible ideal of kQ 2 ❼ Λ admits a grading Λ = � i ≥ 0 Λ i , Λ 0 is isomorphic to k or copies of k and has a minimal graded free resolution. 11
❼ ❼ A canonical construction of a projective resolution for Koszul quiver algebras Let L − → Λ 0 be a minimal projective resolution of Λ 0 as a right Λ-module, L ❼ contains all the necessary information needed to construct a minimal projective resolution of Λ 0 as a left Λ-module ❼ contains all the necessary information to construct a minimal projective resolution of Λ over the enveloping algebra Λ e . 12
A canonical construction of a projective resolution for Koszul quiver algebras Let L − → Λ 0 be a minimal projective resolution of Λ 0 as a right Λ-module, L ❼ contains all the necessary information needed to construct a minimal projective resolution of Λ 0 as a left Λ-module ❼ contains all the necessary information to construct a minimal projective resolution of Λ over the enveloping algebra Λ e . ❼ There exist integers { t n } n ≥ 0 and elements { f n i } t n i =0 in R = kQ such that L can be given in terms of a filtration of right ideals t n − 1 t n t 0 � � f n − 1 � f 0 f n · · · ⊆ i R ⊆ R ⊆ · · · ⊆ i R = R i i =0 i =0 i =0 ❼ The f n can be choosen so that they satisfy a comultiplicative i structure. 12
A result of E.L. Green, G. Hartman, E. Marcos, Ø. Solberg [2] Theorem 1 Let Λ = kQ / I be a Koszul algebra. Then for each r , with 0 ≤ r ≤ n , and i , with 0 ≤ i ≤ t n , there exist elements c pq ( n , i , r ) in k such that for all n ≥ 1 , t n − r t r f n � � c pq ( n , i , r ) f r p f n − r i = ( comultiplicative structure ) q p =0 q =0 Theorem 2 Let Λ = kQ / I be a Koszul algebra. The resolution ( K , d ) is a minimal projective resolution of Λ with Λ e -modules t n � Λ o ( f n i ) ⊗ k t ( f n K n = i )Λ i =0 with each K n having free basis elements { ε n i } t n i =0 and they are given explicitly by ε n i = (0 , . . . , 0 , o ( f n i ) ⊗ k t ( f n i ) , 0 , . . . , 0). 13
Homotopy lifting maps
Making sense of Equation (1) using homotopy lifting Definition µ → Λ be a projective resolution of Λ as Λ e -module. Let Let K − ∆ : K − → K ⊗ Λ K be a chain map lifting the identity map on Λ and η ∈ Hom Λ e ( K n , Λ) a cocycle. A module homomorphism ψ η : K − → K [1 − n ] is called a homotopy lifting map of η with respect to ∆ if d ψ η − ( − 1) n − 1 ψ η d = ( η ⊗ 1 − 1 ⊗ η )∆ and (2) ( − 1) n − 1 ηψ µψ η is cohomologous to (3) for some ψ : K → K [1] for which d ψ − ψ d = ( µ ⊗ 1 − 1 ⊗ µ )∆. Remark. For Koszul algebras Equation (3) holds. 14
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