The Gerstenhaber bracketon Hochschild cohomology

Sarah WitherspoonTexas A&M University

Definition of Hochschild cohomology

R - algebra over a field k, Re = R⊗Rop, ⊗ = ⊗kHH∗(R) = Ext∗R⊗Rop(R,R)

There are binary operations ^ and [ , ] on HH∗(R) defined via the barresolution:Let Pn = R⊗ · · · ⊗R (n+ 2 factors),dn(r0 ⊗ · · · ⊗ rn+1) =

∑ni=0(−1)ir0 ⊗ · · · ⊗ riri+1 ⊗ · · · rn+1.

Let f ∈ HomRe(R⊗(m+2), R) ∼= Homk(R⊗m, R), g ∈ Homk(R⊗n, R)

(f ^ g)(r1⊗ · · · ⊗ rm+n) = (−1)mnf(r1⊗ · · · ⊗ rm)g(r1⊗ · · · ⊗ rn),

[f, g] = f ◦ g − (−1)(m−1)(n−1)g ◦ f where

(f ◦ g)(r1 ⊗ · · · ⊗ rm+n−1) =m∑i=1

(−1)(n−1)(i−1)·

f(r1 ⊗ · · · ⊗ ri−1 ⊗ g(ri ⊗ · · · ⊗ ri+n−1)⊗ ri+n ⊗ · · · ⊗ rm+n−1)

Definition of Hochschild cohomology

R - algebra over a field k, Re = R⊗Rop, ⊗ = ⊗kHH∗(R) = Ext∗R⊗Rop(R,R)

There are binary operations ^ and [ , ] on HH∗(R) defined via the barresolution:Let Pn = R⊗ · · · ⊗R (n+ 2 factors),dn(r0 ⊗ · · · ⊗ rn+1) =

∑ni=0(−1)ir0 ⊗ · · · ⊗ riri+1 ⊗ · · · rn+1.

Let f ∈ HomRe(R⊗(m+2), R) ∼= Homk(R⊗m, R), g ∈ Homk(R⊗n, R)

(f ^ g)(r1⊗ · · · ⊗ rm+n) = (−1)mnf(r1⊗ · · · ⊗ rm)g(r1⊗ · · · ⊗ rn),

[f, g] = f ◦ g − (−1)(m−1)(n−1)g ◦ f where

(f ◦ g)(r1 ⊗ · · · ⊗ rm+n−1) =m∑i=1

(−1)(n−1)(i−1)·

f(r1 ⊗ · · · ⊗ ri−1 ⊗ g(ri ⊗ · · · ⊗ ri+n−1)⊗ ri+n ⊗ · · · ⊗ rm+n−1)

Definition of Hochschild cohomology

R - algebra over a field k, Re = R⊗Rop, ⊗ = ⊗kHH∗(R) = Ext∗R⊗Rop(R,R)

There are binary operations ^ and [ , ] on HH∗(R) defined via the barresolution:Let Pn = R⊗ · · · ⊗R (n+ 2 factors),dn(r0 ⊗ · · · ⊗ rn+1) =

∑ni=0(−1)ir0 ⊗ · · · ⊗ riri+1 ⊗ · · · rn+1.

Let f ∈ HomRe(R⊗(m+2), R) ∼= Homk(R⊗m, R), g ∈ Homk(R⊗n, R)

(f ^ g)(r1⊗ · · · ⊗ rm+n) = (−1)mnf(r1⊗ · · · ⊗ rm)g(r1⊗ · · · ⊗ rn),

[f, g] = f ◦ g − (−1)(m−1)(n−1)g ◦ f where

(f ◦ g)(r1 ⊗ · · · ⊗ rm+n−1) =m∑i=1

(−1)(n−1)(i−1)·

f(r1 ⊗ · · · ⊗ ri−1 ⊗ g(ri ⊗ · · · ⊗ ri+n−1)⊗ ri+n ⊗ · · · ⊗ rm+n−1)

Definition of Hochschild cohomology

R - algebra over a field k, Re = R⊗Rop, ⊗ = ⊗kHH∗(R) = Ext∗R⊗Rop(R,R)

There are binary operations ^ and [ , ] on HH∗(R) defined via the barresolution:Let Pn = R⊗ · · · ⊗R (n+ 2 factors),dn(r0 ⊗ · · · ⊗ rn+1) =

∑ni=0(−1)ir0 ⊗ · · · ⊗ riri+1 ⊗ · · · rn+1.

Let f ∈ HomRe(R⊗(m+2), R) ∼= Homk(R⊗m, R), g ∈ Homk(R⊗n, R)

(f ^ g)(r1⊗ · · · ⊗ rm+n) = (−1)mnf(r1⊗ · · · ⊗ rm)g(r1⊗ · · · ⊗ rn),

[f, g] = f ◦ g − (−1)(m−1)(n−1)g ◦ f where

(f ◦ g)(r1 ⊗ · · · ⊗ rm+n−1) =m∑i=1

(−1)(n−1)(i−1)·

f(r1 ⊗ · · · ⊗ ri−1 ⊗ g(ri ⊗ · · · ⊗ ri+n−1)⊗ ri+n ⊗ · · · ⊗ rm+n−1)

Gerstenhaber algebra structure

^ and [ , ] induce operations

HHm(R)×HHn(R)^−−→ HHm+n(R) = Ext∗Re(R,R)

HHm(R)×HHn(R)[ , ]−−→ HHm+n−1(R)

HH∗(R) is a Gerstenhaber algebra:• under ^, it is associative and graded commutative• under [ , ], it is a graded Lie algebra• ∀α, [ , α] is a graded derivation with respect to ^

Gerstenhaber algebra structure

^ and [ , ] induce operations

HHm(R)×HHn(R)^−−→ HHm+n(R) = Ext∗Re(R,R)

HHm(R)×HHn(R)[ , ]−−→ HHm+n−1(R)

HH∗(R) is a Gerstenhaber algebra:• under ^, it is associative and graded commutative• under [ , ], it is a graded Lie algebra• ∀α, [ , α] is a graded derivation with respect to ^

ObservationOf the two bilinear operations on Hochschild cohomology HH∗(R),[ , ] has been harder to understand than ^

At the same time, [ , ] is important in algebraic deformation theory,formality, Deligne’s Conjecture . . .

ProblemUnderstand [ , ] better

ObservationOf the two bilinear operations on Hochschild cohomology HH∗(R),[ , ] has been harder to understand than ^

At the same time, [ , ] is important in algebraic deformation theory,formality, Deligne’s Conjecture . . .

ProblemUnderstand [ , ] better

ProblemDefine [ , ] on HH∗(R) := ⊕n≥0HHn(R)

in terms of an arbitrary projective resolution P of R as R⊗Rop-module

Some results on this and related problems• Schwede 1998; Hermann 2014

realized [ , ] as loops in extension categories• Keller 2004

HH∗(R) is the Lie algebra of the derived Picard group of R• Negron-W 2016

[ , ] defined under some conditions on P• Suarez-Alvarez 2017

[ , ] defined on HH1(R)×HHn(R), arbitrary P• Volkov 2019

[ , ] defined on HHm(R)×HHn(R), arbitrary P

ProblemDefine [ , ] on HH∗(R) := ⊕n≥0HHn(R)

in terms of an arbitrary projective resolution P of R as R⊗Rop-module

Some results on this and related problems• Schwede 1998; Hermann 2014

realized [ , ] as loops in extension categories• Keller 2004

HH∗(R) is the Lie algebra of the derived Picard group of R• Negron-W 2016

[ , ] defined under some conditions on P• Suarez-Alvarez 2017

[ , ] defined on HH1(R)×HHn(R), arbitrary P• Volkov 2019

[ , ] defined on HHm(R)×HHn(R), arbitrary P

ProblemDefine [ , ] on HH∗(R) := ⊕n≥0HHn(R)

in terms of an arbitrary projective resolution P of R as R⊗Rop-module

Some results on this and related problems• Schwede 1998; Hermann 2014

realized [ , ] as loops in extension categories• Keller 2004

HH∗(R) is the Lie algebra of the derived Picard group of R• Negron-W 2016

[ , ] defined under some conditions on P• Suarez-Alvarez 2017

[ , ] defined on HH1(R)×HHn(R), arbitrary P• Volkov 2019

[ , ] defined on HHm(R)×HHn(R), arbitrary P

ProblemDefine [ , ] on HH∗(R) := ⊕n≥0HHn(R)

in terms of an arbitrary projective resolution P of R as R⊗Rop-module

Some results on this and related problems• Schwede 1998; Hermann 2014

realized [ , ] as loops in extension categories• Keller 2004

HH∗(R) is the Lie algebra of the derived Picard group of R• Negron-W 2016

[ , ] defined under some conditions on P• Suarez-Alvarez 2017

[ , ] defined on HH1(R)×HHn(R), arbitrary P• Volkov 2019

[ , ] defined on HHm(R)×HHn(R), arbitrary P

ProblemDefine [ , ] on HH∗(R) := ⊕n≥0HHn(R)

in terms of an arbitrary projective resolution P of R as R⊗Rop-module

Some results on this and related problems• Schwede 1998; Hermann 2014

realized [ , ] as loops in extension categories• Keller 2004

HH∗(R) is the Lie algebra of the derived Picard group of R• Negron-W 2016

[ , ] defined under some conditions on P• Suarez-Alvarez 2017

[ , ] defined on HH1(R)×HHn(R), arbitrary P• Volkov 2019

[ , ] defined on HHm(R)×HHn(R), arbitrary P

ProblemDefine [ , ] on HH∗(R) := ⊕n≥0HHn(R)

in terms of an arbitrary projective resolution P of R as R⊗Rop-module

Some results on this and related problems• Schwede 1998; Hermann 2014

realized [ , ] as loops in extension categories• Keller 2004

HH∗(R) is the Lie algebra of the derived Picard group of R• Negron-W 2016

[ , ] defined under some conditions on P• Suarez-Alvarez 2017

[ , ] defined on HH1(R)×HHn(R), arbitrary P• Volkov 2019

[ , ] defined on HHm(R)×HHn(R), arbitrary P

Toward new techniques for [ , ]

Remark Often one computes HH∗(R) using a “nice” resolution, and thencomputes [ , ] via explicit chain maps to and from the bar resolutionwhere [ , ] is defined. This is hard.

An alternate approachAssume P has a coalgebra structure, that is, there is a chain map∆ : P→ P⊗R P lifting the identity map on R ∼= R⊗R R that is• coassociative (i.e. (∆⊗ 1) ◦∆ = (1⊗∆) ◦∆) and• counital (i.e. (ε⊗ 1) ◦∆ = 1 = (1⊗ ε) ◦∆)

Remark If R is Koszul, then its Koszul resolution satisfies thisassumption (Buchweitz-Green-Snashall-Solberg ’08)

Toward new techniques for [ , ]

Remark Often one computes HH∗(R) using a “nice” resolution, and thencomputes [ , ] via explicit chain maps to and from the bar resolutionwhere [ , ] is defined. This is hard.

An alternate approachAssume P has a coalgebra structure, that is, there is a chain map∆ : P→ P⊗R P lifting the identity map on R ∼= R⊗R R that is• coassociative (i.e. (∆⊗ 1) ◦∆ = (1⊗∆) ◦∆) and• counital (i.e. (ε⊗ 1) ◦∆ = 1 = (1⊗ ε) ◦∆)

Remark If R is Koszul, then its Koszul resolution satisfies thisassumption (Buchweitz-Green-Snashall-Solberg ’08)

Toward new techniques for [ , ]

Remark Often one computes HH∗(R) using a “nice” resolution, and thencomputes [ , ] via explicit chain maps to and from the bar resolutionwhere [ , ] is defined. This is hard.

An alternate approachAssume P has a coalgebra structure, that is, there is a chain map∆ : P→ P⊗R P lifting the identity map on R ∼= R⊗R R that is• coassociative (i.e. (∆⊗ 1) ◦∆ = (1⊗∆) ◦∆) and• counital (i.e. (ε⊗ 1) ◦∆ = 1 = (1⊗ ε) ◦∆)

Remark If R is Koszul, then its Koszul resolution satisfies thisassumption (Buchweitz-Green-Snashall-Solberg ’08)

Toward new techniques for [ , ]

Remark Often one computes HH∗(R) using a “nice” resolution, and thencomputes [ , ] via explicit chain maps to and from the bar resolutionwhere [ , ] is defined. This is hard.

An alternate approachAssume P has a coalgebra structure, that is, there is a chain map∆ : P→ P⊗R P lifting the identity map on R ∼= R⊗R R that is• coassociative (i.e. (∆⊗ 1) ◦∆ = (1⊗∆) ◦∆) and• counital (i.e. (ε⊗ 1) ◦∆ = 1 = (1⊗ ε) ◦∆)

Remark If R is Koszul, then its Koszul resolution satisfies thisassumption (Buchweitz-Green-Snashall-Solberg ’08)

Diagonal map and algebraic structure of HH∗(R)

The cup product f ^ g can be given as a composition of maps

P ∆−→ P⊗R P f⊗g−−→ R⊗R R∼−→ R

Theorem (Negron-W 2016) Assume P µ−→ R has a coalgebra structure.Define a circle product f ◦ g to be a composition of maps

P ∆(2)−−→ P⊗R P⊗R P 1⊗g⊗1−−−−→ P⊗R P φ−→ P f−→ R,

where d(φ) = µ⊗1−1⊗µ. Then [f, g] = f ◦ g− (−1)(m−1)(n−1)g ◦ frepresents the Gerstenhaber bracket on Hochschild cohomology.

Diagonal map and algebraic structure of HH∗(R)

The cup product f ^ g can be given as a composition of maps

P ∆−→ P⊗R P f⊗g−−→ R⊗R R∼−→ R

Theorem (Negron-W 2016) Assume P µ−→ R has a coalgebra structure.Define a circle product f ◦ g to be a composition of maps

P ∆(2)−−→ P⊗R P⊗R P 1⊗g⊗1−−−−→ P⊗R P φ−→ P f−→ R,

where d(φ) = µ⊗1−1⊗µ. Then [f, g] = f ◦ g− (−1)(m−1)(n−1)g ◦ frepresents the Gerstenhaber bracket on Hochschild cohomology.

Application to polynomial rings extended by finite groups

S = k[x1, . . . , xn] with action of a finite group GR = S oG, a ring with elements

∑g∈G sgg and multiplication

(sgg) · (rhh) := sg(g · rh)ghHH∗(R) ↪→

⊕g∈G

HH∗g(S)

α, β ∈ HH∗(R) may be written α =∑g∈G

αg, β =∑h∈G

βh

Theorem (Negron-W 2017) On HH∗(S oG), the bracket [ , ] is given by

[α, β] =∑

g,h∈Gpgh{αg, βh}

where { , } is the Schouten bracket on the space of multivector fieldsS ⊗

∧(V ∗) ∼= HH∗(S) and pgh is projection onto the gh-component,

where V is the vector space with basis x1, . . . , xn

Application to polynomial rings extended by finite groups

S = k[x1, . . . , xn] with action of a finite group GR = S oG, a ring with elements

∑g∈G sgg and multiplication

(sgg) · (rhh) := sg(g · rh)ghHH∗(R) ↪→

⊕g∈G

HH∗g(S)

α, β ∈ HH∗(R) may be written α =∑g∈G

αg, β =∑h∈G

βh

Theorem (Negron-W 2017) On HH∗(S oG), the bracket [ , ] is given by

[α, β] =∑

g,h∈Gpgh{αg, βh}

where { , } is the Schouten bracket on the space of multivector fieldsS ⊗

∧(V ∗) ∼= HH∗(S) and pgh is projection onto the gh-component,

where V is the vector space with basis x1, . . . , xn

Application to polynomial rings extended by finite groups

S = k[x1, . . . , xn] with action of a finite group GR = S oG, a ring with elements

∑g∈G sgg and multiplication

(sgg) · (rhh) := sg(g · rh)ghHH∗(R) ↪→

⊕g∈G

HH∗g(S)

α, β ∈ HH∗(R) may be written α =∑g∈G

αg, β =∑h∈G

βh

Theorem (Negron-W 2017) On HH∗(S oG), the bracket [ , ] is given by

[α, β] =∑

g,h∈Gpgh{αg, βh}

where { , } is the Schouten bracket on the space of multivector fieldsS ⊗

∧(V ∗) ∼= HH∗(S) and pgh is projection onto the gh-component,

where V is the vector space with basis x1, . . . , xn

Application to polynomial rings extended by finite groups

S = k[x1, . . . , xn] with action of a finite group GR = S oG, a ring with elements

∑g∈G sgg and multiplication

(sgg) · (rhh) := sg(g · rh)ghHH∗(R) ↪→

⊕g∈G

HH∗g(S)

α, β ∈ HH∗(R) may be written α =∑g∈G

αg, β =∑h∈G

βh

Theorem (Negron-W 2017) On HH∗(S oG), the bracket [ , ] is given by

[α, β] =∑

g,h∈Gpgh{αg, βh}

where { , } is the Schouten bracket on the space of multivector fieldsS ⊗

∧(V ∗) ∼= HH∗(S) and pgh is projection onto the gh-component,

where V is the vector space with basis x1, . . . , xn

Application to twisted tensor product algebras

Cap-Schichl-Vanzura 1995:

A, B - algebras over a field k, ⊗ = ⊗k

A k-linear map τ : B ⊗A→ A⊗B is a twisting map if the composition

A⊗B ⊗A⊗B 1⊗τ⊗1−−−−−→ A⊗A⊗B ⊗B mA⊗mB−−−−−→ A⊗B

defines an associative multiplication on A⊗B.

In this case, write A⊗τB, the twisted tensor product algebra

Examples: skew group algebras, smash/crossed products with Hopfalgebras, Ore extensions

Application to twisted tensor product algebras

Cap-Schichl-Vanzura 1995:

A, B - algebras over a field k, ⊗ = ⊗k

A k-linear map τ : B ⊗A→ A⊗B is a twisting map if the composition

A⊗B ⊗A⊗B 1⊗τ⊗1−−−−−→ A⊗A⊗B ⊗B mA⊗mB−−−−−→ A⊗B

defines an associative multiplication on A⊗B.

In this case, write A⊗τB, the twisted tensor product algebra

Examples: skew group algebras, smash/crossed products with Hopfalgebras, Ore extensions

Application to twisted tensor product algebras

Cap-Schichl-Vanzura 1995:

A, B - algebras over a field k, ⊗ = ⊗k

A k-linear map τ : B ⊗A→ A⊗B is a twisting map if the composition

A⊗B ⊗A⊗B 1⊗τ⊗1−−−−−→ A⊗A⊗B ⊗B mA⊗mB−−−−−→ A⊗B

defines an associative multiplication on A⊗B.

In this case, write A⊗τB, the twisted tensor product algebra

Examples: skew group algebras, smash/crossed products with Hopfalgebras, Ore extensions

Gerstenhaber brackets for twisted tensor product algebras

Let P,Q be Ae, Be-proj. resolutions of A,B, resp. Under somehypotheses, there is a twisted product resolution P ⊗τ Q of A⊗τ B.

Theorem (Karadag-McPhate-Ocal-Oke-W) Under some hypotheses,there is a map σ : (P ⊗τ Q)⊗A⊗τB (P ⊗τ Q)→ (P ⊗AP )⊗τ (Q⊗BQ)

for which φ := (φP ⊗ µQ ⊗ 1Q + 1P ⊗ µP ⊗ φQ)σ gives theGerstenhaber bracket via formula f ◦ g = fφ(1⊗ g ⊗ 1)∆(2).

Remark The hypotheses hold for bar resolutions, Koszul resolutions, andothers. The formula is used to compute Gerstenhaber brackets for:• Twisting given by bicharacter (Grimley-Nguyen-W 2017)• Skew group algebras (Shepler-W, arXiv 2019)• Jordan plane (Karadag-McPhate-Ocal-Oke-W, in preparation) Cf.Lopes-Solotar, arXiv 2019.

Gerstenhaber brackets for twisted tensor product algebras

Let P,Q be Ae, Be-proj. resolutions of A,B, resp. Under somehypotheses, there is a twisted product resolution P ⊗τ Q of A⊗τ B.

Theorem (Karadag-McPhate-Ocal-Oke-W) Under some hypotheses,there is a map σ : (P ⊗τ Q)⊗A⊗τB (P ⊗τ Q)→ (P ⊗AP )⊗τ (Q⊗BQ)

for which φ := (φP ⊗ µQ ⊗ 1Q + 1P ⊗ µP ⊗ φQ)σ gives theGerstenhaber bracket via formula f ◦ g = fφ(1⊗ g ⊗ 1)∆(2).

Remark The hypotheses hold for bar resolutions, Koszul resolutions, andothers. The formula is used to compute Gerstenhaber brackets for:• Twisting given by bicharacter (Grimley-Nguyen-W 2017)• Skew group algebras (Shepler-W, arXiv 2019)• Jordan plane (Karadag-McPhate-Ocal-Oke-W, in preparation) Cf.Lopes-Solotar, arXiv 2019.

Gerstenhaber brackets for twisted tensor product algebras

Let P,Q be Ae, Be-proj. resolutions of A,B, resp. Under somehypotheses, there is a twisted product resolution P ⊗τ Q of A⊗τ B.

Theorem (Karadag-McPhate-Ocal-Oke-W) Under some hypotheses,there is a map σ : (P ⊗τ Q)⊗A⊗τB (P ⊗τ Q)→ (P ⊗AP )⊗τ (Q⊗BQ)

for which φ := (φP ⊗ µQ ⊗ 1Q + 1P ⊗ µP ⊗ φQ)σ gives theGerstenhaber bracket via formula f ◦ g = fφ(1⊗ g ⊗ 1)∆(2).

Remark The hypotheses hold for bar resolutions, Koszul resolutions, andothers. The formula is used to compute Gerstenhaber brackets for:• Twisting given by bicharacter (Grimley-Nguyen-W 2017)• Skew group algebras (Shepler-W, arXiv 2019)• Jordan plane (Karadag-McPhate-Ocal-Oke-W, in preparation) Cf.Lopes-Solotar, arXiv 2019.

Gerstenhaber brackets for twisted tensor product algebras

Let P,Q be Ae, Be-proj. resolutions of A,B, resp. Under somehypotheses, there is a twisted product resolution P ⊗τ Q of A⊗τ B.

Theorem (Karadag-McPhate-Ocal-Oke-W) Under some hypotheses,there is a map σ : (P ⊗τ Q)⊗A⊗τB (P ⊗τ Q)→ (P ⊗AP )⊗τ (Q⊗BQ)

for which φ := (φP ⊗ µQ ⊗ 1Q + 1P ⊗ µP ⊗ φQ)σ gives theGerstenhaber bracket via formula f ◦ g = fφ(1⊗ g ⊗ 1)∆(2).

Remark The hypotheses hold for bar resolutions, Koszul resolutions, andothers. The formula is used to compute Gerstenhaber brackets for:• Twisting given by bicharacter (Grimley-Nguyen-W 2017)• Skew group algebras (Shepler-W, arXiv 2019)• Jordan plane (Karadag-McPhate-Ocal-Oke-W, in preparation) Cf.Lopes-Solotar, arXiv 2019.

Gerstenhaber brackets for twisted tensor product algebras

Let P,Q be Ae, Be-proj. resolutions of A,B, resp. Under somehypotheses, there is a twisted product resolution P ⊗τ Q of A⊗τ B.

Theorem (Karadag-McPhate-Ocal-Oke-W) Under some hypotheses,there is a map σ : (P ⊗τ Q)⊗A⊗τB (P ⊗τ Q)→ (P ⊗AP )⊗τ (Q⊗BQ)

for which φ := (φP ⊗ µQ ⊗ 1Q + 1P ⊗ µP ⊗ φQ)σ gives theGerstenhaber bracket via formula f ◦ g = fφ(1⊗ g ⊗ 1)∆(2).

Remark The hypotheses hold for bar resolutions, Koszul resolutions, andothers. The formula is used to compute Gerstenhaber brackets for:• Twisting given by bicharacter (Grimley-Nguyen-W 2017)• Skew group algebras (Shepler-W, arXiv 2019)• Jordan plane (Karadag-McPhate-Ocal-Oke-W, in preparation) Cf.Lopes-Solotar, arXiv 2019.

A generalization

Theorem (Volkov 2019)Let P be an arbitrary projective resolution of R as R⊗Rop-module, letf ∈ HomR⊗kRop(Pm, R) and g ∈ HomR⊗kRop(Pn, R). There existf1 ∈ HomR⊗kRop(Pm+n−1, Pn) and g1 ∈ HomR⊗kRop(Pm+n−1, Pm)

such that[f, g] = fg1 − (−1)(m−1)(n−1)gf1

represents the Lie bracket on HH∗(R).

Remark The map f1 corresponds to a homotopy lifting of f , i.e. a mapf for whichd f + (−1)mf d = (f ⊗R 1− 1⊗R f)∆ and f0 satisfies an initialcondition Similarly g1.

A generalization

Theorem (Volkov 2019)Let P be an arbitrary projective resolution of R as R⊗Rop-module, letf ∈ HomR⊗kRop(Pm, R) and g ∈ HomR⊗kRop(Pn, R). There existf1 ∈ HomR⊗kRop(Pm+n−1, Pn) and g1 ∈ HomR⊗kRop(Pm+n−1, Pm)

such that[f, g] = fg1 − (−1)(m−1)(n−1)gf1

represents the Lie bracket on HH∗(R).

Remark The map f1 corresponds to a homotopy lifting of f , i.e. a mapf r for whichd(f r) = (f ⊗R 1− 1⊗R f)∆ and f0 satisfies an initial condition.Similarly g1.

More on homotopy liftings (Volkov 2019)

A homotopy lifting of f ∈ HomRe(Pm, R) is mapf r : HomRe(P, P [1−m]) for which

d(f r) = (f ⊗R 1− 1⊗R f)∆

and fψ r+ (−1)mµPf r ∼ 0

where ψ r∈ HomRe(P, P [1]) satisfies d(ψ) = (µ⊗R 1− 1⊗R µ)∆.

Open problems

• Improve techniques for computation and theoretical understanding of[ , ] on HH∗(R)

• Find connections between recent results on [ , ] and- Schwede’s exact sequence interpretation of [ , ]

- Keller’s derived Picard interpretation of [ , ]

• Look for insight into questions of formality, i.e. when does [ , ] onHH∗(R) extend to [ , ] on HomR⊗Rop(B, R)?

Open problems

• Improve techniques for computation and theoretical understanding of[ , ] on HH∗(R)

• Find connections between recent results on [ , ] and- Schwede’s exact sequence interpretation of [ , ]

- Keller’s derived Picard interpretation of [ , ]

• Look for insight into questions of formality, i.e. when does [ , ] onHH∗(R) extend to [ , ] on HomR⊗Rop(B, R)?

Open problems

• Improve techniques for computation and theoretical understanding of[ , ] on HH∗(R)

• Find connections between recent results on [ , ] and- Schwede’s exact sequence interpretation of [ , ]

- Keller’s derived Picard interpretation of [ , ]

• Look for insight into questions of formality, i.e. when does [ , ] onHH∗(R) extend to [ , ] on HomR⊗Rop(B, R)?

...

Twisted product of resolutionsM an A-module with projective res. N a B-module with projective res.

0←M←P0←P1← P2←· · · 0←N←Q0←Q1←Q2←· · ·

Shepler-W 2019: Define their twisted tensor product by...��

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P0 ⊗Q2

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P1 ⊗Q2oo

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P2 ⊗Q2oo

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· · ·oo

P0 ⊗Q1

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P1 ⊗Q1oo

��

P2 ⊗Q1oo

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· · ·oo

P0 ⊗Q0 P1 ⊗Q0oo P2 ⊗Q0

oo · · ·oo

• Take the total complex (i.e. add up along diagonals)•Want (1) an A⊗τB-module structure on M ⊗N and on Pi ⊗Qj

and (2) Pi ⊗Qj to be projective—there are conditions for this• Then P· ⊗Q· is a resolution of M ⊗N

Twisted product of resolutionsM an A-module with projective res. N a B-module with projective res.

0←M←P0←P1← P2←· · · 0←N←Q0←Q1←Q2←· · ·

Shepler-W 2019: Define their twisted tensor product by...��

...��

...��

P0 ⊗Q2

��

P1 ⊗Q2oo

��

P2 ⊗Q2oo

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· · ·oo

P0 ⊗Q1

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P1 ⊗Q1oo

��

P2 ⊗Q1oo

��

· · ·oo

P0 ⊗Q0 P1 ⊗Q0oo P2 ⊗Q0

oo · · ·oo

• Take the total complex (i.e. add up along diagonals)•Want (1) an A⊗τB-module structure on M ⊗N and on Pi ⊗Qj

and (2) Pi ⊗Qj to be projective—there are conditions for this• Then P· ⊗Q· is a resolution of M ⊗N

Twisted product of resolutionsM an A-module with projective res. N a B-module with projective res.

0←M←P0←P1← P2←· · · 0←N←Q0←Q1←Q2←· · ·

Shepler-W 2019: Define their twisted tensor product by...��

...��

...��

P0 ⊗Q2

��

P1 ⊗Q2oo

��

P2 ⊗Q2oo

��

· · ·oo

P0 ⊗Q1

��

P1 ⊗Q1oo

��

P2 ⊗Q1oo

��

· · ·oo

P0 ⊗Q0 P1 ⊗Q0oo P2 ⊗Q0

oo · · ·oo

• Take the total complex (i.e. add up along diagonals)•Want (1) an A⊗τB-module structure on M ⊗N and on Pi ⊗Qj

and (2) Pi ⊗Qj to be projective—there are conditions for this• Then P· ⊗Q· is a resolution of M ⊗N

Twisted product of resolutionsM an A-module with projective res. N a B-module with projective res.

0←M←P0←P1← P2←· · · 0←N←Q0←Q1←Q2←· · ·

Shepler-W 2019: Define their twisted tensor product by...��

...��

...��

P0 ⊗Q2

��

P1 ⊗Q2oo

��

P2 ⊗Q2oo

��

· · ·oo

P0 ⊗Q1

��

P1 ⊗Q1oo

��

P2 ⊗Q1oo

��

· · ·oo

P0 ⊗Q0 P1 ⊗Q0oo P2 ⊗Q0

oo · · ·oo

• Take the total complex (i.e. add up along diagonals)•Want (1) an A⊗τB-module structure on M ⊗N and on Pi ⊗Qj

and (2) Pi ⊗Qj to be projective—there are conditions for this• Then P· ⊗Q· is a resolution of M ⊗N

Twisted product of resolutionsM an A-module with projective res. N a B-module with projective res.

0←M←P0←P1← P2←· · · 0←N←Q0←Q1←Q2←· · ·

Shepler-W 2019: Define their twisted tensor product by...��

...��

...��

P0 ⊗Q2

��

P1 ⊗Q2oo

��

P2 ⊗Q2oo

��

· · ·oo

P0 ⊗Q1

��

P1 ⊗Q1oo

��

P2 ⊗Q1oo

��

· · ·oo

P0 ⊗Q0 P1 ⊗Q0oo P2 ⊗Q0

oo · · ·oo

• Take the total complex (i.e. add up along diagonals)•Want (1) an A⊗τB-module structure on M ⊗N and on Pi ⊗Qj

and (2) Pi ⊗Qj to be projective—there are conditions for this• Then P· ⊗Q· is a resolution of M ⊗N

Advantages of twisted product resolution construction

• Twisted product resolutions unify many constructions in the literature:- Gopalakrishnan-Sridharan 1966 (Ore extensions)- Guccione-Guccione 2002 (crossed products with Hopf algebras)- Bergh-Oppermann 2008 (twisting by bicharacter on grading groups)- Shepler-W 2014, Walton-W 2014 (smash products)

• Explicit construction can facilitate calculations and yield insight

Advantages of twisted product resolution construction

• Twisted product resolutions unify many constructions in the literature:- Gopalakrishnan-Sridharan 1966 (Ore extensions)- Guccione-Guccione 2002 (crossed products with Hopf algebras)- Bergh-Oppermann 2008 (twisting by bicharacter on grading groups)- Shepler-W 2014, Walton-W 2014 (smash products)

• Explicit construction can facilitate calculations and yield insight

Advantages of twisted product resolution construction

• Twisted product resolutions unify many constructions in the literature:- Gopalakrishnan-Sridharan 1966 (Ore extensions)- Guccione-Guccione 2002 (crossed products with Hopf algebras)- Bergh-Oppermann 2008 (twisting by bicharacter on grading groups)- Shepler-W 2014, Walton-W 2014 (smash products)

• Explicit construction can facilitate calculations and yield insight

Advantages of twisted product resolution construction

• Twisted product resolutions unify many constructions in the literature:- Gopalakrishnan-Sridharan 1966 (Ore extensions)- Guccione-Guccione 2002 (crossed products with Hopf algebras)- Bergh-Oppermann 2008 (twisting by bicharacter on grading groups)- Shepler-W 2014, Walton-W 2014 (smash products)

• Explicit construction can facilitate calculations and yield insight

Advantages of twisted product resolution construction

• Twisted product resolutions unify many constructions in the literature:- Gopalakrishnan-Sridharan 1966 (Ore extensions)- Guccione-Guccione 2002 (crossed products with Hopf algebras)- Bergh-Oppermann 2008 (twisting by bicharacter on grading groups)- Shepler-W 2014, Walton-W 2014 (smash products)

• Explicit construction can facilitate calculations and yield insight

Advantages of twisted product resolution construction

• Twisted product resolutions unify many constructions in the literature:- Gopalakrishnan-Sridharan 1966 (Ore extensions)- Guccione-Guccione 2002 (crossed products with Hopf algebras)- Bergh-Oppermann 2008 (twisting by bicharacter on grading groups)- Shepler-W 2014, Walton-W 2014 (smash products)

• Explicit construction can facilitate calculations and yield insight

Summary

• Many known algebras are twisted tensor product algebras• Resolutions for their modules are constructed explicitly from resolutions

for component parts• These twisted product resolutions unify many known such constructions• They have many applications: explicit calculations of cohomology and

theoretical understanding of the structure of cohomology anddeformations of algebras

Summary

• Many known algebras are twisted tensor product algebras• Resolutions for their modules are constructed explicitly from resolutions

for component parts• These twisted product resolutions unify many known such constructions• They have many applications: explicit calculations of cohomology and

theoretical understanding of the structure of cohomology anddeformations of algebras

Summary

• Many known algebras are twisted tensor product algebras• Resolutions for their modules are constructed explicitly from resolutions

for component parts• These twisted product resolutions unify many known such constructions• They have many applications: explicit calculations of cohomology and

theoretical understanding of the structure of cohomology anddeformations of algebras

Summary

• Many known algebras are twisted tensor product algebras• Resolutions for their modules are constructed explicitly from resolutions

for component parts• These twisted product resolutions unify many known such constructions• They have many applications: explicit calculations of cohomology and

theoretical understanding of the structure of cohomology anddeformations of algebras