Formality properties of finitely generated groups and Lie algebras

Suciu, Alexander I.; Wang, He

doi:10.1515/forum-2018-0098

Cited by 24 publications

(97 citation statements)

References 82 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We start by recalling some notions from [, Appendix A]; see also . A Malcev Lie algebra is a Lie algebra

frakturm

over a field

double-struckk

of characteristic 0, endowed with a decreasing, complete

double-struckk

‐vector space filtration

F = {F_{i}}_{i ⩾ 1}

such that

F_{1} = m

and

false[F_{i}, F_{j} false] \subset F_{i + j}

, for all

i, j

, and with the property that the associated graded Lie algebra,

gr false(frakturm false) = ⨁_{i ⩾ 1} F_{i} / F_{i + 1}

, is generated in degree 1.…”

Section: Malcev Lie Algebras Derived Series and Hall–reutenauer Basesmentioning

confidence: 99%

“…We recall from that a group

π

is said to be filtered formal if

m false(π false) ≅ \hat{L}

as filtered Lie algebras, where

L

is a graded Lie algebra generated in degree 1, of the form

L = L / J

, with

double-struckL

a finitely generated, free, graded Lie algebra and

J \subseteq {double-struckL}^{⩾ 2}

a graded Lie ideal generated in degrees 2 and higher.…”

Section: Malcev Lie Algebras Derived Series and Hall–reutenauer Basesmentioning

confidence: 99%

“…We denote by

{false(\cdot false)}^{false(k false)}

the

k

th term of the derived series of groups and Lie algebras. It follows from [, Theorem 3.5; , Theorem 1.7] that the groups

{double-struckF}_{n} / {double-struckF}_{n}^{(k)}

are filtered formal, with corresponding Lie algebras

L = {double-struckL}_{n} / {double-struckL}_{n}^{(k)}

, where

L_{n}

denotes the free Lie algebra on

n

generators and

F_{n}

is the free group on

n

generators.…”

Section: Malcev Lie Algebras Derived Series and Hall–reutenauer Basesmentioning

confidence: 99%

“…In the above example,

h (A)

still admits the structure of a graded Lie algebra, with

x_{1}

and

x_{2}

in degree 1, and

x_{3}

in degree 2. Nevertheless, using a construction from , we may define a minimal, finite

sans-serifcdga

A

for which

h (A)

does not admit any grading compatible with the lower central series filtration. Example Let

A = ⋀ (a_{1}, \dots, a_{5})

, with differential given by

d a_{4} = a_{1} \land a_{3}

d a_{5} = a_{1} \land a_{4} + a_{2} \land a_{3}

, and

d a_{i} = 0

, otherwise.…”

Section: A Functorial 1‐minimal Model Mapmentioning

confidence: 99%

“…Example Let

A = ⋀ (a_{1}, \dots, a_{5})

, with differential given by

d a_{4} = a_{1} \land a_{3}

d a_{5} = a_{1} \land a_{4} + a_{2} \land a_{3}

, and

d a_{i} = 0

, otherwise. Then, as shown in [, Example 10.5],

h (A)

is not isomorphic to

gr (h (A))

, the associated graded Lie algebra with respect to the LCS filtration.…”

Section: A Functorial 1‐minimal Model Mapmentioning

confidence: 99%

See 4 more Smart Citations

Infinitesimal finiteness obstructions

Papadima

Suciu

2018

J. London Math. Soc.

Self Cite

View full text Add to dashboard Cite

Does a space enjoying good finiteness properties admit an algebraic model with commensurable finiteness properties? In this note, we provide a rational homotopy obstruction for this to happen. As an application, we show that the maximal metabelian quotient of a very large, finitely generated group is not finitely presented. Using the theory of 1‐minimal models, we also show that a finitely generated group π admits a connected 1‐model with finite‐dimensional degree 1 piece if and only if the Malcev Lie algebra m(π) is the lower central series completion of a finitely presented Lie algebra.

show abstract

“…We start by recalling some notions from [, Appendix A]; see also . A Malcev Lie algebra is a Lie algebra

frakturm

over a field

double-struckk

of characteristic 0, endowed with a decreasing, complete

double-struckk

‐vector space filtration

F = {F_{i}}_{i ⩾ 1}

such that

F_{1} = m

and

false[F_{i}, F_{j} false] \subset F_{i + j}

, for all

i, j

, and with the property that the associated graded Lie algebra,

gr false(frakturm false) = ⨁_{i ⩾ 1} F_{i} / F_{i + 1}

, is generated in degree 1.…”

Section: Malcev Lie Algebras Derived Series and Hall–reutenauer Basesmentioning

confidence: 99%

“…We recall from that a group

π

is said to be filtered formal if

m false(π false) ≅ \hat{L}

as filtered Lie algebras, where

L

is a graded Lie algebra generated in degree 1, of the form

L = L / J

, with

double-struckL

a finitely generated, free, graded Lie algebra and

J \subseteq {double-struckL}^{⩾ 2}

a graded Lie ideal generated in degrees 2 and higher.…”

Section: Malcev Lie Algebras Derived Series and Hall–reutenauer Basesmentioning

confidence: 99%

“…We denote by

{false(\cdot false)}^{false(k false)}

the

k

th term of the derived series of groups and Lie algebras. It follows from [, Theorem 3.5; , Theorem 1.7] that the groups

{double-struckF}_{n} / {double-struckF}_{n}^{(k)}

are filtered formal, with corresponding Lie algebras

L = {double-struckL}_{n} / {double-struckL}_{n}^{(k)}

, where

L_{n}

denotes the free Lie algebra on

n

generators and

F_{n}

is the free group on

n

generators.…”

Section: Malcev Lie Algebras Derived Series and Hall–reutenauer Basesmentioning

confidence: 99%

“…In the above example,

h (A)

still admits the structure of a graded Lie algebra, with

x_{1}

and

x_{2}

in degree 1, and

x_{3}

in degree 2. Nevertheless, using a construction from , we may define a minimal, finite

sans-serifcdga

A

for which

h (A)

does not admit any grading compatible with the lower central series filtration. Example Let

A = ⋀ (a_{1}, \dots, a_{5})

, with differential given by

d a_{4} = a_{1} \land a_{3}

d a_{5} = a_{1} \land a_{4} + a_{2} \land a_{3}

, and

d a_{i} = 0

, otherwise.…”

Section: A Functorial 1‐minimal Model Mapmentioning

confidence: 99%

“…Example Let

A = ⋀ (a_{1}, \dots, a_{5})

, with differential given by

d a_{4} = a_{1} \land a_{3}

d a_{5} = a_{1} \land a_{4} + a_{2} \land a_{3}

, and

d a_{i} = 0

, otherwise. Then, as shown in [, Example 10.5],

h (A)

is not isomorphic to

gr (h (A))

, the associated graded Lie algebra with respect to the LCS filtration.…”

Section: A Functorial 1‐minimal Model Mapmentioning

confidence: 99%

See 3 more Smart Citations

Infinitesimal finiteness obstructions

Papadima

Suciu

2018

J. London Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

The Pure Braid Groups and Their Relatives

Suciu

Wang

2017

Perspectives in Lie Theory

Self Cite

View full text Add to dashboard Cite

In this mostly survey paper, we investigate the resonance varieties, the lower central series ranks, and the Chen ranks, as well as the residual and formality properties of several families of braid-like groups: the pure braid groups P n , the welded pure braid groups wP n , the virtual pure braid groups vP n , as well as their 'upper' variants, wP + n and vP + n . We also discuss several natural homomorphisms between these groups, and various ways to distinguish among the pure braid groups and their relatives.

show abstract

On the universal ellipsitomic KZB connection

Calaque

Gonzalez

2020

Sel. Math. New Ser.

View full text Add to dashboard Cite

Formality properties of finitely generated groups and Lie algebras

Cited by 24 publications

References 82 publications

Infinitesimal finiteness obstructions

Infinitesimal finiteness obstructions

The Pure Braid Groups and Their Relatives

On the universal ellipsitomic KZB connection

Contact Info

Product

Resources

About