Homological invariants in category $\mathcal {O}$ for the general linear superalgebra

Coulembier, Kevin; Serganova, Vera

doi:10.1090/tran/6891

Cited by 12 publications

(42 citation statements)

References 32 publications

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“…This follows from . Another proof can be obtained by using [, Theorem 3.11] and . Weight spaces with respect to the diagonal subalgebra

h \subset sl (\infty)

correspond to the complexified reduced Grothendieck groups of the blocks of

O_{m false| n}^{Z}

.…”

Section: Sl(∞)‐modules Arising From Category Scripto For Gl(m|n)mentioning

confidence: 91%

“…Let

P_{m false| n}

denote the semisimple subcategory of

O_{m false| n}^{Z}

which consists of projective

gl (m | n)

‐modules, and let

P_{m false| n}

denote the reduced Grothendieck group of

P_{m false| n}

. The

sl (\infty)

‐module

{boldP}_{m, n} : = P_{m | n} \otimes_{double-struckZ} C

is the socle of

T_{m, n}

[, Theorem 3.11]. Note that for any projective module

P \in {scriptP}_{m | n}

the functor

{prefixHom}_{gl (m | n)} false(P, \cdot false)

O_{m false| n}^{Z}

is exact, and for any module

M \in {scriptF}_{m | n}

the functor

{prefixHom}_{gl (m | n)} false(\cdot, M false)

P_{m false| n}

is exact.…”

Section: Sl(∞)‐modules Arising From Category Scripto For Gl(m|n)mentioning

confidence: 99%

“…Herein, we will identify

K_{m false| n}

with its image

Ψ false(K_{m false| n} false) = span false{⟨ l_{λ}, \cdot ⟩ ∣ λ \in normalΦ false}

, where

l_{λ} : = false[L (λ) false]

. Now

soc false(Γ_{frakturg, frakturk} ({boldP}_{m | n}^{\lor}) false) = {boldP}_{m | n}

, since

P_{m false| n}

is semisimple, and

soc {boldT}_{m | n} = {boldP}_{m | n}

by [, Theorem 3.11]. Therefore, since

{boldT}_{m | n} \subset {boldK}_{m | n} \subset {normalΓ}_{g, k} false(P_{m false| n}^{\lor} false)

, we have

soc {boldK}_{m | n} = {boldP}_{m | n}

.…”

Section: Sl(∞)‐modules Arising From Category Scripto For Gl(m|n)mentioning

confidence: 99%

“…Let

Q_{m false| n}

denote the additive subcategory of

F_{m false| n}^{Z}

which consists of projective finite‐dimensional

gl (m | n)

‐modules, and let

Q_{m false| n}

denote the reduced Grothendieck group of

Q_{m false| n}

. It was proven in [, Theorem 3.11] that

{boldQ}_{m | n} : = Q_{m | n} \otimes_{double-struckZ} C

is the socle of the module

Λ_{m false| n}

, implying that

{boldQ}_{m | n} ≅ {boldV}^{{(m)}^{⊥}, {(n)}^{⊥}}

, where

⊥

indicates the conjugate partition. Corollary implies the following.…”

Section: Sl(∞)‐modules Arising From Category Scripto For Gl(m|n)mentioning

confidence: 99%

“…Let

O_{m false| n}^{i n d}

be the category whose objects are direct limits of objects in

O_{m false| n}

. Then by [, Lemma 5.2] the restriction of

D S_{x}

O_{m false| n}

is a well‐defined functor

D S_{x} : {scriptO}_{m | n} \to {scriptO}_{m - k | n - k}^{i n d} .

Lemma The functor

D S_{x} : {scriptO}_{m | n}^{double-struckZ} \to {({scriptO}_{m - k | n - k}^{double-struckZ})}^{i n d}

commutes with translation functors.…”

Section: Sl(∞)‐modules Arising From Category Scripto For Gl(m|n)mentioning

confidence: 99%

See 4 more Smart Citations

Integrable sl(∞)‐modules and category O for gl(m|n)

Hoyt

Penkov

Serganova

2018

J. London Math. Soc.

Self Cite

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We introduce and study new categories Tfrakturg,frakturk of integrable g=sl(∞)‐modules which depend on the choice of a certain reductive in frakturg subalgebra k⊂g. The simple objects of Tfrakturg,frakturk are tensor modules as in the previously studied category Tg [Dan‐Cohen, Penkov and Serganova, Adv. Math. 289 (2016) 250–278]; however, the choice of frakturk provides for more flexibility of nonsimple modules in Tfrakturg,frakturk compared to Tg. We then choose frakturk to have two infinite‐dimensional diagonal blocks, and show that a certain injective object Kmfalse|n in Tfrakturg,frakturk realizes a categorical sl(∞)‐action on the category Omfalse|nZ, the integral category scriptO of the Lie superalgebra gl(m|n). We show that the socle of Kmfalse|n is generated by the projective modules in Omfalse|nZ, and compute the socle filtration of Kmfalse|n explicitly. We conjecture that the socle filtration of Kmfalse|n reflects a ‘degree of atypicality filtration’ on the category Omfalse|nZ. We also conjecture that a natural tensor filtration on Kmfalse|n arises via the Duflo–Serganova functor sending the category Omfalse|nZ to Om−1false|n−1Z. We prove a weaker version of this latter conjecture for the direct summand of Kmfalse|n corresponding to finite‐dimensional gl(m|n)‐modules.

show abstract

“…This follows from . Another proof can be obtained by using [, Theorem 3.11] and . Weight spaces with respect to the diagonal subalgebra

h \subset sl (\infty)

correspond to the complexified reduced Grothendieck groups of the blocks of

O_{m false| n}^{Z}

.…”

Section: Sl(∞)‐modules Arising From Category Scripto For Gl(m|n)mentioning

confidence: 91%

“…Let

P_{m false| n}

denote the semisimple subcategory of

O_{m false| n}^{Z}

which consists of projective

gl (m | n)

‐modules, and let

P_{m false| n}

denote the reduced Grothendieck group of

P_{m false| n}

. The

sl (\infty)

‐module

{boldP}_{m, n} : = P_{m | n} \otimes_{double-struckZ} C

is the socle of

T_{m, n}

[, Theorem 3.11]. Note that for any projective module

P \in {scriptP}_{m | n}

the functor

{prefixHom}_{gl (m | n)} false(P, \cdot false)

O_{m false| n}^{Z}

is exact, and for any module

M \in {scriptF}_{m | n}

the functor

{prefixHom}_{gl (m | n)} false(\cdot, M false)

P_{m false| n}

is exact.…”

Section: Sl(∞)‐modules Arising From Category Scripto For Gl(m|n)mentioning

confidence: 99%

“…Herein, we will identify

K_{m false| n}

with its image

Ψ false(K_{m false| n} false) = span false{⟨ l_{λ}, \cdot ⟩ ∣ λ \in normalΦ false}

, where

l_{λ} : = false[L (λ) false]

. Now

soc false(Γ_{frakturg, frakturk} ({boldP}_{m | n}^{\lor}) false) = {boldP}_{m | n}

, since

P_{m false| n}

is semisimple, and

soc {boldT}_{m | n} = {boldP}_{m | n}

by [, Theorem 3.11]. Therefore, since

{boldT}_{m | n} \subset {boldK}_{m | n} \subset {normalΓ}_{g, k} false(P_{m false| n}^{\lor} false)

, we have

soc {boldK}_{m | n} = {boldP}_{m | n}

.…”

Section: Sl(∞)‐modules Arising From Category Scripto For Gl(m|n)mentioning

confidence: 99%

“…Let

Q_{m false| n}

denote the additive subcategory of

F_{m false| n}^{Z}

which consists of projective finite‐dimensional

gl (m | n)

‐modules, and let

Q_{m false| n}

denote the reduced Grothendieck group of

Q_{m false| n}

. It was proven in [, Theorem 3.11] that

{boldQ}_{m | n} : = Q_{m | n} \otimes_{double-struckZ} C

is the socle of the module

Λ_{m false| n}

, implying that

{boldQ}_{m | n} ≅ {boldV}^{{(m)}^{⊥}, {(n)}^{⊥}}

, where

⊥

indicates the conjugate partition. Corollary implies the following.…”

Section: Sl(∞)‐modules Arising From Category Scripto For Gl(m|n)mentioning

confidence: 99%

“…Let

O_{m false| n}^{i n d}

be the category whose objects are direct limits of objects in

O_{m false| n}

. Then by [, Lemma 5.2] the restriction of

D S_{x}

O_{m false| n}

is a well‐defined functor

D S_{x} : {scriptO}_{m | n} \to {scriptO}_{m - k | n - k}^{i n d} .

Lemma The functor

D S_{x} : {scriptO}_{m | n}^{double-struckZ} \to {({scriptO}_{m - k | n - k}^{double-struckZ})}^{i n d}

commutes with translation functors.…”

Section: Sl(∞)‐modules Arising From Category Scripto For Gl(m|n)mentioning

confidence: 99%

See 3 more Smart Citations

Integrable sl(∞)‐modules and category O for gl(m|n)

Hoyt

Penkov

Serganova

2018

J. London Math. Soc.

Self Cite

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show abstract

The classification of blocks in BGG category $${\mathcal {O}}$$

Coulembier

2019

Math. Z.

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The Duflo–Serganova Functor, Vingt Ans Après

Gorelik

Hoyt²,

Serganova³

et al. 2022

J Indian Inst Sci

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Homological invariants in category $\mathcal {O}$ for the general linear superalgebra

Cited by 12 publications

References 32 publications

Integrable sl(∞)‐modules and category O for gl(m|n)

Integrable sl(∞)‐modules and category O for gl(m|n)

The classification of blocks in BGG category $${\mathcal {O}}$$

The Duflo–Serganova Functor, Vingt Ans Après

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