An odd categorification of $U_q (\mathfrak{sl}_2)$

Ellis, Alexander P.; Lauda, Aaron D.

doi:10.4171/qt/78

Cited by 24 publications

(35 citation statements)

References 86 publications

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“…Suppose that

frakturg

is odd

{sl}_{2}

, that is,

I

is an odd singleton. Then the 2‐category

{\overset{\underset{̲}{frakturU}}{true}}_{q, π} false(frakturg false)

is 2‐equivalent to the 2‐category introduced . We do not think that this is an important result going forward, so we will only give a rough sketch of its proof in the next paragraph.…”

Section: Introductionmentioning

confidence: 92%

“…For the quiver with one odd vertex, the quiver Hecke superalgebra is the odd nilHecke algebra defined independently in ; see also [, § 3.3] which introduced the closely related degenerate spin affine Hecke algebras. In this case, a super analog of the Kac–Moody 2‐category was defined and studied already in . We will work here in the setting of 2‐ supercategories following , since it leads to some conceptual simplifications compared to the approach of .…”

Section: Introductionmentioning

confidence: 99%

“…Then, the appropriate 2‐functor in the direction

{frakturU}_{E L} \to {\overset{\underset{̲}{frakturU}}{true}}_{q, π} false(frakturg false)

is the identity on the object set

P

. It sends the generating 1‐morphisms

E 1_{λ}, F 1_{λ}

and

Π 1_{λ}

from [, § 3.2.1] to our 1‐morphisms

E 1_{λ}, {normalΠ}^{\bar{λ} + \bar{1}} F 1_{λ}

and

π_{λ}

, respectively. On the generating 2‐morphisms from [, § 3.2.2], it goes as follows:

We leave it to the reader to compare the relations in with our relations, and to construct a quasi‐inverse 2‐functor in the other direction.…”

Section: Introductionmentioning

confidence: 99%

“…On the generating 2‐morphisms from [, § 3.2.2], it goes as follows:

We leave it to the reader to compare the relations in with our relations, and to construct a quasi‐inverse 2‐functor in the other direction. In fact, when doing this carefully, one uncovers some inconsistencies in the relations of ; for example, the relation [, (3.1)] is wrong in the case

λ = 0

(due to an error in the last sentence of the proof of [, Lemma 5.1] related to the nilpotency of the odd bubble).…”

Section: Introductionmentioning

confidence: 99%

“…Turning to the odd case, the Decategorification Conjecture for odd

{sl}_{2}

is proved in [, Theorem 8.4]. The only additional finite type possibilities come from odd

b_{n}

, that is, type

b_{n}

with the element of

I

corresponding to the short simple root chosen to be odd.…”

Section: Introductionmentioning

confidence: 99%

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Super Kac–Moody 2‐categories

Brundan

Ellis

2017

Proc. London Math. Soc.

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“…Suppose that

frakturg

is odd

{sl}_{2}

, that is,

I

is an odd singleton. Then the 2‐category

{\overset{\underset{̲}{frakturU}}{true}}_{q, π} false(frakturg false)

is 2‐equivalent to the 2‐category introduced . We do not think that this is an important result going forward, so we will only give a rough sketch of its proof in the next paragraph.…”

Section: Introductionmentioning

confidence: 92%

Section: Introductionmentioning

confidence: 99%

“…Then, the appropriate 2‐functor in the direction

{frakturU}_{E L} \to {\overset{\underset{̲}{frakturU}}{true}}_{q, π} false(frakturg false)

is the identity on the object set

P

. It sends the generating 1‐morphisms

E 1_{λ}, F 1_{λ}

and

Π 1_{λ}

from [, § 3.2.1] to our 1‐morphisms

E 1_{λ}, {normalΠ}^{\bar{λ} + \bar{1}} F 1_{λ}

and

π_{λ}

, respectively. On the generating 2‐morphisms from [, § 3.2.2], it goes as follows:

We leave it to the reader to compare the relations in with our relations, and to construct a quasi‐inverse 2‐functor in the other direction.…”

Section: Introductionmentioning

confidence: 99%

“…On the generating 2‐morphisms from [, § 3.2.2], it goes as follows:

λ = 0

(due to an error in the last sentence of the proof of [, Lemma 5.1] related to the nilpotency of the odd bubble).…”

Section: Introductionmentioning

confidence: 99%

“…Turning to the odd case, the Decategorification Conjecture for odd

{sl}_{2}

is proved in [, Theorem 8.4]. The only additional finite type possibilities come from odd

b_{n}

, that is, type

b_{n}

with the element of

I

corresponding to the short simple root chosen to be odd.…”

Section: Introductionmentioning

confidence: 99%

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Super Kac–Moody 2‐categories

Brundan

Ellis

2017

Proc. London Math. Soc.

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Branes and Supergroups

Mikhaylov

Witten

2015

Commun. Math. Phys.

135

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Extending previous work that involved D3-branes ending on a fivebrane with θ YM = 0, we consider a similar two-sided problem. This construction, in case the fivebrane is of NS type, is associated to the three-dimensional Chern-Simons theory of a supergroup U(m|n) or OSp(m|2n) rather than an ordinary Lie group as in the one-sided case. By S-duality, we deduce a dual magnetic description of the supergroup Chern-Simons theory; a slightly different duality, in the orthosymplectic case, leads to a strong-weak coupling duality between certain supergroup Chern-Simons theories on R 3 ; and a further T -duality leads to a version of Khovanov homology for supergroups. Some cases of these statements are known in the literature. We analyze how these dualities act on line and surface operators.

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The Differential Graded Odd NilHecke Algebra

Ellis

2016

Commun. Math. Phys.

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An odd categorification of $U_q (\mathfrak{sl}_2)$

Cited by 24 publications

References 86 publications

Super Kac–Moody 2‐categories

Super Kac–Moody 2‐categories

Branes and Supergroups

The Differential Graded Odd NilHecke Algebra

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