Odd Khovanov's arc algebra

Naisse, Grégoire; Vaz, Pedro

doi:10.4064/fm328-6-2017

Cited by 7 publications

(39 citation statements)

References 30 publications

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“…Finally, we show that the ‘odd’ center

O Z (O H_{k}^{n - k})

of the odd arc algebra coincides with the cohomology of the real

(n - k, k)

‐Springer fiber, using the same arguments as in the

(n, n)

‐case considered in [22].…”

Section: Introductionmentioning

confidence: 82%

“…Our goal now will be to show the following theorem, which is a generalization to the

(n - k, k)

‐case of one of the main results in [22]: Theorem There is an isomorphism of rings

h : O H^{*} false(B_{k}^{n - k} (C) false) \overset{≃}{\to} H^{*} false(T_{k}^{n - k} false) .

…”

Section: Cohomology Of Real Two‐row Springer Fibersmentioning

confidence: 99%

“…The topological real Springer fiber is the union of certain diagonal subsets in

T^{n}

labeled by crossingless matchings. It is an extension of the construction of the

(n, n)

topological real Springer fiber from [22], and follows the topological description of the complex two‐row Springer fiber but replacing 2‐spheres by 1‐spheres.…”

Section: A Topological Description Of Real Two‐row Springer Fibersmentioning

confidence: 99%

“…It arises from a chronological TQFT

, where

boldChCob

is the category of chronological cobordisms as developed by Putyra [25], and

Z - g m o d

is category of graded abelian groups. This chronological TQFT was used by Vaz and the second author [22] in their construction of an odd arc algebra

O H_{n}^{n}

, and by Putyra and the second author in [21] in their extension of odd Khovanov homology from links to tangles.…”

Section: Introductionmentioning

confidence: 99%

“…As a first step in this direction, [22] introduced a topological space

T_{n}^{n}

similar to the topological complex Springer fiber, but with 2‐spheres

S^{2}

replaced with 1‐spheres

S^{1}

. The authors of [22] also constructed isomorphisms between the graded center of

O H_{n}^{n}

, the cohomology ring

H^{*} false(T_{n}^{n} false)

and

O H^{*} false(B_{n}^{n} (C) false)

, where

O H^{*} false(B_{n}^{n} (C) false)

is the ‘oddification’ of

H^{*} false(B_{n}^{n} (C) false)

defined by Lauda–Russell [18].…”

Section: Introductionmentioning

confidence: 99%

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Real Springer fibers and odd arc algebras

Eberhardt

Naisse

Wilbert

2020

J. London Math. Soc.

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We give a topological description of the two‐row Springer fiber over the real numbers. We show its cohomology ring coincides with the oddification of the cohomology ring of the complex Springer fiber introduced by Lauda–Russell. We also realize Ozsváth–Rasmussen–Szabó's odd TQFT from pullbacks and exceptional pushforwards along inclusion and projection maps between hypertori. Using these results, we construct the odd arc algebra as a convolution algebra over components of the real Springer fiber, giving an odd analog of a construction of Stroppel–Webster.

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“…Finally, we show that the ‘odd’ center

O Z (O H_{k}^{n - k})

of the odd arc algebra coincides with the cohomology of the real

(n - k, k)

‐Springer fiber, using the same arguments as in the

(n, n)

‐case considered in [22].…”

Section: Introductionmentioning

confidence: 82%

“…Our goal now will be to show the following theorem, which is a generalization to the

(n - k, k)

‐case of one of the main results in [22]: Theorem There is an isomorphism of rings

h : O H^{*} false(B_{k}^{n - k} (C) false) \overset{≃}{\to} H^{*} false(T_{k}^{n - k} false) .

…”

Section: Cohomology Of Real Two‐row Springer Fibersmentioning

confidence: 99%

“…The topological real Springer fiber is the union of certain diagonal subsets in

T^{n}

labeled by crossingless matchings. It is an extension of the construction of the

(n, n)

topological real Springer fiber from [22], and follows the topological description of the complex two‐row Springer fiber but replacing 2‐spheres by 1‐spheres.…”

Section: A Topological Description Of Real Two‐row Springer Fibersmentioning

confidence: 99%

“…It arises from a chronological TQFT

, where

boldChCob

is the category of chronological cobordisms as developed by Putyra [25], and

Z - g m o d

is category of graded abelian groups. This chronological TQFT was used by Vaz and the second author [22] in their construction of an odd arc algebra

O H_{n}^{n}

, and by Putyra and the second author in [21] in their extension of odd Khovanov homology from links to tangles.…”

Section: Introductionmentioning

confidence: 99%

“…As a first step in this direction, [22] introduced a topological space

T_{n}^{n}

similar to the topological complex Springer fiber, but with 2‐spheres

S^{2}

replaced with 1‐spheres

S^{1}

. The authors of [22] also constructed isomorphisms between the graded center of

O H_{n}^{n}

, the cohomology ring

H^{*} false(T_{n}^{n} false)

and

O H^{*} false(B_{n}^{n} (C) false)

, where

O H^{*} false(B_{n}^{n} (C) false)

is the ‘oddification’ of

H^{*} false(B_{n}^{n} (C) false)

defined by Lauda–Russell [18].…”

Section: Introductionmentioning

confidence: 99%

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Real Springer fibers and odd arc algebras

Eberhardt

Naisse

Wilbert

2020

J. London Math. Soc.

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Odd two-variable Soergel bimodules and Rouquier complexes

Khovanov,

Putyra,

Vaz

2024

Algebraic and Topological Aspects of Representation Theory

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We consider the odd analogue of the category of Soergel bimodules. In the odd case and already for two variables, the transposition bimodule cannot be merged into the generating Soergel bimodule, forcing one into a monoidal category with a larger Grothendieck ring compared to the even case. We establish biadjointness of suitable functors and develop graphical calculi in the 2-variable case for the odd Soergel category and the related singular Soergel 2-category. We describe the odd analogue of the Rouquier complexes and establish their invertibility in the homotopy category. For three variables, the absence of a direct sum decomposition of the tensor product of generating Soergel bimodules presents an obstacle for the Reidemeister III relation to hold in the homotopy category.

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Not even Khovanov homology

Vaz

2020

Pacific J. Math.

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Odd Khovanov's arc algebra

Cited by 7 publications

References 30 publications

Real Springer fibers and odd arc algebras

Real Springer fibers and odd arc algebras

Odd two-variable Soergel bimodules and Rouquier complexes

Not even Khovanov homology

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