Quiver Hecke superalgebras

Kang, Seok-Jίn; Kashiwara, Masaki; Tsuchioka, Shunsuke

doi:10.1515/crelle-2013-0089

Cited by 33 publications

(62 citation statements)

References 16 publications

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“…For objects

i = i_{n} \otimes \dots \otimes i_{1} \in I^{\otimes n}

and

j = j_{m} \otimes \dots \otimes j_{1} \in I^{\otimes m}

, there are no non‐zero morphisms

i \to j

scriptH

unless

m = n

. The graded endomorphism superalgebra

H_{n} : = ⨁_{i, j \in I^{\otimes n}} {prefixHom}_{scriptH} false(boldi, boldj false)

is the quiver Hecke superalgebra from . Let

H_{q, π}

be the

(Q, Π)

‐envelope of the monoidal supercategory

scriptH

, which is defined like in Definition remembering that monoidal supercategories are 2‐supercategories with one object; see also [, Definition 1.16].…”

Section: Surjectivity Of γmentioning

confidence: 99%

“…The quiver Hecke supercategory H is the (strict) monoidal supercategory generated by objects I and morphisms and of parities |i| and |i||j|, respectively, subject to the relations (1.7)-(1.9) (omitting the label λ from these diagrams). For objects i = i n ⊗ · · · ⊗ i 1 ∈ I ⊗n and j = j m ⊗ · · · ⊗ j 1 ∈ I ⊗m , there are no non-zero morphisms i → j in H unless m = n. The graded endomorphism superalgebra H n := i,j∈I ⊗n Hom H (i, j) (11.1) is the quiver Hecke superalgebra from [18]. Let H q,π be the (Q, Π)-envelope of the monoidal supercategory H, which is defined like in Definition 1.6 remembering that monoidal supercategories are 2-supercategories with one object; see also [4,Definition 1.16].…”

Section: Surjectivity Of γmentioning

confidence: 99%

“…The main purpose of this article is to extend the computations made in to include super Kac–Moody 2‐ categories . We will define these shortly following Rouquier's approach, starting from certain underlying quiver Hecke superalgebras which were introduced already by Kang, Kashiwara and Tsuchioka . 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.…”

Section: Introductionmentioning

confidence: 99%

“…Definition The Kac–Moody 2‐ supercategory is the 2‐supercategory

U (g)

with objects

P

, generating 1‐morphisms

E_{i} 1_{λ} : λ \to λ + α_{i}

and

F_{i} 1_{λ} : λ \to λ - α_{i}

for each

i \in I

and

λ \in P

, and generating 2‐morphisms

x : E_{i} 1_{λ} \to E_{i} 1_{λ}

of parity

| i |

τ : E_{i} E_{j} 1_{λ} \to E_{j} E_{i} 1_{λ}

of parity

| i | | j |

η : 1_{λ} \to F_{i} E_{i} 1_{λ}

of parity

\overset{0}{true}

and

ε : E_{i} F_{i} 1_{λ} \to 1_{λ}

of parity

\overset{0}{true}

, subject to certain relations. To record the relations among these generators, we switch to diagrams, representing the identity 2‐morphisms of

E_{i} 1_{λ}

and

F_{i} 1_{λ}

by and , respectively, and the other generators by

We denote the

n

th power of

x

(under vertical composition) by

First, we have the quiver Hecke superalgebra relations from …”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Super Kac–Moody 2‐categories

Brundan

Ellis

2017

Proc. London Math. Soc.

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show abstract

“…For objects

i = i_{n} \otimes \dots \otimes i_{1} \in I^{\otimes n}

and

j = j_{m} \otimes \dots \otimes j_{1} \in I^{\otimes m}

, there are no non‐zero morphisms

i \to j

scriptH

unless

m = n

. The graded endomorphism superalgebra

H_{n} : = ⨁_{i, j \in I^{\otimes n}} {prefixHom}_{scriptH} false(boldi, boldj false)

is the quiver Hecke superalgebra from . Let

H_{q, π}

be the

(Q, Π)

‐envelope of the monoidal supercategory

scriptH

, which is defined like in Definition remembering that monoidal supercategories are 2‐supercategories with one object; see also [, Definition 1.16].…”

Section: Surjectivity Of γmentioning

confidence: 99%

Section: Surjectivity Of γmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Definition The Kac–Moody 2‐ supercategory is the 2‐supercategory

U (g)

with objects

P

, generating 1‐morphisms

E_{i} 1_{λ} : λ \to λ + α_{i}

and

F_{i} 1_{λ} : λ \to λ - α_{i}

for each

i \in I

and

λ \in P

, and generating 2‐morphisms

x : E_{i} 1_{λ} \to E_{i} 1_{λ}

of parity

| i |

τ : E_{i} E_{j} 1_{λ} \to E_{j} E_{i} 1_{λ}

of parity

| i | | j |

η : 1_{λ} \to F_{i} E_{i} 1_{λ}

of parity

\overset{0}{true}

and

ε : E_{i} F_{i} 1_{λ} \to 1_{λ}

of parity

\overset{0}{true}

, subject to certain relations. To record the relations among these generators, we switch to diagrams, representing the identity 2‐morphisms of

E_{i} 1_{λ}

and

F_{i} 1_{λ}

by and , respectively, and the other generators by

We denote the

n

th power of

x

(under vertical composition) by

First, we have the quiver Hecke superalgebra relations from …”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Super Kac–Moody 2‐categories

Brundan

Ellis

2017

Proc. London Math. Soc.

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“…The odd nilHecke algebras [19] and quiver Hecke(-Clifford) superalgebras of [22] should satisfy the (super analogue of) the axioms of B-quasihereditary for a class B of algebras which are built out of polynomial and Clifford algebras.…”

Section: 4mentioning

confidence: 99%

Affine highest weight categories and affine quasihereditary algebras

Kleshchev

2015

Proceedings of the London Mathematical Society

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Abstract. Koenig and Xi introduced affine cellular algebras. Kleshchev and Loubert showed that an important class of infinite dimensional algebras, the KLR algebras R(Γ) of finite Lie type Γ, are (graded) affine cellular; in fact, the corresponding affine cell ideals are idempotent. This additional property is reminiscent of the properties of quasihereditary algebras of Cline-Parshall-Scott in a finite dimensional situation. A fundamental result of Cline-Parshall-Scott says that a finite dimensional algebra A is quasihereditary if and only if the category of finite dimensional A-modules is a highest weight category. On the other hand, S. Kato and Brundan-Kleshchev-McNamara proved that the category of finitely generated graded R(Γ)-modules has many features reminiscent of those of a highest weight category. The goal of this paper is to axiomatize and study the notions of an affine quasihereditary algebra and an affine highest weight category. In particular, we prove an affine analogue of the Cline-Parshall-Scott Theorem. We also develop stratified versions of these notions.

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Type C blocks of super category $$\mathcal {O}$$

Brundan

Davidson

2019

Math. Z.

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Quiver Hecke superalgebras

Cited by 33 publications

References 16 publications

Super Kac–Moody 2‐categories

Super Kac–Moody 2‐categories

Affine highest weight categories and affine quasihereditary algebras

Type C blocks of super category $$\mathcal {O}$$

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