Simplicity of heads and socles of tensor products

Kang, Seok Jin; Kashiwara, Masaki; Kim, Myung Ho; Oh, Se Jin

doi:10.1112/s0010437x14007799

Cited by 83 publications

(105 citation statements)

References 15 publications

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“…They imply that

F false(L (a_{1}) false) \nabla F false(S_{0} false) ≃ F false(L (a_{1}) false) \nabla F false(S_{0}^{'} false)

. By [, Corollary 3.14], we have

F false(S_{0} false) ≃ F false(S_{0}^{'} false)

, as desired.

□

…”

Section: Quantum Affine Algebra Of Type B and Duality Functorsmentioning

confidence: 69%

“…For simple modules M and N , we set [14]. Let M and N be simple modules in R-gmod, and assume that one of them is real.…”

Section: R-matrices For Quiver Hecke Algebramentioning

confidence: 99%

“…is equal to the specialization of R ren [14]. Let V, W be simple modules in C g and assume that one of them is real.…”

Section: R-matrices For Quantum Affine Algebrasmentioning

confidence: 99%

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Monoidal categories of modules over quantum affine algebras of type A and B

Kashiwara

2018

Proc. London Math. Soc.

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We construct an exact tensor functor from the category scriptA of finite‐dimensional graded modules over the quiver Hecke algebra of type A∞ to the category CBn(1) of finite‐dimensional integrable modules over the quantum affine algebra of type Bnfalse(1false). It factors through the category T2n, which is a localization of scriptA. As a result, this functor induces a ring isomorphism from the Grothendieck ring of T2n (ignoring the gradings) to the Grothendieck ring of a subcategory CBn(1)0 of CBn(1). Moreover, it induces a bijection between the classes of simple objects. Because the category T2n is related to categories CA2n−1(t)0 (t=1,2) of the quantum affine algebras of type A2n−1false(tfalse), we obtain an interesting connection between those categories of modules over quantum affine algebras of type A and type B. Namely, for each t=1,2, there exists an isomorphism between the Grothendieck ring of CA2n−1(t)0 and the Grothendieck ring of CBn(1)0, which induces a bijection between the classes of simple modules.

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“…They imply that

F false(L (a_{1}) false) \nabla F false(S_{0} false) ≃ F false(L (a_{1}) false) \nabla F false(S_{0}^{'} false)

. By [, Corollary 3.14], we have

F false(S_{0} false) ≃ F false(S_{0}^{'} false)

, as desired.

□

…”

Section: Quantum Affine Algebra Of Type B and Duality Functorsmentioning

confidence: 69%

“…For simple modules M and N , we set [14]. Let M and N be simple modules in R-gmod, and assume that one of them is real.…”

Section: R-matrices For Quiver Hecke Algebramentioning

confidence: 99%

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Monoidal categories of modules over quantum affine algebras of type A and B

Kashiwara

2018

Proc. London Math. Soc.

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show abstract

“…The KLR algebra

R (j)

is symmetric in the sense of [, Definition 1.3]. In particular, there is an

R

‐matrix

r_{L_{j}, M} : L_{j} \circ M ⟶ M \circ L_{j}

for every

R (ν)

‐module

M

.…”

Section: Crystal Operatorsmentioning

confidence: 99%

“…Proof By [, Theorem 3.2], if

M

is simple, then the image of the

R

‐matrix can be identified with both the head of

L (j) \circ M

and the socle of

M \circ L (j)

, and furthermore this image is simple. Consider the diagram:

The unlabelled morphisms in the top row are, from left to right, the canonical surjection, the canonical inclusion, the canonical isomorphism and the circle product of the morphisms