I Use a similar approach to get the quantum group to work. (Needs probably some mixed Howe duality `a la Queffelec–Sartori.)
I q-monoidal categories are probably very useful to study representation categories of coideal subalgebras. An abstract formulation `a la Brundan–Ellis (“super monoidal”) should be useful.
I Coideal subalgebras are amenable to categorification, cf. Ehrig–Stroppel or Bao–Shan–Wang–Webster. Similarly, their representation categories should be amenable to categorification.
I Formulate everything in a “2-q-monoidal language”. (Again, `a la Brundan–Ellis.)
∗=n−1/2for typeB,
∗=n/2for typeD.
This should give the quantum group story, but it is much trickier since e.g.
Vsoq ∼=Vglq ⊕(Vqgl)∗⊕C asUq(gl∗)-modules in typeB.
Thus, the above is not the usualU(gl∗)-U(glk) duality.
a a b b
=q∗
a a b b
q-interchange law
∗=some power depending ona,b c
b a+b a a+b
+c
Singular cobordisms (“foams”,
`
a la Khovanov–Rozansky and Mackaay–Stoˇsi´c–Vaz) categorify webs.
Theq-monoidal property has to be smartly encoded.
2-q-monoidal foams. (Maybe connected to Beliakova–Putyra–Wehrli
whose pictures I shamelessly stole.)
Further directions
I Use a similar approach to get the quantum group to work. (Needs probably some mixed Howe duality `a la Queffelec–Sartori.)
I q-monoidal categories are probably very useful to study representation categories of coideal subalgebras. An abstract formulation `a la Brundan–Ellis (“super monoidal”) should be useful.
I Coideal subalgebras are amenable to categorification, cf. Ehrig–Stroppel or Bao–Shan–Wang–Webster. Similarly, their representation categories should be amenable to categorification.
I Formulate everything in a “2-q-monoidal language”. (Again, `a la Brundan–Ellis.)
∗=n−1/2for typeB,
∗=n/2for typeD.
This should give the quantum group story, but it is much trickier since e.g.
Vsoq ∼=Vglq ⊕(Vqgl)∗⊕C asUq(gl∗)-modules in typeB.
Thus, the above is not the usualU(gl∗)-U(glk) duality.
a a b b
=q∗
a a b b
q-interchange law
∗=some power depending ona,b c
b a+b a a+b
+c
Singular cobordisms (“foams”,
`
a la Khovanov–Rozansky and Mackaay–Stoˇsi´c–Vaz) categorify webs.
Theq-monoidal property has to be smartly encoded.
2-q-monoidal foams. (Maybe connected to Beliakova–Putyra–Wehrli
whose pictures I shamelessly stole.)
Further directions
I Use a similar approach to get the quantum group to work. (Needs probably some mixed Howe duality `a la Queffelec–Sartori.)
I q-monoidal categories are probably very useful to study representation categories of coideal subalgebras. An abstract formulation `a la Brundan–Ellis (“super monoidal”) should be useful.
I Coideal subalgebras are amenable to categorification, cf. Ehrig–Stroppel or Bao–Shan–Wang–Webster. Similarly, their representation categories should be amenable to categorification.
I Formulate everything in a “2-q-monoidal language”. (Again, `a la Brundan–Ellis.)
∗=n−1/2for typeB,
∗=n/2for typeD.
This should give the quantum group story, but it is much trickier since e.g.
Vsoq ∼=Vglq ⊕(Vqgl)∗⊕C asUq(gl∗)-modules in typeB.
Thus, the above is not the usualU(gl∗)-U(glk) duality.
a a b b
=q∗
a a b b
q-interchange law
∗=some power depending ona,b
c b a+b a a+b
+c
Singular cobordisms (“foams”,
`
a la Khovanov–Rozansky and Mackaay–Stoˇsi´c–Vaz) categorify webs.
Theq-monoidal property has to be smartly encoded.
2-q-monoidal foams. (Maybe connected to Beliakova–Putyra–Wehrli
whose pictures I shamelessly stole.)
Further directions
I Use a similar approach to get the quantum group to work. (Needs probably some mixed Howe duality `a la Queffelec–Sartori.)
I q-monoidal categories are probably very useful to study representation categories of coideal subalgebras. An abstract formulation `a la Brundan–Ellis (“super monoidal”) should be useful.
I Coideal subalgebras are amenable to categorification, cf. Ehrig–Stroppel or Bao–Shan–Wang–Webster. Similarly, their representation categories should be amenable to categorification.
I Formulate everything in a “2-q-monoidal language”. (Again, `a la Brundan–Ellis.)
∗=n−1/2for typeB,
∗=n/2for typeD.
This should give the quantum group story, but it is much trickier since e.g.
Vsoq ∼=Vglq ⊕(Vqgl)∗⊕C asUq(gl∗)-modules in typeB.
Thus, the above is not the usualU(gl∗)-U(glk) duality.
a a b b
=q∗
a a b b
q-interchange law
∗=some power depending ona,b
c b a+b a a+b
+c
Singular cobordisms (“foams”,
`
a la Khovanov–Rozansky and Mackaay–Stoˇsi´c–Vaz) categorify webs.
Theq-monoidal property has to be smartly encoded.
2-q-monoidal foams. (Maybe connected to Beliakova–Putyra–Wehrli
whose pictures I shamelessly stole.)
Further directions
I Use a similar approach to get the quantum group to work. (Needs probably some mixed Howe duality `a la Queffelec–Sartori.)
I q-monoidal categories are probably very useful to study representation categories of coideal subalgebras. An abstract formulation `a la Brundan–Ellis (“super monoidal”) should be useful.
I Coideal subalgebras are amenable to categorification, cf. Ehrig–Stroppel or Bao–Shan–Wang–Webster. Similarly, their representation categories should be amenable to categorification.
I Formulate everything in a “2-q-monoidal language”. (Again, `a la Brundan–Ellis.)
∗=n−1/2for typeB,
∗=n/2for typeD.
This should give the quantum group story, but it is much trickier since e.g.
Vsoq ∼=Vglq ⊕(Vqgl)∗⊕C asUq(gl∗)-modules in typeB.
Thus, the above is not the usualU(gl∗)-U(glk) duality.
a a b b
=q∗
a a b b
q-interchange law
∗=some power depending ona,b c
b a+b a a+b
+c
Singular cobordisms (“foams”,
`
a la Khovanov–Rozansky and Mackaay–Stoˇsi´c–Vaz) categorify webs.
Theq-monoidal property has to be smartly encoded.
2-q-monoidal foams.
(Maybe connected to Beliakova–Putyra–Wehrli whose pictures I shamelessly stole.)
A pioneer of representation theory
Schur’sremarkable relationship betweenglnand the symmetric groupSk:
Schur∼1901.LetV=Vgl=Cn. There are commuting actions U(gln)V⊗ · · · ⊗V| {z }
ktimes C[Sk] generating each other’s centralizer. TheU(gln)-C[Sk]-bimodule decomposes as
M λ∈P
L(gln, λ)⊗L(Sk, λT).
Theλ’s are partitions (Young diagrams) ofkwith at mostnrows.
First statement
Second statement Third statement
The precise form does not matter for today. It is only important that one can make it explicit.
Daniel Tubbenhauer Webs andq-Howe dualities in typesBCD April 20173 / 14
Another pioneer of representation theory
Brauer’sremarkable relationship betweengn=son,spnand the Brauer algebraBrkn:
Brauer∼1937.LetV=Cn. There are commuting actions U(gn)V⊗ · · · ⊗V|{z }
ktimes Brkn generating each other’s centralizer. TheU(gn)-Brkn-bimodule decomposes as
M λ∈P
L(gn, λ)⊗L(Brkn, λT).
Theλ’s are partitions ofk,k−2,k−4, . . .whose precise form depend ongn.
Be careful: One needs to work withonin typeD. Today, I silently stay withson, and thus, in typeB.
Daniel Tubbenhauer Webs andq-Howe dualities in typesBCD April 20178 / 14
Still alive: The diagrammatic presentation machine U(gln) V•V⊗ · · · ⊗V•V U(glk)
Howe’s11/2th statementdefinesadiagrammaticcategoryWebsuch that
˙
commutes. In particular,Webis athickversion of the Brauer algebra.
Restricting the action on one side
Increases the centralizer on the other
Hence, we get
“old diagram generators” and
“new diagram generators”
Daniel Tubbenhauer Webs andq-Howe dualities in typesBCD April 201711 / 14
Some delicate quantizations Uq(gln) V•
Quantum skew Howe duality: Lehrer–Zhang–Zhang∼2009. (But its quite easy and not their main point.)
Does not quantize! Quantizes easily
No action at all. Action unclear.
The quantum dimension ofVgl qis [n].
The quantum dimension ofVso qis [n−1]+1.
Hence,Vso qdoes not come fromVgl
q! This “flaw” propagates all the way trough:
Va
qVsoqhave “weird” quantum dimensions.
The quantum dimension ofVso5↑ q
Above: Kuperberg’sB2web relations∼1995.
We wanted to generalize Kuperberg’s
results. We failed because quantization is hard outside
of typeA.
But let me explain what we can do.
Using a coideal subalgebra does the trick.
The action is constructed using the unquantized diagrammatics.
Daniel Tubbenhauer Webs andq-Howe dualities in typesBCD April 201712 / 14
Some delicate quantizations Uq(gln) V•
Using aq-monoidaldiagrammaticcategoryWebq,qnwe candefinea full Howe functor ΦextBDsuch that we get a commuting diagram
U˙q(so2k) Rep0q(son)
Hereby,Rep0q(son) is theq-monoidal representation category ofU0q(son), and Brq,qnis Molev’sq-Brauer category (∼2002).
Quantum skew Howe duality: Lehrer–Zhang–Zhang∼2009. (But its quite easy and not their main point.)
Does not quantize! Quantizes easily
No action at all. Action unclear.
The quantum dimension ofVglqis [n]. The quantum dimension ofVsoqis [n−1]+1.
Hence,Vso qdoes not come fromVgl
q! This “flaw” propagates all the way trough: Va
qVso qhave “weird” quantum dimensions.
The quantum dimension ofVso5 q
↑
Above: Kuperberg’sB2web relations∼1995. We wanted to
generalize Kuperberg’s results. We failed because quantization is hard outside
of typeA.
But let me explain what we can do. Using a coideal
subalgebra does the trick.
The action is constructed using the unquantized diagrammatics.
Daniel Tubbenhauer Webs andq-Howe dualities in typesBCD April 201712 / 14
Dual pair ModuleM q-version andwebcalculi
U(gln)-U(glk) V•(Cn⊗Ck) Cautis–Kamnitzer–Morrison∼2012 U(gl1|1)-U(glk) V•(C1|1⊗Ck) Sartori∼2013, Grant∼2014
Up to quantization, all of this (and more) is basically already in Howe’s paper.
TypeAis in a fairly good shape: The story partially works “integrally” (Elias∼2015). Applications to link polynomials (e.g. Wedrich–Vaz and coauthors∼2015).
Partially categorified (e.g. Huerfano–Khovanov∼2002, Mackaay∼2009). Applications to link homologies (e.g. Lauda–Queffelec–Rose∼2012). Applications to canonical bases and geometry (e.g. Cautis–Kamnitzer∼2016).
...
TypesBCDare not really understood.
Back
Monoidal generators ofWeb:
a Relations are the typeArelations and e.g.:
aabb
No orientations needed in typesBCD.
Recall:
q-Monoidal generators ofWebq,qn:
a Relations are the typeArelations and e.g.:
aabb
=q∗
aabb
q-interchange law
∗=some power depending ona,b , Quantization ofU(son)
“Nice quantum numbers”
“Nice topology”
Connected to Peng’s talk yesterday:θ=ω, the Chevalley involution ω(Ei) =−Fi,ω(Fi) =−Ei,ω(Hi) =−Hi.
U0q(son) is a (left) coideal:
∆ :U0q(son)→Uq(gln)⊗U0q(son).
Hence,Rep0q(son) is onlyq-monoidal and carries a left action ofRepq(gln).
5
A-web new part
Back
Daniel Tubbenhauer Webs andq-Howe dualities in typesBCD April 20171 / 1
There is stillmuchto do...
Thanks for your attention!
A pioneer of representation theory
Schur’sremarkable relationship betweenglnand the symmetric groupSk:
Schur∼1901.LetV=Vgl=Cn. There are commuting actions U(gln)V⊗ · · · ⊗V| {z }
ktimes C[Sk] generating each other’s centralizer. TheU(gln)-C[Sk]-bimodule decomposes as
M λ∈P
L(gln, λ)⊗L(Sk, λT).
Theλ’s are partitions (Young diagrams) ofkwith at mostnrows.
First statement
Second statement Third statement
The precise form does not matter for today. It is only important that one can make it explicit.
Daniel Tubbenhauer Webs andq-Howe dualities in typesBCD April 20173 / 14
Another pioneer of representation theory
Brauer’sremarkable relationship betweengn=son,spnand the Brauer algebraBrkn:
Brauer∼1937.LetV=Cn. There are commuting actions U(gn)V⊗ · · · ⊗V|{z }
ktimes Brkn generating each other’s centralizer. TheU(gn)-Brkn-bimodule decomposes as
M λ∈P
L(gn, λ)⊗L(Brkn, λT).
Theλ’s are partitions ofk,k−2,k−4, . . .whose precise form depend ongn.
Be careful: One needs to work withonin typeD. Today, I silently stay withson, and thus, in typeB.
Daniel Tubbenhauer Webs andq-Howe dualities in typesBCD April 20178 / 14
Still alive: The diagrammatic presentation machine U(gln) V•V⊗ · · · ⊗V•V U(glk)
Howe’s11/2th statementdefinesadiagrammaticcategoryWebsuch that
˙
commutes. In particular,Webis athickversion of the Brauer algebra.
Restricting the action on one side
Increases the centralizer on the other
Hence, we get
“old diagram generators” and
“new diagram generators”
Daniel Tubbenhauer Webs andq-Howe dualities in typesBCD April 201711 / 14
Some delicate quantizations Uq(gln) V•
Quantum skew Howe duality: Lehrer–Zhang–Zhang∼2009. (But its quite easy and not their main point.)
Does not quantize! Quantizes easily
No action at all. Action unclear.
The quantum dimension ofVgl qis [n].
The quantum dimension ofVso qis [n−1]+1.
Hence,Vso qdoes not come fromVgl
q! This “flaw” propagates all the way trough:
Va
qVsoqhave “weird” quantum dimensions.
The quantum dimension ofVso5↑ q
Above: Kuperberg’sB2web relations∼1995.
We wanted to generalize Kuperberg’s
results. We failed because quantization is hard outside
of typeA.
But let me explain what we can do.
Using a coideal subalgebra does the trick.
The action is constructed using the unquantized diagrammatics.
Daniel Tubbenhauer Webs andq-Howe dualities in typesBCD April 201712 / 14
Some delicate quantizations Uq(gln) V•
Using aq-monoidaldiagrammaticcategoryWebq,qnwe candefinea full Howe functor ΦextBDsuch that we get a commuting diagram
U˙q(so2k) Rep0q(son)
Hereby,Rep0q(son) is theq-monoidal representation category ofU0q(son), and Brq,qnis Molev’sq-Brauer category (∼2002).
Quantum skew Howe duality: Lehrer–Zhang–Zhang∼2009. (But its quite easy and not their main point.)
Does not quantize! Quantizes easily
No action at all. Action unclear.
The quantum dimension ofVglqis [n]. The quantum dimension ofVsoqis [n−1]+1.
Hence,Vso qdoes not come fromVgl
q! This “flaw” propagates all the way trough: Va
qVso qhave “weird” quantum dimensions.
The quantum dimension ofVso5 q
↑
Above: Kuperberg’sB2web relations∼1995. We wanted to
generalize Kuperberg’s results. We failed because quantization is hard outside
of typeA.
But let me explain what we can do. Using a coideal
subalgebra does the trick.
The action is constructed using the unquantized diagrammatics.
Daniel Tubbenhauer Webs andq-Howe dualities in typesBCD April 201712 / 14
Dual pair ModuleM q-version andwebcalculi
U(gln)-U(glk) V•(Cn⊗Ck) Cautis–Kamnitzer–Morrison∼2012 U(gl1|1)-U(glk) V•(C1|1⊗Ck) Sartori∼2013, Grant∼2014
Up to quantization, all of this (and more) is basically already in Howe’s paper.
TypeAis in a fairly good shape: The story partially works “integrally” (Elias∼2015). Applications to link polynomials (e.g. Wedrich–Vaz and coauthors∼2015).
Partially categorified (e.g. Huerfano–Khovanov∼2002, Mackaay∼2009). Applications to link homologies (e.g. Lauda–Queffelec–Rose∼2012). Applications to canonical bases and geometry (e.g. Cautis–Kamnitzer∼2016).
...
TypesBCDare not really understood.
Back
Monoidal generators ofWeb:
a Relations are the typeArelations and e.g.:
aabb
No orientations needed in typesBCD.
Recall:
q-Monoidal generators ofWebq,qn:
a Relations are the typeArelations and e.g.:
aabb
=q∗
aabb
q-interchange law
∗=some power depending ona,b , Quantization ofU(son)
“Nice quantum numbers”
“Nice topology”
Connected to Peng’s talk yesterday:θ=ω, the Chevalley involution ω(Ei) =−Fi,ω(Fi) =−Ei,ω(Hi) =−Hi.
U0q(son) is a (left) coideal:
∆ :U0q(son)→Uq(gln)⊗U0q(son).
Hence,Rep0q(son) is onlyq-monoidal and carries a left action ofRepq(gln).
5
A-web new part
Back
Daniel Tubbenhauer Webs andq-Howe dualities in typesBCD April 20171 / 1
There is stillmuchto do...
Thanks for your attention!
Figure:Two of the main players for today: Schur and Brauer.
Curtis, C.W.Pioneers of representation theory: Frobenius, Burnside, Schur, and Brauer.
Figure:Quotes from “Theory of Groups of Finite Order” by Burnside. Top: first edition (1897); bottom: second edition (1911).
Monoidal generator ofS:
: 2→2.
Relations e.g.:
=
interchange law
, = , =
“Reidemeister relations”
Back
Dual pair ModuleM q-version andwebcalculi U(gln)-U(glk) V•(Cn⊗Ck) Cautis–Kamnitzer–Morrison∼2012 U(gl1|1)-U(glk) V•(C1|1⊗Ck) Sartori∼2013, Grant∼2014
U(gln)-U(glk) Sym•(Cn⊗Ck) Rose and coauthors∼2015 U(glm|n)-U(glk) V•(Cm|n⊗Ck) Queffelec–Sartori, Grant∼2015 U(glm|n)-U(gll|k) V•(Cm|n⊗Cl|k) Vaz–Wedrich and coauthors∼2015
U(son)-U(so2k) V•(Cn⊗Ck)
U(son)-U(sp2k) Sym•(Cn⊗Ck) Sartori U(spn)-U(sp2k) V•(Cn⊗Ck) and coauthors∼2017 U(spn)-U(so2k) Sym•(Cn⊗Ck)
Up to quantization, all of this (and more) is basically already in Howe’s paper.
TypeAis in a fairly good shape:
The story partially works “integrally” (Elias∼2015).
Applications to link polynomials (e.g. Wedrich–Vaz and coauthors∼2015). Partially categorified (e.g. Huerfano–Khovanov∼2002, Mackaay∼2009).
Applications to link homologies (e.g. Lauda–Queffelec–Rose∼2012). Applications to canonical bases and geometry (e.g. Cautis–Kamnitzer∼2016).
...
Types BCDare not really understood.
Dual pair ModuleM q-version andwebcalculi U(gln)-U(glk) V•(Cn⊗Ck) Cautis–Kamnitzer–Morrison∼2012 U(gl1|1)-U(glk) V•(C1|1⊗Ck) Sartori∼2013, Grant∼2014
U(gln)-U(glk) Sym•(Cn⊗Ck) Rose and coauthors∼2015 U(glm|n)-U(glk) V•(Cm|n⊗Ck) Queffelec–Sartori, Grant∼2015 U(glm|n)-U(gll|k) V•(Cm|n⊗Cl|k) Vaz–Wedrich and coauthors∼2015
U(son)-U(so2k) V•(Cn⊗Ck)
U(son)-U(sp2k) Sym•(Cn⊗Ck) Sartori U(spn)-U(sp2k) V•(Cn⊗Ck) and coauthors∼2017 U(spn)-U(so2k) Sym•(Cn⊗Ck)
Up to quantization, all of this (and more) is basically already in Howe’s paper.
TypeAis in a fairly good shape:
The story partially works “integrally” (Elias∼2015).
Applications to link polynomials (e.g. Wedrich–Vaz and coauthors∼2015).
Partially categorified (e.g. Huerfano–Khovanov∼2002, Mackaay∼2009).
Applications to link homologies (e.g. Lauda–Queffelec–Rose∼2012).
Applications to canonical bases and geometry (e.g. Cautis–Kamnitzer∼2016).
...
Types BCDare not really understood.
Dual pair ModuleM q-version andwebcalculi U(gln)-U(glk) V•(Cn⊗Ck) Cautis–Kamnitzer–Morrison∼2012 U(gl1|1)-U(glk) V•(C1|1⊗Ck) Sartori∼2013, Grant∼2014
U(gln)-U(glk) Sym•(Cn⊗Ck) Rose and coauthors∼2015 U(glm|n)-U(glk) V•(Cm|n⊗Ck) Queffelec–Sartori, Grant∼2015 U(glm|n)-U(gll|k) V•(Cm|n⊗Cl|k) Vaz–Wedrich and coauthors∼2015
U(son)-U(so2k) V•(Cn⊗Ck)
U(son)-U(sp2k) Sym•(Cn⊗Ck) Sartori U(spn)-U(sp2k) V•(Cn⊗Ck) and coauthors∼2017 U(spn)-U(so2k) Sym•(Cn⊗Ck)
Up to quantization, all of this (and more) is basically already in Howe’s paper.
TypeAis in a fairly good shape:
The story partially works “integrally” (Elias∼2015).
Applications to link polynomials (e.g. Wedrich–Vaz and coauthors∼2015).
Partially categorified (e.g. Huerfano–Khovanov∼2002, Mackaay∼2009).
Applications to link homologies (e.g. Lauda–Queffelec–Rose∼2012).
Applications to canonical bases and geometry (e.g. Cautis–Kamnitzer∼2016).
...
Types BCDare not really understood.
Monoidal generators ofWeb :
a a+b
b
:a⊗b→a+b and
a
a+b b
:a+b→a⊗b.
Relations e.g.:
a a+b
b a
a+b
b
=
a a+b
b a
a+b
b interchange law
,
a b c
a+b+c
=
c b a
a+b+c
Associativity
One needs orientations in typeA, but I am going to ignore them.
Back
Monoidal generators ofWeb :
One needs orientations in typeA, but I am going to ignore them.
Back
Root conventions is typeA:
· · ·
α1 α2 αk−2 αk−1
ε1−ε2 ε2−ε3 εk−2−εk−1 εk−1−εk
Thus, because of statement 11/2, we should set
Ei1λ7−→
Root conventions is typeA:
· · ·
α1 α2 αk−2 αk−1
ε1−ε2 ε2−ε3 εk−2−εk−1 εk−1−εk
Thus, because of statement 11/2, we should set
Ei1λ7−→
βA:S →Web
7−→
1 1
1 1
= −
1 1
1 1
+
1 1
1 1
C[Sk]−→∼= EndWeb (1⊗k)
Back
Monoidal generators ofBrn:
, :∅ →2 , : 2→ ∅.
Relations e.g.:
=
interchange law
, = ±n.
circle removal
From “Brauer, R.On algebras which are connected with the semisimple continuous groups. Ann. of Math. (2) 38 (1937), no. 4, 857–872.”
Back
Monoidal generators ofBrn:
, :∅ →2 , : 2→ ∅.
Relations e.g.:
=
interchange law
, = ±n.
circle removal
From “Brauer, R.On algebras which are connected with the semisimple continuous groups.
Ann. of Math. (2) 38 (1937), no. 4, 857–872.”
Back
Monoidal generators ofWeb :
typeAgenerators
,
Relations are the typeArelations and e.g.:
a a b b
No orientations needed in typesBCD.
Recall:
Monoidal generators ofWeb :
typeAgenerators
,
Relations are the typeArelations and e.g.:
a a b b
No orientations needed in typesBCD.
Recall:
Root conventions is typeD:
· · ·
α1 α2 αk−3 αk−2
αk−1
αk εk−1+εk
Thus, because of statement 11/2, we should set
Ek1λ7−→
a1 a1
. . . . . . . . .
ak−2 ak−2
ak−1 ak−1+1
ak ak+1
Fk1λ7−→
a1 a1
. . . . . . . . .
ak−2 ak−2
ak−1 ak−1−1
ak ak−1
FE1(−n/2,−n/2)7−→
F
E
Root conventions is typeD:
· · ·
α1 α2 αk−3 αk−2
αk−1
αk εk−1+εk
Thus, because of statement 11/2, we should set
Ek1λ7−→
a1 a1
. . . . . . . . .
ak−2 ak−2
ak−1 ak−1+1
ak ak+1
Fk1λ7−→
a1 a1
. . . . . . . . .
ak−2 ak−2
ak−1 ak−1−1
ak ak−1
FE1(−n/2,−n/2)7−→
F
E
β :Brn→Web
7−→
1 1
1 1
= −
1 1
1 1
+
1 1
1 1
7−→
1 1
, 7−→
1 1
Brkn −→∼= EndWeb (1⊗k)
Back
q-Monoidal generators ofWebq,qn:
typeAgenerators
,
a a
,
a a
new generators
Relations are the typeArelations and e.g.:
a a b b
=q∗
a a b b
q-interchange law
∗=some power depending ona,b
,
TypeAcrossing:
1
q-Monoidal generators ofWebq,qn:
typeAgenerators
,
a a
,
a a
new generators
Relations are the typeArelations and e.g.:
a a b b
=q∗
a a b b
q-interchange law
∗=some power depending ona,b
,
TypeAcrossing:
1
Via restriction, we see that theUq(gln)-intertwiners a+ba,b and a,ba+b are U0q(son)-equivariant as well.
Note thatV⊗V contains a copy of the trivialU(son)-module. One shows that the same holds withqand one gets inclusions and projections
:Cq→Vq⊗Vq, :Vq⊗Vq→Cq.
As before, use these to quantize Howe’s duality.
Back
Uq(son) U0q(son) Subalgebra ofUq(gln)
Hopfalgebra Quantization ofU(son)
“Nice quantum numbers”
“Nice topology”
Connected to Peng’s talk yesterday: θ=ω, the Chevalley involution ω(Ei) =−Fi, ω(Fi) =−Ei, ω(Hi) =−Hi.
U0q(son) is a (left) coideal:
∆ :U0q(son)→Uq(gln)⊗U0q(son).
Hence,Rep0q(son) is onlyq-monoidal and carries a left action ofRepq(gln).
5 8
3 4
3 9
3 6
3 1
1 1 1 1
7
A-web new part
Uq(son) U0q(son) Subalgebra ofUq(gln)
Hopfalgebra Quantization ofU(son)
“Nice quantum numbers”
“Nice topology”
Connected to Peng’s talk yesterday: θ=ω, the Chevalley involution ω(Ei) =−Fi, ω(Fi) =−Ei, ω(Hi) =−Hi.
U0q(son) is a (left) coideal:
∆ :U0q(son)→Uq(gln)⊗U0q(son).
Hence,Rep0q(son) is onlyq-monoidal and carries a left action ofRepq(gln).
5 8
3 4
3 9
3 6
3 1
1 1 1 1
7
A-web new part