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Some delicate quantizations

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 ofVglq is [n]. The quantum dimension ofVsoq is [n−1]+1.

Hence,Vsoq does not come fromVglq ! This “flaw” propagates all the way through:Va

qVsoq have “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.

Some delicate quantizations

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 ofVglq is [n]. The quantum dimension ofVsoq is [n−1]+1.

Hence,Vsoq does not come fromVglq ! This “flaw” propagates all the way through:Va

qVsoq have “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.

Some delicate quantizations

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 ofVglq is [n]. The quantum dimension ofVsoq is [n−1]+1.

Hence,Vsoq does not come fromVglq ! This “flaw” propagates all the way through:Va

qVsoq have “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.

Some delicate quantizations

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 ofVglq is [n]. The quantum dimension ofVsoq is [n−1]+1.

Hence,Vsoq does not come fromVglq ! This “flaw” propagates all the way through:Va

qVsoq have “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.

Some delicate quantizations

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 ofVglq is [n]. The quantum dimension ofVsoq is [n−1]+1.

Hence,Vsoq does not come fromVglq ! This “flaw” propagates all the way through:Va

qVsoq have “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.

Some delicate quantizations

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 ofVglq is [n].

The quantum dimension ofVsoq is [n−1]+1.

Hence,Vsoq does not come fromVglq ! This “flaw” propagates all the way through:Va

qVsoq have “weird” quantum dimensions.

The quantum dimension ofVso5

q

Above: Kuperberg’sB web 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.

Some delicate quantizations

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 ofVglq is [n].

The quantum dimension ofVsoq is [n−1]+1.

Hence,Vsoq does not come fromVglq ! This “flaw” propagates all the way through:Va

qVsoq have “weird” quantum dimensions.

The quantum dimension ofVso5

q

Above: Kuperberg’sB web 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.

Some delicate quantizations

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 ofVglq is [n]. The quantum dimension ofVsoq is [n−1]+1.

Hence,Vsoq does not come fromVglq ! This “flaw” propagates all the way through:Va

qVsoq have “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.

Some delicate quantizations

Using aq-monoidal diagrammatic category Webq,qn we can define a full Howe functor ΦextBD such that we get a commuting diagram

q(so2k) Rep0q(son)

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 ofVglq is [n]. The quantum dimension ofVsoq is [n−1]+1.

Hence,Vsoq does not come fromVglq ! This “flaw” propagates all the way through:Va

qVsoq have “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.