Tutorial: center of the small Quantum group¶

This page appears both rendered to HTML in the docs, and as an interactive notebook in the /examples folder in the repository.

Introduction¶

This tutorial will show how to compute the center of the small quantum group, as described in the two papers by Annach Lachowska and You Qi:

https://arxiv.org/abs/1604.07380v3

https://arxiv.org/abs/1703.02457v3

Let $G$ be a complex simple Lie group, and let $P$ be a parabolic subgroup. Then we consider the cotangent bundle of the associated partial flag variety:

\[\tilde{\mathcal N}_P :=T^*(G/P)\]

We are then interested in computing

\[HH^s(\tilde{\mathcal N}_P)\cong \bigoplus_{i+j+k=s}H^i(\tilde{\mathcal N}_P,\wedge^jT\tilde{\mathcal N}_P)^k\]

This can be computed by using the BGG resolution. We define the following module:

\[M_j^k = \bigoplus_r \operatorname{Sym}^{j-r+k/2}\mathfrak u_P\otimes \wedge^r\mathfrak g\otimes \wedge^{j-r}\mathfrak n_P\]

Let $\Delta\colon\mathfrak p\to \mathfrak g\oplus \mathfrak u_P\otimes\mathfrak n_P$ be given by the inclusion in the first component and in the second component by the adjoint action (after identifying $\operatorname{End}(\mathfrak n_P)$ with $\mathfrak u_P\otimes \mathfrak n_P$). Then $\Delta$ induces a map $M_{j-1}^k\to M_j^k$. We define the module

\[E_j^k = M_j^k\big/\Delta(M_{j-1}^k)\]

Then the cohomology of the BGG resolution of $E_j^k$ in degree $i$ with respect to a dominant weight $\mu$ computes the multiplicity of $\mu$ of $H^i(\tilde{\mathcal N}_P,\wedge^jT\tilde{\mathcal N}_P)^k$.

Example¶

We begin by loading the required packages.

[1]:

from bggcohomology.la_modules import BGGCohomology
from bggcohomology.bggcomplex import BGGComplex
from bggcohomology.quantum_center import *
from bggcohomology.quantum_center import _compute_kernel, _compute_kernel2
from bggcohomology.weight_set import WeightSet
from bggcohomology.cohomology import compute_diff

Now let us compute a simple example and set $G=A_2$. We will compute $H^0(\tilde{\mathcal N}_0,\wedge^2T\tilde{\mathcal N}_0)^{-2}$. Here the subscript $0$ indicates the principal block. The principal block corresponds to the parameter subset=[]. The other parameters are s,i,j,k, and they satisfy i+j+k=s and j>=i.

If we want to change the parabolic subalgebra $P$ then we can change subset to e.g. subset=[1] for

\[\mathfrak p = \mathfrak b\oplus\mathbb C\langle e_1\rangle=\mathbb C\langle e_1,f_1,f_2,f_{12} \rangle,\qquad \mathfrak u_P = \mathbb C\langle e_{12},e_2\rangle,\qquad \mathfrak n_P = \mathbb C\langle f_{12},f_2\rangle\]

[4]:

BGG = BGGComplex('A2')
s = 1
i = 1
j = 2
k = s-i-j
print('i+j=%d, j-i=%d, k=%d\n'%(i+j,j-i,k))

subset = [1]
mjk = Mjk(BGG,j,k,subset=subset)
cohom = BGGCohomology(BGG, mjk, coker=Eijk_basis(BGG,j,k,subset=subset))
cohom.cohomology_LaTeX(i=i,only_non_zero=False,print_modules=True,print_betti=True,complex_string=r'\tilde{\mathcal N}_0,\wedge^2T\tilde{\mathcal N}_0)^{-2}')

i+j=3, j-i=1, k=-2

$\displaystyle \mathrm H^{1}(\tilde{\mathcal N}_0,\wedge^2T\tilde{\mathcal N}_0)^{-2})=\mathbb{C}$

$\displaystyle \mathrm b^{1}(\tilde{\mathcal N}_0,\wedge^2T\tilde{\mathcal N}_0)^{-2})=1$

CPU times: user 20.6 ms, sys: 58 µs, total: 20.7 ms
Wall time: 20.7 ms

Bigraded tables¶

The program can also produce bigraded tables. We decompose impose a bigrading on the total cohomology $HH^s(\tilde{\mathcal N}_P)$ by $a=i+j$ and $b = j-i$. Since the computation can take a while a progress bar is provided.

Due to the $\mathfrak{sl}_2$ symmetry in the bigraded table, we only actually need to compute half of it. We can choose to either compute the whole table, or only half if it through the half_only parameter, and extend by symmetry. It’s default value is False.

The bigraded table can be rendered in LaTeX through the display_ab_dic function, which takes an additional parameter extend_half which computes the second half of the table from the first half through symmetry considerations. The produced LaTeX code can easily be copied. The code below also saves the tables to a LaTeX document.

The parameter compact determines how the weight decomposition of the cohomology is displayed. E.g. with compact=False it would display $\mathbb C\oplus L(\alpha_1+\alpha_2)^2$ and with compact=True it gives $\mathbb C L_{1,1}^2$.

[ ]:

import pickle
import os
import itertools
from tqdm.auto import tqdm
os.makedirs('pickles', exist_ok=True)
os.makedirs('tables', exist_ok=True)

# Only compute cohomology for particular highest weight module
mu = None
#mu=(0,0)

# the parameters we actually want to change
diagram = 'A4'
BGG = BGGComplex(diagram)
subset=[]

# compute only half of the table, extend by symmetry
half_only = True
extend_half = half_only

# Exclude the top-left to bottom-right diagonal. If s=0, these should all be the trivial rep.
exclude_diagonal = True

# Display in full form
compact = True

# Load results if already computed
load_pickle = True

# Increase max memory size of the pari stack
# Set this as high as possible.
pari.allocatemem(10^6,30*10^9)

s=0
#for method in [0,1]:
#for s in itertools.count():
method = 0
for s in [0]:
    picklefile = os.path.join('pickles',f'{diagram}-s{s}-{subset}.pkl')
    if load_pickle and os.path.isfile(picklefile):
        previous_cohom = pickle.load(open(picklefile, 'rb'))
    else:
        previous_cohom = None
    texfile = os.path.join('tables',f'{diagram}-s{s}-{subset}.tex')
    cohom_dic = dict()
    with tqdm(all_abijk(BGG,s=s,subset=subset,half_only=half_only)) as inner_pbar:
        with tqdm(leave=None) as outer_pbar:
            map_pbar = tqdm()
            for a,b,i,j,k in inner_pbar:
                if previous_cohom is not None and (a,b) in previous_cohom:
                    cohom_dic[(a,b)]=previous_cohom[(a,b)]
                    inner_pbar.update()
                    continue
                if exclude_diagonal and s==0 and (a==b):
                    cohom_dic[(a,b)]=[((0,)*BGG.rank,1)]
                    inner_pbar.update()
                    continue
                inner_pbar.set_description('i+j= %d, j-i = %d'%(a,b))
                #coker = Eijk_basis(BGG,j,k,subset=subset,pbar=outer_pbar)
                mjk = Mjk(BGG,j,k,subset=subset)
                outer_pbar.set_description('Initializing cohomology')
                coker=Eijk_basis(BGG,j,k,subset=subset,method=method)
                cohom = BGGCohomology(BGG, mjk,
                                      coker=coker,pbars = [outer_pbar,map_pbar])
                outer_pbar.set_description('Computing cohomology')
                cohom_list = cohom.cohomology(i, mu=mu)
                cohom_dic[(a,b)] = cohom_list
                with open(picklefile, 'wb') as f:
                    pickle.dump(cohom_dic,f)
    print('-'*50)
    print(f'type {diagram}, table for s={s}:')
    cohom = BGGCohomology(BGG)
    cohom_dic = extend_from_symmetry(cohom_dic)
    latex_dic = {k:cohom.cohom_to_latex(c, compact=compact) for k,c in cohom_dic.items()}
    betti_dic = {k:cohom.betti_number(c) for k,c in cohom_dic.items()}
    tab1 = display_bigraded_table(latex_dic)
    tab2 = display_bigraded_table(betti_dic)
    tab3 = display_cohomology_stats(cohom_dic, BGG)
    with open(texfile, 'w') as f:
        f.write(prepare_texfile([tab1,tab2,tab3],title=f'type {diagram}, s={s}, subset={subset}'))

PARI stack size set to 1000000 bytes, maximum size set to 30000001024

[ ]: