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Invariant weights associated to algebraic quantum groups

Consider an algebraic quantum group $(A, \Delta)$ in the sense of Van Daele, i.e. a regular multiplier Hopf $^*$-algebra with a positive left integral $\varphi$ and a positive right integral $\psi$. ...
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Tensor product of operator values weights (in the theory of locally compact quantum groups)

Let $M$ be a von Neumann algebra and $\psi$ a normal faithful semifinite weight on $M$. Then one should be able to form the object $$\iota \otimes \psi: (M\overline{\otimes} M)_+ \to \widehat{M_+}.$$ ...
Andromeda's user avatar
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Integral weight filtration on cohomology with no compact support

In "Descent, motives and K-theory", Gillet and Soule define a weight filtration on integral cohomology $H^{*}_{c}(X, \mathbb{Z})$ of a complex variety with compact support. They write that ...
Piotr Pstrągowski's user avatar
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Every locally compact group gives rise to a locally compact quantum group

A locally compact quantum group (in the sense of Vaes-Kustermans) consists of the data $(M, \Delta, \varphi, \psi)$ with $M$ a von Neumann algebra, $\Delta: M \to M \overline{\otimes} M$ a normal ...
Andromeda's user avatar
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Takesaki: question about lemma in section "Left Hilbert algebras and weights"

To make this question relatively self-contained, this post is quite long, but the question itself is rather short. Consider the following fragments in Takesaki's second volume "Theory of operator ...
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Disconnected reductive algebraic groups in Sage

All simply connected split simple groups have been implement on Sage and it is possible to find their highest roots, fundamental weights, Dynkin diagrams or compute the tensor of two of their ...
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Takesaki II "Connes cocycle derivative"

Consider the following fragments from Takesaki's second volume "Theory of operator algebras" (chapter VIII $\oint 3$, Modular Automorphism groups, p107-108: Why are the second and third ...
Andromeda's user avatar
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Sum of weights of an irreducible representation of $U(N)$

Let $R$ be a finite-dimensional irreducible representation of $U(N)$, with the set of weights $W_R$. Each element of $W_R$ is a vector of length $N$ with integer entries. Firstly, I would like to know ...
Blind Miner's user avatar
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What's the intuition for weighted limits?

I am reading Fosco's Coend Calculus and Emily Riehl's Categorical Homotopy Theory, Riehl's book motivates it in the following way, Abstraction 1: Classical limits in terms of cones: Cones from an ...
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Takesaki lemma 1.16 (volume II, chapter VII)

I am trying to understand the proof of the implication $(i)\implies (ii)$ in Takesaki's book "Theory of operator algebras II", chapter VII, which says the following: The relevant setting ...
Andromeda's user avatar
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How to recalculate the weights for an event that happens multiple times and requires all outcumes to be unique?

I think it's easiest to explain with an example. I have a weighted probability list A : 0.15 B : 0.15 C : 0.15 D : 0.1 E : 0.1 F : 0.1 G : 0.1 H : 0.075 I : 0.075 ...
Darius Takacs's user avatar
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Takesaki volume II chapter VII lemma 1.15

Consider the following fragments from Takesaki's second volume "Theory of operator algebras" in chapter VII on weight theory, lemma 1.15. Here $\mathcal{M}$ is a von Neumann algebra and $\...
Andromeda's user avatar
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Weighted limit calculus

In coend calculus by Fosco Loregian, it is mentioned that Lawvere conjectured that co/ends constitute a categorification of logical calculus. In his own words: The somewhat far-fetched conjecture ...
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Decomposition of an element as a difference of positive elements in the definition subalgebra of a weight (Takesaki)

I originally asked this on MSE, but did not get an answer there. Let $M$ be a von Neumann algebra. Let $\varphi: M_+ \to [0, \infty]$ be a weight on $M$. Consider \begin{align*}&\mathfrak{p}_\...
Andromeda's user avatar
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When the Leray spectral sequences for nice compactifications give the Deligne's weight ones?

Assume that $X$ is a proper smooth variety over an algebraically closed field $k$, $U=X\setminus (\cup D_i))$ where $D_i$ are closed subvarieties such that the set-theoretic intersections of all sets ...
Mikhail Bondarko's user avatar
4 votes
1 answer
398 views

Comparison of weight filtration on cohomology of complex manifold

Let $X$ be a smooth scheme of finite type over $\mathbb{Z}$ (or let's say a finitely generated $\mathbb{Z}$ algebra). To each prime $p \in \mathbb{Z}$ we can consider the $\mathbb{F}_p$ variety $$X_{\...
Tommaso Scognamiglio's user avatar
1 vote
1 answer
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inequivalent vertex weights on finite poset

Let $m\geq1$ and $P$ be an arbitrary poset with vertex set $V=\{v_1,\dots,v_n\}$, edge set $E,$ and set $O$ of orbits under $\text{Aut}(P).$ Can we efficiently generate all inequivalent nonnegative ...
mathematrucker's user avatar
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932 views

Nearby cycles without a function

Suppose that: $X$ is a smooth complex algebraic variety, $f : X \to D$ is a proper map to a small disc, smooth away from 0, $Z_\epsilon = f^{-1}(\epsilon)$, and $Z = Z_0$. Then there is a procedure (...
Geordie Williamson's user avatar
4 votes
1 answer
129 views

Functoriality of weighted limits

Let $C$ be a complete category, let $I$ be a small category, let $F,G:I\to C$ be functors, and let $W,U:C\to\mathrm{Set}$ be also functors, which we view as "weights". The weighted limits ...
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Effective weight-monodromy conjecture

$\DeclareMathOperator\Gr{Gr}$Let $G$ be the absolute Galois group of a finite extension of $\mathbb{Q}_p$ with inertia subgroup $I$, and let $V$ be an $\ell$-adic representation of $G$. Grothendieck's ...
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Where general mixed Galois representations are defined?

I am interested in etale cohomology of varieties, and respectively, in mixed $\mathbb Q_{\ell}$-adic Galois representations over finitely generated fields. What is the canonical reference for this ...
Mikhail Bondarko's user avatar
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Is a $\tau$-mixed Weil sheaf always a direct summand of a $\tau$-real sheaf?

Fix an isomorphism $\tau: \overline{\mathbf{Q}_\ell} \cong \mathbf{C}$ and consider Weil sheaves on a scheme $X$ over $\mathbf{F_q}$. It is a theorem that a $\tau$-real sheaf $\mathscr{F}$ is $\tau$-...
Kim's user avatar
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What is an example of a lisse Weil sheaf which is not mixed?

Fix a base field $\mathbf{F}_q$. What is a simple example of a lisse Weil sheaf which is not $\tau$-mixed for any identification $\tau: \overline{\mathbf{Q}_\ell} \cong \mathbf{C}$? Addendum: it ...
Kim's user avatar
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2 votes
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Weights of restriction to a Levi subgroup

Let us consider the group $G = \mathrm{GSp_{2g}/\mathbf{Q}}$ with respect to the symplectic pairing given by the matrix $\begin{pmatrix} & s \\ -s & \end{pmatrix}$, where $s$ is the $g\times ...
franck's user avatar
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Which fields and schemes "have enough finite residue fields"?

I am looking for assumptions on the spectrum $S$ of a field $K$ that ensure the following: there exists an excellent noetherian finite dimensional (integral) scheme $S'$ such that $S$ is its generic ...
Mikhail Bondarko's user avatar
1 vote
1 answer
1k views

Significance of the Eigenvalues of the adjacency matrix of a weighted di-graph

I'm currently running a simulation on a bunch of randomly generated points, each with two randomly selected 'partners' from the set of points. In the simulation the points try to move such that they ...
rysuds's user avatar
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About the geometry of the set of weights that is strongly linked to $\lambda$

Define $\eta\uparrow\lambda$ if $\eta=\lambda$ or $\eta=s_\alpha\cdot\lambda<\lambda$ for some $\alpha\in\Phi^+$. More generally, $\eta\uparrow\lambda$ if $\eta=\lambda$ or $\eta=s_{\alpha_1}s_{\...
James Cheung's user avatar
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About weights in $\mathfrak{h}^*$

On p.24 of the book "Representations of Semisimple Lie Algebras in the BGG Category $\mathcal{O}$". I quote: "For example, the fixed point $-\rho$ under the dot action lies in a linkage class by ...
James Cheung's user avatar
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1 answer
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Highest weight of a representation of a Lie Algebra [closed]

Given a Lie Algebra $\mathfrak{g}$, its Cartan Matrix $A$ and a finite representation $R$, is there a way of determining its highest weight $\Lambda$ in a simple way? In my course, we consider $\...
th-phy-student's user avatar
2 votes
1 answer
90 views

Show that if $\Theta$ is an infinitesimal weight of a real $T$-module $W$ ($T$ is a torus) then $-\Theta$ is also a weight

Show that if $\Theta$ is an infinitesimal weight of a real $T$-module $W$ ($T$ is a torus) then $-\Theta$ is also a weight. It is an exercise of Bröcker's book on Representations of Compact Lie ...
Gomes93's user avatar
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Barycentric weighting of a point in a mesh

I know there is a way to get a barycentric weighting of a point $p$ inside a convex polygon; for weighting $\omega_n$ of component $q_n$ with adjacent angles $\gamma_n$ and $\delta_n$: $$\omega_n = \...
rhavin's user avatar
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1 answer
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About regular infinitesimal character

Let $Z(\mathfrak{g})$ be the centre of $U(\mathfrak{g})$ and let $\chi_\lambda$ be an algebra homomorphism $Z(\mathfrak{g}) \to \mathbb{C}$ such that $ z \cdot v = \chi_\lambda(z)v $ for all $z \in Z(\...
James Cheung's user avatar
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1 vote
1 answer
98 views

About finite direct sum of full subcategory of category $\mathcal{O}^\mathfrak{p}$

As is shown in Representations of Semisimple Lie Algebras in the BGG Category $\mathcal{O}$, every nonzero module $M \in \mathcal{O}^\mathfrak{p}$ has a finite filtration with nonzero quotients, each ...
James Cheung's user avatar
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Associated subgroup of Weyl group

Let $\Phi$ be a root system. For a weight $\lambda\in\mathfrak{h}^*$, start by defining $ \Phi_{[\lambda]}:=\{\alpha\in \Phi \ | \ \langle \lambda,\alpha^{\lor}\rangle\in\mathbb{Z} \} $ and $ W_{[\...
James Cheung's user avatar
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Simplicity Criterion for Verma module

In Representations of Semisimple Lie Algebras in the BGG Category $\mathcal{O}$, $\lambda\in\mathfrak{h}^*$ is antidominant if $\langle \lambda + \rho, \alpha^\lor\rangle \not\in \mathbb{Z}^{>0}$ ...
James Cheung's user avatar
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0 votes
2 answers
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Some confusion about weights and roots in parabolic root systems

I was reading James Arthur's book An Introduction to the Trace Formula and had a couple of questions. Here $A_0$ is a maximal split torus of a reductive group $G$, $P_0 \supset A_0$ is a minimal ...
D_S's user avatar
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2 votes
1 answer
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About Extension group and weights in $\mathcal{O}^\mathfrak{p}$

Denote $M_I(\lambda)$ be the generalized Verma module with highest weight $\lambda$ and $L(\mu)$ is the simple highest weight module with highest weight $\mu$. Suppose $\text{Ext}_{\mathcal{O}^\...
James Cheung's user avatar
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1 vote
1 answer
209 views

About Hom and weight space of nilpotent Lie algebra cohomology

Let $\mathfrak{g}$ be a complex semisimple Lie algebra. Denote by $\Phi$ the root system of $(\mathfrak{g},\mathfrak{h})$ and denote by $\mathfrak{g}_\alpha$ the root subspace of $\mathfrak{g}$ ...
James Cheung's user avatar
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1 answer
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Minimizing the set of "wrong" edges in $K_\omega$ with $\{0,1\}$-weights

For any set $X$, let $[X]^2 = \big\{\{x,y\}:x\neq y\in X\big\}$. Let $f:[\omega]^2\to\{0,1\}$ be a function. The principal goal is to find a partition of $\omega$ such that if $m\neq n\in \omega$ ...
Dominic van der Zypen's user avatar
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About Extension group in Category $\mathcal{O}^\mathfrak{p}$

Let $M_I(\lambda)$ be the generalized Verma module with highest weight $\lambda$, $L(\lambda)$ be the simple highest weight module with highest weight $\lambda$. Suppose $\mu\le \lambda\le \nu$, does ...
James Cheung's user avatar
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about weight decomposition of U(sl3) [closed]

the universal enveloping algebra is $\mathbb{Z}^2$-graded algebra. $U(sl_3)= \bigoplus_{(i_1,i_2) \in \mathbb{Z}^2 } U_{i_1\alpha_1+i_2\alpha_2}$ $($the weight decomposition$)$ where $\alpha_1$ and $...
John's user avatar
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2 answers
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About block $\mathcal{O}_\lambda$ of Category $\mathcal{O}$

The blocks of $\mathcal{O}$ are precisely the subcategories consisting of modules whose composition factors all have highest weights linked by $W_{[\lambda]}$ to an antidominant weight $\lambda$. I ...
James Cheung's user avatar
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1 vote
1 answer
268 views

About subcategory of parabolic Category $\mathcal{O}^\mathfrak{p}$

It follows from Exercise 1.13 in Humphreys' Category $\mathcal{O}$ book that $M\in\mathcal{O}^\mathfrak{p}_{\chi_\lambda}$ has a direct sum decomposition $M=\oplus M_i$ such that all weights of each $...
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About weight of $M\in\mathcal{O}_{\chi_\lambda}$ in the Category $\mathcal{O}$

Given $M\in\mathcal{O}_{\chi_\lambda}$, how does the weights of $M$ relate to $\lambda$? Is it something about $W\cdot\lambda$?
James Cheung's user avatar
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2 votes
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About block of category $\mathcal{O}$

In the book "Representation of Semisimple Lie Algebra in the BGG Category $\mathcal{O}$". Exercise 1.13. Suppose $\lambda\not\in\Lambda$, so the linkage class $W\cdot\lambda$ is the disjoint union of ...
James Cheung's user avatar
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Highest weight category and weight structures

In various branches of representation theory, there is a notion of highest weight category. On the other hand there is a notion of weight structure on a triangulated category $C$ (introduced by ...
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About Morita equivalent and regular block of category $\mathcal{O}^\mathfrak{p}$

In the following paper: Representation type of the blocks of category $\mathcal{O}_S$ https://www.sciencedirect.com/science/article/pii/S0001870804002853 On p.196, it states that "When $\mu$ is ...
James Cheung's user avatar
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4 votes
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Good range and fair range

Let $G$ be a noncompact simple Lie group with complexified Lie algebra $\mathfrak{g}$. Fix a Cartan involution $\theta$, which defines a maximal compact subgroup $K$ of $G$. Take a $\theta$-stable ...
Hebe's user avatar
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1 vote
2 answers
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Achieving every possible ranking by rearranging weights

This problem actually arose as a question in the real world (see the paragraph "Origin of the problem" below). Let $\mathbb{N}$ denote the set of the positive integers and let $n\in\mathbb{N}$. If $...
Dominic van der Zypen's user avatar
9 votes
3 answers
969 views

Decomposition of tensor power of symmetric square

Studying some representation theory I came up with the following problem. We work over a field of characteristic $0$. Let $V$ be the standard representation of $\mathrm{GL}_n$ and let $W$ be the ...
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