All Questions
931 questions with no upvoted or accepted answers
1
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60
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Comments/references on an obscure category of "rudimentary representations"
Consider a finite toset (cool word) $A=\{a_1<\ldots<a_N\}$. Let an ordered monomial be a finite formal product $b_1\ldots b_M$ with $b_i\in A$ and $b_i\le b_{i+1}$.
Consider the following ...
- 3,513
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0
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251
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A property of the Weyl vector of an irreducible root system
Let $R$ be one of the $A,B,C,D,E,F$ root systems, let $\Lambda$ be the lattice generated by the roots, $R^+$ a system of poitive roots and $\rho$ and $\alpha$ be the Weyl vector and the highest root ...
1
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0
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77
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Weyl's formula and Cartan decompositiom of semisimple lie algebras
I'm working on the article of Benkart and Osborn " Flexible Lie-admissible algebras", specially, i'm working on the lemma 3.1, this lemma represent the dimension of L-module homomorphisms of $L\...
- 11
1
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0
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140
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Some questions about $\rho^{\vee}$ in Lie theory
Let $\mathfrak{g}$ be a semisimple Lie algebra and $I$ its vertices of Dynkin diagram. The weight $\rho$ is defined by $\rho = \sum_{i \in I} \omega_i = \frac{1}{2} \sum_{\alpha \in \Phi^+} \alpha$, ...
- 6,211
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116
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Lie Algebra Module Decomposition in GAP
Let $\mathfrak{g}$ be a complex finite-dimensional Lie algebra and let $V$ be a finite-dimensional $\mathfrak{g}$-module. Is there a way for me to check in GAP or some other software package whether $...
user117370
1
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0
answers
73
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Decompose elements in $SL_2$ as a pair of elements in $SL_2^*$.
I have a question about decomposing elements in $SL_2$ as a pair of elements in $SL_2^*$. Here $SL_2^*$ is the dual Poisson Lie group of $SL_2$ which is defined as follows.
Let $G$ be a Poisson-Lie ...
- 6,211
1
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0
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65
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Is there a decomposition exists for $e^{c(K_++K_-)^2}$
In the usual $SU(1,1)$ group:
$$[K_+,K_-]=-2K_z,\quad [K_z,K_\pm]=\pm K_\pm.$$
Is there a decomposition exist for $e^{c(K_++K_-)^2}$?
Of course there won't exist a decomposition to $e^{K_+},e^{K_-},...
user120129
1
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0
answers
177
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Isogenies of type A_n, basis of cocharacter lattice
The isogenies of type $A_n$ are indexed by the subgroups of $\mathbb{Z} / (n+1) \mathbb{Z}$, i.e. by the positive divisors of $n+1$. If $a$ is a positive integer and $a \mid (n+1)$, then the ...
- 123
1
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0
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87
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Maximum Number of Skew-Symmetric matrices
I want to count the maximum number of rank 2 matrices in a space of certain dimension but I am stuck at some point. Any help/ suggestions are appreciated. Here is the question.
Let $\mathbb{M}_m$ be ...
- 179
1
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254
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Adjoint cohomology of Lie algebra commutes with direct sum?
The Witt algebra (denoted by $W$) is an infinite dimensional Lie algebra as:
$[L_{m},L_{n}]=(m-n)L_{m+n}; \,\,\,\ m, n\in \mathbb{Z}$.
I am looking for second adjoint cohomology $H^{2}(W_{1}\oplus ...
1
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0
answers
201
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The order of the antipode in a Hopf algebra
As a result of Radford, any finite-dimensional Hopf algebra an antipode of finite order.
My question: How can we classify all finite-dimensional Hopf algebras whose antipode is identity?
Here are ...
1
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0
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235
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Connectedness of symmetric subgroup of simply connected Lie group
Let $G$ be a connected real simple Lie group, and $\tau$ be an involutive automorphism of $G$. Then $\tau$ defines a symmetric subgroup $G^\tau$ of $G$. In general, $G^\tau$ is not necessarily ...
- 951
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0
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348
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rigid analytic geometry positive characteristic
I am a beginning graduate student. I have the following basic question I am very confused about:
Suppose $C$ is a smooth geometrically irreducible curve over a finite field $\mathbb{F}_q$, $q=p^m$, $...
- 147
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162
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Lie algebras over $\mathbb{C}(t)$
Are there naturally-arising Lie algebras(or superalgebras) over $\mathbf{C}(t)$?
I have such an algebra, and I want to compute its cohomology–but I'm out of ideas. Are there known methods or theory ...
- 11
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173
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Levi decompositions of k-rational points of linear algebraic groups
Let $k$ be a field with characteristic zero and $G$ be a (connected or not) linear algebraic group defined over $k$. We know that $G$ has a Levi decomposition i.e., $G=R_u(G)\rtimes L$, where $R_u(G)$ ...
- 1,702
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187
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Unitary dual of the Heisenberg group over non-archimedean local fields of characteristic two
What is the unitary dual of the Heisenberg group over non-archimedean local fields k of characteristic two? This is well-known for the real Heisenberg group and in fact, when local fields have ...
- 1,702
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0
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135
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Equivalence between blocks of BGG categories $\mathcal{O}$ which does not preserve highest weight structure
Let $\mathfrak{g}$ be a finite-dimensional (complex) semisimple Lie algebra. Then we denote the BGG category for $\mathfrak{g}$ by $\mathcal{O}$ as usual. It is well-known that $\mathcal{O}$ as well ...
- 159
1
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0
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149
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Smoothability of stable curves in mixed characteristic
Let $R$ be a complete DVR with residue field $k$ algebraically closed of characteristic $p$ and fraction field $K$ of characteristic zero. Let $C$ be a stable curve (in the sense of Mumford-Deligne) ...
- 2,323
1
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0
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460
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Orthonormal basis of matrices
I am asking if somebody knows how to do or is aware of the following construction:
Let $n \in \mathbb{N}$ be given, then take an arbitrary matrix $A \in \mathbb{C}^{n \times n}$. Then, there are maps ...
- 21
1
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0
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399
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Center of matrices
I encountered a neat problem in a problem in particle physics
So given $n$ skew symmetric matrices $A_1,...,A_n$ in $\mathbb{C}^{d \times d}.$
I would like to call this the commutator property: $...
1
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0
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201
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Laplacian on two Lie groups have the same Lie algebra
I know that if $G$ is a Lie group and $\mathfrak g = span\{X_i, 1\leq i \leq n\}$ be its Lie algebra, where $\{X_i\}$ are the vector fields of $G$. Then, the Casimir-Laplacian of $G$ is given by
$$\...
- 650
1
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0
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125
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About Composition diamond lemma
Composition Diamond lemma for Lie algebra over a field $F$ has already been investigated in several papers :
L.Bokut and Y.Q.Chen Groebner-Shirshov bases for Lie algebras
and
A.I Shirshov, ...
- 367
1
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0
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173
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Generating $\mathfrak{so}(7)$
Short version: Let $V$ be a 7-dimensional linear space of (real) square matrices. Suppose further that $[V,V]$ (the linear space spanned $[X,Y]$, $X,Y\in V$) is actually a subalgebra isomorphic to $\...
- 369
1
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0
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81
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Symplectic gradients whose span doesn't intersect Lie group orbits
I asked a similar question some hours ago. But thinking about my problem, I found a loophole in my arguments, so that my question wasn't the right one. This one is what I wanted to ask:
Let $G$ be a ...
- 501
1
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0
answers
84
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Integrable modules and comodules
Let $G$ be a semisimple Lie group and $\mathfrak{g}$ its Lie algebra. Do we have the following result: $V$ is an integrable $U(\mathfrak{g})$-module if and only if $V$ is a $\mathbb{C}[G]$-comodule? ...
- 6,211
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213
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Why the sub-Laplacian $\Delta_{sub}$ on the Heisenberg group $H^3 $ is sub-elliptic but not elliptic?
I want to know why the sub-Laplacian $\Delta_{sub}= X^2 + Y^2$ on the Heisenberg group $H^3 = \mathbb C \times \mathbb R$ is sub-elliptic but not elliptic, where $X$ and $Y$ are the left-invariant ...
- 650
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0
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71
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Low-dimensional classical r-matrices
Let $g= gl_2$. Suppose that $r \in g \otimes g$ satisfies the following properties:
(1) $r_{12} + r_{21} \in g \otimes g$ is $g$-invariant, $r_{12} = r$, $r_{21} = \tau \ r_{12}$.
(2) $[r_{12}, r_{...
- 6,211
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0
answers
99
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Generalized Gaussian Decomposition
Let $G$ be a connected complex semisimple Lie group. Let $H$ be a maximal torus of $G$, let $W$ be the Weyl group of $G$, and let $N_\pm$ be a pair of opposite maximal unipotent subgroups. For each ...
- 540
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0
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84
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Extra-Lorentzian Kac-Moody algebras
My question is about Kac-Moody (KM) algebras of finite rank with symmetrized Cartan matrices $B = C A$ ($A$ is Cartan matrix) of signatures $(-,-,+,...,+)$, $(-,-,-,+,...,+)$, etc. i.e. with $2$,...
- 371
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0
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51
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Iwasawa decomposition and Non-Abelian Centraliser of A
I'm studying Knapp's book "Representation Theory of Semisimple Groups" and am trying to understand the structure theory of non-compact groups. Namely, let $G=KAN$ be the Iwasawa decomposition, $\...
- 51
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114
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A simple Lie algebra with modules of a particular type
I’m trying to copy the following construction of P. Forster in group theory for Lie algebras. He takes a non-abelian simple group E which has an FpE-module V such that R = Rad(V ) is faithful and ...
- 310
1
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0
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189
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Poincaré inequality for connected Lie groups
Let $G$ be a compactly generated second countable locally compact group, and let $\mu$ be a probability measure which is:
symmetric,
adapted (in the sense that there is no proper subgroup $H$ such ...
- 527
1
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0
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801
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Differential and pre-differential of Jacobi identity
Let M be a manifold.
To what extent all Lie algebra structures with tensorial property on $\chi^{\infty}(M)$ are studied?
That is a Lie algebra structure for which $[X,fY]=f[X,Y]$.
(For every ...
- 366
1
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0
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108
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General quantum highest-weights dimension formulas
The formulas hold modulo typos :-)
It is well known (tl;dr fun fact: not well enough for me, I forgot where I saw it so I guess-computed it from the data in the Hayashi paper; promptly I found it in ...
- 4,829
1
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0
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33
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artinian quotients of U(g)
Suppose that $G$ is a connected Lie Group with Lie algebra $\mathfrak{g}$ and $\Gamma$ is a cocompact lattice, and you choose a left-invariant metric on $G/\Gamma$ and let $\Delta$ be the Laplacian. ...
- 2,125
1
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0
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63
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Reference for using an algebra of meromorphic functions to extend a Lie algebra
For example, let $\mathfrak{g}=\mathfrak{sl}_{2}\left(\mathbb{C}\right)$, let $s_{0}=1$, $s_{1}=-1$, $s_{2}$=0, $s_{3}=\infty$ in $\mathbb{P}_{1}\left(\mathbb{C}\right)$ and $\mathcal{R}$ is the ...
- 357
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184
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How the exceptional simple Lie groups/ algebras were first discovered and by whom?
I am wondering whether exceptional simple Lie groups/ algebras were first discovered in order to obtain a complete list of such objects, or they appeared as answers to completely different questions.
...
- 21.8k
1
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123
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How to define the determinant of a morphism between graded Lie algebras?
I have the following question. Suppose $g_1$ and $g_2$ are two finite dimensional, nilpotent, stratified Lie algebras and $A:g_1\to g_2$ is a morphism of the graded Lie algebra. I wonder whether there ...
- 1,881
1
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0
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277
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List of irreducible representations whose weights are in a single Weyl group orbit
Let $\mathfrak g$ be a (finite dimensional) simple (nonabelian) Lie algebra
over $\mathbb C$. I need a complete list of irreducible (nontrivial) representations $V$
of $\mathfrak g$ such that the Weyl ...
- 853
1
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0
answers
240
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A Lie algebra associated with a one dimensional foliation
A non vanishing vector field $X$ on a manifold is called "well behaved" if for every non vanishing smooth function $f$ we have $$C(X)\simeq C(fX)$$ This means that the centralizer Lie algebras $C(...
- 366
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0
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279
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Testing the faithfulness of group homomorphisms by testing on the level of induced Lie Algebras
Let $G$ be a group and let $\Gamma_G(k)$ be the $k$th term of the
lower central series of $G$. For each $k\geq 1$, set
$\mathrm{gr}_k(G)=\Gamma_G(k)/\Gamma_G(k+1)$ and
$$\mathrm{gr}_*(G):=\...
- 1,108
1
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0
answers
187
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Examples of Lie subalgebras of universal enveloping algebras
I'm looking for non-trivial examples of triples $(\mathfrak g, L, \psi)$, where $\mathfrak g$ and $L$ are finite-dimensional non-abelian Lie algebras over field $\Bbbk$ and $\psi\colon U_{\mathfrak g}\...
- 1,422
1
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0
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217
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Representations of $\mathfrak{so}(3)$ ($\mathfrak{so}(2,1)$) and $SO(3)$ ($SO(2,1)$)
(Apologies if this question is too basic!) I have explicit 5-dimensional real representations of $\mathfrak{so}(3)$ and $\mathfrak{so}(2,1)$, and I want to know whether it's necessarily true that the ...
- 818
1
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0
answers
324
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Solving $T^2 = -\kappa\, \mathrm{Tr}\, (\log(e^{i T \hat{H}_0} \hat{O}) )^2$ equation
Is there a way to solve the equation: $T^2 = -\kappa\, \mathrm{Tr}\, (\log(e^{i T \hat{H}_0} \hat{O}) )^2$ for $T$?
Here $\kappa$ is an arbitrary positive constant, $\hat{H}_0 \in \mathfrak{su}(N)$ ...
- 2,099
1
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0
answers
66
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A canonical map Aut$_{\mathsf{Lie}_R}(\mathfrak{n} \rtimes_\pi \mathfrak{g}) \to$ Aut$_{\mathsf{Lie}_R}(\mathfrak{n})$
Let $\mathfrak{n}$, $\mathfrak{g} \in \mathsf{Lie}_R$ be two Lie algebras over a commutative ring $R$, s.t. $\mathfrak{g}$ acts on $\mathfrak{n}$ as a derivation: $\pi:\mathfrak{g} \to \text{Der}(\...
- 19
1
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0
answers
85
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Analytic vectors of elliptic elements of the enveloping algebra in a strongly continuous representation of a Lie group
Let $G$ be a semi-simple real Lie group, and $\rho$ a strongly continuous unitary representation of $G$ in a Hilbert space $H$. Let $\mathfrak{g}$ be its Lie algebra and $d\rho$ be the infinitesimal ...
- 1,582
1
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0
answers
120
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Vanishing theorems that work in positive characteristic
Let $X$ be a smooth projective variety over a field of characteristic $p>0$ of dimension at least $2$. I am looking for some examples when $H^2(\mathcal{O}_X)$ vanishes. Is there any standard way ...
- 1,981
1
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0
answers
185
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Exact sequence of L-infinity-algebras
We call a sequence of $L_\infty$-algebras (weak) maps
$$0\to L\xrightarrow{f} M\xrightarrow{g} N\to 0$$
is exact if it is exact on the the underlying chain complexes level.
Thought I don't know ...
- 1,271
1
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0
answers
75
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Is it possible to compute the Iwahori Decomposition using the Chavalley Commutator Formulas?
Ideally, I would like a constructive, algorithmic proof of this fact. I have convinced myself that it is true, but my "proof" is not pretty. I would like to know if a more attractive or intuitive ...
- 11
1
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0
answers
216
views
polynomial representation of $sl_{2}(k)$
Let $k$ be an algebraic closed field of characteristic 0. We write
$$X=\left(
\begin{array}{ccc}
0 & 1\\
0 & 0\\
\end{array}
\right),~~
Y=\left(
\begin{array}{ccc}
0 & 0\\
1 & 0\\
\...