All Questions
2,633 questions
2
votes
0
answers
72
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Dimensions of centralizers in quantum Lie algebras associated to $\mathfrak{sl}_n$
Following ideas of Woronowicz, Lyubaschenko and Sudbery defined in Quantum Lie algebras of type $A_n$ the notion of a quantum Lie algebra $\mathfrak{sl}_n$. Let me focus on the case where $q$ is not a ...
- 13k
5
votes
3
answers
1k
views
Reg the motivation behind Lusztig-Vogan bijection
Let $G$ be an algebraic group. Choose a Borel subgroup $B$ and
a maximal Torus $T \subset B$. Let $\Lambda$ be the set of weights wrt $T$ and let $\mathfrak{g}$ be the lie algebra of $G$.
Now, ...
- 2,821
1
vote
0
answers
86
views
Isomorphism between $T_{[g,X]} (G \times _H \mathfrak{g}/\mathfrak{h})$ and $\mathfrak{g}/\mathfrak{h} \times \mathfrak{g}/\mathfrak{h}$
Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}$ and $H$ be a Lie subgroup of $G$ with Lie algebra $\mathfrak{h}$. Consider the action of $H$ on $G$ by right multiplication and the ...
- 139
6
votes
1
answer
507
views
Degree-2 étale covers of curves in characteristic 2 vs torsion points on the Jacobian
It can be found, at Hartshorne exercise 4.2.7 for example, that in the case where $\operatorname{char}(k) \neq 2$ we have a nice correspondence between etale degree 2 covers of a curve $C$ and 2-...
- 13k
3
votes
0
answers
169
views
The subalgebras of $\mathfrak{su}(2^n)$
$\DeclareMathOperator\su{\mathfrak{su}}$I want to find out all the subalgebras of $\su(N)$, in particular, $N=2^n$, which is the Lie algebra of $n$-qubits.
I don't know whether this is a hard question ...
- 63.9k
41
votes
2
answers
2k
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Implications of non-negativity of coefficients of arbitrary Kazhdan-Lusztig polynomials?
In their seminal 1979 paper Representations of Coxeter groups and Hecke algebras (Invent. Math. 53, doi:10.1007/BF01390031),
Kazhdan and Lusztig studied an arbitrary Coxeter group $(W,S)$ and the ...
- 35.5k
15
votes
2
answers
554
views
Annihilate a simple Lie algebra using two commutators
Let $\mathfrak{g}$ be a simple finite-dimensional Lie algebra over an arbitrary field $K$. For any nonzero $x\in\mathfrak{g}$ we must have $[\mathfrak{g},x]\neq\{0\}$, or else we violate simplicity.
...
- 13k
3
votes
1
answer
297
views
Showing the positivity of the determinant of $\mathfrak{sp}(n)$ without making use of diagonalization
Let $\mathfrak{sp}(n)$ be the lie algebra of compact symplectic group $\mathrm{SP}(n)$, regarded as a compact form of $\mathfrak{sp}(2n,\mathbb{C})$, so we can talk about its (complex) determinant.
...
- 18.7k
3
votes
0
answers
114
views
A certain Lie algebra associated to a bialgebra
Let $A$ be a bialgebra over a field $k$. The space $A^* = \operatorname{Hom}(A, k)$ possesses an algebra structure, given by the convolution product $f*g = f \otimes g \circ \Delta$.
Let $\gamma \in A^...
- 338
2
votes
0
answers
676
views
Derivative of the projection map $\pi: G \times \mathfrak{g} \rightarrow G \times_K \mathfrak{g}$
Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}.$ Let $K$ be a closed subgroup of $G$ with Lie algebra $\mathfrak{k}.$ We define the manifold $$\mathcal{E}:= G \times_K \mathfrak{g}$$
to ...
- 139
3
votes
0
answers
149
views
What direction does the derivation of an inseparable algebraic variable point in?
I've been thinking about the geometry of inseparable field extensions lately, since I'm studying smoothness in commutative rings in an advanced topics course this semester. I've generally come to the ...
- 1,969
5
votes
0
answers
123
views
Product of $U^+_q(\mathfrak{sl}_2)_i$ in $U_q(\mathfrak{g})$ according to some reduced expression
Let $\mathfrak{g}$ be some simple Lie algebra, $\alpha_1,\alpha_2,\cdots,\alpha_n$ be its simple roots. Let $U^+_q(\mathfrak{sl}_2)_i$ be the subalgebra of $U_q(\mathfrak{g})$ generated by $E_i,K_i$. ...
- 91
17
votes
2
answers
947
views
Are the unipotent and nilpotent varieties isomorphic in bad characteristics?
In characteristic 0 or good prime characteristic, there are standard ways to relate the unipotent variety $\mathcal{U}$ of a simple algebraic group $G$ and the nilpotent variety $\mathcal{N}$ of its ...
- 13k
3
votes
1
answer
494
views
Calculating fusion rules for $\operatorname{Rep}(G)$ and $G_{k}$ [reference request]
Could somebody please direct me to textbooks / literature that (perhaps lay the foundations for and) detail the method for determining the fusion rules for categories such as
$\operatorname{Rep}(G)$ ...
0
votes
0
answers
229
views
How do I detect whether a representation is (or is not) the adjoint representation?
Let $(\mathfrak{g},[\cdot,\cdot])$ be a Lie algebra. There is a God-given representation of $\mathfrak{g}$, namely, the adjoint representation $\operatorname{ad} : \mathfrak{g} \to \operatorname{Der}(\...
- 13k
3
votes
1
answer
129
views
Schur multiplier of finite-dimensional simple Lie algebras in positive characteristic
The Schur multipliers of finite simple groups are known and easily accessible:
https://en.wikipedia.org/wiki/List_of_finite_simple_groups
Moreover, as a consequence of the second Whitehead's Lemma, if ...
- 5,878
2
votes
0
answers
69
views
Abelian category for $(\mathfrak{g},T)$ modules with nontrival Grothendieck group
Let $G$ be a reductive Lie group over $\mathbb{C}$, and write $\mathfrak{g}$ for its Lie algebra. Let $T\subseteq B\subseteq G$ be a maximal torus and Borel subgroup, where $\operatorname{Lie}B=\...
- 13k
5
votes
1
answer
1k
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What are some interesting examples of quotients by Lie group actions?
I’ve been going through Lee’s Introduction to Differential Geometry. It’s a great book but I feel it lacks examples and concrete applications of the ideas presented. I want to work out a list of ...
- 111
1
vote
1
answer
360
views
Unipotent orbits and intersection with Levi and pseudo-Levi subgroups
Given a simple complex Lie group $G$ (I might say upfront that I am mostly interested with exceptional Lie algebras) and a nilpotent orbit $\mathcal{O}\subset G$ I would like to describe the ...
- 1,336
1
vote
0
answers
103
views
Infinite-dimensional Lie group corresponding to $U\mathfrak{g}$?
Let $\mathfrak{g}$ be a Lie algebra. The universal enveloping algebra $U\mathfrak{g}$ is then an infinite-dimensional associative algebra which can be endowed with the structure of a Lie algebra. Is ...
- 63.9k
6
votes
2
answers
448
views
Homogeneous symplectic manifolds
I have often heard/read a statement (see, e.g., this MathOverflow question) equivalent to the following:
Let $G$ be a connected Lie group and $(M,\omega)$ a connected and simply-connected symplectic ...
- 2,289
5
votes
1
answer
183
views
Find all finite dimensional simple Lie algebras satisfying certain conditions
Let $\mathfrak{g}$ be a finite dimensional simple Lie algebra over $\mathbb{C}$. Suppose that $\mathfrak{g}$ can be generated by five nonzero elements $x,y,x',y',h\in \mathfrak{g}$, which satisfy the ...
CommunityBot
- 1
6
votes
1
answer
643
views
Classification of simple Lie algebras over finite fields
Classification of simple (or simple-restricted) Lie algebras over algebraically closed fields in positive characteristic is studied for a long time. Today, we know all finite-dimensional simple (or ...
- 5,878
1
vote
1
answer
207
views
Character formula for the fundamental representations of $\frak{sl}_n$
For the Lie algebra $\frak{sl}_{n+1}$ we denote its fundamental irreducible representations by $V(\pi_i)$, with $i=1, \dots, n$. Where can I find a table of the character formula (in other words a ...
- 8,597
3
votes
1
answer
222
views
Is there a name for a noncommutative generalization of Poisson algebra?
Is there a name for an associative algebra which is further endowed with a Lie algebra structure such that the Leibniz rule holds, i.e.,
$$[a, b \circ c]=[a,b]\circ c+b\circ [a,c], \hspace{15mm}(*)$$ ...
- 16.9k
2
votes
0
answers
171
views
Monogenic function fields
Recall that a number field $K$ is said to be monogenic if its ring of integers is of the form $\mathbb{Z}[\alpha]$, or equivalently, if it has a power integral basis. There are many references one can ...
- 103
3
votes
4
answers
474
views
Nilradical of a Lie algebra associated to a associative algebra
Let $A$ be a finite dimensional (unital) $K$-Algebra. By $A^{\circ}$ we denote the associated $K$-Lie-algebra of $A$ with respect to the product $a\circ b:=ab-ba$. In addition, we denote by $rad(A^{\...
- 1
27
votes
4
answers
3k
views
Have people successfully worked with the full ring of differential operators in characteristic p?
This question is inspired by an earlier one about the possibility of using the full ring of differential operators on a flag variety to develop a theory of localization in characteristic $p$. (Here ...
- 1,228
1
vote
0
answers
149
views
Why do such a birational map exists? And why it is unique?
Let $G$ be a complex linear algebraic group which is connected and reductive and let $\mathfrak g$ be its Lie algebra.
Suppose that $H \subset G$ is a 1-dimensional torus such that
the action of $H$ ...
- 381
2
votes
1
answer
1k
views
Lie derivative on Lie group in the direction of an element of Lie algebra
I want a reference to the definition of the Lie derivative of a smooth function $f:G \to \mathbb R$ on a Lie group $G$ in the direction of an element $\theta$ of the Lie algebra $\mathfrak G$.
I can ...
- 126
0
votes
1
answer
169
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Representations of simply connected Lie groups [closed]
Let $G$ be a simply connected Lie group. Is it true that any finite dimensional representation of its Lie algebra is the differential of a representation of $G$?
A reference would be helpful.
Sorry if ...
- 126
0
votes
1
answer
130
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Number of reduced decompositions of the dihedral group $D_6$ [closed]
The Weyl group of $\frak{g}_2$ is the dihedral $D_6$. Let us denote its longest element by $w_0$. How many reduced decompositions does $w_0$ have?
- 24.2k
30
votes
4
answers
3k
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A mysterious Heisenberg algebra identity from Sylvester, 1867
I am trying to understand two papers by James Joseph Sylvester:
P92: "Note on the properties of the test operators which occur in the calculus of invariants, their derivatives, analogues, and laws of ...
- 10.5k
24
votes
6
answers
7k
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Introduction to W-Algebras/Why W-algebras?
Does anyone know of an introduction and motivation for W-algebras?
Edit: Okay, sorry I try to add some more background. W algebras occur, for example when you study nilpotent orbits: Take a nice ...
- 4,713
6
votes
0
answers
154
views
Sections of $\mathcal{L}_{\lambda}$ on intersections of open cover on a flag variety
Let $G$ be a reductive complex algebraic group, $P$ a parabolic subgroup, $\mathbb{C}_{\lambda}$ a one-dimensional representation of $P$ and $\mathcal{L}_{\lambda}$ the corresponding line bundle on $G/...
- 1,077
3
votes
2
answers
651
views
Representations of $\mathrm{SO}_n$ versus representations of $\mathrm{Spin}_n$ [duplicate]
$\DeclareMathOperator\SO{SO}\DeclareMathOperator\Spin{Spin}$Since $\Spin_n$ is a compact simply connected simple Lie group, its irreducible representations are equivalent to the irreducible ...
- 954
1
vote
0
answers
269
views
The Lie group corresponding to the Lie algebra $\frak{so}_n$ [closed]
As is standard knowledge - there is a bijective correspondence between compact simply connected simple Lie groups and complex simple Lie algebras. So what is the The Lie group corresponding to the Lie ...
- 327
4
votes
1
answer
238
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Exponential map and Lie correspondence within a Hopf algebra setting
The Cartier-Konstant-Milnor-Moore (et al.) theorem for Hopf algebras states that a cocommutative Hopf algebra over $\mathbb{C}$ is isomorphic to a smash product of a universal enveloping algebra of a ...
- 1,144
2
votes
0
answers
102
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Submodules of direct sums of Verma modules for the Virasoro algebra
Let $M_1=M(c, h_1)$, $M_2 = M(c, h_2)$ be two Verma modules for the Virasoro algebra. Let $N$ be a submodule of the direct sum $M_1 \oplus M_2$. I am wondering if there is any classification results ...
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1
vote
1
answer
170
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Question about regular elements in a Lie subalgebra
Let $G$ be a compact connected Lie group and $T$ is a maximal torus of $G$. Let $K$ be a non trivial connected Lie subgroup of $G$.
We say that $r \in \mathfrak{g}$ is a regular element of the Lie ...
- 13k
4
votes
1
answer
198
views
Simple restricted but not restricted simple Lie algebras
Let $F$ be a field which has a positive characteristic $p \ge 2$ and $(\mathfrak{g},[p])$ be a restricted Lie algebras over a field $F$ where $[p]$ is a $p$-th power map on $\mathfrak{g}$. $(\mathfrak{...
- 5,878
2
votes
1
answer
261
views
Semisimple Lie algebra modules with $1$-dimensional weight spaces
Given a semisimple complex Lie algebra $\frak{g}$ of rank $r$, with Chevally generators $E_i,F_i,K_i$. Let $V$ be a finite dimensional representation of $\mathfrak{g}$ such that each weight space of $...
- 8,597
6
votes
2
answers
380
views
Rank one adjoint operators on a Lie algebra
Let $\mathfrak{g}$ be a (finite dimensional) semi-simple Lie algebra over a field $k$ and let $x \in \mathfrak{g}$. By definition, we have the equivalence:
$$ \mathrm{rk}(\mathrm{ad}_x) = 0 \iff x = 0,...
- 63.9k
3
votes
0
answers
268
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The double exponential map and the Baker–Campbell–Hausdorff–Dynkin series
$\DeclareMathOperator\Exp{Exp}$A. Gavrilov has several nice works studying the double exponential map and its properties ([1] and references therein).
Given a complete Riemannian manifold $(M,g)$ and ...
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9
votes
2
answers
1k
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Examples of Richardson orbit closures not having a symplectic resolution?
This is a follow-up to a recent question asked by Peter Crooks here. The answer by Ben Webster includes a helpful link to the corrected arXiv version of Baohua Fu's 2003 Invent. Math. paper ...
- 4,713
2
votes
1
answer
185
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Finite, normal subgroups of reductive groups in positive characteristic
Consider the following statement about a connected, reductive group $G$ over a field $k$:
Every finite, normal subgroup $N$ of $G$ is central.
In characteristic $0$, this is true, and the proof is ...
- 13k
6
votes
1
answer
1k
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Centralizers of nilpotent elements in semisimple Lie algebras
Let $G$ be a connected, simply-connected, complex, semisimple Lie group with Lie algebra $\frak{g}$, and let $\xi\in\frak{g}$ be a nilpotent element. I am interested in understanding the structure of $...
- 63.9k
6
votes
1
answer
351
views
Necessary and sufficient conditions for Littlewood Richardson coefficients to be non zero
Is there any necessary and sufficient conditions for $V(\tau)$ to be an irreducible component of the tensor product of two irreducible representations $V(\lambda)$ and $V(\mu)$ of a simple lie algebra ...
- 24.2k
11
votes
1
answer
1k
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How is this (Tannakian) de Rham fundamental group calculated?
$\newcommand{\dR}{\mathrm{dR}}$Edit: I originally asked this question on MSE, but migrated it to MO after a long period of inactivity and a recommendation from another user.
Let $X$ be a complex ...
- 63.9k
1
vote
0
answers
188
views
Is the group law for SO(2n, R) encoded in so(2n,R)?
Note that this is a partial duplicate of my math.stackexchange question here. In this post I am asking something slightly broader. Note that I am a mathematical physicist and not a representation ...
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