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Particular decomposition of $SU(n)$

Given $a,b \in \mathfrak{su(n)}$ which generate the full algebra, it is possible to write and $G \in SU(n)$ as: $G = \exp(\alpha_1 a)\exp(\beta_1 b) \ldots \exp(\alpha_m a)\exp(\beta_m b)$ for some ...
Benjamin's user avatar
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3 votes
0 answers
162 views

Kac-Weisfeiler conjecture for Cartan type Lie algebras

Is there an analogue of Kac-Weisfeiler conjecture (a theorem of Premet) on the power of p dividing dimension of an irreducible representation for Lie algebras of Cartan type? Benkart and Feldvoss ...
Roman's user avatar
  • 1,526
3 votes
0 answers
111 views

Simple $\mathfrak{g}$-modules preserved by twisting

Let $G$ be a semi-simple lie group (simply connected for simplicity), $\mathfrak{g}$ its lie algebra. Write $\overline{G}=Inn(\mathfrak{g})$ for the adjoint form of $G$ which we identify here with ...
freeRmodule's user avatar
  • 1,077
3 votes
0 answers
307 views

Isotrivial factors of Jacobian

Let $k$ be an algebraically closed field of positive characteristic that it is not the algebraic closure of a finite field. Fix a smooth proper $k$-curve $C$ and write $J_C$ for its Jacobian abelian ...
Emiliano Ambrosi's user avatar
3 votes
0 answers
113 views

Cohomologies of $[V/GL_n]$ in characteristic $p$ for a representation $V$ of $GL_n$

Let $V$ be a representation of $G=GL_n$ (or more generally any reductive group $G$) over an algebraically closed field $\mathbb k$ of characteristic $p$. Let $[V/G]$ be the corresponding quotient ...
user42024's user avatar
  • 790
3 votes
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142 views

Is there a precisely formulable obstruction for the tangent bundle being a Lie algebra bundle?

Although vector fields (which are sections of the tangent bundle) form Lie algebras, the bundle itself, as far as I know, almost never carries Lie algebra structure; that is, in general, I believe, ...
მამუკა ჯიბლაძე's user avatar
3 votes
0 answers
112 views

Find logarithm of a matrix containing a constrained set of basis elements

Let $U$ be a unitary matrix, and let $H$ be an Hermitian matrix. I want to know if there is a $t \in\mathbb R$ such that $\exp(i t H) = U$. A connected question is: given a set $\{g_1, g_2, ..., g_N\}...
glS's user avatar
  • 342
3 votes
0 answers
96 views

Parametrization of the admissible dual

I have found mention of the fact that the spherical dual of a (real reductive quasisplit) group can be canonically parametrized by a suitable finite-dimensional complex vector space. More precisely, ...
Desiderius Severus's user avatar
3 votes
0 answers
156 views

How large is the intersection of the root system of a subalgebra of a compact Lie algebra with the original root system?

Let $\mathfrak{g}$ be a finite-dimensional real compact Lie algebra and $\mathfrak{t}\subset \mathfrak{g}$ a maximal abelian subalgebra. Let $\Delta(\mathfrak{g}_\mathbb{C},\mathfrak{t}_\mathbb{C})\...
B K's user avatar
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116 views

Algebraic (g,B) modules

Let $\mathfrak{g}$ be a semisimple lie algebra over a field of characteristic $0$, $G$ its adjoint group, with Borel group $B$. I am trying to understand the theory of algebraic $(g,B)$ modules as ...
C.Niculescu's user avatar
3 votes
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108 views

Is the exp map for the Lie group of positive functions on a manifold a global diffeo?

I asked this question on math stack exchange but figured I might get a quicker answer here. I included a bit more information at the end. Thank you for your help! As I understand it, for the Lie ...
X-Naut PhD's user avatar
3 votes
0 answers
51 views

Maximizing a function on $SU(4)$ similar to Von Neumann Trace Inequality

Given arbitrary $X,Y \in \mathfrak{su}(4)$, I want to maximize either of the following functions: $\max_{U,V \in SU(2)} \Re(\text{Tr}(X^\dagger (U^{\dagger} \otimes V^{\dagger})Y (U \otimes V)))$ ...
Benjamin's user avatar
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3 votes
0 answers
282 views

How do I obtain Vandermonde identity from Weyl's denominator formula?

Let $\Phi$ be a root system with Weyl group $\operatorname{Weyl}(\Phi)$, let $\Phi^+$ be a set of positive roots for $\Phi$ and $\rho$ be the half sum of the elements of $\Phi^+$. Then the Weyl's ...
Stabilo's user avatar
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3 votes
0 answers
155 views

Is the set of rays through imaginary roots of a Kac-Moody algebra dense in the imaginary cone?

Let $\mathfrak{g}(A)$ be the Kac-Moody Lie algebra associated to an indecomposable generalized Cartan matrix $A$, and let $Z$ be the convex hull in $\mathfrak{h}^*_{\mathbb{R}}$ of the set $\Delta_+^{...
Ben Davison's user avatar
3 votes
0 answers
137 views

Lagrangian subvariety in coadjoint orbit

In Chriss and Ginzburg's Representation Theory and Complex Geometry, Theorem 3.3.6 says that "Let $\mathbb{O}$ be a coadjoint orbit in $\mathfrak{g}^*$. Let $x\in \mathbb{O}$ be such that $x|_{\...
Daps's user avatar
  • 540
3 votes
0 answers
116 views

Extension of representations of certain compact Lie groups

Let $G$ be a real, connected, non-compact, semisimple Lie group with finite center and real rank $1$. Let $G=KAN$ be an Iwasawa decomposition, then $K\subset G$ is a maximal compact subgroup, and $\...
B K's user avatar
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3 votes
0 answers
130 views

About the purpose of introducing '"groups of Heisenberg type"

I would like to know, can we say that the "groups of Heisenberg type" where introduced by A. Kaplan in "Kaplan, A. (1980). Fundamental solutions for a class of hypoelliptic PDE generated by ...
Z. Alfata's user avatar
  • 650
3 votes
0 answers
132 views

References and Enlightenment for Kazhdan-Lusztig Composition Factor Multiplicities

I am studying the BGG category $\mathcal{O}$. There is one problem I couldn't get out of my mind and I don't know enough to find a good reference. Let $\mathfrak{g}$ be a finite-dimensional ...
Batominovski's user avatar
3 votes
0 answers
110 views

Gelfand Kirillov dimension and associated graded algebras

Given a filtered module $M$ over a Lie algebra $\mathfrak{g}$, can one describe the Gelfand Kirillov dimension of $M$ in terms of Gelfand Kirillov dimension of its associated graded $grM$? If so, ...
mkim17's user avatar
  • 31
3 votes
0 answers
406 views

Semisimple Lie algebras and the commutator algebra

Suppose $A$ is a associative unital $k$-algebra, where $\operatorname{char}k=0$. As is well-known, $A$ becomes a Lie algebra with respect to the commutator bracket $[x, y] = xy-yx$ for $x,y \in A$. ...
Paul Gilmartin's user avatar
3 votes
0 answers
153 views

Classical Yang-Baxter equation for Lie algebras and Lie superalgebras

The classical Yang-Baxter equation is \begin{align} [r_{12}, r_{13}] + [r_{12}, r_{23}] + [r_{13}, r_{23}] = 0. \quad (1) \end{align} What are the differences between this equation in the case of Lie ...
Jianrong Li's user avatar
  • 6,211
3 votes
0 answers
189 views

Non-invariant Lagrangian on SU(n)

I have a Lagrangian on $SU(n)$, which is not invariant. Given the Lagrangian $\mathcal{L}[U_t, \dot{U}_t] = \langle \dot{U}_t, \nabla J \big|_{U_t} \rangle$ I need to find the curves of stationary ...
Benjamin's user avatar
  • 2,099
3 votes
0 answers
255 views

Roots of exceptional complex reflection groups

I am looking to do a case-by-case check of a conjecture I have about Shephard groups, which are a subclass of complex reflection groups. These were classified by Shephard and Todd and there is one ...
andrewBee's user avatar
  • 273
3 votes
0 answers
275 views

Invariant functions on the dual Lie algebra

Let $G$ be a real Lie group and $\mathfrak{g}$ the corresponding Lie algebra. Let $\mathfrak{g}^*$ be the dual of the Lie algebra. Then we have the coadjoint action of $G$ on $\mathfrak{g}^*$. ...
Feanoris's user avatar
3 votes
0 answers
84 views

Are there nilpotent Manin Triples?

Let $\mathfrak{g}$ be a Lie bialgebra and denote by $\mathfrak{d}$ the double of $\mathfrak{g}$, i.e. $\mathfrak{d}$ is a Manin triple. Are there known examples or conditions on $\mathfrak{g}$ for ...
Wernli's user avatar
  • 31
3 votes
0 answers
73 views

False optima for control on Lie groups?

Consider the equation $\frac{d Y_t}{dt} = (A + w(t)B) Y_t$ evolving on a compact semi-simple Lie group $G$ where $\frak{g}$ is the Lie algebra of $G$ and $A,B \in \frak{g}$ and: $J:G \rightarrow [0,...
Benjamin's user avatar
  • 2,099
3 votes
0 answers
142 views

Automorphism group of Lie algebra of bounded operators

What is the automorphism group of the complex Lie algebra of bounded operators on a complex Hilbert space, with the commutator as Lie bracket? What for the real Lie algebra of bounded antihermitian ...
Arnold Neumaier's user avatar
3 votes
0 answers
126 views

Irreducible representations in BGG category $\mathcal{O}$ over (finitely) direct sum of general linear Lie superalgebra

Let $\mathfrak{g} = \oplus_i^k\mathfrak{gl}(m_i|n_i)$ be a direct sum of general linear Lie superalgebras $\mathfrak{gl}(m_i|n_i)$'s with the Cartan subalgebra $\mathfrak{h} = \oplus_i^k \mathfrak{h}...
Steven's user avatar
  • 159
3 votes
0 answers
204 views

Exponential analogue of formal connections

Let $F=\mathbb{C}((t))$. Let $G=GL_n$. Then $G(F)$ acts on $\mathfrak{g}(F)$ by gauge transformation: $$ g.x:=gxg^{-1} + \dot{g}g^{-1},\quad \quad g\in G(F), \quad x\in \mathfrak{g}(F). $$ Here, $\...
Dr. Evil's user avatar
  • 2,751
3 votes
0 answers
236 views

Deligne-Simpson problem for classical groups

Additive Deligne-Simpson problem was partially prooved by Kostov. Also there is Crawley-Boevey's approach to the question. The problem is about existence of a solution of the equation $$ A_1 +...+A_n =...
quantum's user avatar
  • 181
3 votes
0 answers
70 views

Attainability of Global Optima In Optimal Control

Given a manifold $M$ and a set of smooth functions of one real variable $\mathcal{A}$ and a 'control system' type first order differential equation: $\frac{d x(t)}{dt} = F(x,u)$ one can consider the ...
Benjamin's user avatar
  • 2,099
3 votes
0 answers
176 views

Double loop groups and cohomology

Let $G$ be a connected reductive group over $\mathbb{C}$ of Lie algebra $\mathfrak{g}$. What is the value of $H^{3}(\mathfrak{g}((t))((s)),\mathbb{C})$?
prochet's user avatar
  • 3,472
3 votes
0 answers
253 views

Can the product of a simple and a non-simple indecomposable representation be semisimple?

Consider two (possibly infinite-dimensional) representations $\rho$, $\pi$ of a semisimple Lie algebra $\mathfrak{g}$, with $\rho$ irreducible and $\pi$ indecomposable but not irreducible (i.e., not ...
Giuseppe Sellaroli's user avatar
3 votes
0 answers
170 views

The special embedding $\mathfrak{so}(7)\subset\mathfrak{so}(8)$

It is commonly known that we have a chain of embeddings $$SU(4)\subset Spin(7)\subset SO(8)$$ (there is more than one possible $Spin(7)$, just take one). Which is the explicit analog for the Lie ...
Jjm's user avatar
  • 2,091
3 votes
0 answers
215 views

scalar multiple of Young symmetrizer

The following is a lemma from Fulton and Harris' book -Representation theory,a first course (page 53): Lemma: For all $x\in \mathbb{C}\mathfrak{S}_r$, $c_{\lambda}\cdot x\cdot c_{\lambda}= scalar \;...
yang's user avatar
  • 181
3 votes
0 answers
154 views

Homology of derivations of Differential Graded Lie algebra

Let $(L,d)$ be a Differential Graded Lie Algebra ($L=\bigoplus L_i$ and $d:L_i \to L_{i-1}$ satisfying the graded Leibniz rule). On the algebra $\mathrm{Der}L$ of derivations of $L$ define a grading ...
Itai's user avatar
  • 131
3 votes
0 answers
404 views

rational representation of semisimple algebraic group

Let $G$ be a connected semisimple algebraic group defined over $\mathbb Q$. Could some expert give me a complete classification of finite dimensional $\mathbb Q$-irreducible representations of $G$? ...
ronggang's user avatar
  • 853
3 votes
0 answers
285 views

Can the commuting condition in Jordan-Chevalley decomposition be replaced with this global criterion?

Let $G$ be a reductive linear algebraic group defined over an algebraically closed field $k$ of arbitrary characteristic, and write $\mathfrak{g}$ for its Lie algebra. The Jordan-Chevalley ...
Jason's user avatar
  • 53
3 votes
0 answers
440 views

Decomposition of a representation of SU(N) into representations of SU(N-1)

Let $\omega_k$ be the highest weight of the $k$-th antisymmetric representation of $\mathfrak{su}(N)$. Consider an irreducible representation of $\mathfrak{su}(N)$, characterized by its highest ...
Bruno Le Floch's user avatar
3 votes
0 answers
3k views

Mathematica package for Lie algebra computations?

I am interested in performing Lie algebra computations in Mathematica. I did a bit of searching and found several packages (LieART, KILLING, SuperLie, maybe more), and wondered if anyone would ...
Idempotent's user avatar
3 votes
0 answers
141 views

Examples of divisible Lie algebras

We say that a nonzero Lie algebra $L$ is divisible, if for all elements $a$ and $b$ with $a\neq 0$, there exists $x\in L$ such that $[a, x]=b$. What are examples of divisible Lie algebras?
Sh.M1972's user avatar
  • 2,233
3 votes
0 answers
235 views

The fundamental in the tensor square of a complex representation of $SO(N)$

I would like to figure out whether there is an irreducible complex (in the sense non-self-conjugate) representation of a group $SO(N)$, $N>2$, whose tensor square contains the fundamental ...
D M's user avatar
  • 173
3 votes
0 answers
742 views

Explicit description of O^{cris}_n in Fontaine/Messing

Let $k$ be a perfect field of characteristic $p$, $W(k)$ the Witt ring and $K$ its quotient field. In their article "$p$-adic periods and $p$-adic etale cohomology" Fontaine and Messing give in II.1.4 ...
Matthias Kümmerer's user avatar
3 votes
0 answers
234 views

Generators and relations for the enveloping algebra of a unipotent radical

Let $G$ be a semisimple algebraic group over an algebraically closed field $k$ of characteristic 0 and let $B \subseteq G$ be a Borel subgroup with unipotent radical $U$. Let $P \supseteq B$ be a ...
Chuck Hague's user avatar
  • 3,637
3 votes
0 answers
308 views

Invertible Hasse-Witt for non-ordinary curves

Assume that $S$ is a smooth curve of over a field $k$ of characteristic $p>0$ and $f\colon X\to S$ is a relative curve over $S$ (i.e., the fibers are curves). It is well-known that when all the ...
Cyrus's user avatar
  • 395
3 votes
0 answers
281 views

What would be a characteristic-$p$ analogue for $C^{\infty}$-fiber bundles?

I'd like to know a notion for a morphism between algebraic varieties in characteristic $p$ that plays the role of a $C^{\infty}$-fiber bundle. It should be, in particular, flat. I'm not assuming the ...
shenghao's user avatar
  • 4,265
3 votes
0 answers
359 views

Does Branching in the Weight Diagram affect an embedding?

All groups here are compact semisimple Lie groups. Out of laziness I will use $B_7$ to mean $Spin(15)$. Suppose that one has a group $H$ and a subgroup $G$. The embedding determines the decomposition ...
ARupinski's user avatar
  • 5,191
3 votes
0 answers
423 views

Cohomologies associated to residually torsion-free nilpotent groups

This question is related to my previous question: Relationship between the cohomology of a group and the cohomology of its associated Lie algebra. A group $G$ is ${\it residually \ torsion \ free \ ...
Peter Goetz's user avatar
3 votes
0 answers
264 views

Is it possible to construct category $\mathcal{O}^{\mathfrak{p}}$ with non-standard parabolic subalgebra

The usual definition of the parabolic category $\mathcal{O}^{\mathfrak{p}}$ is the following. We consider Lie algebra $\mathfrak{g}$ of the rank $r$ with the root system $\Delta$ and the set of ...
Anton Nazarov's user avatar
3 votes
0 answers
487 views

Is the Cartan matrix a complete invariant of a Kac-Moody algebra?

In chapter 1 of Kac's book "Infinite dimensional Lie algebras" it is mentioned that two Kac-Moody algebras are isomorphic if and only if their Cartan matrices are isomorphic (i.e. they are the same up ...
user717's user avatar
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