All Questions
931 questions with no upvoted or accepted answers
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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 ...
3
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0
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162
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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 ...
3
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111
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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 ...
3
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307
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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 ...
3
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113
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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 ...
3
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142
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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, ...
3
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112
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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\}...
3
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96
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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, ...
3
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156
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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})\...
3
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116
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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 ...
3
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108
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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 ...
3
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51
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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:
$\max_{U,V \in SU(2)} \Re(\text{Tr}(X^\dagger (U^{\dagger} \otimes V^{\dagger})Y (U \otimes V)))$
...
3
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282
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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 ...
3
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155
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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_+^{...
3
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137
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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|_{\...
3
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116
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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 $\...
3
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130
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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 ...
3
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132
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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 ...
3
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110
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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, ...
3
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406
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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 $x,y \in A$. ...
3
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153
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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 ...
3
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189
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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 ...
3
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0
answers
255
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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 ...
3
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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 $\mathfrak{g}^*$.
...
3
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84
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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 ...
3
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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,...
3
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142
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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 ...
3
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126
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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}...
3
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0
answers
204
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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, $\...
3
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answers
236
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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
$$
A_1 +...+A_n =...
3
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70
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Attainability of Global Optima In Optimal Control
Given a manifold $M$ and a set of smooth functions of one real variable $\mathcal{A}$ and a 'control system' type first order differential equation:
$\frac{d x(t)}{dt} = F(x,u)$
one can consider the ...
3
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answers
176
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Double loop groups and cohomology
Let $G$ be a connected reductive group over $\mathbb{C}$ of Lie algebra $\mathfrak{g}$.
What is the value of $H^{3}(\mathfrak{g}((t))((s)),\mathbb{C})$?
3
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253
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Can the product of a simple and a non-simple indecomposable representation be semisimple?
Consider two (possibly infinite-dimensional) representations $\rho$, $\pi$ of a semisimple Lie algebra $\mathfrak{g}$, with $\rho$ irreducible and $\pi$ indecomposable but not irreducible (i.e., not ...
3
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0
answers
170
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The special embedding $\mathfrak{so}(7)\subset\mathfrak{so}(8)$
It is commonly known that we have a chain of embeddings
$$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 ...
3
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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 \;...
3
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answers
154
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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 ...
3
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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 $G$?
...
3
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285
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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 ...
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440
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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 ...
3
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3k
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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 ...
3
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141
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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?
3
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235
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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 ...
3
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742
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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 ...
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234
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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 ...
3
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308
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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 ...
3
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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 ...
3
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0
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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 ...
3
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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 \ ...
3
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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 ...
3
votes
0
answers
487
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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 ...