Questions tagged [lie-algebras]

Lie algebras are algebraic structures which were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" (after Sophus Lie) was introduced by Hermann Weyl in the 1930s. In older texts, the name "infinitesimal group" is used. Related mathematical concepts include Lie groups and differentiable manifolds.

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How to define cohomology of algebraic structures?

I learned that the Hochschild cohomology of an associative algebra $A$ with a bimodule $M$ is defined using the cochain \begin{align*} \cdots \rightarrow C^n(A,M) \stackrel{d^n}{\longrightarrow} C^{n+...
Xiaosong Peng's user avatar
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Set of equivalence classes of a Lie algebra under the action of the automorphism group

I recently became interested in the following question: Given a Lie algebra $\mathfrak{g}$, define two elements $x,y\in\mathfrak{g}$ to be equivalent if there exists an automorphism $\phi\in\...
Joakim Arnlind's user avatar
1 vote
1 answer
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On Euler angles decomposition of $\mathrm{SU}(N)$

$\DeclareMathOperator\SU{SU}$I am looking for a (generalized) Euler angles decomposition for $\SU(N)\ (N>1)$ in the following fashion: $$ \SU(N)\ni m = a\, u \, b $$ where $a,b$ are independent ...
IgnoranteX's user avatar
4 votes
1 answer
291 views

Verma modules and Borel–Weil

Let $\mathfrak{g}$ be a semisimple Lie algebra and fix a root system. Let $\mathfrak{b}:=\mathfrak{h}\oplus\bigoplus_{\alpha\in R^+}\mathfrak{g}_\alpha$. The complex irreducible representation of $\...
lw h's user avatar
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6 votes
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Does the isometry group determine the Riemannian metric?

Suppose $G \subset \text{Iso}(M)$ is a Lie group acting smoothly on a (pseudo-)Riemannian manifold $(M, g)$. Then $G$ induces a Lie algebra of Killing vectors on $M$. In this paper by Goenner and ...
Katerina's user avatar
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Conjugacy classes of Cartan subspaces in parahermitian symmetric spaces

Are there any good tables of the numbers of conjugacy classes of Cartan subspaces in pseudo-Riemannian symmetric spaces? Or a good method to count them? In particular, I am interested in the ...
Callum's user avatar
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Reference on "infinite dimensional Lie algebras" from a mathematical physics point of view

It happens that I stumbled on a class of infinite dimensional Lie algebras that are not Kac-Moody algebras and for which I was not really prepared for. I know some general results on infinite ...
Dac0's user avatar
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What is the Molien series of the SO(2)-invariant ring on the plane (sometimes written C[X]^{SO(2)} )?

Let SO(2) be the group of rotations in the plane. What is the Molien series (sometimes called the Hilbert-Poincare series) of the SO(2)-invariant ring of polynomials? N.B. The main goal being to ...
Victoria's user avatar
1 vote
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Integrable modules of the affine Lie algebra $\mathfrak{su}(2)_k$

I am a physicist studying conformal field theory (CFT), so what I state can be not precise. In physics literature, the affine Lie algebra $\mathfrak{su}(2)_k$ (here $k$ is the level) has only finitely ...
Laplacian's user avatar
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A possible gap in Faltings note to prove the Tate conjecture for finitely generated field over $\mathbb{Q}$

Qeustion: Given a Lie algebra $\mathfrak{g}$ over $\mathbb{Q}_\ell$ with an ideal $\mathfrak{g}^O$ and a subalgebra $\mathfrak{h}$, such that $\mathfrak{g}=\mathfrak{g}^O+\mathfrak{h}$. Now given a ...
Yu LUO's user avatar
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Semisimple Lie algebra and convexity

There is any relation between semisimple lie algebras and symmetric cones? I'm saying this because the classification of the euclidean Jordan algebras, dual by Koecher-Vinberg theorem to homogeneous ...
Nicolas Medina Sanchez's user avatar
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What is the Lie superalgebra generated by permutations?

Consider the group algebra of the symmetric group $\mathbb{C}S_n$. Then there is a corresponding Lie algebra $\mathfrak{L}(S_n)$ defined by $$[\sigma, \tau] = \sigma\circ\tau - \tau\circ\sigma,$$ ...
WunderNatur's user avatar
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When does the null-cone consist entirely of eigenvectors?

Let $V$ be a finite-dimensional representation of a complex reductive Lie algebra $\mathfrak g$. For our purposes, we may define the null-cone like this: $v\in V$ belongs to the null-cone if and only ...
მამუკა ჯიბლაძე's user avatar
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Lie algebra action Whittaker model

Let $(\pi, H)$ be an irreducible unitary generic representation of $G=\operatorname{GL}(r,\mathbb{C})$ and let $H^{\infty}$ be its subspace of smooth vectors. Let $W :G\to\mathbb{C} $ be the Whittaker ...
Akash Yadav's user avatar
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Orthosymplectic superalgebra

Let $V=V_0 \oplus V_1$ be a $\mathbb Z_2$-graded vector space over $\mathbb C$. Suppose $V$ has an even non-degenerate bilinear form $(-, -)$ which is symmetric on $V_0$, skew symmetric on $V_1$, and ...
jack's user avatar
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Is the restriction of the Cartan 3-form on conjugacy classes exact?

Let $G$ be a complex semisimple group and $\mathcal{O} \subset G$ a conjugacy class, i.e. $\mathcal{O} = \{gag^{-1} : g \in G\}$ for some $a \in G$. Let $\Omega$ be the Cartan 3-form on $G$ defined by ...
Lorenz Haber's user avatar
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A quantity computed from weights of representations -- Have you seen it?

The following quantity has come up in some work my collaborators and I are doing on equivariant D-modules, and in that particular context it seems to be very significant (i.e. it's the only "...
Avi Steiner's user avatar
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7D simple Lie algebras over $\mathbb{F}_3$

Up to isomorphism, what are all the seven-dimensional simple Lie algebras over the field with three elements?
Daniel Sebald's user avatar
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Associativity of the Campbell-Baker-Hausdorff operation on a Banach-Lie algebra

Let $(\mathfrak{g}, [\cdot,\cdot]_\mathfrak{g}, \Vert \cdot \Vert_\mathfrak{g})$ be an infinite-dimensional Banach-Lie algebra, and let us define for any $a,b \in \mathfrak{g}$ the series $$~ Z^\...
Marcos Gonzalez's user avatar
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Real Representation ring of $U(n)$ and the adjoint representation

I have two questions: It is well known that the complex representation ring $R(U(n))=\mathbb{Z}[\lambda_1,\cdots,\lambda_n,\lambda_n^{-1}]$, where $\lambda_1$ is the natural representation of $U(n)$ ...
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1 answer
280 views

Degenerate representation

Let $\rho : \mathbb{R}^n\to \mathfrak{so}(2m)$ be a faithful representation of the commutative Lie algebra $\mathbb{R}^n$ into the Lie algebra of skew-symmetric matrices. There is an orthonormal basis ...
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A representation of $\frak{sl}_n$ as partial derivatives on polynomials

As is known to all, the Lie algebra $\frak{sl}_2$ admits a very nice representation on $$ \mathbb{K}[X,Y] $$ the polynomials in two variables, given by $$ E \mapsto X\frac{\partial }{\partial Y}, ~~ F ...
Jake Wetlock's user avatar
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Are isotypic components of $S(\mathfrak{g})$ finite-dimensional?

Let $\mathfrak{g}$ be a complex simple Lie algebra. Let $S(\mathfrak{g})$ be the algebra of polynomial functions on $\mathfrak{g}$, viewed as a $\mathfrak{g}$-representation. Are the isotypic ...
Lorenz Haber's user avatar
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65 views

Non-proper orthant automorphisms

Given a real vector space $V$ of dimension $d$, it is known that the automorphism Lie group of the nonnegative orthant $\mathcal{O}^+_d$ can be described just as $$\mathrm{Aut}(\mathcal{O}^+_d)=\...
Nicolas Medina Sanchez's user avatar
2 votes
0 answers
100 views

Kac-Peterson modular forms and shifted theta functions

Let $\Lambda$ be the root lattice corresponding to an ADE root system $R$ of rank $n$. With the ADE assumption, the weight lattice is simply the dual lattice $\Lambda^{\vee}$. Given any weight vector $...
Benighted's user avatar
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2 votes
1 answer
255 views

Complete $2$-step solvable Lie algebras

A Lie algebra is complete if its center is zero and all its derivations are inner. I would like to study a class of Lie algebras, in particular Let $C$ be the class of finite dimensional $2$-step ...
user56980's user avatar
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Is the homogeneous coordinate ring of a flag variety a UFD?

I was wondering if $G$ is a semisimple complex algebraic group, then is the homogeneous coordinate ring of a flag variety a UFD or not?
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Extension of an involution on $G$ to an involution on $G_\mathbb{C}$

I asked this question on MSE https://math.stackexchange.com/questions/4475382/extension-of-an-involution-on-g-to-an-involution-on-g-mathbbc but didn't receive any answer so far. My question is the ...
Mira's user avatar
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Right-invariant metrics on the unitary groups and embeddings in the complexification

Let $G = SU(n)$ and $G_c = SL(n, \mathbb{C})$. Let $g$ be a right-invariant metric on $G$ and let $g_k$ be the Killing metric on $G_c$. Define the map $p$ from $G_c$ to $G$ which maps $h \in G_c$ to $$...
Malkoun's user avatar
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2 votes
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Question on a remark in Speh's paper

I am reading Birgit Speh's paper entitled "Unitary representations of Gl(n,R) with nontrivial (g,K)-cohomology" in Invent. Math. 71 (1983), no. 3, 443–465. In Remark 1.2.2.(b), it says that &...
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0 answers
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What is the importance of Cartan decomposition of a semi-simple Lie algebra?

I just started learning about Cartan decomposition of semi-simple Lie algebras, and I'm curious to know what are their applications in studying semi-simple Lie algebras. My guess was that it might be ...
Mira's user avatar
  • 129
2 votes
0 answers
159 views

Root systems and subroot systems

Given the root system $E_{6}$ with basis $\alpha_{1},\dotsc,\alpha_{6}$. How would I find all subroot systems of $E_{6}$ (up to Weyl equivalence) where I can write the basis of each subroot system in ...
PSHINH2's user avatar
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1 vote
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When is the zero weight space of an irreducible $\frak{sl}_{n+1}$-module non-trivial?

Take the complex semisimple Lie algebra $\frak{sl}_{n+1}$, with space of dominant integral weights $P(\frak{sl}_{n+1})$. For $V(\lambda)$ the irreducible representation corresponding to $\lambda \in P(...
Dave Shulman's user avatar
2 votes
0 answers
325 views

The Weyl dimension formula for fundamental weights

The Weyl dimension formula is an equation to calculate the dimension of a simple $\frak{g}$-module $V_{\lambda}$, of highest weight $\lambda$, for $\frak{g}$ a complex semisimple Lie algebra. ...
Dave Shulman's user avatar
3 votes
0 answers
219 views

The geometry of the group of automorphisms of a manifold

Given a manifold $M$, the group $Aut(M)$ is made of diffeomorphisms $M\to M$. Since the complete vector fields on $M$ form an infinite dimensional Lie algebra, and each generates a 1 dimensional Lie ...
Carles Gelada's user avatar
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Classification problem of equivariant vector bundles on affine space

We work over $\mathbb{C}$. Let $\mathfrak{k}$ be a Lie algebra, and let $V,W$ be finite-dimensional $\mathfrak{k}$-modules. Consider the $\mathfrak{k}$-algebra $A:=S^\bullet V$. Write $\mathfrak{m}$...
freeRmodule's user avatar
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2 votes
0 answers
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What is the intersection of all maximal abelian subalgebras of a compact simple Lie algebra, containing a given element?

Let $\mathfrak g$ be a compact simple real Lie algebra and let $x\in\mathfrak g$. What is the intersection of all maximal abelian subalgebras of $\mathfrak g$ which contain $x$? For instance, in the ...
Fridrich Valach's user avatar
5 votes
1 answer
776 views

Constructing a Kac-Moody group as a quotient of the free product of its root subgroups

The paper "Regular Functions on Certain Infinite-dimensional Groups" by Kac and Peterson describes the construction of a group associated to the datum of a Kac-Moody algebra in a way I haven'...
fklein's user avatar
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Gross-Hopkins duality

$\DeclareMathOperator\Spf{Spf}\DeclareMathOperator\Pic{Pic}\DeclareMathOperator\Mod{Mod}$One can deduce the invertibility of the Gross-Hopkins dualizing spectrum from purely algebro-geometric ...
taf's user avatar
  • 448
2 votes
1 answer
331 views

Lie algebroid in algebraic geometry

When I did net-surfing at home, I met some geometric backgrounds of Lie algebras and encountered the concept of Lie algebroids. In differential geometry, a Lie algebroid seems to be defined as ...
Frank Voigt's user avatar
1 vote
1 answer
437 views

Centralizer of an element in a matrix Lie group whose Jordan form is given

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\ad{ad}\DeclareMathOperator\Ad{Ad}$Let $G\subset \GL_n(\mathbb{C})$ be a complex matrix Lie group, ...
mhahthhh's user avatar
  • 381
3 votes
0 answers
81 views

Closedness of subgroup corresponding to semi-simple real Lie subalgebra

I have an impression (but could be wrong) that I heard that for any semi-simple (real) Lie subalgebra $\mathfrak{k}$ of $\mathfrak{gl}(n,\mathbb{R})$ there exists a connected closed Lie subgroup $K\...
asv's user avatar
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2 votes
1 answer
79 views

The universal envelope U(L) is a PI-algebra iff L is abelian

Let $k$ be a field with $\operatorname{char} k = 0$. Let $L$ be a Lie $k$-algebra. Then the universal envelope $U(L)$ is a PI-algebra iff $L$ is abelian. Remark: PI-algebra means polynomial identity ...
Functor's user avatar
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5 votes
2 answers
340 views

What is Pressley and Segal's "basic inner product" for compact simple Lie algebras of types B and C?

In Pressley and Segal's book Loop Groups, they define a "basic inner product" $\langle-,-\rangle$ on a simple Lie algebra to be (minus) the Killing form scaled so that $\langle h_\alpha,h_\...
David Roberts's user avatar
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2 votes
0 answers
145 views

Solvability and nilpotency for Banach algebras

Do we have topological counterparts of solvability and nilpotency, which are central concepts for (finite-dimensional) Lie algebras, for infinite dimensional Banach algebras with the commutator ...
Onur Oktay's user avatar
  • 2,263
4 votes
2 answers
658 views

Conjugacy classes in the automorphism group of a simple Lie algebra

A lower bound of the number of conjugacy classes in the automorphism group of a simple Lie algebra $\mathfrak{s}$, of finite dimension over an arbitrary field $\mathbb{F}$, can be the size of the ...
Youness EL KHARRAF's user avatar
2 votes
1 answer
84 views

Non-isomorphic direct products of a solvable and a semisimple Lie algebra

Given a solvable Lie algebra $\frak{a}$ and a semisimple Lie algebra $\frak{g}$ we can take their semidirect product $\frak{a} \rtimes \frak{g}$, with respect to a Lie algebra map $\frak{g} \to \...
Boris Henriques's user avatar
2 votes
0 answers
127 views

Fusion rules for the Lie algebra $\frak{so}_{2n+1}$

For the Lie algebra $\mathfrak{so}_{2n+1}$ where can I find a description of the fusion rules of it fundamental representations? In more detail: For $\pi_i$ and $\pi_j$ two fundamental weights of $\...
Boris Henriques's user avatar
6 votes
1 answer
127 views

Does the centralizer of a regular element in a semisimple Lie algebra act by polynomials?

Let $\mathfrak g$ be a semisimple Lie algebra over $\mathbb C$, $\rho : \mathfrak g \to \operatorname{End}(V)$ a finite-dimensional irreducible representation and $x \in \mathfrak g$ regular with ...
Bart Michels's user avatar
1 vote
0 answers
120 views

Irreducible unitary representations of $\mathrm{SL}(n,\mathbb R)$ from those of $\mathrm{GL}(n,\mathbb R)$

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$In the case of a non-Archimedean local field $\mathbb F$, one may reduce the representation theory of $\SL(n,\mathbb F)$ to that of $\GL(n,\...
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