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 ...

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10
votes
2answers
444 views

Geodesics on $SU(4)$

Are the geodesics of the following metrics on $SU(4)$ known or easy (in a way not known to me!) to find? In the adjoint representation, one can express the Killing form as a matrix and consider it as ...
18
votes
1answer
344 views

Why should affine lie algebras and quantum groups have equivalent representation theories?

Let $\mathfrak{g}$ be a simple lie algebra over $\mathbb{C}$ and let $\hat{\mathfrak{g}}$ be the Kac-Moody algebra obtained as the canonical central extension of the algebraic loop algebra ...
0
votes
0answers
48 views

Involution on $\chi^{\infty}(\mathbb{R}^{n})$

Is there a Lie algebraic involution $\theta$ on $\chi^{\infty}(\mathbb{R}^{n})$ which restriction to linear vector fields is the involution $\theta(A)=-A^{tr}$ for $A\in M_{n}(\mathbb{R})$? ...
0
votes
0answers
98 views

A linear/Lie algebra problem

Let $\mathfrak{g}$ be a complex linear Lie algebra of dimension $n$. If there exists a basis $\{e_1,\dots,e_n\}$ of $\mathfrak{g}$ such that ...
23
votes
2answers
1k views

Why do Lie algebras pop up, from a categorical point of view?

Groups pop up as automorphism groups in any category. Rings pop up as endomorphism rings in any additive category. Is there a similar way to attach a Lie algebra to an object in a category of a ...
3
votes
2answers
167 views

A Lie algebra identity

Let $G$ be a Lie group and $\mathfrak{g}$ its Lie algebra. Fix a basis $e_1,\dots,e_n$ of $\mathfrak{g}$ and let $e^1,\dots,e^n$ be its dual basis. We also use $e^i$ to denote the left-invariant ...
7
votes
3answers
664 views

nth term in the Baker-Campbell-Hausdorff formula

I am trying to prove a result for which I need the nth term of the Baker-Campbell-Hausdorff formula. I came at this particular result (which is not of significance for the question, but mentioning for ...
1
vote
2answers
201 views

A question on involutions on the Lie algebra of vector fields

Edite According to the essential comment of Ian Agol I revise the question as follows For a smooth manifold $M$, is there a non identity involution $\theta$ on the lie algebra $\chi^{\infty}(M)$ ...
2
votes
0answers
62 views

Springer Isomorphisms for Adjoint Simple Exceptional Groups

I'm trying to understand explicitly a construction of Springer isomorphisms for adjoint exceptional groups given by Bardsley and Richardson. Their construction is as follows. Let $G$ be an adjoint ...
5
votes
1answer
312 views

The surjectivity of the exponential map for the isometry group

Little is known on general conditions guaranteeing that the exponential map between a Lie algebra and an associated Lie group is surjective. Let $M$ be a noncompact connected Riemann manifold, and ...
2
votes
1answer
42 views

Termination conditions for matrix Lie alebra basis generation via “P. Hall algorithm”

Suppose $g_1,...,g_n\in\mathbb{M}_{d\times d}(\mathbb{C})$ are matrices and we are interested in finding the smallest matrix Lie algebra containing them, that is, the matrix Lie algebra generated by ...
0
votes
0answers
103 views

Non-semisimple Lie algebra tensors

Let $\mathfrak{L}$ be a non-semisimple Lie algebra. Let $T_i$ be its generators. As usual, define the structure constants $C_{ij}^k$ by $[T_i,T_j]=C_{ij}^kT_k$ and the metric tensor $g_{jm}$ by ...
7
votes
2answers
303 views

Quadratic Casimir of fundamental irreps of simply-laced Lie algebras

I have the following question, motivated by the expression for the character of level 1 highest weight integrable representations of simply-laced affine algebras (in terms of the string function). It ...
5
votes
0answers
143 views

Generators for invariant tensors of lie algebras

EDITED FOR (hopeful) CLARITY: For a simple Lie algebra $\mathfrak{g}$ (over $\mathbb{C}$) and its adjoint group G, the $G$-invariant polynomials on $\mathfrak{g}$ are linear combinations of products ...
3
votes
1answer
170 views

Lie group GL(4) representation decomposition

Let $V$ be the defining representation of $GL(4,\mathbb C)\to GL(V)$ with $V=\mathbb{C}^4$. Let $Ext\;V$ be the exterior square of $V$ which is a 6-dim repsentation. My question: How does $$V\otimes ...
4
votes
1answer
125 views

PBW basis and canonical basis

Consider the example of $\mathfrak{g} = sl_3$. Then $$ \mathfrak{g} = \mathfrak{n} \oplus \mathfrak{h} \oplus \mathfrak{n}^{-}, $$ where $\mathfrak{n}$ is generated by $E_{12}, E_{13}, E_{23}$, ...
4
votes
1answer
151 views

About the construction of the Universal Enveloping Lie Algebroid

Let $X$ be a reasonable smooth scheme over some base $S$. The tangent sheaf $T_X$ is a Lie algebroid, locally free as a $\mathcal{O}_X$ module, and its Universal Enveloping Lie Algebroid ...
1
vote
0answers
73 views

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 ...
1
vote
1answer
68 views

Smallest subalgebra of $\mathfrak{su}(4)$ arrising from a control problem on $SU(4)$

What is the smallest subalgebra of $\mathfrak{su}(4)$ containing the span of the set $A = \{A, B_1, B_2\}$ where: $A = i (J^x \sigma_x \otimes \sigma_x + J^y \sigma_y \otimes \sigma_y + J^z \sigma_z ...
2
votes
2answers
136 views

Which linear combinations of simple roots are roots

Let $\Delta$ be the root system of a complex simple Lie algebra, $\Delta^+$ be positive roots and $\Pi$ be simple roots. We view $\Pi$ as nodes of the Dynkin diagram. Then for any two simple roots ...
6
votes
1answer
104 views

Number of Richardson orbits in simple Lie algebras of types $E_n$?

This is a follow-up to my question about nilpotent orbits here asked in connection with an earlier discussion of symplectic resolutions. Leaving aside the connections with algebraic geometry and ...
1
vote
2answers
140 views

Understanding lie bracket of simple Lie algebra $W(2)$ [closed]

Please accept my apologies in advance for my simple question. Let $W(2)$ be a simple Lie algebra over $\mathrm{GF}(2)$. We know that it has a basis with three elements like ${w_1,w_2,w_3}$. I cannot ...
0
votes
1answer
75 views

A property of compact involutions of semi-simple Lie algebras?

I need to prove the statement below. Since my background on Lie theory is rather weak, I post it here. Let $\frak{g}$ be a complex semi-simple Lie algebra. Fix a Cartan subalgebra $\frak{h}$ with ...
2
votes
0answers
44 views

Locus maximizing the holomorphic sectional curvature in a non-compact Hermitian symmetric space

Is there a quick way to prove the following statement, if possible without resorting to the classification of simple Lie groups? Let $G$ be a simple Lie group of non-compact Hermitian type of rank ...
7
votes
1answer
282 views

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 ...
8
votes
4answers
422 views

Computing $\int_0^T e^{itA}Be^{-itA} dt$ without an infinite series

I'm hoping to compute the following integral: $\int_0^T e^{itA}Be^{-itA} dt$ where $iA, iB$ are traceless anti-Hermitian matrices (i.e. $\mathfrak{su}(n)$). I have found the following form for the ...
1
vote
0answers
61 views

Adjoint action of semi-direct product

Let $G$ and $H$ be Lie groups with associated Lie algebras $\mathfrak{g}:=\text{Lie}(G)$ and $\mathfrak{h}:=\text{Lie}(H)$ and adjoint actions $\text{Ad}^G:G \to \text{Aut}_\text{Lie}(\mathfrak{g})$ ...
2
votes
1answer
187 views

R-linear representations of sl(2,C)

Is there some good reference for the classification of finite-dimensional ${\mathbb R}$-linear (as opposed to ${\mathbb C}$-linear) representations of $\mathfrak{sl}_2{\mathbb C}$? Equivalently, what ...
1
vote
1answer
73 views

A canonical G_m (or G) action on the Slodowy slice

Question By Slodowy slice I mean a transverse slice at a subregular nilpotent orbit in a simple Lie algebra $\mathfrak{g}$ (in particular I am not intersecting with the nilpotent cone). Consider the ...
3
votes
0answers
303 views

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, $\hat{H}_0 \in \mathfrak{su}(N)$ ...
3
votes
1answer
251 views

Special Riemannian connections?

Assume that $E$ is a bundle of Lie Algebras. Let $g$ be an invariant metric on $E$, that is for all $p\in M$, $$g_p([x,y],z)+g_p(y,[x,z])=0,$$ where $x,y,z\in E_p$ are arbitrary. Is there a ...
5
votes
3answers
636 views

How many three dimensional real Lie algebras are there?

The main point of the question is: I would like to know whether there are only finitely many, countable infinitely many or even uncountable many isomorphism classes of $3$-dimensional real lie ...
0
votes
0answers
43 views

Special family of Metrics on Transitive Lie Algebroids?

Let $\rho:E\longrightarrow TM$ is a transitive Lie Algebroid, then $L=ker\rho$ is bundle of lie algebras. Suppose $\Gamma:TM\longrightarrow E$ be a linear splitting. Define $$\nabla_X ...
4
votes
0answers
97 views

When does finite presentability of the associated graded Lie algebra of a group imply the group is finitely presented?

Let $G$ be a finitely generated group; let $L(G)$ denote the graded Lie algebra (over $\mathbb{Q}$) associated to the lower central series of $G$. I would like to know conditions for when the finite ...
3
votes
1answer
104 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 ...
1
vote
1answer
88 views

tensor product of two irreducibles having same maximal weight

Is there any explicit decomposition of tensor product of two finite dimensional irreducible modules of simple lie algebras whose highest weights are same?
4
votes
2answers
534 views

A question about Marsden-Weinstein reduction theory

Let $G$ ba a compat Lie group and $\frak g$ be its Lie algebra, then by Marsden-Weinstein reduction theory we know that if we take $M=T^*G$ and $J:M\to \frak g^*$ be its moment map then the reduced ...
2
votes
0answers
77 views

Automorphisms of Nilmanifolds

Let $\mathfrak{g}$ be an n-dimensional, rational, nilpotent Lie algebra with simply connected that lie group $G$. It is stated in some papers that if $A$ is an automorphism of $\mathfrak{g}$ which is ...
2
votes
1answer
154 views

“Quantum Littlewood-Richardson” Rule?

Let $\frak{g}$ be a complex semi-simple Lie algebra, and $\lambda,\mu \in P^+$ two positive dominant weights with corresponding irreducible representations $V(\lambda)$ and $V(\mu)$. The tensor ...
4
votes
2answers
246 views

Lie's Theorem in characteristic $p$

Let $K$ be an algebraically closed field with characteristic $0$ and $V$ be a Lie sub-algebra of $M_n(K)$, the $n\times n$ matrices over $K$. If $V$ is solvable, then, according to Lie's theorem, $V$ ...
2
votes
2answers
194 views

Lie group about the quantum harmonic oscillator [closed]

We konw that in quantum harmonic oscillator $H=a^\dagger a$, $a^\dagger$, $a$, $1$ will span a Lie algebra, where $a, a^\dagger$ is annihilation and creation operator, $H$ is the Hamiltonian operator. ...
0
votes
0answers
77 views

Explicit calculation of module of derivations on noncommutative polynomial ring

Let $R$ be a commutative unital associative ring and set $R<x,y>$ to be the $R$-algebra of non-commuting polynomials in two variables over $R$. Explicitly how would one go about computing ...
1
vote
0answers
94 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 ...
-2
votes
1answer
102 views

Action of automorphism group on Lie algebra [closed]

I want to know whether an automorphism group of a simple Lie algebra over $GF(2)$, acts transitively on non-zero elements of Lie algebra or not? How can I check this property?
1
vote
2answers
196 views

Are two distinct Weyl chambers always disjoint?

Let $G$ be a real semisimple Lie group; we suppose $G$ is connected and centerless. Let $\mathfrak{g}$ be its Lie algebra, $\mathfrak{a}$ a Cartan subspace of $\mathfrak{g}$ (i. e. a maximal abelian ...
10
votes
1answer
254 views

Cohomological Proof of Serre Relations for a Symmetrizable Kac-Moody Algebra

In 'Infinite Dimensional Lie Algebras, 3rd edition', Kac mentions at the top of p. 170 that 'a simple cohomological proof of Theorem 9.11 was found by O. Mathieu (unpublished)'. Does anyone know how ...
1
vote
1answer
207 views

homogeneous algebras

Let $A$ be a finite dimensional algebra over finite field (not necessarily associative). Then $A$ is said to be homogeneous if $Aut(A)$ acts transitively on the one-dimensional subspace of A. If A is ...
0
votes
1answer
120 views

Comparison of two infinite dimensional Lie Algebras

Is there an example of a real analytic (compact) manifold $M$ such that the following two lie algebras are isomorphic Lie algebras: $\chi^{\infty}(M)$, the Lie algebra of all smooth vector ...
1
vote
0answers
48 views

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 ...
4
votes
2answers
200 views

Dimension of the nilpotent centralizer of a nilpotent matrix

Fix a natural number $n$ and an algebraically closed field $k$. Let $\mathfrak{g}=\mathfrak{gl}_n(k)$. For any partition of $n$, $\lambda=(\lambda_1,\ldots,\lambda_r)$, let $A_{\lambda}$ be the ...