For questions about Lie algebras, an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds.

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2
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1answer
19 views

Let $X_i, Y_i$ be vector fields on the manifolds M and N. $X_i\oplus Y_j$ on $M\times N$. $[X_1\oplus Y_1,X_2\oplus Y_2]=[X_1,Y_1]\oplus [X_2,Y_2]$

Let $M$ and $N$ be two differentiable manifolds and $X_1,X_2$ be two vector fields on $M$ and $Y_1, Y_2$ on $N$. Using the fact that $T_p(M)\oplus T_q(N)$ is naturally isomorphic to $T_{(p,q)}(M\times ...
1
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0answers
30 views

Reference Request: Dynkin Basis

I am currently working with the set of commutators of $A$ and $B$. Due to special properties of these, I am interested in working with a basis of these commutators (due to anticommutativity, not all ...
1
vote
1answer
19 views

How do you define the inverse of an (exponential Lie) operator?

I know this is a fairly general question, but I would like to know anything I can about obtaining the inverse of an exponential of a lie operator. More specifically, I want to know how one can ...
2
votes
1answer
81 views

For which category (if any) are Lie algebras the algebras of a monad?

I was reading about monads recently, and it came to me that the purpose of the category of algebras of a monad seems to be to switch to a "representation" which is easier for computations. Soon after ...
2
votes
1answer
26 views

Can I integrate the Lie-algebra (body angular velocity) of a quaternion?

This is my first mathematics question here. So I am trying to model a 3-d rotation rigid body by Euler's equation. Of course quaternion is the place to go. If in each time step I receive the body-...
1
vote
1answer
31 views

Deformation complex of Lie algebra structures

I am learning about deformation theory, e.g. through The unbearable lightness of deformation theory by Szendröi. There the standard example of deformations of a structure of associative algebra, ...
0
votes
1answer
15 views

Conjugacy of Cartan subalgebras

This is probabably a very silly question, stemming from some fundamental misunderstanding I have of the relevant definitions, but I am stumped by it. I know that any two Cartan subalgebras of $\...
3
votes
1answer
53 views

How to explain this contradiction about Weyl group of $SL_n(K)$?

I have some difficulties in understanding why the Weyl group of algebraic group $SL_n(K)$ is isomorphic to symmetric group $S_n$. Let $G=SL_n(K)$ be the simply-connected algebraic group over the ...
0
votes
1answer
17 views

Table of e8 representations

I want to understand the representation theory for the (complex-valued) $e8$ exceptional Lie algebra. An ideal answer to this question would contain a link to a text file (or any other format) ...
2
votes
3answers
71 views

Baker-Campbell-Hausdorff/Zassenhaus formula to first order in one matrix

Is there a closed-form expression for the term of $e^{t(c \hat{X} + d \hat{Y})}$ that is first-order in $d$, where $t$, $c$, and $d$ are scalars and $\hat{X}$ and $\hat{Y}$ are finite-dimensional ...
0
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1answer
19 views

Tensor product of representations of a Lie algebra (or Lie Superalgebra)

Let $V$ and $W$ be finite dimensional irreducible representations of a Lie Algebra or a Lie Superalgebra. If $V$ is one dimensional, is $V\otimes W$ necessarily irreducible? I know this to be true ...
4
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2answers
75 views

Trace identities for $\text{SO}(n)$

The Green-Schwarz mechanism in Type I string theory involves certain identities relating traces in the vector and adjoint representations of $\text{SO}(n)$ of dimension $n$ and $n(n - 1)/2$ ...
1
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0answers
10 views

Multiple of roots in symmetric spaces

Fix a Cartan subalgebra $\mathfrak{h}$ on a (compact simple) Lie algebra $\mathfrak{g}$ and consider the associated root system. If $\alpha$ is a root, it is well-known that $k\alpha$ is also a root ...
0
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0answers
20 views

Explicit matrix representation of $\mathfrak{sl}_3$

Given a semisimple Lie algebra $\mathfrak{g}$ and a dominant integral weight $\lambda$ (and all the other necessary data), I want to be able to write down a matrix representation for $V(\lambda)$, the ...
2
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0answers
52 views

Is the assignment of a root system to a semisimple Lie algebra functorial?

As described here, we have a category of root systems, where a morphism from a root system $\Phi$ in a Euclidean space $E$ to a root system $\Phi'$ in $E'$ is given by a linear map $f: E \to E'$ such ...
0
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0answers
28 views

Commutators in the context of local Lie groups.

Let $G$ be a local Lie group in the neighbourhood $V \subseteq \mathbb{C}^d$ with identity element denoted by $e \in G$. Also, let $$ t \mapsto f(t) = (f_1(t), \dots, f_d(t)) \quad \forall t \in \...
2
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0answers
34 views

Deriving an Element of the Lorentz Group SO(1, 3)

We know that $SO(1, 3)$ is isomorphic to $SU(2) \otimes SU(2)$: $$SO(1, 3) \cong SU(2) \otimes SU(2)$$ We also know that $$ \left(\frac{1}{2}, \frac{1}{2}\right) = \left(\frac{1}{2}, 0\right) \...
1
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2answers
39 views

Non Inner Automorphism of Lie Algebras

I have seen some examples of inner automorphisms of Lie algebras. Can anyone please give me an example of an automorphism of Lie algebras that is not inner (with proof). Note - An automorphism is said ...
0
votes
0answers
19 views

Constructing element of the Weyl Group

Let $\mathfrak{g}$ be a semisimple Lie algebra, $\mathfrak{h}$ a CSA with root system $\Phi$, base $\Delta$, and Weyl group $W$. Then there exists a unique element $\sigma\in W$ such that $\sigma(\...
0
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0answers
20 views

When does the unit of universal enveloping algebra adjunction fail to be injective?

The Poincaré–Birkhoff-Witt (PBW) theorem implies that if $K$ is a commutative ring and the Lie $K$-algebra $\mathfrak{g}$ is a free $K$-module, then the unit $\eta_{\mathfrak{g}}: \mathfrak{g} \to U (\...
2
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0answers
26 views

Finite Dimensional Representation of Lie Algebra.

Let $V, W, U$ be finite dimensional representations of a lie algebra $\mathfrak{g}$. Show that $\hom(V \otimes W, U) \cong \hom (V, U \otimes W^*)$. I think I have to use the enveloping algebra of ...
0
votes
0answers
25 views

Irreducible modules of dimension $\leq d$ is finite

Let $L$ be a finite dimensional semisimple Lie algebra and $V(\lambda)$ denote the unique irreducible (upto isomorphism of $L$ modules) standard cyclic module of highest weight $\lambda$.For each $p\...
1
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1answer
61 views

Maurer Cartan Form of the Heisenberg group

I'm trying to understand meaning and application of the Maurer Cartan Form, but I'm still not quite there. I'm then trying to do some examples and trying understand how it works. I begun with the ...
1
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0answers
14 views

Levi decomposition of infinite Lie algebra for KP equation.

My query is related to infinite Lie algebra of KP equation having commutation relations are given by $[X(f_1), X(f_2)] = X(f_1\dot{f_{2}}-f_2\dot{f_{1}})$ $[X(f), Y(g)] = Y(f\dot{g}-\frac{2}{3}\...
1
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1answer
43 views

Is the universal enveloping algebra functor exact?

The universal enveloping algebra is a functor from Lie algebras to unital associative algebras, and is left adjoint to the functor which sends a unital associative algebra to a Lie algebra with ...
-1
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0answers
14 views

infinitesimal generators of the conformal inversions

More broadly speaking, this question involves the subgroups of the conformal Lie groups on Euclidean space. With some insight, one will know that these consist of the infinitesimal rotations, ...
2
votes
1answer
32 views

How to find a Cartan subalgebra of $so(3)$.

Let $so(3)$ be the Lie algebra given by $$ so(3) = \{X \in \text{Mat}_{3 \times 3}: X^T = - X \}. $$ Here $\text{Mat}_{3 \times 3}$ is the set of all $3 \times 3$ matrices and $X^T$ is the transpose ...
0
votes
1answer
27 views

Regular Representation of Lie Algebras

I have a basic understanding of Lie Algebra and it may be naive but is there a regular representation of lie algebras as in case of Finite Groups ? Do the generators form a representation ?
0
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0answers
18 views

How to define a map on Leibniz algebras?

fee Leibniz algebras are defined as follows: Let $X$ be a set and $F(X)$ be a non associative algebra on that and let $I$ be two sided ideal generated by $[a,[b,c]]-[[a,b],c]-[[a,c],b]$ for $a,b,...
0
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0answers
21 views

An example of semi direct product of Lie algebras

Let $L$ be Lie algebra and $d: L \to L $ a derivation. Let we have a one-dimensional Lie Algebra generated by element $t$. What will be the semi direct product of $L$ with one-dimensional Lie algebra ...
0
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0answers
55 views

Lie algebra equivalent definition

I am reading the paper "Affine projections of polynomials" by Neeraj Kayal. I need a clarification regarding the equivalence of two definitions of lie algebra : Let $f\in\mathbb{F}[x_1,\ldots,x_n]$ ...
1
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0answers
15 views

How can we show that $U(L)/U(A)$ is free left $U(A)$-module?

Let $U(L)$ and $U(A)$ be the universal enveloping algebras of Lie algebra $L$ and its given subalgebra $A$. We consider $U(L)/U(A)$. It is clear that $U(L)/U(A)$ is a left $U(A)$-module. But I want to ...
0
votes
0answers
28 views

Is $\mathfrak{su}_2 \simeq \mathbb{R}^3 \simeq \textrm{Im}\mathbb{H}? $

From what I've heard we have the following identifications: $\mathfrak{su}_2 \simeq \mathbb{R}^3$: $\left(x_1, x_2,x_3\right) \in \mathbb{R}^3 \leftrightarrow -\frac{i}{2}\begin{pmatrix} -x_3 &...
1
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0answers
18 views

Are these Lie commutation relations really closed?

Suppose we have a finite dimensional Lie algebra with basis: $X_1 = \partial_t,\;X_2 = \beta_2\,\partial_T-\beta_1\,\partial_C,\;X_3 = x_3\,\partial_p+\frac{1}{\beta_2\rho_0 g}\,\partial_C$ $X_4 = ...
1
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0answers
60 views

The root datum of a connected algebraic group, a few questions

$T$ is a maximal torus of $G$, and $P$ is the set of characters $\beta$ of $T$ for which the weight space $$\mathfrak g_{\beta} = \{ X \in \mathfrak g : \textrm{Ad } t(X) = \beta(t)X, \textrm{ for all ...
0
votes
0answers
24 views

Types of simple Lie algebra [duplicate]

What exactly is Simple algebra of type $A_2$? I found that it has something to do with root systems, which I also don't really know what those are. Any idea? Thanks!
1
vote
3answers
54 views

Faithful representation of the Heisenberg group

I have been trying to solve a problem concerning the Heisenberg Lie group $H$. Show that there does not exist a faithful representation $\rho:H\to\text{GL}(2,\mathbb{R})$. Any ideas about how to ...
6
votes
2answers
165 views

understanding relevance of Lie vs topological groups

A silly easy to state question. When dealing with topological groups, I'm trying to understand more profoundly the advantages of having a Lie group structure against just a topological one. Can ...
0
votes
0answers
42 views

Nilpotent Lie algebra question

Can someone please help me to prove the following statement: In $\mathbb{R}^{d+1}$ we consider the Lie algebra generated by the vectors field: $X_j=∂_{q_j},\hspace{1cm} Y_j=∂_{q_j}V(q)iτ_0,\;j=1,...,...
0
votes
1answer
29 views

Prove that $\text{exp}(tX)=\alpha_X(t)$ for all $t\in\mathbb{R}$?

Let $G$ be a Lie group and $\mathfrak{g}$ be the lie algebra of $G$. We know that for any $X\in\mathfrak{g}$ there exists an unique $\alpha_X:(\mathbb{R},+)\longrightarrow (G,\cdot)$ one-parameter ...
1
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0answers
37 views

What is the rigorous way for a Lie group (SU(n)) element to be “near” another element?

Statement of the problem I'm working with a function $\lambda : SU(n)\times SU(n)\times SU(n) \rightarrow \mathbb{C}$. Given $U_1, U_2, U_3 \in SU(n)$, I'd like to know how to calculate $\lambda (\...
1
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0answers
35 views

references for concrete computations in Lie groups for abstract toplogical concepts

A Lie group is a smooth manifold whose tangent space at its origin is its Lie algebra. Taking an example for lie group such as SL(2), and due to above facts we should then be able to translate the ...
2
votes
0answers
34 views

Inner-product on skew-hermitian matrices

Let $$\mathfrak{u}(n)=\{X\in M(n,\Bbb C):X+X^*=0\}$$ where $X^*$ is the conjugate transpose. Then, $\mathfrak{u}(n)$ is a real vector space. Problem. Show that $\langle X,Y\rangle=\...
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0answers
40 views

Reference Request: Lie Theory For Quantum Field Theory

I have encountered the section on non-Abelian gauge theories in Peskin and Schroeder's QFT textbook, and although I am comfortable with the derivation of the Yang-Mills Lagrangian they present, the ...
1
vote
0answers
44 views

Trace of the product of a Lie algebra and Lie group element

Take $U \in SU(n)$ and $X \in \mathfrak{su}(n)$. What is known about \begin{align} \text{Tr} (UX) \end{align} In particular Are there any useful identities that apply here? When does $\text{Tr} (...
2
votes
0answers
51 views

Is the commutator subgroup $[G,G]$ isomorphic to $G/Z_G$?

Let $G$ be a connected reductive Lie group with Lie algebra $\mathfrak{g}$. That means that $\mathfrak{g}=Z_\mathfrak{g}\oplus[\mathfrak{g},\mathfrak{g}]$, and $[\mathfrak{g},\mathfrak{g}]$ is ...
6
votes
0answers
136 views

Generators of so(7)

Short version: Let $V$ be a 7-dimensional linear space of (real) square matrices. Suppose further that $[V,V]$ (the linear space spanned $[X,Y]$, $X,Y\in V$) is isomorphic to $\mathfrak{so}(7)$. Can ...
1
vote
1answer
25 views

What is the root system and the Weyl group of the group spin$(2n)$?

Reading on root systems and Weyl groups, unfortunately I am highly confused when it comes to the spin-groups (the two-fold universal cover of SO$(2n, \mathbb{C})$, realizable as a quotiënt in a ...
1
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0answers
34 views

The relation between Weyl character formula and Frobenius characteristic map

Let $\mathfrak{gl}(n)$ be the general linear Lie algebra of rank $n$, and $\mathfrak{S}_d$ be the symmetric group of rank $d$. It is well-known that the Schur-Weyl duality provide a equivalence ...
0
votes
1answer
62 views

Cartan decomposition diffeomorphism at the level of (compact) groups

I am trying to understand the Cartan decomposition in the case of a compact group $G$ with subgroup $K$, such that their respective Lie algebras $(\mathfrak{g},\mathfrak{k})$ correspond to a Cartan ...