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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How Levi decompositon is useful in classification by an inner automorphism?

Suppose I have a 9-dim Lie algebra with non zero commutation: $[{ e1},{ e4}]=-{ e4},[{ e1},{ e6}]=-{ e6},[{ e1} ,{ e7}]=-{ e7},[{ e1},{ e8}]=-{ e8},[{ e2},{ e3}] ={ e5},[{ e2},{ e4}]={ e6},[{ e2},{ ...
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0answers
16 views

How to to write 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}\...
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1answer
24 views

Question about graded Lie algebra

From Wikipedia, a graded Lie algebra is defined as a direct sum of vector spaces $$ \mathfrak{g} = \bigoplus_{i \in \mathbb{Z}} \mathfrak{g}_i \tag{1} $$ such that the Lie bracket satisfies $$ [\...
6
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1answer
56 views

How do I use the fact that $\langle \gamma, \lambda \rangle > 0$?

If I post all the details of my question up to the point where I am stuck, no one is going to want to read all that. So I hope there is someone reading this question who is familiar with Weyl ...
2
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0answers
44 views

Properties of the complexification functor for Lie algebras

The complexification of Lie algebras determines a functor from real Lie algebras to complex Lie algebras, whose right adjoint is the restriction of scalars functor. Thus, we know that complexification ...
1
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1answer
49 views

In what sense do roots span a vector space?

If I am in two dimensional space, the meaning I have for the span is the usual one from linear algebra. But I do not know what it means to say the roots in a root system, R, span the inner product ...
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11 views

Schreier Variety of algebras

Let $V$ be a variety of algebra (over field $F$) which satisfies the property: an arbitrary subalgebra of a free algebra of $V$ is free algebra of $V$. Then $V$ is called Schreier variety. How can we ...
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1answer
33 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, ...
3
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1answer
59 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 ...
2
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1answer
20 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 ...
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0answers
32 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 ...
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1answer
22 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
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1answer
28 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-...
2
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1answer
86 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
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3answers
76 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 ...
2
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1answer
453 views

tensor product of modules of Lie algebras

Let $\mathfrak{g}$ be a semisimple Lie algebra and $M, N$ be two modules of $\mathfrak{g}$. Is it true that $M \otimes N \cong N \otimes M$? If $\mathfrak{g}$ is replaced by other algebras, $M \otimes ...
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1answer
16 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 $\...
0
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1answer
18 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) ...
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 ...
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0answers
54 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 ...
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2answers
79 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 $n(n - 1)/2$ ...
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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 ...
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0answers
21 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 ...
9
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1answer
151 views

Are there common inequivalent definitions of Cartan subalgebra of a real Lie algebra?

I'm confused about seemingly different notions of a Cartan subalgebra of a real semisimple Lie algebra, and I'm wondering if there are common inequivalent definitions. In the book Lie Groups: Beyond ...
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4answers
2k views

How should I show that the Lie algebra so(6) of SO(6) is isomorphic to the Lie algebra su(4) of SU(4)?

As far as I can see, an isomorphism of Lie algebras is a bijective map which preserves the Lie bracket. I need to show that $\mathfrak{so}(6)$ (the Lie algebra of SO(6)) is isomorphic to the $\...
15
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2answers
245 views

Linear Algebra : Invertible Matrix Proof

I was doing some linear algebra exercises and came across the following tough problem : Let $M_{n\times n}(\mathbf{R})$ denote the set of all the matrices whose entries are real numbers. Suppose $\...
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0answers
30 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 \...
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2answers
40 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 ...
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) \...
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154 views

Relationship between hyperalgebra (algebra of distributions) of an affine group scheme to its cohomology

Let $G$ be an affine group scheme, and $\mathrm{Dist}(G)$ its hyperalgebra. I am wondering what is the relationship between $\mathrm{Dist}$(G) and $G$ interms of Cohomology? Is there a cohomology ...
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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(\...
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0answers
21 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 (\...
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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\...
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 ...
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3answers
231 views

Are Structure Constants of a Lie Algebra always Totally Antisymmetric?

Are the Structure Constants $c^a_{bc}$ of a Lie Algebra always totally antisymmetric so, $$ c_{abc} = c_{bca} = c_{cab} $$ Or is this just the case for semi-simple algebras?
2
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1answer
113 views

Commutator formula in infinite dimensions

The commutator formula states that for $A,B$ elements of a Lie algebra, $$ \lim_{n\to \infty}\left\{ \exp\left(-A\tfrac{t}{n}\right)\exp\left(-B\tfrac{t}{n}\right)\exp\left(A\tfrac{t}{n}\right)\exp\...
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 ...
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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 ...
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5answers
4k views

Jacobi identity - intuitive explanation

I am really struggling with understanding the Jacobi Identity. I am not struggling with verifying it or calculating commutators.. I just can't see through it. I can't see the motivation behind it (as ...
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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
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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 ...
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1answer
617 views

On the relationship between the commutators of a Lie group and its Lie algebra

I was trying to teach myself some basic Lie theory, and I came across this statement on Mathworld, relating the commutator of a group, $\alpha\beta\alpha^{-1}\beta^{-1}$, to the commutator of its Lie ...
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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 ?
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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 ...
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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,...
1
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1answer
51 views

Isomorphic Lie algebras have isomorphic centers

I think that if two Lie algebras are isomorphic, then their centers should be isomorphic - is this true? I am sure the answer is obvious to those in the know! Here is my attempt at a proof which looks ...
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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 ...
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0answers
56 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 $f\in\mathbb{F}[x_1,\ldots,x_n]$ ...
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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 ...
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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 &...