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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5
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2answers
321 views

Representations of Direct Sum of Lie Algebras

I'm trying to prove the following. Let $\frak{g}$ and $\frak{h}$ be (semisimple) Lie algebras. Then every representation $d$ of $\frak{g}\oplus\frak{h}$ is the tensor product of representations $d^1$ ...
3
votes
1answer
354 views

Tensor product Lie algebras

How can I show that $\mathfrak{so}(3, \mathbb{R}) \otimes \mathbb{C} \simeq \mathfrak{sl}(2)$? Is there a way that doesn't involve systems of roots?
2
votes
1answer
138 views

Isomorphism with Lie algebra $\mathfrak{sl}(2)$

Let $L$ be a Lie algebra on $\mathbb{R}$. We consider $L_{\mathbb{C}}:= L \otimes_{\mathbb{R}} \mathbb{C}$ with bracket operation $$ [x \otimes z, y \otimes w] = [x,y] \otimes zw $$ far all $x,y \in ...
1
vote
1answer
332 views

Lie Algebra of the Lorentz Group $SO(1,3)^{\uparrow}$

I'm trying to get my head around the Lie algebra of the Lorentz group once and for all, but have got tied up in knots. Where is my error in the following? The universal covering group of the Lorentz ...
6
votes
1answer
168 views

$\mathfrak{sl}(2,\mathbb C)$ real v. complex

I'm a Lie theory novice, so please bear with me. My understanding is that the Lie algebra $\mathfrak g$ of a matrix Lie group $G$ is the pair $(V, [\cdot, \cdot ])$ where $V$ is the real vector space ...
3
votes
3answers
327 views

Free Lie algebra.

Let $F$ be the free Lie algebra on $\{x,y,z\}$ and $L$ the quotient of $F$ by the ideal $I$ generated by brackets that involve at least three free generators. I have to prove that $dim(L)$ is $6$ and ...
2
votes
1answer
98 views

Complexificantion of a Lie algebra.

Let $L$ be a real Lie algebra. We can consider $L_{\mathbb{C}}:=L \otimes_{\mathbb{R}} \mathbb{C}$ the complexification of $L$. So on $L$ we can define the bracket operation: $[x \otimes z, y \otimes ...
0
votes
1answer
63 views

Enveloping algebra of a Lie algebra

Let $U(L)$ be the enveloping algebra of a Lie algebra $L$. How can I prove that $U(L)$ hasn't zero divisiors (e.g. if $xy=0$, $x,y \in U(L)$ then $x=0$ or $y=0$)?
2
votes
1answer
137 views

Nilpotent Lie algebra

I have to find an example of non nilpotent Lie algebra $L$ and an ideal $I$ of $L$ such that $L/I$ is nilpotent. So we can take the algebra of $ 2 \times2$ matrix upper triangular and with null trace. ...
5
votes
1answer
486 views

Invariant Inner Product on Lie Algebra

Let $G$ be a Lie group, $\frak{g}$ its Lie algebra. Suppose $\mathcal{D}$ a representation of $G$ on $V$, $d$ the associated Lie algebra representation. Suppose $V$ is endowed with an inner product. ...
1
vote
1answer
103 views

Semisimplicity, simplicity and type of symplectic algebra $sp(2n)$

Let $sp(2n)$ be the symplectic algebra. I have to prove that $sp(2n)$ is a simple algebra and its type is $C_{n}$. In order to prove the semisimplicity we can consider the this theorem: Let $V$ be a ...
4
votes
1answer
111 views

The set of complete vector fields

The set of all complete vector fields in $\mathbb R^{n}$ is closed under Lie bracket? is this set a $D$-module where $D$ is the ring of bounded smooth funcions? Can anyone recomend me a book on the ...
3
votes
1answer
161 views

Induced metric via $\mathbb C P^n \cong SU(n + 1)/S(U(n) \times U(1))$

I was wondering if the homeomorphism above gives me the Fubini Study metric on $\mathbb C P^n$. More precisely: Consider $\mathbb C P^n$ equipped with the metric induced by the standard construction ...
1
vote
0answers
297 views

jacobian involving SO(3) exponential map: $\log(R \exp(m))$

I would like to compute the 3 × 3 Jacobian of $$ \log(R \exp(m)) $$ with respect to the 3-vector $m$, evaluated at $m=0$. In the above, $\exp$ is the exponential map from so(3) to SO(3), $\log$ is ...
10
votes
2answers
2k views

On surjectivity of exponential map for Lie groups

A recent question made me realize I didn't know any proof that exponential of a Lie algebra $\mathfrak g$ of a compact connected Lie group $G$ is surjective. After a bit of thinking I've come up with ...
2
votes
1answer
132 views

Spinor Mapping is Surjective

I'm (still) trying to prove that $SL(2,\mathbb{C})$ is the universal covering group the the proper orthochronous Lorentz group $L$. I have completed the following steps. (1) Prove that the vector ...
6
votes
2answers
375 views

Universal Covering Group of $SO(1,3)^{\uparrow}$

I'm trying to prove that $SL(2,\mathbb{C})$ is the universal covering group for the proper orthochronous Lorentz group $SO(1,3)^{\uparrow}$. The standard way goes as follows. (1) Exhibit a real ...
0
votes
1answer
247 views

How to verify the Jacobi identity for the semidirect product Lie algebra

I've been trying to check the claim that the vector space direct sum $L \oplus D$ is a Lie algebra, and I'm having a lot of trouble with verifying the Jacobi identity. It's defined where $L$ is a Lie ...
2
votes
1answer
58 views

Representation of non-Abelian, dimension 2 Lie algebra

Let $k$ be a field and $\mathfrak{g}=kx\oplus ky$ with $[x,y]=y$. Show that $\rho(x)=t\,\frac{d}{dt}$ and $\rho(y)=t\cdot$ (mult. by $t$) define a representation $\rho:\mathfrak{g}\to ...
4
votes
1answer
122 views

Centre of $\mathfrak {sl}_{3}\mathbb{C} $ Lie algebra

$\mathfrak {sl}_{3}\mathbb{C}$ is the Lie algebra of $3\times3$ matrices with complex entries and trace $0$ and Lie bracket $[X,Y] = XY-YX \hspace{3mm} \forall \hspace{3mm} X,Y\in \mathfrak ...
5
votes
1answer
597 views

Lie algebra and Killing form

Let $L$ be a Lie algebra. I have to prove that if $L$ is a simple Lie algebra every bilinear associative form (e.g. $([x,y],z)= (x,[y,z])$ for all $x,y,z \in L$) is a multiple of Killing form.
1
vote
0answers
115 views

Root system of a Lie Algebra

Could anybody help me to solve this problem with roots system? Be $\Phi$ an irreducible root system. $\Phi^{+}$ a choice of positives roots in $\Phi$. Prove that if $(\alpha,\beta)\ge0$ $\forall ...
0
votes
1answer
73 views

Differences between a Cartan subalgebra and a Levi subalgebra?

Let $\mathfrak{h}$ be a Cartan subalgebra and $\mathfrak{l}$ be a Levi subalgebra of $\mathfrak{gl_n}$, where $\mathfrak{h}$ and $\mathfrak{l}$ are both semisimple subalgebras. This is a simple ...
4
votes
1answer
228 views

Computing Lie algebra homomorphism from Lie group homomorphism

I'm pretty much stuck on the following question (taken from the book Lie groups and introduction to linear groups by Rossman W.): I've found some clues, but I think I lack proper understanding of ...
1
vote
1answer
100 views

surjective Lie algebra homomorphism preserves center

If $\phi: L_1 \rightarrow L_2"$ is a surjective Lie algebra homomorphism, is it true that $\phi (Z(L_1))=Z(L_2)$. I see that $\phi (Z(L_1))$ is in $Z(L_2)$, but if $\phi^{-1}(0)$ is not $0$, i.e ...
1
vote
1answer
122 views

Question on definition: primitives in the enveloping algebra of a Lie algebra.

Let $C$ be a coalgebra, and take $c\in C$. Then $c$ is group-like if $\Delta c=c\otimes c$ and $\epsilon(c)=1_k$, and the set of group-like elements is denoted $G(C)$. For $g,h\in G(C)$, $c$ ...
0
votes
0answers
50 views

Map algebras between scheme and Lie algebra

Kindly asking for any hints about the following questions: Suppose, $X$ be an arbitrary scheme over an algebraically closed field $k$. 1- In general, what is the structure of $A= ...
0
votes
0answers
59 views

Lie algebra homomorphism as an scheme morphism

Kindly asking for any hints about the following questions: Assume $g$ is a finite-dimensional Lie algebra. We denote the group of Lie algebra automorphisms of $g$ by $\rm Aut_k g$. Any Lie algebra ...
5
votes
2answers
172 views

Classification of irreducible representations via Casimirs

Physicists almost always label irreducible representations via Casimirs (e.g., characterizing the irreducible representations of $SO(3)$ by spin). I've been looking far and wide to see the general ...
0
votes
1answer
117 views

Lie group reps induced by Lie algebra reps

Let $G$ be a Lie group and $\mathfrak g$ its Lie algebra. Suppose that $\rho_\mathfrak{g}$ is a representation of $g$ on a vector space $V$. Is it true that the mapping $\rho$ from the identity ...
1
vote
1answer
37 views

U(m) as a subgroup of SO(2m)

We know $U(m)$ is one the subgroups of $SO(2m)$ acting transitively on the sphere $S^{2m-1}$ (one of the groups in the Borel's list). What is the explicit formula of this embedding (or it's action)?
0
votes
1answer
117 views

Orthogonal subspace relative to the Killing form

I'm following a book in Humphrey's Introdutction to Lie Algebra's and Representation Theory. I'm reading the proof that a semisimple Lie algebra is the direct sum of simple modules. It uses the ...
1
vote
3answers
207 views

Why is it so important for the characteristic value of the field of a lie algebra to not be two for many propositions?

In reading my Lie algebra text, I see a lot of propositions starting with, "If char F does not equal 2, then..." For example, if char F does not equal 2, then o(n,F) is a subalgebra of sl(n,F). I am ...
3
votes
1answer
71 views

Existence of a Lie subgroup

Let $G=SU(k)\times T^1$, $S$ a subgroup of the center of $SU(k)$ ($Z(SU(k)\cong \mathbb{Z}_k$) and $\eta$ a homomorphism from $S$ into $T^1$. Suppose $(S, \eta)$ denotes the subgroup of $G$ contains ...
1
vote
0answers
41 views

Symmetric algebra and complex polynomial on Lie algebra

Let $G$ a Lie group and $\mathfrak{G}$ its Lie algebra. How can I identify the symmetric algebra on $\mathfrak{G}$ ($S(\mathfrak{G})$) with the algebra of complex polynomials on $\mathfrak{G}$, that I ...
4
votes
0answers
228 views

Representations of non-semisimple Lie algebras

Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}$, and suppose $\mathfrak{g}$ is semisimple. An integral weight for $G$ is an element $\lambda \in \mathfrak{t}^*$ with ...
1
vote
1answer
76 views

Specific help in showing that Poisson Bracket is part of this Lie Algebra

Given in this exercise is the following set: $U = \{f(z) = z^TCz\ \vert\ C \in \textrm{Mat}_{2n}(\mathbb R),\ C^T=C\}$ is a Lie Algebra with $\left\{\cdot , \cdot \right\}$ where $$ \left\{f,g ...
2
votes
1answer
260 views

What is the relationship between semisimple lie algebras and semisimple elements?

A Lie algebra $\mathfrak{g}$ is said to be semisimple if its radical is zero. An element $x \in \mathfrak{g}$ is said to be semisimple if $\text{ad} x$ is diagonalizable. A complex semisimple Lie ...
2
votes
1answer
79 views

Non split extension of lie algebra?

Is there a Lie Algebra $\mathfrak g$ so that the extension $0\xrightarrow{}\mathfrak h\xrightarrow{}\mathfrak g\xrightarrow{}\mathfrak q\xrightarrow{}0$ does not split, i.e. $\mathfrak g$ is not a ...
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vote
0answers
61 views

Invariants of representation theory of Lie groups

How to compute the determinant of a representation of an element of the special linear group? How do I argue that it doesn't change? (@Marek: @rschwieb: Yes well, given one represenation (with ...
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vote
2answers
108 views

Action of $\mathfrak{sl}({V})$ in tensor spaces

what is the natural action of $\mathfrak{sl}({V})$ in tensor spaces ?
0
votes
1answer
199 views

Special Case of Lie-algebra

Suppose $\Bbb{R}^3$ with $[u,v]=u\times v$, thus the cross product of $u$ and $v$ and suppose also $\mathfrak{so}(n)$, the space of skew symmetric $n\times n$-matrices with $[a,b]=ab-ba$. Then i have ...
3
votes
2answers
226 views

Orthonormal basis of Cartan subalgebra relative to Killing form

I'm trying to understand a step in a proof: Let $\mathfrak{g}$ be semi-simple (finite dimensional) Lie-algebra over $\mathbb{C}$, $\mathfrak{h}\subset\mathfrak{g}$ a Cartan subalgebra and let ...
2
votes
0answers
85 views

$x$ regular $\Leftrightarrow$ $x$ is in exactly one CSA

Here's a statement and a proof given in a Lie Algebra course (in the tutorial): Let $L$ be a semisimple Lie algebra over a field $F$ with $\text{char} F=0$. Let $x\in L$ be a semisimple element. ...
1
vote
1answer
230 views

Dimension of Lie algebra according to root system

I was wondering how is it possible to find the dimension of a semi-simple lie algebra $L$ if its corresponding root system is (lets make it simple) of type $B_2$. We can find the number of roots and ...
3
votes
2answers
120 views

Nilpotent Lie Group that is not simply connect nor product of Lie Groups?

I have been trying to find for days an non-abelian nilpotent Lie Group that is not simply connect nor product of Lie Groups, but haven't been able to succeed. Is there an example of this, or hints to ...
2
votes
2answers
107 views

The Lie algebras $\mathfrak{o}_3(\mathbb{C})$ and $\mathfrak{sl}_2(\mathbb{C})$

The adjoint representation of $\mathfrak{sl}_2(\mathbb{C})$ under a natural basis, it is given by $$\text{ad}: \mathfrak{sl}_2(\mathbb{C})\to\mathfrak{gl}_3(\mathbb{C})$$ ...
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0answers
37 views

Lie Derivative in Projective Hilbert Space

In considering a projective Hilbert space, $P(H)$, for linear maps (tensors) of vectors in the space, $A^{a}_{b}v_{a}=u_b$, is there a natural definition for the Lie Derivative for such linear maps? ...
2
votes
2answers
352 views

Length of root strings is at most 4

Let $\Phi$ be a root system. In his Introduction to Lie algebras and Representation Theory, J. Humphreys proves that if $$\beta-p\alpha,\dots,\beta,\dots,\beta+q\alpha$$ is the $\alpha$-root string ...
1
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1answer
56 views

Example ideal of $\mathfrak{sl}(2,\mathbb{C})$

I need an example about ideal from lie algebra $\mathfrak{sl}(2,\mathbb{C})$ except trivial ideal and $\mathfrak{sl}(2,\mathbb{C})$ itself, can someone help me? I try to make ideal except trivial ...