1
vote
0answers
26 views

regular representation of algebras

Let suppose we have universal enveloping algebra, what is the meaning of the notion of the right regular representation of that? How can we determine the right regular representation of universal ...
4
votes
1answer
34 views

Construction of the simply connected Lie group of a given Lie algebra

Given a finite dimensional real Lie algebra $\mathfrak{g}$, I am trying to obtain a concrete realization of its simply connected Lie group $G$, with $\mathrm{Lie}(G) \cong \mathfrak{g}$. Let us ...
0
votes
0answers
25 views

Codimension of $\ker $ $\alpha $

Can someone explain why the codimension of $\ker $ $\alpha $ is $1$ in $ H $, with complement $ Fh_\alpha $? Is this because $ h_\alpha $ when $ \alpha $ is simple is part of the dual basis to ...
0
votes
0answers
37 views

Dual spaces: Roots and Cartan subalgebra

Can someone show that the roots and the Cartan subalgebra are dual vector spaces? I don't see how simple roots acting on non-corresponding indices of a Cartan basis produce 0 and a simple root ...
0
votes
2answers
43 views

Associative Lie algebra without Jacobi identity

1) Is there a name for associative Lie algebra that does not require Jacobi identity to hold? 2) Can such algebra exist, and if it does exist, can this algebra contain infinitely many elements? 3) ...
0
votes
0answers
24 views

two Roots questions

Just two questions on roots... 1) Can the length of roots only be defined relatively? And does length only come about because of the dot product and cartan integers? 2) This might be a weird ...
2
votes
2answers
43 views

Closure of a Fundamental Weyl Chamber

Can someone explain what a "closure" of a Fundamental Weyl Chamber means? I assume it is related to an algebraic closure, but I don't see how. In addition, how does the Weyl group act on it and why ...
1
vote
1answer
36 views

killing form and the dot product

When going from talking about roots as functionals to talking about roots as vectors in a Euclidian space (root system), does the killing form become the dot product? Are the killing form and dot ...
1
vote
0answers
40 views

How to visualise the Killing form of a Lie algebra

Given a Lie algebra $\mathfrak{g}$, we can define its Killing form $$K(x,y) = \mathrm{Tr}(ad_x\circ ad_y)$$for $x, y\in \mathfrak g$. Whilst I understand that the Cartan decomposition $$\mathfrak g ...
1
vote
0answers
43 views

Root space decomposition

Regarding the direct sum of vector spaces/algebras, the dimensions of the parts of the sum should equal the whole. With the root decomp, the cartan sub algebra seems to have a dimension as high as the ...
1
vote
1answer
33 views

Lie ideals of $gl_n(K)$

I am looking for some reference where I can find a detailed study of the Lie ideals of the general linear Lie algebra $gl_n(K)$ with the bracket $[A,B]=AB-BA$, where $K$ is a field (if there are ...
2
votes
1answer
74 views

explicit matrix example of irreducible representation of s0(3)

Can someone give me a concrete or an explicit example of an irreducible representation of the Lie algebra so$(3)$? I know they are given by the Wigner D matrices but I want an explicit example of such ...
0
votes
0answers
21 views

Trace functionals as invariant elements of $R[\mathfrak{g}]$ under $G$

Let $\mathfrak{g}$ be a semisimple Lie algebra over $\mathbb{C}$ and let $G$ be its inner automorphism group. Then $G$ acts on $R[\mathfrak{g}]\cong S(\mathfrak{g}^*)$ via $(\sigma\cdot f)(x) = ...
2
votes
1answer
33 views

Two actions of $U(\mathfrak{h})$ on $U(\mathfrak{g})$ where $\mathfrak{h}\hookrightarrow\mathfrak{g}$

Let $\mathfrak{h}$ be a Lie subalgebra of $\mathfrak{g}$, then by PBW theorem we know $U(\mathfrak{h})\hookrightarrow U(\mathfrak{g})$. Let $\{x_i, y_i\}$ be an ordered basis of $\mathfrak{g}$ where ...
2
votes
0answers
30 views

Angles between adjacent roots in a reduced root system.

Let $R$ be a reduced root system. ($R$ is a finite set spanning $V$, $\alpha \in R \rightarrow -k\alpha \in R$ iff $k=1$, $s_{\alpha}(R)=R$, $s_{\alpha}(\beta)-\beta=k\alpha$ whit $k$ integer). ...
1
vote
1answer
35 views

Confusion regarding PBW theorem

I was reading up Humphrey's Introduction to Lie Algebras and Representation Theory and have a confusion regarding a consequence of PBW. First some notations: Let $L$ be a Lie algebra over ...
0
votes
1answer
42 views

Lie rings: reference request

Dear friends: I am looking for a modern reference for Lie rings (In particular, I would like to have nice references for the structure of Lie ideals), let it be lecture notes or a book, in the sense ...
1
vote
0answers
53 views

Regular elements of a module is open and dense

Let $\mathfrak{g}$ be an algebraic Lie algebra and $V$ be a $\mathfrak{g}$-module, then for each $v\in V$, define $\mathfrak{g}^v = \{x\in\mathfrak{g}:xv = 0\}$. Let $V_{reg}$ be the set of all $v$ ...
2
votes
1answer
30 views

Computing a differential on a derivation

Let $\varphi:G\to G'$ be a morphism of algebraic groups over an algebraically closed field $k$, so that $d\varphi:\mathscr{L}(G)\to\mathscr{L}(G')$ is a morphism of Lie algebras. Here I view ...
8
votes
1answer
244 views

Proving that there exists a saturated set with given highest weight

This is an question about an exercise in Humphreys book on Lie algebras. First of all a bunch of definitions and notation, see §13 in Humphreys for details. Let $\Phi$ be a root system, $\Delta$ a ...
2
votes
1answer
39 views

Is $sp(4)$ a subalgebra of $su(5)$?

Is $sp(4)$ a subalgebra of $su(5)$? And how can I prove/disprove this? I know already that it cannot be a regular maximal subgroup of $su(5)$ since the Dynkin diagram (which has two roots of unequal ...
2
votes
1answer
51 views

Differentiating a representation

I'm reading the paper Presenting Schur algebras as quotients of the universal enveloping algebra of $\mathfrak{gl_2}$. It describes a representation of the group algebra ...
4
votes
2answers
74 views

Is there a definition of a dual Lie algebra?

Let $L$ be a Lie algebra. For vector spaces, modules, Banach spaces, etc. we have the notion of a dual. Question: Is it possible to define naturally a Lie algebra $L^*$ that is in some sense dual to ...
0
votes
1answer
35 views

invariants of a Lie algebra

What does it mean by "constructing invariants" in algebraic topology or algebra in general? How to define a "invariant" in algebra? What does it mean by the "invariant of a Lie algebra"?
1
vote
1answer
43 views

Lifts of embeddings of Lie algebras to their universal enveloping algebras

Let $k$ be an algebraically closed field, and let $(\mathfrak{h},[\;,\;])$ be a finite dimensional abelian Lie algebra $k$. Let $(\mathfrak{g},[\;,\;])$ be a finite dimensional Lie algebra over $k$ ...
2
votes
1answer
43 views

Is every element of a complex semisimple Lie algebra a commutator?

Let $L$ be a (finite-dimensional) complex semisimple Lie algebra. Then we know that $L = [L,L]$. Is it true that every element of $L$ must be a commutator? Since a complex semisimple Lie algebra is ...
2
votes
0answers
35 views

Recovering a restricted Lie algebra from its restricted enveloping algebra

Let $k$ be an algebraically closed field of characteristic $p>0$, and let $\mathfrak{g}$ be a restricted Lie algebra, with restricted enveloping algebra $u(\mathfrak{g})$. We can place a Hopf ...
2
votes
0answers
21 views

Are there always nontrivial primitive elements in a Hopf algebra?

Let $k$ be an algebraically closed field of arbitrary characteristic. Let $H$ be a Hopf algebra over $k$. We say $x\in H$ is a primitive element if $\Delta(x)=1\otimes x+x\otimes 1$, where $\Delta$ ...
1
vote
1answer
33 views

Obtaining representations of $G$ from $\mathrm{Lie}(G)$.

Suppose $\mathfrak{g}$ is a semisimple Lie algebra over $\mathbb{C}$, and $\tilde{G}$ is the unique connected, simply connected Lie group whose Lie algebra is $\mathfrak{g}$. Let $C$ be any discrete ...
2
votes
0answers
41 views

A construction of $\mathfrak{e}_8$ in Fulton and Harris

In section $22.4$ of "Representation Theory: A First Course" by Fulton and Harris, the exceptional Lie algebra $\mathfrak{e}_8$ is constructed using a method of Freudenthal. For background, I will ...
5
votes
2answers
68 views

$\mathfrak{so}(n)$ has trivial center when $n\geq 3$

Is there a nice way to show that $$\mathfrak{so}(n)=\{A \in M(n,\mathbb{R}): A+A^t=0\} $$ has zero center for $n \geq 3$?
0
votes
1answer
24 views

Is the restricted enveloping algebra local?

Let $k$ be a field of characteristic $p>0$, and let $(\mathfrak{g},[\cdot,\cdot],(\cdot)^{[p]})$ be a finite-dimensional restricted Lie algebra over $k$. Let $u(\mathfrak{g})$ be the restricted ...
1
vote
1answer
40 views

For an element $x$ in an algebraic group $G$, why do we have $\mathscr{L}(C_G(x))\subset\mathfrak{c}_{\mathfrak{g}}(x)$?

I'm reading Humphreys' Linear Algebraic Groups, trying to understand the following argument found on the top of pg. 76. Let $G$ be an algebraic group over some field $k$, with $x\in G$. Let ...
0
votes
0answers
36 views

Ideals in the unitary group

What would be examples of one-dimensional ideals in the lie algebra of the unitary group? Moreover, how would one show that it is in the tangent space of the center of the unitary group and that the ...
3
votes
1answer
213 views

Baker–Campbell–Hausdorff formula for [exp(x),exp(y)]

Can someone provide a explicit (the first priority with leading orders, then the secondary consider as complete as possible, or) expansion like Baker–Campbell–Hausdorff formula for the commutator: ...
1
vote
2answers
50 views

Simple Lie algebra is a matrix algebra?

Wedderburn's Theorem. Let $A$ be a simple finite $k$-algebra. Then $A$ is a matrix algebra over a finite $k$-algebra $K$ which is a skew field. (Here matrix algebra means $A=M_n(K)$ for some $n$.) ...
1
vote
0answers
59 views

Explicit Weyl group invariant polynomials

Quoting this post, "Let $\mathfrak{g}$ be a simple complex Lie algebra, and let $\mathfrak{h} \subset\mathfrak{g}$ be a fixed Cartan subalgebra. Let $W$ be the Weyl group associated to g. Let ...
0
votes
1answer
62 views

Kernel of homomorphism on unit circle S1

Let $f : S^1 \to S^1$ be defined such that $f(z) = z^2$, where $z$ is a complex number. It's easy to check that this is a homomorphism on $S^1$. However, how would you find the kernel and the coset ...
1
vote
1answer
56 views

Models for Lie algebra E8 and octonions

I've heard that one can construct the exceptional Lie algebra $E_8$ as the Lie algebra of the group of isometries of projective plane over octonions, or something of this form. Unfortunately, I do not ...
3
votes
1answer
55 views

— Cartan matrix for a semisimple Lie algebra with an extension

The question is a modified one inspired by this post: What is the Cartan matrix for this Lie algebra below? (for this semisimple Lie algebra $g(X) \oplus h(Y)$,) $$ [X_i, X_j] = f_{ij}{}^k X_k ...
1
vote
1answer
42 views

Borel subalgebras contain solvable radical

Let $L$ be a Lie algebra and let $B$ be a Borel subalgebra (a maximal solvable subalgebra) of $L$. I want to understand why $\operatorname{Rad} L \subseteq B$. In his proof, Humphreys ...
1
vote
1answer
94 views

Problem 9.7 - Lie Algebras - Humphreys

Let $\alpha,\beta\in\Phi$ span a subspace $E'$ of $E$. Prove that $E'\cap\Phi$ is a root system in $E'$. Prove similarly that $\Phi\cap(\mathbb{Z}\alpha+\mathbb{Z}\beta)$ is a root system in E' (must ...
3
votes
0answers
61 views

Highest weights of irreducible components of tensor product of irreducible sl(3)-module.

I am study the representation theory of $sl(3)$ and I have a question about the tensor representation of irreducible $sl(3)$-modules as follows: For each weight $\mu$, let $L(\mu)$ be the irreducible ...
1
vote
0answers
46 views

Lie algebra of the unipotent radical of a standard parabolic subgroup in $GL_n$

Let $k$ be a field, and consider the algebraic group $G=GL_n(k)$. For any partition $n_1+n_2+\ldots+n_m=n$, we have a parabolic subgroup of the form ...
0
votes
0answers
39 views

Extend to a homomorphism.

My question is regarding a step in "p-Automorphisms of Finite p-Groups" by Evgenii I. Khukhro (p. 117 line 7) and would like some response to my argumentation/understanding of it. p-Automorphisms of ...
0
votes
1answer
66 views

The decomposition of the exterior of the symmetric square over Lie algebra sl(3)

I am studying the representation theory of finite dimensional modules over the simple Lie algebra $\operatorname{sl}(3)$. I know some basics facts about the decomposition of some construction of ...
2
votes
0answers
73 views

Coxeter numbers for semisimple and reductive algebraic groups

I'd like to know how to define the coxeter number for semisimple and reductive algebraic groups. I know that for a simple algebraic group $G$, we can fix a maximal torus $T\subset G$, which acts on ...
3
votes
0answers
39 views

Correspondence between unipotent and nilpotent elements

Let $G$ be a simple algebraic group over an algebraically closed field $k$ of characteristic $p$. Let $\mathcal{U}(G)$ be the closed subvariety of unipotent elements of $G$, i.e., all elements whose ...
1
vote
1answer
43 views

Invertible derivation of $[L, \operatorname{Rad}(L)]$

Suppose $L$ is a finite-dimensional Lie algebra over the field of characteristic $0$. Let $\operatorname{Rad}(L)$ denote the radical of $L$. My question is: Does there always exist an invertible ...
3
votes
1answer
298 views

How to determine the matrix of adjoint representation of Lie algebra?

My questions will concern two pages: http://mathworld.wolfram.com/AdjointRepresentation.html and http://mathworld.wolfram.com/KillingForm.html In the first page, we know the basis of four matrix ...