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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25
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6answers
3k views

Why the Killing form?

I'm teaching a short summer course on algebraic groups and it's time to talk about the Killing form on the Lie algebra. The students are all undergrads of varying levels of inexperience, and I try to ...
26
votes
9answers
3k views

What is significant about the half-sum of positive roots?

I apologize for the somewhat vague question: there may be multiple answers but I think this is phrased in such a way that precise answers are possible. Let $\mathfrak{g}$ be a semisimple Lie algebra ...
40
votes
3answers
2k views

Is “semisimple” a dense condition among Lie algebras?

The "Motivation" section is a cute story, and may be skipped; the "Definitions" section establishes notation and background results; my question is in "My Question", and in brief in the title. Some ...
4
votes
1answer
217 views

The existence of a finite dimensional Lie algebra with a given symmetric invariant metric

The question is motivated by a more broad perspective in another MO post and here, but here we would like to understand a specific case (our question potentially connects to / is motivated b Quantum ...
4
votes
1answer
221 views

Finite dimensional Lie algebra with non-degenerate invariant bilinear forms $\Omega_{ab}$

Firstly, my apology to MO experts that I am in a more science/physics background (a PhD). So please feel free refine/modify/comment my language if I have different math accents than yours. From ...
3
votes
2answers
331 views

Complete classification of six dimensional non-semi simple Lie algebra

I would aim to know the complete classification of 6 dimensional non-semi simple Lie algebra (here the dimension stands for the generators; or the dimension $\leq 6$). In this paper, in page 7, it ...
4
votes
2answers
369 views

Chevalley Groups over an arbitrary ring.

My question is simply about the Chevalley groups over rings. In many books, including Carter's book on "Simple groups of Lie types", the groups are considered over fields. I have checked the ...
1
vote
2answers
193 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 $\chi^{\infty}(M)$ ...
-1
votes
1answer
244 views

Heisenberg Lie algebras

Dear forum, I would like to ask if $H(m)$ is the Heisenberg Lie algebra of dimension $2m+1$ and $M$ is an ideal of $H(m)$. Can we say that $M$ has a complement in $H(m)$?
46
votes
10answers
11k views

why study Lie algebras?

I don't mean to be rude asking this question, I know that the theory of Lie groups and Lie algebras is a very deep one, very aesthetic and that has broad applications in various areas of mathematics ...
16
votes
17answers
8k views

Learning about Lie groups

Can someone suggest a good book for teaching myself about Lie groups? I study algebraic geometry and commutative algebra, and I like lots of examples. Thanks.
9
votes
3answers
1k views

How to Compute the coordinate ring of flag variety?

Let $G$ be algebraic group over $\mathbb(C)$(semisimple), $B$ be Borel subgroup. consider flag variety $G/B$. It is known that $G/B$ is projective variety. So one consider the homogeneous coordinate ...
16
votes
2answers
2k views

Cohomology of Lie groups and Lie algebras

The length of this question has got a little bit out of hand. I apologize. Basically, this is a question about the relationship between the cohomology of Lie groups and Lie algebras, and maybe ...
18
votes
1answer
812 views

What's the status of the following relationship between Ramanujan's $\tau$ function and the simple Lie algebras?

Qiaochu asked this in the comments to this question. Since this is really his question, not mine, I will make this one Community Wiki. In MR0522147, Dyson mentions the generating function $\tau(n)$ ...
24
votes
1answer
1k views

Does the quantum subgroup of quantum su_2 called E_8 have anything at all to do with the Lie algebra E_8?

The ordinary McKay correspondence relates the subgroups of SU(2) to the affine ADE Dynkin diagrams. The correspondence is that the vertices correspond to irreducible representations of the subgroup, ...
17
votes
8answers
2k views

Is every finite-dimensional Lie algebra the Lie algebra of an algebraic group?

Harold Williams, Pablo Solis, and I were chatting and the following question came up. In Lie group land (where you're doing differential geometry), given a finite-dimensional Lie algebra g, you can ...
8
votes
4answers
1k views

Basis-free definition of Casimir element?

Let $V$ be a finite-dimensional vector space and let $\mathfrak g \subset \mathfrak{gl}(V)$ be a representation of a semisimple Lie algebra on $V$. Let $e_1, \dots, e_n$ be a basis for $V$. Let $e_1', ...
7
votes
2answers
400 views

What is a Homotopy between $L_\infty$-algebra morphisms

A $L_\infty$-algebra can be defined in many different ways. One common way, that gives the 'right' kind of morphisms, is that a $L_\infty$-algebra is a graded cocommutative and coassociative ...
14
votes
3answers
756 views

What is the Zariski closure of the space of semisimple Lie algebras?

Given Leonid Positselski's excellent answer and comments to this question, I expect that the present one is a hard question. Recall that the Lie algebra structures on a (finite-dimensional over ...
10
votes
2answers
308 views

Which nilpotent Lie algebras appear as nilradicals of parabolic subalgabras?

I am interested to identify (ideally classify) nilpotent Lie algebras that occur as nilradicals of parabolic subalgebras in (say) reductive Lie algebras. For example, all Heisenberg Lie algebras ...
7
votes
1answer
428 views

Uniform proof of dimension formula for minimal special nilpotent orbit?

Given a simple Lie algebra over an algebraically closed field of good characteristic such as $\mathbb{C}$, its subvariety $\mathcal{N}$ of nilpotent elements has dimension $2N$ (where $N$ is the ...
2
votes
1answer
157 views

A class of Lie groups with $f^{abc} \neq -f^{acb}$ (not fully anti-symmetrized) or $f^{abc} \neq f^{bca}$ (not-cyclic)

With the motivation to understand the Lie group structure constraint on a non-Abelian Chern-Simons theory, could some experts give a class of Lie groups with structure constants cannot fully ...
12
votes
4answers
392 views

Ternary “Lie structure”

One of the motivation of the theory of Lie Algebras is that every associative algebra $A$ is a LA when the bracket is defined by $[a,b]=ab-ba$ : this is skew-symmetric and satisfies the Jacobi ...
6
votes
4answers
1k views

Quotient space of $\mathbb{C}^5$ under the action of $SL(2,\mathbb{C})$

One sees that given the $SL(2,\mathbb{C})$ action on $\mathbb{C}^5$, thought of as the space of polynomials of the form, $$a_0 x^4 + 4a_1 x^3 y + 6a_2x^2y^2 + 4a_3xy^3 + a_4 y^4$$ the ring of ...
5
votes
3answers
784 views

Simple question in the representation of SL(2,C)

Let $V$ the standard two dimensional representation of SL(2,C). The Fulton's book in representation theory say in pag 156 that $Sym^3(Sym^2V)=Sym^6(V) \oplus Sym^2(V)$. In the excercises 11.23, the ...
3
votes
3answers
1k views

On the Weyl character formula

So let $G$ be a compact real Lie group. Let $\rho:G\rightarrow GL_n(\mathbb{C})$ be an irreducible representation of $G$ and let $\chi_{\rho}$ be the character associated to $\rho$. Let ...
3
votes
1answer
271 views

Why are relations of degree 3 or less enough in a presentation of the polynomial current Lie algebra g[t]?

Let $\mathfrak{g}$ be a finite dimensional simple Lie algebra over $\mathbb{C}$. The polynomial current Lie algebra $\mathfrak{g}[t] = \mathfrak{g} \otimes \mathbb{C} [t]$ has the bracket $$[xt^r, ...
2
votes
1answer
220 views

Characterization of the weight orbit in the projective space via second order Casimir.

This is the spin-off of the question I previously asked. First, let me remind you some notation from that question: $G_0$ - compact, simply connected Lie group giving rise (by complexification) ...
14
votes
3answers
566 views

Can one show the equivalence of the abstract and classical Jordan decompositions for simple Lie algebras without complete reducibility?

The following fact is basic in the theory of complex Lie algebras: Theorem. Let ${\mathfrak g} \subset {\mathfrak gl}_n({\bf C})$ be a simple Lie algebra, and let $x \in {\mathfrak g}$. Let $x = ...
7
votes
1answer
274 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 ...
3
votes
3answers
494 views

Finite Order Automorphisms on Complex Simple Lie Algebras

Let $L$ be a finite dimensional complex simple Lie algebra, and let $F(L)$ be the set of all finite order automorphisms on $L$. Suppose that we declare $f,h \in F(L)$ to be equivalent if there exists ...
1
vote
2answers
206 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 ...
10
votes
3answers
589 views

For $\mathfrak g$ A Lie algebra of type $ E_7 $, $\mathfrak h $ a Cartan subalgebra and $\Delta$ the resulting root system, does $ Aut(\mathfrak g,\mathfrak h)\rightarrow Aut(\Delta) $ split over the Weyl group?

Given a complex simple Lie algebra $ \mathfrak g $ of type $E_7$, Cartan subalgebra $ \mathfrak h $ and simple roots $\alpha_1,…\alpha_n $, suppose $\pi $ is an involution of the extended Dynkin ...
5
votes
2answers
468 views

Failure of Jacobson Morozov in positive characteristics

The Jacobson-Morozov theorem that any nilpotent $e$ in the lie algebra of a simple algebraic group $G$ can be embedded in an $sl_2$-triple, has a restriction (in terms of the coxeter number) on the ...
4
votes
1answer
480 views

Explicit convergence of Baker-Campbell-Hausdorff

Let g be a finite dimensional simple Lie algebra over C. The Baker-Campbell-Hausdorff series defines a (multivariable) analytic function from a neighborhood of 0 in g \times g \to g. What is the ...