Questions tagged [lie-algebras]

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 groups and differentiable manifolds.

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3
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0answers
41 views

The double cover in the classical limit of $U_q(\frak{sl}_2)$

I am trying to learn about Drinfeld--Jimbo quantum groups and I am having trouble with the classical limit of $U_q(\frak{sl}_2)$. When properly expressed the limit makes sense as $q\to 1$ - see for ...
2
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0answers
198 views

Why does Kashiwara define $U_q(\mathfrak{g})$ over $\mathbb{Q}(q)$ rather than $\mathbb{C}(q)$?

When defining crystal bases, why do we typically view $U_q(\mathfrak{g})$ as an algebra over $\mathbb{Q}(q)$ rather than $\mathbb{C}(q)$? In Kashiwara's original paper introducing crystal bases, he ...
6
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0answers
181 views

Which representations of $\mathfrak{sl}(2)$ are tensor products of powers of irreducible representations?

The so called plethysm problem, from what I understand, deals with decomposing tensor products of representations as direct sums of irreducible representations. I kind of would like to go in the other ...
2
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0answers
28 views

Lie Groups and Lie algebras related to Jordan algebras

Let $J$ be a Jordan algebra. I knew three relative Lie groups/Lie algebras to $J$. In the paper "The Capelli Identity, Tube Domains, and the Generalized Laplace Transform" Jacobson [J] has ...
4
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1answer
95 views

Classification of root lattice embeddings in $E_{10}$

There is a well-known classification result which gives a complete list of all root lattices which can embed into the lattice $E_8$. Moreover, each of these root lattices admits a unique embedding ...
0
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0answers
52 views

Levi decomposition for associative algebras

Does there exist an analogue of Lie algebra Levi decomposition for some/any class of associative algebras?
1
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0answers
49 views

What subspace of $\operatorname{SU}(4)$ group keeps an element of the $\mathfrak{su}(2)$ subalgebra within $\mathfrak{su}(2)$ upon adjoint action?

Consider the Lie group $G_4=\operatorname{SU}(4)$ with (15) generators $T^a$. A basis for the latter is $$\{\sigma^j \times 1_2, \quad \quad \sigma^i \times \sigma^j, \quad \quad 1_2 \times \sigma^j\},...
2
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0answers
38 views

Primitive elements in Hopf algebras over the integers

Let $H$ be a Hopf algebra over $\mathbb Z$, and assume that $H$ is cocommutative, graded, generated in degree $1$, and connected (its degree-$0$ part is $\mathbb Z$). Are there nice, natural ...
4
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0answers
75 views

The coherence property of center of universal enveloping algebra for reductive Lie algebra?

Let $G' \subset G$ be two reductive Lie groups over $\mathbb{R}$ and $\mathfrak{g}_{\mathbb{C}}' \subset \mathfrak{g}_{\mathbb{C}}$ be their complexified reductive Lie algebra over $\mathbb{C}$, ...
2
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0answers
129 views

Spherical harmonics, $\frak{sl}_2$, and algebra gradings

Let $S^2$ be the usual $2$-sphere considered as the quotient $S^3/S^1$, and denote by $\operatorname{Pol}(S^2)$ the algebra of polynomial functions on $S^2$. We can decompose $\operatorname{Pol}(S^2)$ ...
9
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1answer
579 views

Which infinite-dimensional Lie algebras have realizations as algebras of global sections of vector bundles with special structure?

I would not be surprised by downvotes since this question is at the same time very naïve, very vague and might be asking about things well known for decades. Specifically vagueness comes from the word ...
4
votes
3answers
264 views

Root system of fixed point Lie sub-algebra

It is known that a non-simply laced simple root system can be constructed from the simply-laced root system by folding the Dynkin diagram and hence the corresponding non-simply-laced Lie algebra can ...
8
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0answers
116 views

Cohomology algebra of the maximal nilpotent subalgebra of a semisimple Lie algebra

The answer to this question may be well-known, but I failed to locate it in any obvious source. From the results of Bott (Ann. Math. 66 (1957), 203-248) and Kostant (Ann. Math. 74 (1961), 329-387), it ...
3
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0answers
92 views

The product of $Z(\mathfrak{g})$-finite functions is also $Z(\mathfrak{g})$-finite?

Let $G$ be a classical group defined over $\mathbb{Q}$. Let $\mathfrak{g}$ be the Lie algebra of $G(\mathbb{R})$ and $U(\mathfrak{g}_{\mathbb{C}})$ its universal enveloping algebra of $\mathfrak{g}_{\...
10
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2answers
350 views

A specific coset decomposition of $\mathrm{GL}_n(\mathbb{C})$

Disclaimer: I am a theoretical chemist (not a mathematician). I have tried asking this question at Math SE with no luck (https://math.stackexchange.com/questions/4080696/a-specific-coset-decomposition-...
1
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0answers
99 views

What is the analogue of Leibniz's rule for universal enveloping algebra?

Let $G$ be a reductive group over $\mathbb{R}$ and $\mathfrak{g}$ its complexitied Lie algebra. Let $U(\mathfrak{g})$ be the universal enveloping algebra and $Z(\mathfrak{g})$ is the center of $U(\...
1
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0answers
87 views

How can we prove a specific isomorphism between graded Lie algebra and graded universal enveloping algebra?

$\DeclareMathOperator\gr{gr}$Let $L$ be a Lie algebra over field of characteristic different from $2$, and let $L_n$ be its descending central series. Therefore, we consider the associated graded Lie ...
2
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0answers
62 views

Explicit example of prime ideal not an intersection of maximal ideals, in universal enveloping algebra

Let $A$ be a $\mathbb k$-algebra. If $A$ is affine commutative, by Nullstellensatz, then every prime ideal of $A$ is an intersection of maximal ideals. To justify the notion of being primitive in ...
4
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0answers
84 views

Examples of indefinite Einstein non-Ricci-flat metric on solvmanifolds

A solvmanifold (resp. nilmanifold) is a compact quotient of a solvable (risp. nilpotent) Lie group $G$. Usually one consider a co-compact lactice $\Gamma$ on $G$ and the solvmanifold is $G/\Gamma$. ...
10
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1answer
391 views

What's the advantage of defining Lie algebra cohomology using derived functors?

The way I learned Lie algebra cohomology in the context of Lie groups was a direct construction: one defines the Chevalley-Eilenberg complex with coefficients in a vector space $V$ (we assume the real ...
2
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0answers
41 views

Canonical parabolics vs Levi subgroups

Let $G$ be a reductive group over a field $k$ of characteristic zero. The Jacobson-Morozov theorem gives a method of embedding any unipotent element into an $\mathfrak{sl}_2$ triple, which in turn ...
5
votes
1answer
94 views

Reference request for indecomposable representations of $\mathfrak{sl}(2)$ over an algebraically closed field of characteristic $p > 0$

Is there a good reference on the classification of indecomposable representations of the Lie algebra $\mathfrak{sl}(2)$ over an algebraically closed field of characteristic $p > 0$ (of course, in ...
2
votes
1answer
108 views

Cartan subspace of graded Lie algebras

Suppose $\mathfrak{g}$ is a complex reductive Lie algebra and $\theta$ is an automorphism of order $2$. Let $\mathfrak{g} = \mathfrak{g_0} \oplus \mathfrak{g}_1$ be the corresponding $\mathbb Z_2$-...
3
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0answers
110 views

Singular del Pezzo surfaces and degeneration of root systems

Let $S$ be a smooth del Pezzo surface of degree $d$ and $K_S^*$ the anticanonical class. It is well known that the set of classes $$R(S)=\{\alpha\in H^2(S,\mathbb Z)|\alpha^2=-2,\alpha\cdot K_S^*=0\},$...
2
votes
0answers
63 views

Jordan decomposition for simple Lie algebra

Let $\mathfrak{g}$ be a simple complex Lie algebra. Let us fix a Cartan, a Borel, and generators $x_\alpha$ of negative simple roots. Then $N:=\sum x_\alpha$ is a principal (=regular) nilpotent ...
4
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0answers
190 views

Derivative of a representation

I'm learning about Maass--Shimura operators, and there's a term that I'm not sure how to generalize nicely. Let $\mathfrak{h}$ be the upper half-plane with parameter $z= x + iy$, and write $s = \frac{...
8
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0answers
113 views

Alternate proof of easiest special case of PBW theorem

Let $L$ be a Lie algebra over a field $k$ of characteristic $0$ (I'm happy for that field to be $\mathbb{C}$) and let $U(L)$ be its universal enveloping algebra. One of the standard consequences of ...
1
vote
1answer
267 views

Are smooth Schubert varieties Kähler? [closed]

Schubert variety $V$ is a special type of (possibly singular) subvarieties of a Grassmannian. Since the Grassmannians are Kähler manifolds (in fact projective varieties) are we able to conclude that ...
2
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0answers
75 views

Commutators and brackets in nilpotent Lie algebras

Let $\mathfrak{g}$ be a finite-dimensional nilpotent Lie algebra over an algebraically closed field $k$ of characteristic zero. Throughout, let $x,y$ and $z$ be elements of $\mathfrak{g}$. The Baker-...
2
votes
0answers
71 views

Elements of the Hall basis described via permutations

Good morning, Suppose that $\mathfrak{g}$ is a free graded Lie algebra generated by the elements $1,\dots, n$, i.e. assume that $\mathfrak{g}_1=\mathrm{span}\{1,\dots, n\}$. Let us focus on the Hall ...
7
votes
1answer
61 views

Universal enveloping algebra of Malcev Lie algebra associated to nilpotent group

Let $G$ be a finitely generated torsion-free nilpotent group. The Malcev completion of $G$ is a nilpotent Lie group $N$ into which $G$ embeds as a lattice. One way to construct this is to take the ...
1
vote
0answers
90 views

$G$-equivariant modules and Lie algebra cohomology

$\DeclareMathOperator\Id{Id}\DeclareMathOperator\Ad{Ad}$Is there a link between $G$-equivariant modules and Lie algebra cohomology? Tell me if I'm mistaken: On one side, if $p:E\longrightarrow M$ is ...
7
votes
1answer
110 views

When is a nilpotent Lie algebra isomorphic to the associated graded of its lower central series?

All Lie algebras in this question are finite-dimensional and defined over a field $k$ of characteristic $0$ which I'm happy to take to be $\mathbb{R}$ or $\mathbb{C}$. $\DeclareMathOperator\gr{gr}$Let ...
1
vote
2answers
200 views

Dimensions of $\frak{sl}_n$-representations

The dimension of any irreducible $\frak{sl}_n$-representation $V$ is clearly equal to the dimension of its dual representation $V^*$. Does the dimension of an irreducible $\frak{sl}_n$-representation ...
5
votes
2answers
285 views

Derivation on $SO(3)$

Let $$u:\mathbb{R}\ni t \mapsto u(t)\in\mathcal{S}, \quad v:\mathbb{R}\ni t \mapsto v(t)\in\mathcal{S}$$ where $\mathcal{S}$ is the unit sphere of $\mathbb{R}^3$. Consider \begin{align} R:\ \mathcal{S}...
2
votes
1answer
72 views

Rigidity of Borel Lie algebras

Let $\mathfrak b$ be a Borel subalgebra of dimension $n$ in a real semisimple Lie algebra $\mathfrak g$. I am trying to reconcile two facts about $\mathfrak b$: $\mathfrak b$ is rigid, that is, the ...
0
votes
0answers
111 views

Special orthonormal frame on $\mathbb{S}^3$

Does there exist an orthonormal frame $X_1,X_2,X_3$ on $\mathbb{S}^3$ such that $\nabla_{X_i}X_j$ is collinear with $X_i$ for all $1\leq i\neq j\leq3$?
2
votes
0answers
36 views

Equivalence classes of Hermitian elements in a central simple algebra with an involution of second kind

Let $k_0$ be a field of characteristic 0, $k/k_0$ be a quadratic extension, and $A/k$ be a central simple algebra over $k$ of dimension $9=3^2$ with an involution of second kind $\sigma$. Then $^\...
3
votes
0answers
23 views

Find all finite dimensional simple quotients of a possibly infinite dimensional Lie algebra generated by three elements

Let $\mathfrak{g}$ be the (possibly infinite dimensional) Lie algebra over $\mathbb{C}$ with three generators $a,b,c\in \mathfrak{g}$ and defining relations $$[a,[a,b]]=b, ~~[b,[b,a]]=a \tag{1}$$ $$[b,...
2
votes
0answers
41 views

Find all finite dimensional simple Lie algebras satisfying certain conditions

Let $\mathfrak{g}$ be a finite dimensional simple Lie algebra over $\mathbb{C}$. Suppose that $\mathfrak{g}$ can be generated by five nonzero elements $x,y,x',y',h\in \mathfrak{g}$, which satisfy the ...
3
votes
1answer
202 views

Showing the positivity of the determinant of $\mathfrak{sp}(n)$ without making use of diagonalization

Let $\mathfrak{sp}(n)$ be the lie algebra of compact symplectic group $\mathrm{SP}(n)$, regarded as a compact form of $\mathfrak{sp}(2n,\mathbb{C})$, so we can talk about its (complex) determinant. ...
1
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0answers
72 views

Algebraic K-theory of enveloping algebras and PBW-algebras - a reference request

Let for simplicity $k$ be a field of characteristic zero, let $A$ be a finitely generated $k$-algebra which is regular and let $\alpha: L\rightarrow \operatorname{Der}_k(A)$ be a Lie-Rinehart algebra (...
4
votes
1answer
92 views

Non-cosemisimple duals of pointed Hopf algebras

I take the following quote from an answer to this question A Hopf algebra is called pointed if all its simple left (or right) comodules are one-dimensional. The quantized enveloping algebras and ...
2
votes
0answers
67 views

The centralizer and normalizer of products of (SU(n) $\times$ SU(p) $\times$ …) in U(m)

$\DeclareMathOperator\SU{SU}\DeclareMathOperator\U{U}\DeclareMathOperator\Spin{Spin} $Consider the special unitary group $\SU(n)$ and the unitary group $\U(m)$. Below I specify a specfic way to embed $...
2
votes
1answer
116 views

Action of the Casimir on highest weight modules for Kac-Moody algebra

Let $g$ be a Kac-Moody algebra with a symmetrizable Cartan matrix, and let $\{u_j\}$ and $\{u^j\}$ be bases of $g$ dual with respect to a nondegenerate invariant bilinear form $(\cdot|\cdot)$ on $g$, ...
3
votes
0answers
159 views

Isometry groups of symmetric spaces

Let $M=G/K$ be a symmetric space where $G=\mathrm{Isom}(M)$ and $K$ is the isotropy at some point $o\in M$. Moreover, let $\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$ be the Cartan decomposition of $\...
5
votes
0answers
124 views

Group-like elements of universal enveloping algebra

Suppose $\mathfrak{g}$ is a finite-dimensional Lie algebra over $\mathbb C$. Take $A=U(\mathfrak g[[t]])$, a universal enveloping algebra of $\mathfrak g[[t]]$ over $\mathbb C[[t]]$. Then we may ...
3
votes
1answer
169 views

Schubert cells in G/P for reductive G

All literature on the Schubert cells of the generalized flag varieties $G/P$ ("generalized" here means that $P$ is an arbitrary parabolic) assumes that $G$ is a semisimple complex group. I ...
1
vote
0answers
108 views

Why is this operator independent of the choice of basis

I asked this question in MSE but I received no answer https://math.stackexchange.com/questions/4009524/why-is-the-following-operator-independent-of-the-choice-of-basis/4013636#4013636 Let $G$ be a lie ...
11
votes
1answer
270 views

Rational homology of $\Omega^2 ( \mathbb CP^n \vee S^d)$

What is the rational homology of $\Omega^2 ( \mathbb CP^n \vee S^d)$? Here $\Omega$ denotes based loop space.

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