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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Gross-Hopkins duality

$\DeclareMathOperator\Spf{Spf}\DeclareMathOperator\Pic{Pic}\DeclareMathOperator\Mod{Mod}$One can deduce the invertibility of the Gross-Hopkins dualizing spectrum from purely algebro-geometric ...
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Lie algebroid in algebraic geometry

When I did net-surfing at home, I met some geometric backgrounds of Lie algebras and encountered the concept of Lie algebroids. In differential geometry, a Lie algebroid seems to be defined as ...
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Centralizer of an element in a matrix Lie group whose Jordan form is given

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\ad{ad}\DeclareMathOperator\Ad{Ad}$Let $G\subset \GL_n(\mathbb{C})$ be a complex matrix Lie group, ...
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Closedness of subgroup corresponding to semi-simple real Lie subalgebra

I have an impression (but could be wrong) that I heard that for any semi-simple (real) Lie subalgebra $\mathfrak{k}$ of $\mathfrak{gl}(n,\mathbb{R})$ there exists a connected closed Lie subgroup $K\...
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The universal envelope U(L) is a PI-algebra iff L is abelian

Let $k$ be a field with $\operatorname{char} k = 0$. Let $L$ be a Lie $k$-algebra. Then the universal envelope $U(L)$ is a PI-algebra iff $L$ is abelian. Remark: PI-algebra means polynomial identity ...
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What is Pressley and Segal's "basic inner product" for compact simple Lie algebras of types B and C?

In Pressley and Segal's book Loop Groups, they define a "basic inner product" $\langle-,-\rangle$ on a simple Lie algebra to be (minus) the Killing form scaled so that $\langle h_\alpha,h_\...
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Solvability and nilpotency for Banach algebras

Do we have topological counterparts of solvability and nilpotency, which are central concepts for (finite-dimensional) Lie algebras, for infinite dimensional Banach algebras with the commutator ...
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Conjugacy classes in the automorphism group of a simple Lie algebra

A lower bound of the number of conjugacy classes in the automorphism group of a simple Lie algebra $\mathfrak{s}$, of finite dimension over an arbitrary field $\mathbb{F}$, can be the size of the ...
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Non-isomorphic direct products of a solvable and a semisimple Lie algebra

Given a solvable Lie algebra $\frak{a}$ and a semisimple Lie algebra $\frak{g}$ we can take their semidirect product $\frak{a} \rtimes \frak{g}$, with respect to a Lie algebra map $\frak{g} \to \...
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Fusion rules for the Lie algebra $\frak{so}_{2n+1}$

For the Lie algebra $\mathfrak{so}_{2n+1}$ where can I find a description of the fusion rules of it fundamental representations? In more detail: For $\pi_i$ and $\pi_j$ two fundamental weights of $\...
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Does the centralizer of a regular element in a semisimple Lie algebra act by polynomials?

Let $\mathfrak g$ be a semisimple Lie algebra over $\mathbb C$, $\rho : \mathfrak g \to \operatorname{End}(V)$ a finite-dimensional irreducible representation and $x \in \mathfrak g$ regular with ...
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Irreducible unitary representations of $\mathrm{SL}(n,\mathbb R)$ from those of $\mathrm{GL}(n,\mathbb R)$

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$In the case of a non-Archimedean local field $\mathbb F$, one may reduce the representation theory of $\SL(n,\mathbb F)$ to that of $\GL(n,\...
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Decomposition of tensor product of two representations of $so(10,\mathbb{C})$

There exist two 16-dimensional irreducible non-isomorphic representations of $so(10,\mathbb{C})$. Consider the tensor products of each of them with the standard (10-dimensional) representation. What ...
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10-dimensional irreducible representations of $so(10,\mathbb{C})$

Are there 10-dimensional irreducible representations of the Lie algebra $so(10,\mathbb{C})$ which are not isomorphic to the standard representation?
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Is there a classifying space for transitive Lie algebroids? If so, what is it?

Let $M$ be a manifold. The data of a Lie groupoid over $M$ is equivalent to the data of a singular foliation $M=\sqcup\mathcal{F}_i$ and, for each $i$, a map (mod homotopy) $f_i:F_i\to BG_i$ (where $...
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Equivalence of categories of modules over $U(\mathfrak{g})$ on which $Z(\mathfrak{g})$ acts by central characters

$\newcommand{\g}{\mathfrak{g}}$Setting: $\mathfrak{g}$ is a semisimple complex Lie algebra. Here $\chi_\lambda$ denotes the central character corresponding to the action of $Z(\g)$ on a highest weight ...
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Primitive elements in the universal enveloping algebra of Lie superalgebra

Let $\mathfrak{g}$ be a Lie superalgebra over $\mathbb{C}$. Denote by $U(\mathfrak{g})$ the universal enveloping algebra of $\mathfrak{g}$. We know that there is a natural super Hopf algebra structure ...
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Generalized commutator

A well-known generalization of the commutator for operators is the so-called q-commutator defined as $$[A,B]_q=AB-qBA.$$ I was wondering if the case where $q$ is not a number but other operator has ...
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Element conjugate to a maximal torus

It is well known that any element in a compact connected Lie group is conjugate to an element in a maximal torus. Let G be a Lie group that is not necessarily compact and/or connected. Let $x\in G$ ...
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Question about adjoint orbits

I am looking for a proof or a reference of the following claim: Let $G$ be a real connected semi-simple Lie group and let $\mathfrak{g}=\mathfrak{k}+\mathfrak{p}$ be the Cartan decomposition for its ...
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Invariants of Lie superalgebras

Let $V=V_0 \oplus V_1$ be a $\mathbb Z_2$-graded vector space over $\mathbb C$. Suppose $V$ has an even non-degenerate bilinear form $(-, -)$ which is symmetric on $V_0$, skew symmetric on $V_1$, and ...
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PBW basis for the quantized enveloping Lie algebra of $\mathfrak{g}_2$

I would like to know if you have any reference where I can find the canonical PBW basis for $U_q(\mathfrak{g}_2),$ computed using the action of the braid group as defined by Luzstig. Alternatively I ...
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Dimensions of centralizers in quantum Lie algebras associated to $\mathfrak{sl}_n$

Following ideas of Woronowicz, Lyubaschenko and Sudbery defined in Quantum Lie algebras of type $A_n$ the notion of a quantum Lie algebra $\mathfrak{sl}_n$. Let me focus on the case where $q$ is not a ...
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Spectral sequence from standard/Verma filtration/flag to compute Lie algebra cohomology of tensor product with respect to $\mathfrak{n}$

I'm not sure this question fully qualifies as a research-level math question, but from my (limited) past experience on stackexchanged I feared this question might not get an answer there. Setting: the ...
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An upper bound on the dimension of a subalgebra of $\mathfrak{so}(p,q)$ with non-trivial centre

Let $\mathfrak{so}(p,q)$ be the real definite/indefinite orthogonal Lie algebra, $p,q\ge0$, $p+q=n\in\mathbb{N}$, and $L\subset\mathfrak{so}(p,q)$ a Lie subalgebra with non-trivial centre, $\mathrm{Z}(...
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Isomorphism between $T_{[g,X]} (G \times _H \mathfrak{g}/\mathfrak{h})$ and $\mathfrak{g}/\mathfrak{h} \times \mathfrak{g}/\mathfrak{h}$

Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}$ and $H$ be a Lie subgroup of $G$ with Lie algebra $\mathfrak{h}$. Consider the action of $H$ on $G$ by right multiplication and the ...
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The subalgebras of $\mathfrak{su}(2^n)$

$\DeclareMathOperator\su{\mathfrak{su}}$I want to find out all the subalgebras of $\su(N)$, in particular, $N=2^n$, which is the Lie algebra of $n$-qubits. I don't know whether this is a hard question ...
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The identity connected component of centralizers of unipotent orbits

This is, in a way, a follow up question to Unipotent orbits and intersection with Levi and pseudo-Levi subgroups. I was reading "A generalisation of the Bala–Carter theorem for nilpotent orbits&...
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Derivative of the projection map $\pi: G \times \mathfrak{g} \rightarrow G \times_K \mathfrak{g}$

Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}.$ Let $K$ be a closed subgroup of $G$ with Lie algebra $\mathfrak{k}.$ We define the manifold $$\mathcal{E}:= G \times_K \mathfrak{g}$$ to ...
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A certain Lie algebra associated to a bialgebra

Let $A$ be a bialgebra over a field $k$. The space $A^* = \operatorname{Hom}(A, k)$ possesses an algebra structure, given by the convolution product $f*g = f \otimes g \circ \Delta$. Let $\gamma \in A^...
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22 votes
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Lie groups generated by finitely many Lie algebra elements

Let $G$ be a connected Lie group with Lie algebra $\mathfrak{g}$. A standard fact is that $G$ is generated by $\exp(\mathfrak{g})$, i.e. every $g \in G$ can be written as $g=\exp(x_1)\cdots\exp(x_n)$ ...
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How to calculate $\frac{d}{dy} \Bigg|_{t=0} [g.e^{-t(\beta + x +h)};x,y] $?

I'm working through some lecture notes and there is a calculation done by the author which I didn't understand, however I can't give precisely the data using in this notes because it will be very long ...
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14 votes
2 answers
487 views

Annihilate a simple Lie algebra using two commutators

Let $\mathfrak{g}$ be a simple finite-dimensional Lie algebra over an arbitrary field $K$. For any nonzero $x\in\mathfrak{g}$ we must have $[\mathfrak{g},x]\neq\{0\}$, or else we violate simplicity. ...
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Product of $U^+_q(\mathfrak{sl}_2)_i$ in $U_q(\mathfrak{g})$ according to some reduced expression

Let $\mathfrak{g}$ be some simple Lie algebra, $\alpha_1,\alpha_2,\cdots,\alpha_n$ be its simple roots. Let $U^+_q(\mathfrak{sl}_2)_i$ be the subalgebra of $U_q(\mathfrak{g})$ generated by $E_i,K_i$. ...
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How do I detect whether a representation is (or is not) the adjoint representation?

Let $(\mathfrak{g},[\cdot,\cdot])$ be a Lie algebra. There is a God-given representation of $\mathfrak{g}$, namely, the adjoint representation $\operatorname{ad} : \mathfrak{g} \to \operatorname{Der}(\...
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Abelian category for $(\mathfrak{g},T)$ modules with nontrival Grothendieck group

Let $G$ be a reductive Lie group over $\mathbb{C}$, and write $\mathfrak{g}$ for its Lie algebra. Let $T\subseteq B\subseteq G$ be a maximal torus and Borel subgroup, where $\operatorname{Lie}B=\...
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4 votes
1 answer
287 views

What are some interesting examples of quotients by Lie group actions?

I’ve been going through Lee’s Introduction to Differential Geometry. It’s a great book but I feel it lacks examples and concrete applications of the ideas presented. I want to work out a list of ...
2 votes
1 answer
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Calculating fusion rules for $\operatorname{Rep}(G)$ and $G_{k}$ [reference request]

Could somebody please direct me to textbooks / literature that (perhaps lay the foundations for and) detail the method for determining the fusion rules for categories such as $\operatorname{Rep}(G)$ ...
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1 vote
1 answer
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Unipotent orbits and intersection with Levi and pseudo-Levi subgroups

Given a simple complex Lie group $G$ (I might say upfront that I am mostly interested with exceptional Lie algebras) and a nilpotent orbit $\mathcal{O}\subset G$ I would like to describe the ...
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Infinite-dimensional Lie group corresponding to $U\mathfrak{g}$?

Let $\mathfrak{g}$ be a Lie algebra. The universal enveloping algebra $U\mathfrak{g}$ is then an infinite-dimensional associative algebra which can be endowed with the structure of a Lie algebra. Is ...
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6 votes
1 answer
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How are Sheffer polynomials related to Lie theory?

Sheffer polynomials are orthogonal polynomial sets $\{P_n(x)\}$ having generating function $P(x,t) = \sum_{n=0}^{\infty}P_n(x)t^n=A(t)e^{xu(t)}$. This form reminds me of the Lie group–Lie algebra ...
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1 vote
1 answer
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Character formula for the fundamental representations of $\frak{sl}_n$

For the Lie algebra $\frak{sl}_{n+1}$ we denote its fundamental irreducible representations by $V(\pi_i)$, with $i=1, \dots, n$. Where can I find a table of the character formula (in other words a ...
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3 votes
1 answer
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Schur multiplier of finite-dimensional simple Lie algebras in positive characteristic

The Schur multipliers of finite simple groups are known and easily accessible: https://en.wikipedia.org/wiki/List_of_finite_simple_groups Moreover, as a consequence of the second Whitehead's Lemma, if ...
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5 votes
1 answer
255 views

Classification of simple Lie algebras over finite fields

Classification of simple (or simple-restricted) Lie algebras over algebraically closed fields in positive characteristic is studied for a long time. Today, we know all finite-dimensional simple (or ...
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Why do such a birational map exists? And why it is unique?

Let $G$ be a complex linear algebraic group which is connected and reductive and let $\mathfrak g$ be its Lie algebra. Suppose that $H \subset G$ is a 1-dimensional torus such that the action of $H$ ...
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2 votes
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Lie derivative on Lie group in the direction of an element of Lie algebra

I want a reference to the definition of the Lie derivative of a smooth function $f:G \to \mathbb R$ on a Lie group $G$ in the direction of an element $\theta$ of the Lie algebra $\mathfrak G$. I can ...
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Representations of simply connected Lie groups [closed]

Let $G$ be a simply connected Lie group. Is it true that any finite dimensional representation of its Lie algebra is the differential of a representation of $G$? A reference would be helpful. Sorry if ...
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Number of reduced decompositions of the dihedral group $D_6$ [closed]

The Weyl group of $\frak{g}_2$ is the dihedral $D_6$. Let us denote its longest element by $w_0$. How many reduced decompositions does $w_0$ have?
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2 answers
276 views

Representations of $\mathrm{SO}_n$ versus representations of $\mathrm{Spin}_n$ [duplicate]

$\DeclareMathOperator\SO{SO}\DeclareMathOperator\Spin{Spin}$Since $\Spin_n$ is a compact simply connected simple Lie group, its irreducible representations are equivalent to the irreducible ...
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1 vote
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The Lie group corresponding to the Lie algebra $\frak{so}_n$ [closed]

As is standard knowledge - there is a bijective correspondence between compact simply connected simple Lie groups and complex simple Lie algebras. So what is the The Lie group corresponding to the Lie ...
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