# Tagged Questions

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### Invariant tensors in adjoint representation

Suppose we have a simple Lie group $G$ with algebra $\mathfrak{g}=\{X_a\}$, where the generators $X_a$ are in some matrix representation. Is it true that the only invariant rank $n$ tensor in the ...
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### Derived series of a Lie algebra

I've been studying semisimple Lie algebras and solvability and was wondering if someone could explain to me the meaning of the derived series of a Lie algebra L and this part: $$L^{(1)}=[LL]$$ I don't ...
Question: Let $A$, $B$ be two $5 \times 5$ (or $7 \times 7$) skew-symmetric complex matrices (i.e. $A^t = -A$), and suppose that $$\forall t,s \in \mathbb{C}, \quad M(t,s):=(tA+sB)^*(tA+sB) \text{ ... 2answers 37 views ### If HP=PH+P for H,P n×n complex matrices, must H be diagonalizable? If F is a field of characteristic zero, H,P are n\times n matrices over F, 0 \neq \alpha \in F, and HP=PH+\alpha P, then must the minimal polynomial of H be square-free and must P ... 2answers 66 views ### Naive question about the group SU(n)? As usual, let SU(n) represent the set of all the n\times n unitary matrices with determinant 1. It's easy to show that any matrix U takes the form U=e^{iA} (A is a n\times n traceless ... 1answer 36 views ### Skew-Symmetric after base change symmetric? Are there invertible matrices A,B \in \textrm{GL}(\mathbb{C}^3) such that for every skew-symmetric matrix S \in \textrm{Mat}_{3 \times 3} (\mathbb{C}) the matrix A \cdot S \cdot B is symmetric? ... 0answers 15 views ### Dual basis to e_{i+1}-e_{i} \in \ker ((1,1,…1)^\vee\in(\Bbb E^{n+1})^\vee) Studying the root system A_n given by the simple roots v_i:=e_{i+1}-e_i \in \Bbb E^{n+1}/\Bbb R(1,1,...,1) for i = 1,...,n, I came across the following dual basis: v_i^\vee:= ... 1answer 43 views ### Is every element of a complex semisimple Lie algebra a commutator? Let L be a (finite-dimensional) complex semisimple Lie algebra. Then we know that L = [L,L]. Is it true that every element of L must be a commutator? Since a complex semisimple Lie algebra is ... 2answers 68 views ### \mathfrak{so}(n) has trivial center when n\geq 3 Is there a nice way to show that$$\mathfrak{so}(n)=\{A \in M(n,\mathbb{R}): A+A^t=0\} $$has zero center for n \geq 3? 1answer 43 views ### The trace as an integral over a sphere [duplicate] Let V be a real vector space of dimension n and let \langle \, \cdot\, , \,\cdot\, \rangle be an inner product on V. We can define a linear functional on the space of endomorphisms of V by ... 0answers 30 views ### Infinite series of nested commutators I'm trying to show the following: If S_i are a set of three matrices such that$$ [S_i, S_j] = \epsilon_{ijk} S_k $$then$$\exp\big( \alpha_i [S_i, \cdot]\big) S_j = (\exp (M) \vec{S})_j... 2answers 124 views ### traceless matrices The fact that \mathfrak{sl}_2(\mathbb{C}) is a simple Lie algebra implies that every 2 \times 2-matrix A \in \mathbb{C}^{2\times 2} with \mathrm{tr}(A) = 0 can be expressed as a commutator of ... 3answers 142 views ### How to prove that B^\vee is a base for coroots? Let \Phi be a root system in a real inner product space E. Define \alpha^\vee = \frac{2\alpha}{(\alpha, \alpha)}. Then the set \Phi^\vee = \{\alpha^\vee: \alpha \in \Phi \} is also a root ... 1answer 54 views ### Vector Space of Lie Algebra Lie algebra  \mathfrak{g}  for a Lie group  \mathcal{G} is closed under commutation. Also, the elements of Lie Algebra form a Linear Vector Space(LVS). Firstly, when is it allowed to define an ... 0answers 36 views ### Basis of Witt algebra The Witt algebra W(n,m) is defined as the set of element \{\sum f_j D_j such that  f_j ∈ A(n,m)\} with usual Lie bracket. I am a bit confused about basis for W(n,m)? What is the meaning of ... 1answer 18 views ### Uniqueness of the Lie brackets in the quotient space of a Lie algebra Suppose I have a Lie algebra \mathfrak g which is an ideal of \mathfrak a. Then I consider the quotient set \mathfrak g / \mathfrak a which is the set of all equivalence relations of \mathfrak ... 0answers 22 views ### Bilinear form on the space of smooth complex valued functions. Let G be a Lie group and h be the Hermitian bilinear form on smooth complex valued functions then how can we define bilinear form on the space of smooth complex valued functions. 0answers 63 views ### Proof of Horn theorem with moment map Please look at this problem: Let \mathcal{H} be the space of (n,n) hermitian matrix. \phi:\begin{align*} &\mathcal{H} \to \mathfrak{u}(n):=Lie(U(n)) \\&A \mapsto iA \end{align*} ... 1answer 20 views ### Show that (X^*)^*=\epsilon\epsilon 'X A finite dimensional vector space V with a non-degenerate form (,) s.t. (u,v)=\epsilon (v,u) \forall u,v\in V is called a quadratic space of type \epsilon. Let V be a quadratic space of type ... 1answer 36 views ### Why X^* X\in \mathfrak{g}(V). A finite dimensional vector space V with a non-degenerate form (,) s.t. (u,v)=\epsilon (v,u) \forall u,v\in V is called a quadratic space of type \epsilon. Let V be a quadratic space of type ... 1answer 58 views ### What is the smallest Lie subalgebra of  {{\frak{gl}}_{n}}(\mathbb{R})  whose center is the set of  (n \times n) -scalar matrices? We know that the center of the Lie algebra  {{\frak{gl}}_{n}}(\mathbb{R})  of all  (n \times n) -matrices is the Lie subalgebra of all  (n \times n) -scalar matrices. The Lie algebra  ... 1answer 128 views ### Jordan–Chevalley decomposition I'm trying to understand the proof of it in Humphreys(Humphreys 1972, Prop. 4.2, p. 17). And I've not got over which field we are working. The characteristic polynomial may not have roots in the ... 1answer 78 views ### Perturbation in characteristic p, or Why, really, does Lie's theorem fail? While recalling some basics of Lie theory, I found a funny proof of the main lemma in Lie's theorem on triangularity of representations of solvable Lie algebras. It turns out that this proof has a ... 1answer 57 views ### Epimorpsims preserve generalized eigenspaces This is most likely trivial, but I don't get it. In Humphrey's Introduction to Lie algebras, page 82, he says: It is clear that, if \phi \colon L \to L' is epimorphism [of finite dim. Lie ... 1answer 59 views ### Automorphism in the special linear algebra \mathfrak{sl}_2(F) If L=\mathfrak{sl}(n,F), g\in GL(n,F), prove that the map of L to itself defined by x\rightarrow -gx^tg^{-1} (x^t=transpose of x) belongs to \operatorname{Aut}L. When n=2,g=identity ... 1answer 127 views ### Subspace spanned by eigenvectors is a subalgebra Let L be a Lie algebra over an algebraically closed field and let x\in L. Prove that the subspace of L spanned by the eigenvectors of \operatorname{ad}x is a subalgebra. Suppose the ... 1answer 88 views ### Eigenvalues of \operatorname{ad}x Let x\in \operatorname{gl}(n,F) have n distinct eigenvalues a_1,\ldots,a_n in F. Prove that the eigenvalues of \text{ad }x are precisely the n^2 scalars a_i-a_j (1\leq i,j\leq n), ... 2answers 83 views ### Equivalent definitions of Verma modules This is a rather basic question. I was reading some notes on geometric representation theory by Gaitsgory and his defition of Verma module is the following: Let  \lambda  be a weight of  ... 0answers 113 views ### Root-subspaces are \mathfrak{g}-invariant I need some help with technicalities concernig root-subspaces of nilpotent Lie algebras of operators. Let \mathfrak{g} \subseteq \mathfrak{gl}(V) be nilpotent Lie algebra, \alpha \ \colon \ ... 2answers 75 views ### How to work out the inverse matrix A^{-1} ? Suppose A is a matrix over some ring R (might be non-commutative). How to work out the inverse matrix A^{-1}? 2answers 85 views ### V=\{A\in M_{3\times 3}(\mathbb{R}):\text{trace}(A)=0\} is isomorphic to \text{span}\{AB-BA:A,B\in V\} Background: LetV:=\{A\in M_{3\times 3}(\mathbb{R}):\text{trace}(A)=0\}$$be the vector space of 3\times 3 real matrices with vanishing trace, and let [\cdot,\cdot]:V\times V\to V be defined by ... 3answers 115 views ### Show that \exp: \mathfrak{sl}(n,\mathbb R)\to \operatorname{SL}(n,\mathbb R) is not surjective It is well known that for n=2, this holds. The polar decomposition provides the topology of \operatorname{SL}(n,\mathbb R) as the product of symmetric matrices and orthogonal matrices, which can ... 0answers 66 views ### Is algebraic closure required in Weyl's theorem on complete reducibility? (Lie algebras) Weyl's theorem states that finite-dimensional representations of finite dimensional semisimple Lie algebras are completely reducible (expressible as a direct sum of irreducible submodules), with some ... 1answer 44 views ### Why is the coadjoint orbit passing through X determined by the spectrum of X? Let G=SO(n,R) be a Lie group and \mathbb{g} its lie algebra. Take X\in \mathbb{g}. Then why is the coadjoint orbit passing through X determined by the spectrum of X? 1answer 112 views ### Finding the Lie Algebra of a Lie Group I am having a hard time finding the set of all X \in M(n, \mathbb{R}) such that e^{tX^T}Be^{tX} = B for all t \in \mathbb{R}. where b is any matrix in M(n, \mathbb{R}). 2answers 615 views ### Two Definitions of the Special Orthogonal Lie Algebra I am encountering two definitions of the special orthogonal lie algebra, and I would like to know if they are equivalent, and if there are advantages to working with one over the other. If we begin ... 1answer 100 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 ... 2answers 367 views ### Is every skew-adjoint matrix a commutator of two self-adjoint matrices I'm looking to solve some matrix equations. One of the equations involves a commutator, so my question is as follows: let A be a skew-self-adjoint, traceless matrix, does the equation [X,Y] = A ... 1answer 50 views ### preserves eigen spaces? "Let H_0=\begin{pmatrix}i&0\\0&-i\end{pmatrix}, suppose A\in SU(2) commutes with H_0, it must preserves each eigen spaces for H_0, eigen spaces for H_0 are just \mathbb{C}e_1 and ... 1answer 86 views ### Question about 8.4 in Humphreys I am reading section 8.4 in Humphreys' book Introduction to Lie Algebras and Representation Theory. He is showing that the only scalar multiples of a root are 1 and -1, but I have trouble ... 2answers 147 views ### If \exp(itH) A \exp(-itH) = A for all t, do A and H commute? Let H be a self-adjoint n \times n matrix with complex entries. H gives rise to a continuous 1-parameter group of unitaries t \mapsto U_t = \exp(itH) : \mathbb{R} \to U(n). Let A be some ... 1answer 429 views ### Vector space generated by the tensor products of pauli matrices Let \sigma_0,\sigma_x,\sigma_y,\sigma_z stand for the 2\times 2 identity matrix and the well known pauli matrices: ... 1answer 117 views ### What is good about simple Lie algebras? Recently I've been reading Naive Lie Theory by John Stillwell. In the book our aim usually concerns finding whether Lie algebras or Lie groups are simple. I wonder what beautiful properties does a ... 1answer 394 views ### What is the Lie algebra of the indefinite orthogonal group''? For p,q \geq 0 and n=p+q\geq 1, give \mathbb{R}^n the indefinite inner product (written as a matrix)$$ \begin{pmatrix} I_p & \\ & -I_q \end{pmatrix}, $$where I_m is the m \times m ... 1answer 351 views ### Inscrutable proof in Humphrey's book on Lie algebras and representations This is a question pertaining to Humphrey's Introduction to Lie Algebras and Representation Theory Is there an explanation of the lemma in §4.3-Cartan's Criterion? I understand the proof given there ... 0answers 122 views ### Derivative/Chain Rule (for MANLYfolds) Computation Embarrasingly, I can't compute the following derivative. dh(X)=\left.\frac{d}{dt}h(e^{XT}]\right|_{t=0}, where X resides in the lie algebra of \rm SL(3,\Bbb C) [ie \mathfrak{sl}(3,\Bbb C)] ... 2answers 413 views ### Lie Algebra Homomorphism Question So this is a bit of a follow-up to my recent question. I don't mean to inundate the feed with my quandaries, but as I move through the theory I keep hitting stumbling blocks (which y'all so kindly ... 1answer 70 views ### Understanding a statement about the existence of functionals Thm. Let L be a solvable subalgebra of gl(V), V a finite dimensional nonzero vector space. Then V contains a common eigenvector for all the endomorphisms in L. The proof of this theorem is ... 1answer 132 views ### What is the difference between \ker( L \bigwedge L \overset{[-,-]}{\rightarrow} L ) and \langle a \wedge b \big| [a,b]=0\rangle? Let L be a finite dimensional Lie algebra. We view the Lie bracket as a linear map on the exterior square:$$\pi:L \bigwedge L \rightarrow L$$Define$$\bigwedge L := \langle a \wedge b \big| ...
Let $\mathfrak{g}$ be a dimension 3 Lie algebra and $[\quad,\quad]$ be a rank 1 map from $\bigwedge^{2}\mathfrak{g} \rightarrow \mathfrak{g}$. In this case, the kernel of $[\quad,\quad]$ is \$3 - 1 = ...