Representation theory studies (among else) representations of groups by finite matrices. The non-commutative analog of classical Fourier transforms.

learn more… | top users | synonyms (1)

0
votes
0answers
18 views

The $Ad^*$ $G$-Orbits in $\mathcal{g}*$ (for $G$ being the Heisenberg group)

I did not understand why, the $Ad^*$ $G$-Orbits in $\mathcal{g}*$ (for $G$ being the Heisenberg group and $\mathcal{g}*$ is the dual space of its Lie algebra $\mathcal g$) are given by i) The ...
4
votes
0answers
50 views

Proving $End_B(N)$ is Semisimple Algebra

Let $B$ be a semisimple algebra, and $N$ be a finitely generated $B$-module. I am trying to prove $End_B(N)$ is a semisimple algebra. Updated: I changed my attempt based on user26857's and Qiaochu's ...
1
vote
1answer
29 views

The inner product on $\mathfrak{h}^*$ induced by the inner product on $\mathfrak{h}$.

I am reading the book. On page 80, there is a concept the inner product on $\mathfrak{h}^*$ induced by the inner product on $\mathfrak{h}$. Here $\mathfrak{h}$ is a Cartan subalgebra of a Lie algebra ...
0
votes
0answers
18 views

Notation in Belavin-Drinfeld's classification of solutions to classical Yang-Baxter equations.

I am reading the paper, on page 6, equation (3.5), there is a notation $(1 \otimes \alpha)r_0$, where $r_0 \in g \otimes g$, $g$ is a semisimple Lie algebra, $\alpha$ is a root. For example, suppose ...
2
votes
1answer
31 views

Characters of Permutation Group - Example in Serre

The following is from Serre's Linear Representations of Finite Groups. Take for $G$ the group of permutations of three letters. We have $g = 6$ and there are three classes: the element 1, the ...
4
votes
1answer
55 views

When are irreducible projective representations with the same factor system projectively equivalent?

Consider two irreducible, unitary projective representations $\rho$ and $\tau$ of a finite group G onto the same complex matrix space. If these representations are projectively equivalent, ie. $\rho ...
0
votes
0answers
13 views

What is the relation between solutions of classical Yang-Baxter equations and solutions of modified Yang-Baxter equations.

Let $g$ be a Lie algebra. The classical Yang-Baxter equation (CYBE) is: $$ [r_{12}, r_{13}] + [r_{12}, r_{23}] + [r_{13}, r_{23}] = 0. $$ The modified classical Yang-Baxter equation (MCYBE) is: $$ ...
1
vote
0answers
22 views

Fixed-point free representation of a Lie group G, with non-trivial fixed points for proper subgroup

Assume we have a compact Lie group $G$ and a closed proper subgroup $H$. Is there an elementary way to see that there is a finite dimensional representation $V$ of $G$ with only trivial fixed points ...
3
votes
0answers
52 views

Frobenius determinant theorem

can anyone please recommend a paper or a book that gives a detailed proof of the Frobenius determinant theorem? I have read some few papers I saw online but their informations are not sufficient for ...
4
votes
1answer
37 views

Affine scheme obtained from (commutative) group algebra

Let $G$ be a finite abelian group (written multiplicatively), $R$ a commutative ring and let $R [G]$ denote the set of all formal linear combinations of elements of $G$ with coefficients in $R$. Then ...
1
vote
1answer
34 views

If $A$ is a semisimple algebra, then every irreducible $A$-module is cyclic over $A$

Let $A$ be a finite-dimensinoal associative algebra with unit $1$ over $K$. Then $A$ is a $A$-right module (by module I always mean right module) over itself by right multiplication (the so called ...
0
votes
0answers
32 views

Permutation character : what does it mean? and What is free multiplicity ?

I knew a little about representation theory, so, Permutation character: what does it mean? and what is free multiplicity?
4
votes
1answer
46 views

Why do we need the condition “cocommutative” in the definition of a coPoisson Hopf algebra?

In this paper, page 5, Section 3.6, in the definition of a coPoisson Hopf algebra $H$, it is said that: a coPoisson Hopf algebra is a cocommutative Hopf algebra $A$ with a map $\delta: A \to A \otimes ...
1
vote
0answers
38 views

Orthogonal complement of $\mathrm{im}\ T$ for a $G$-invariant transformation $T$.

I am having trouble determining the orthogonal complement of the image of a transformation $T$. I need to show that $(\mathrm{im}\ T)^{\perp}$ is a $G$-invariant subspace. I have a $G$-invariant ...
1
vote
1answer
25 views

Is the space $R^2$ for a ring $R=\mathbb{Z}/p\mathbb{Z}$ the sum of two invariant lines?

I didn't do a very good job of summarising the problem in the title, sorry, but here is the full question: Let us define, for every ring $R$, the set $S(R)=$ {$\rho : \mathbb{Z}/2\mathbb{Z} \to ...
0
votes
0answers
24 views

How to show that $(\Lambda^2(g))^g = H^2(g)$?

Let $g$ be a semisimple Lie algebra and $\Lambda^2(g) = g \wedge g \subset g \otimes g$ the exterior square of $g$. Consider the adjoint action of g on $g \wedge g$ and let $$(\Lambda^2(g))^g = \{x ...
5
votes
1answer
120 views

Composition series for Verma modules.

Let $L$ a Lie Algebra. I need prove that that every Verma module $\Delta(\lambda)$ admits a composition series, i.e a series of submodules with simple factors. I found a proof that is quite short in ...
1
vote
0answers
29 views

Application of Weyl's Theorem

Recall that Weyl's theorem says that any finite dimensional representation of semi simple Lie algebra is completely reducible. I'm trying some examples to understand this theorem properly. Since $\bf ...
4
votes
1answer
72 views

Relation of $G$-invariants and $g$ -invariants.

Let $G$ be a Lie group and $g$ its Lie algebra. Let $H$ a Hopf algebra and $M$ an $H$-module. By definition, $m \in M$ is called invariant if $x.m=\epsilon(x)m$, $\forall x \in H$, where $\epsilon: H ...
1
vote
1answer
46 views

Do we have $(g \wedge g)^g = 0$?

Let $g$ be a simple Lie algebra. Let $(g \wedge g)^g = \{a \wedge b \in g \wedge g: x.(a \wedge b) = [x,a] \wedge b + a \wedge [x,b] = 0\}$ be the set of $g$ invariants under the adjoint action. Do ...
1
vote
0answers
57 views

Tits algebras of E_6

The general construction of Tits algebras of algebraic groups can be found in Knus, Merkurjev, Rost, Tignol - The book of involutions § 27 For every projective, homogeneous G variety $X:=G/P$, with ...
8
votes
1answer
166 views

Proof of the Isomorphism between: $SL(2,\mathbb R) \times SL(2, \mathbb R) \cong SO^+(2,2)$

I want to do a proof that $SL(2,\mathbb R)\times SL(2, \mathbb R) \cong SO^+(2,2)$. My idea was to use the same Argument as in this Question. So I wanted to begin with the Basis of the Lie algebra ...
2
votes
1answer
34 views

What are the $g$-invariants of $g \otimes g \otimes g$ under adjoint representaion?

Let $g$ be a Lie algebra. Consider the adjoint action $g \times g \otimes g \otimes g \to g \otimes g \otimes g$ given by \begin{align} x.(a \otimes b \otimes c) = [x, a] \otimes b \otimes c + a ...
0
votes
0answers
19 views

The sum of two r-matrices.

Let $g$ be a Lie algebra. Suppose that $r \in g \wedge g$ satisfy the condition: $[[r, r]] = [r_{12}, r_{13}]+[r_{12}, r_{23}] + [r_{13}, r_{23}]$ is a non-zero unique, up to scalar multiple, ...
0
votes
0answers
18 views

Uniqueness of a unitary representation in a general Hilbert space

Is it true that if we assume: $$(x,\rho(\alpha) A \rho(\alpha)^{-1} x) = (x, A x) \quad \forall A \in \mathcal{B}(H) $$ for $x$ an element of a Hilbert space $H$, $A$ a bounded operator on $H$, ...
0
votes
0answers
13 views

Representations of affine SO(n)

What are the irreducible representations of affine $SO(n)$ at level 1? At level k? I haven't found any good references for representations of affine $SO(n)$. Any help?
2
votes
0answers
23 views

Deduce that the number of inequivalent degree $1$ complex representations of $G$ are equal to $|G|$.

Describe all the one-dimensional complex representations of a finite abelian group $G$. Deduce that the number of inequivalent degree $1$ complex representations of $G$ are equal to $|G|$. attempt: ...
2
votes
1answer
28 views

Commutator formula in $U(g)$.

Let $g$ be a simple Lie algebra and $U(g)$ the universal enveloping algebra. Let $a,b,c,d \in U(g)$. Then I think that we have $[a \otimes b, c \otimes d] = [a,c] \otimes bd + ca \otimes [b,d]$. Is ...
0
votes
1answer
59 views

Commutator subgroup and bijective representation

Let $G$ be a finite group and $G' = [G,G]$ be its commutator subgroup, which is defined to be the subgroup generated by elements $[g,h] = g^{-1}h^{-1}gh$ for all $g,h \in G$, where $G'$ is a normal ...
0
votes
0answers
10 views

Term describing a restriction of a Lie group to a subset and its representation

I'm looking for the proper term for restrictions of Lie groups to subsets - I'm working with group invariants in data processing, but some data, like pixel representation of fonts is "partially" ...
2
votes
1answer
41 views

The “ring of characters” of a finite group and its automorphisms

Let $G$ be a finite group and let $C(G)$ denote the set of characters of $G$ (in my representation theory course the values these characters take are in $\mathbb{C}$, but this is one point I'd like to ...
1
vote
1answer
41 views

Whitehead's lemma (Lie algebras) for reductive Lie algebras.

I move the question here. Whitehead's lemma (Lie algebras) is: Let $\mathfrak{g}$ be a semisimple Lie algebra over a field of characteristic zero, $V$ a finite-dimensional module over it and $f$: ...
0
votes
0answers
22 views

A representation similar to coadjoing representation

In a project of quantization, I come up with a finite dimensional representation of $so(d)$ that I wish to find some decent references for it. I guess it could have been studied thoroughly in ...
0
votes
1answer
14 views

Show that a 1 dimensional subspace of a CG-module is a submodule.

Good evening, I am trying to finish the following proof from an old exercise sheet I have from my third year of university a few years ago. Let $G$ be a finite group and $V$ be a finite-dimensional ...
1
vote
1answer
22 views

$\mathbb CG$-modules proposition

Another plea for a starting point please! I know what all of the terminology means, but a starting point would be greatly appreciated! Suppose that $V$ is simple of degree $>1$ and $x\in V$. ...
0
votes
0answers
53 views

The character group of $G$ for an abelian group $G$.

Problem Statement: Prove that the one-dimensional characters of a group $G$ form a group under multiplication of functions, i.e. where the group operation is: ...
0
votes
0answers
13 views

Representation of $SL(2,\mathbb{C})$ over Grassmann algebra

I've noticed that when doing the classical Dirac field, sometimes $\psi(x)$ can be treated as a complex-valued spinor field, but when dealing with canonical or path-integral quantization, it should be ...
2
votes
3answers
64 views

Understanding Maschke's Theorem

https://en.wikipedia.org/wiki/Maschke%27s_theorem#Proof I'm trying to understand the need for the condition 'K's characteristic does not divide the order of G' in the statement of the theorem. Where ...
1
vote
1answer
22 views

CG-homorphism proof. Stuck at the end!

I am trying to work on some questions back from my uni days, and one has gotten the better of me at the moment! Let $G$ be a finite group and $V, W$ finite-dimensional $\mathbb{C}G$-modules. Let ...
2
votes
1answer
22 views

Does irreducibility of a representation imply irreduciblity of all restricted representations?

Let $G$ be a group with a subgroup $H$. Then any representation of $G$ can be restricted to $H$. If the $G$ representation is irreducible then should the $H$ representation also be irreducible? If ...
0
votes
0answers
19 views

Relation between Poisson brackets and Poisson bivectors.

I am reading the book a guide to quantum groups. I have some questions about the relation between Poisson brackets and Poisson bivectors. In the end of page 21 and in the beginning of page 22, it is ...
0
votes
1answer
27 views

Relations between Lie algebras and Lie coalgebras.

Let $g^*$ be the dual vector space of a vector space $g$. Suppose that $g^*$ is a Lie algebra and $[,]_{g^*}: \Lambda^2 g^* \to g^*$ satisfies the Jacobi identity. Let $\delta: g \to \Lambda^2 g$ be ...
1
vote
1answer
40 views

Representation of a group and its quotient

Let $G$ be a (finite) group and let $N$ be a normal subgroup of $G$. Suppose that we have a representation $(V,\rho)$ of $N$ and a representation of $(V, \tau)$ of the quotient group $G/N$. Here $V$ ...
1
vote
2answers
31 views

CG-modules: what does this notation mean?

I am trying to solve a question, but I do not know what the notation used means. If anyone could help me out that'd be great! I don't need help doing the proof, just what the notation means would be ...
2
votes
0answers
33 views

Questions on the proof of Beilinson-Bernstein localization theorem

I am trying to understand the Beilinson-Bernstein localization theorem (following the book by Hotta, Takeuchi and Tanisaki). I got stuck at the following two steps. Any help will be greatly ...
2
votes
1answer
36 views

Lifting the projective property through the affine centre

Let $\mathbb{k}$ be an algebraically closed field. There are many interesting examples of $\mathbb{k}$-algebras $R$ which admit a large central subalgebra $Z_0$ such that $R$ is a free $Z_0$-module ...
0
votes
0answers
32 views

How to compute $df(e)$ explicitly?

I am reading the book. On page 244, the formula (9.2.3.4). I would like to compute the bracket on g^* induced from the Poisson bracket on C[G] explicitly in the example of $G=SL_2$. The formula is: ...
0
votes
0answers
35 views

Evaluate the Projection Operator for this Irreducible Representation of Dihedral Group

I am trying to compute the projector for the Dihedral group of order 12 ($D_{12}=D_{2n}$) for a certain Irreducible Representation. The representation is two dimensional and so I need to caculate ...
0
votes
0answers
22 views

How to construct all Indecomposable representations of a quiver (up to isomorphism)?

Can you give me a sketch of how to Construct all (up to isomorphism) indecomposable representations of a quiver of type $E_{6}$ and $D_{6}$. Perhaps using Gabriel theorem. Thank you
1
vote
1answer
51 views

Proving that a group representation is *not* a direct sum of irreducible represenations.

Problem Statement: Let $x$ be a generator of a cyclic group $G$ of order $p$. Sending $x\mapsto \begin{bmatrix} 1 & 1\\ 0 & 1 \end{bmatrix}$ defines a matrix representation $G\rightarrow ...