Questions tagged [representation-of-algebras]

Representation of Algebras is the branch of abstract algebra that studies modules over an associative $R$-algebra $A$ when $R$ is a commutative ring. One of the basic problems in this field is to classify non isomorphic indecomposable representations of a given $R$-algebra $A$

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Why universal enveloping algebra U$(\mathfrak{g})$ has the same representation theory as that of $\mathfrak{g}$?

I am new to concepts from Lie algebra theory, so I was reading Lectures on Lie groups ad Lie algebras by Carter, Macdonald and Segal. At the third chapter we are introducing universal enveloping ...
Mahammad Yusifov's user avatar
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Is $f(\operatorname{rad} A ) \subseteq \operatorname{rad} B$ when $f$ is a not surjective $K$-algebras homomorphism and $K$ is a field?

Let $K$ be a field. For a $K$-algebra $A$ take the definition of the Jacobson radical of $A$ as the intersection of all maximal left ideals of $A$. If $A$ and $B$ are two finite dimensional $K$-...
Hector Blandin's user avatar
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Complete irreducibility of infinitely dimensional representation

This is a step to prove Maschke’s theorem. Let $G$ be a finite group, every indecomposable $G$-module is simple $\iff$ complete reducibility. For finite dimensional representation, we can prove this ...
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Schur's Lemma of the Version for Lie Algebras [closed]

In general, Schur's lemma is stated as follows. (i) Let $(\pi_1,V_1)$ and $(\pi_2,V_2)$ be irreducible representations, and let $T:V_1\rightarrow V_2$ be an intertwining operator. Then either $T$ is ...
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On identifying unitarily inequivalent representations of a C*-algebra

Given a finite-dimensional CCR C*-algebra $\mathcal{A}$ (you can find the details of how CCR algebras are introduced in many books, for example see "Operator algebras and quantum statistical ...
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Can the $\mathbb R$-algebra $M_n(\mathbb C)$ be generated using a set of only $2n$ of its elements?

Can the $\mathbb R$-algebra $M_n(\mathbb C)$ be generated using a set of only $2n$ of its elements? My thoughts are that $M_n(\mathbb C)$ is a simple ring, and we are asking whether there is a ...
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Reducible representations of the Clifford Algebra

I would like to construct reducible representations of the Clifford algebra, that consist of $8\times8$ matrices with purely real or purely imaginary elements. Assume that I am familiar with the Dirac,...
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Repetitive algebra and indecomposable projectives

Let $A$ be a finite dimensional basic algebra over an algebraically closed field $k$. We denote by $D$ the standart duality $D(A) = \text{Hom}_k(A,k)$. The repetitive algebra $\hat{A}$ of $A$ is a ...
Momo1695's user avatar
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Properties of unitary representations for projective representations

I would like to know if there exist concepts of ergodicity and mixing properties for projective representations. If they do, do these properties exhibit similar characteristics to those observed in ...
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Group algebra and the classification of simple modules

This is a question from one of our 'ancient' qualification exams. I am not familiar with representation theory, so the second question baffles me a lot. Let $G=\langle g: g^3=1\rangle$ be the cyclic ...
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Example request of a finite dimensional algebra of infinite global dimension with a module with no self-extensions

I'm looking for an example of a finite dimensional algebra of infinite global dimension with a non projective module $X$ with no self-extensions, that is $\text{Ext}_A^i(X,X) = 0$ for $i>0$. The ...
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$A$ is an associative algebra and $V$ is a representation. Then $\operatorname{End}_{A}(A)=A^{op}?$ (exercise from notes by Etingof)

Let $A$ be an associative algebra. If $V$ is a representation of $A$, write $\operatorname{End}_A(V)$ to denotes the algebra of all homomorphisms of representations $V \to V$ . Show that $\...
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Comparison of speed to generate an algebra between 2 different sets of generators

I'm sorry in advance if my question is unclear (as it is also unclear for me). I'm dealing with 2 different sets of generators, namely $\{p, Q_n\}$ ($n \in [0,N]$ ) with $Q_n = \sum_{n'=0}^n |n'>&...
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Complex representation of Clifford Algebra

In the book Spin Geometry of Lawson was mentioned that a complex representation of $Cl_{r,s}$ is a real representation $\rho:Cl_{r,s}\to\mathrm{Hom}_\mathbb{R}(W,W)$ such that $\rho\circ J(\phi)=J\...
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Irreducible factors of universal enveloping algebra

Let $\mathfrak{g}$ be a simple complex Lie algebra and $U$ be its universal enveloping algebra. We have an action $\mathfrak{g} \to \mathfrak{gl}(U)$ by extending the adjoint action of $\mathfrak{g}$ ...
Henrique Augusto Souza's user avatar
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What is the definition of quasi-equivalent representations by subrepresentations?

N.P. Landsman (2017) defines quasi-equivalent representations as: Two representations $\pi_1,\,\pi_2$ are quasi-equivalent if every subrepresentation of $\pi_1$ has a subrepresentation that is (...
Felipe Dilho's user avatar
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Isomorphism of an endomorphism ring of an algebra representation

Consider a representation $(K^n, J_n(\lambda))$ of an algebra $K[T]$ where $K$ is a field, $\lambda \in \mathbb{R}$ and $J_n$ denotes the $n \times n$ Jordan block. How can we see that $$End_{K[T]}(K^...
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Non-zero longest path in $k(\mathbb{Z}\Delta)$, where $\Delta$ is Dynkin type

I'm reading an article named Selfinjective and simply connected algebras written by Otto Bretscher, Christian L$\ddot{\text{a}}$ser and Christine Riedtmann in 1981. Here is the article: https://link....
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Classify the finite-dimensional indecomposable representations of the quiver $A_n$

I have to classify the finite-dimensional indecomposable representations of the quiver $A_n$ up to isomorphism: 1 $\longrightarrow$ 2 $\longrightarrow \cdots \longrightarrow$ n Any idea of how to ...
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Is it known the $S_n$-module structures of free anti-commutative algebra

I recently started studiyng representation theory, especially I am interested in $S_n$-module structures of free algebras over some variety. I know that the $S_n$-module structures are known for some ...
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Decomposition of an irreducible representation of $A \otimes B$ for $C^{\ast}$-algebras $A$ and $B$.

Let $A$ and $B$ be $C^{\ast}$-algebras and $\pi: A \otimes B \to B(\mathcal{H}) $ be a representation of $A\otimes B$. Then there exists unique commuting representations $\pi_1$ of $A$ and $\pi_2$ of $...
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How does the center of a Path Algebra KQ for a quiver Q look like?

) I am attending a Foundation Course in Representation Theory and I am struggling with the following problem: determine the center of the path algebra KQ for a quiver Q. First of all I've thought that ...
Mikel's user avatar
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Can a finitely generated commutative $\mathbb C$-algebra have an indecomposible representation of dimension $>1$?

Can a finitely generated commutative $\mathbb C$-algebra $A$ have (a) an indecomposible or (b) irreducible representation of dimension $>1$ (over $\mathbb C$)? Without the finite generation ...
Adam's user avatar
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What does $\mathbb{C}[G ]\otimes_{\mathbb{C}[H]} W$ mean?

I am reading the group representation book by Serre. In chapter 7 (which is about induced representation), he introduces the notation $$ \mathbb{C}[G] \otimes_{\mathbb{C}[H]} W .$$ What does this ...
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In a $C^*$ algebra, if $A\geq B\geq 0$ then $A^{\frac{1}{2}}\geq B^{\frac{1}{2}}$

Let $A,B\in\mathfrak{A}$ be two positive elements s.t $A\geq B\geq 0$. Then $A^{\frac{1}{2}}\geq B^{\frac{1}{2}}$. This is an exericse from "$C^*$ algebras by example" by Kenneth Davidson, ...
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How proof The shape of the two hearts in $\mathcal{D}^b(A_n)$ is the same if the graded undergrphas of the Ext quiver of heart of are equal

According to Yu Qiu 's Ext-quivers of hearts of A-type and the orientation of associahedron, We know hearts in $\mathcal{D}^b(A_n)$ one by one corresponds to the Ext quivers and are precisely the ...
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How to show that $|G|=d_1^2+\cdots +d_s^2$ and $G$ is abelian?

In Representation Theory of Finite Groups: An Introductory Approach by Benjamin Steinberg, there is a corollary in which I feel confused. Corollary 6.2.6. Let p be a prime and let $G$ be a group of ...
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1 Dimensional Representation of Fusion Rings

Context: I am a physics grad student studying the fusion ring of topological defect lines in 2d CFTs. Consider a commutative ring with elements $\mathcal{L_1}, \mathcal{L_2}, ... \mathcal{L_n}$ such ...
Yaman Sanghavi's user avatar
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Schur functors are pairwise non isomorphic

In Fulton-Harris Part I Weyl's construction there a characterization of some of the irreducible representation of $GL(V)$, with $V$ a finite complex vector space. In particular, Theorem $6.3$ point $(...
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Why do physicists label irreducible representations of su(2) with half integers?

Im a physics student an I have been studying Lie Groups and Lie algebras for some time from a mathematical point of view mostly following Hall's book. Thing is that the Highest Weight Theorem is ...
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Doubt on trivial representation of the lie algebra $\mathfrak{su}(2)$

First of all, my question is written in $II)$ section. I) Groups and Representations In group theory a representation is a map that is defined as: $$\rho: G \to GL(V) \tag{1}.$$ You take a element ...
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Proper way to represent matrix concatenation with indexed matrix

I'm wondering how to properly represent a matrix concatenation equation. Consider a set of matrix with indices such that $X_1, X_2,\dots X_K$, then I would like to create a concatenated matrix $W$ ...
Kevin's user avatar
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Is there some connection between these two methods counting isomorphic irreducible submodules for a decomposition of regular $A$-module?

Let $A$ be a semisimple algebra over $\mathbb{C}$. Given a decomposition $A^{\circ} = \oplus W_i$ of the regular $A$-module $A^{\circ}$ and an irreducible $A$-submodule $M$, I have seen two ways to ...
zyy's user avatar
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Extending a representation of a $C^*$-algebra to its unitization

I have a basic question about representations of non-unital $C^*$-algebras. Let $A$ be a non-unital $C^*$-algebra, and $\rho\colon A\to\mathcal{B}(H)$ a $*$-representation of $A$ on some separable ...
geometricK's user avatar
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Quaternion Structure on $\mathbb{C}^2\otimes \text{End}(V)$ for a Complex Vector Space $V$

For context (although I will try to phrase this so that it is understandable without the context), I am trying to understand Kronheimer's construction of the ALE (Asymptotically Locally Euclidean) ...
Nick Macleod's user avatar
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Hahn-Banach implies that zero map extends to zero map?

Let $H$ be a complex inner product space and $\hat H$ its Hilbert space completion. Consider the bounded linear functional $\phi : H → \mathbb{C}, h ↦ 0$. Does Hahn-Banach tell us that its extension $\...
Jos van Nieuwman's user avatar
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Characteristic polynomials determine semisimple representation: possible counterexample

Let $G$ be any group and $k$ be any field. While studying representation theory, I saw some theorems which are special cases of the following theorem: $\textbf{Theorem. }$ Let $\rho_1,\rho_2:G\to \...
Daebeom Choi's user avatar
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Decomposing tensor products of $\mathfrak{so}(2n)$-representations

I've been trying to show that the tensor product $V\otimes V$ of a finite dimensional $\mathfrak{so}(2n)$ representation of dimension at least 3 decomposes as a direct sum of at least 3 irreducible ...
Jason V's user avatar
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Set of weights of dual representation $V^*$ in terms of weights of $V$

Let $\frak{g}$ be a finite dimensional semisimple Lie algebra and let $V$ be a $\frak{g}$-module. I've been trying to show that the set of weights $\Psi(V^*)$ of the dual module coincides with $-\Psi(...
Jason V's user avatar
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Isomorphism between irreducible representations and $(g/g')^*$

I have this exercise that I can't do. Let $g$ be a solvable lie algebra. Show that the set of isomorphisms classes os irreducible representations of $g$ is in bijection with $(g/g')^*$ where $g'=[g,g]$...
121212's user avatar
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Characters of irreducible representations of a finite and semisimple algebra

To clarify: I use $\chi_V$ for the character of the finite representation $V$ of $A$ which is an algebra on a generic field $K$. Then you can consider the idel $[A,A]$ generated by commutators, which ...
Don Abbondio's user avatar
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1 answer
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Schroeder-Bernstein theorem for representations of C*-algebras

I am trying to work on an exercise which claims that If two representations ρ and σ, on Hilbert spaces H and G respectively, are each unitarily equivalent to a subrepresentation of the other, then ...
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To show 'dual' of a Projective is projective

I am reading Chapter IV, section 2 in Assem's book Elements of the Representation Theory of Associative Algebras and I am stuck at a claim: Let $A$ be a finite dimensional algebra over $\mathbb{C}$. ...
Subham Jaiswal's user avatar
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A doubt in proof of existence of Auslander Reiten Sequences

I am providing the context. You can skip to the actual doubt below. My doubt doesnt need knowledge of quivers or even almost split sequences. I am reading Theorem 5.4.10 from the book An introduction ...
Subham Jaiswal's user avatar
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A sufficient condition for Automorphism of an exact sequence

I am given the following commutative diagram with non-split exact rows with $L, M, N$ being finite dimensional $A$ modules where $A$ is a finite dimensional $\mathbb{C}$ algebra. where $L$ is an ...
Subham Jaiswal's user avatar
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Irreducible finite-dimensional representations of a commutative algebra

I've just seen a corollary in P.Etingof's book Introducion to Representation Theory which states that every irreducible finite-dimensional representation of a commutative algebra is 1-dimensional: ...
zyy's user avatar
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Dimensions of homomorphisms determines the module

I am reading this paper about degenerations for representations. I am mentioning some context though all of it will not be needed I think for my question. We basically have a finite dimensional ...
Subham Jaiswal's user avatar
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If $A$ is representation finite, then $\{pd(M)|pd(M)<\infty, M\in mod-A\}=max\{pd(M)|M$ indecomposable $\}$?

Let $A$ be finite dimensional $\mathbb{C}$ algebra. Suppose $A$ is representation finite.(i.e. $A$'s has a finite list of isomorphism classes of indecomposables.) Set $mod-A$ to be category of ...
user45765's user avatar
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why does Schur-Weyl duality not hold for $SO(N)$?

I am learning some representation theory now. In the book I am reading (Group theory and physics by Sternberg), the author started with $GL(N, \mathbb{C})$, and then by using the relation, $SU(N) \...
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Unitary representation of $G$ induces representation of $L^1(G)$

I am reading Davidson's '$C^*$ algebras by example'. In chapter VII regarding group $C^*$ algebras, he makes the following claim which I do not understand: When $\pi$ is a unitary representation of a ...
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