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 ...
152 views

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 $K$-...
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1 vote
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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 ...
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1 vote
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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 ...
• 153
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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 ...
• 153
1 vote
40 views

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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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 $g'=[g,g]$...
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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 ...
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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 $\mathbb{C}$. ...
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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 ...
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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 ...
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1 vote
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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: ...
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1 vote
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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 ...
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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 ...
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