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 $A$
Questions tagged [representation-of-algebras]
90 questions
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Examples of actions of algebras on categories
I am trying to learn about actions of groups/algebras on categories. Below is a paragraph from the Preface to "Categorification and higher representation theory", it is the final sentence I wish to understand further:
Categorified representation…
Ted Jh
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- 9
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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 they are unitarily equivalent, which is called…
Yanyu
- 430
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Uniserial modules are determined up to isomorphism by their composition series?
I have a question on uniserial modules. Here, every modules are finitely generated modules over a finite dimensional algebra $A$ over a field $K$.
In the book Elements of the Representation Theory of Associative Algebras: Volume 1 , there is a…
Ji Woong Park
- 438
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The equivalence relation defined by normal form on Von Neumann algebra, its support and the it's representation .
Let M be a von Neumann algebra and $\varphi$ a positve normal form on M.
$N = \lbrace x\in M | \varphi(x^*x)=0\rbrace $ . We denote $M_{\varphi} := M/N$ as the pre-Hilbert space defined by the inner product: $\langle a_{\varphi},b_{\varphi}\rangle…
shestak
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Given a $\mathbb{k}$-algebra $A$ and two $A$-modules $M$ and $N$ find $\dim_{\mathbb{k}}\left(\mathrm{Hom}_{A}(M,N)\right)$
Let $\mathbb{K}$ be a field and $A$ a finite dimensional $\mathbb{k}$-algebra with identity $1_{A}$. For two $A$-modules $M$ and $N$ we have the set of all $A$-linear maps from $M$ to $N$ denoted $\mathrm{Hom}_{A}(M,N)$. This set is not in general…
Hector Blandin
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What's the Virasoro module structure of free boson,. etc?
I'm reading Chapter 2. of "Vertex Algebras and Algebraic Curves". In which they give some examples of 2d CFTs. For example, the Hilbert space of 2d massless free boson are generated by $b_{-n}^{j_n}\dots b_{-1}^{j_i}|{0}\rangle$. In which $b_{-i}$…
Peter Wu
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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 (unitarily) equivalent to some subrepresentation of…
Felipe Dilho
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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 ennunciated for complex semisimple Lie algebras and…
olsrcra
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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 $\rho_1,\rho_2:G\to \text{GL}_n(k)$ be two semisimple representations of…
Daebeom Choi
- 522
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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, $SU(N) \subset SL(N) \subset GL(N)$, he obtained the…
S. Kohn
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2 answers
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 Hausdorff, locally compact group $G$, it induces…
GSofer
- 4,590
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Maschke's Theorem and split extensions
Recall that a short exact sequence of groups like $1\longrightarrow A\longrightarrow E\longrightarrow
G\longrightarrow 1$ is called an extension of $A$ by $G$. Now, let $k$ be a finite field and $G$ a finite group. Let $A$ be a $k[G]$-module. Does…
N. SNANOU
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Does the proof of Ado's theorem in Varadarajan's "Lie groups, Lie Algebras, and their representations" contain a mistake?
I'm working with Varadarajan's book "Lie groups, Lie Algebra and their representations". Specifically on the proof of Ado's theorem page 237.
Ado's theorem : Let $\mathfrak g$ be a Lie algebra over $k$ and $\mathfrak n$ it's nilradical ideal. Then…
Digitallis
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Why $H$ in $\mathfrak{sl}_2$ triple is always semisimple?
I am beginning to learn some materials about representation of Lie algebra. Here I define $\mathfrak{sl}_2$ triple as:
$$
\{H,X,Y\in \text{End}_{\mathbb{C}}(V)|H,X,Y \text{are nonzero},\quad [H, X]=2 X,\quad[H, Y]=-2 Y,\quad[X, Y]=H\}
$$
where $V$…
Edward Z. Miao
- 313
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Non-surjective representation of infinitely-generated simple module (see: Jacobson Density/Burnside's theorems)
The book I'm working through covers (on its way to the Artin-Wedderburn theorem) the Jacobson Density theorem:
Let $L$ be an irreducible left $R$-module, with $D := \text{End}_{R-}(L)^{\text{op}}$ (so $L$ is a $(R,D)$-bimodule).
Let $v_1,\dots,v_n…
