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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4
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1answer
52 views

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 ...
3
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1answer
37 views

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 ...
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1answer
95 views

Existance of certain derivations on the Temperley-Lieb algebra:

Given a formal parameter $\delta$ recall that the Temperley-Lieb algebra $\mathrm{TL}_n(\delta)$ is the unital $\Bbb{C}(\delta)$-algebra generated by symbols $U_1, \dots, U_{n-1}$ subject to the (...
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1answer
27 views

Exact sequences of indecomposable Kronecker-modules

Let $A$ be the Kronecker algebra over an algebraically closed field. I know that the indecomposable $A$-modules fall into the preinjective component $\mathcal{I}$, the postprojective component $\...
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0answers
11 views

Lie group intertwiner is a Lie algebra intertwiner

Let $G<\textrm{GL}_n(\mathbb{C})$ be a linear Lie group, let $\mathfrak{g}$ be its associated Lie algebra, and let $(V_1,\rho_1)$ and $(V_2,\rho_2)$ be two representations of $G$. Then the derived ...
0
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1answer
50 views

Confusion about operators and representations

Physics student here. In my textbook on group theory, there was a matrix $$h_1= \begin{pmatrix} 1/2 & 0 & 0\\ 0 & -1/2 & 0\\ 0 & 0 & 0 \end{pmatrix}$$ Then the author define ...
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18 views

Prove that |A/<e>| = |A/ ann T |, where e ∈ A be a maximal idempotent such that T ⊆ mod(A/<e>).

Let A be a finite dimensional basic algebra over an algebraically closed field k, modA the category of finitely generated left A-modules. Let T be a functorially finite torsion class in modA. Let e ∈ ...
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0answers
28 views

Prove that |Λ/e| = |Λ|-|P|

Let Λ be a finite dimensional basic algebra over an algebraically closed field k. Let P be a finitely generated left Λ module and e an idempotent of Λ such that addP = addΛe. we denote by addP (...
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1answer
38 views

Path algebra $A=kQ$ is not isomorphic to $A^{op}$

This is an exercise from An Introduction to Quiver Representations by Harm Derksen and Jerzy Weyman. Exercise 1.6.3. Let $Q$ be the quiver \begin{equation*} \circ \longrightarrow \circ \...
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2answers
160 views

length of minimal projective resolution and projective dimension

Let $K$ be a field, $A$ be a $K$-algebra, and $M$ be a finite generated $A$-module. How do you show that the length of a minimal projective resolution of $M$ is the projective dimension of $M$? (This ...
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1answer
57 views

Complement of a maximal direct factor is indecomposable

Reading a book I saw the following assertion: Let $R$ be a ring (not necessarilly commutative) and $\varepsilon$ be the poset (w.r.t. inclusion) of all internal direct factors of an $R$-module $M$. ...
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0answers
69 views

Jacobson radical of the $\mathbb{k}$-algebra $\mathbb{k}[x]/\langle x^n\rangle$ where $\mathbb{k}$ is a field

Let $\mathbb{k}$ be a field and $n\geq 1$. Consider the $\mathbb{k}$-algebra $A$ defined as: $$A:=\mathbb{k}[x]/\langle x^n\rangle.$$ Then the Jacobson radical of $A$ (the intersection of all maximal ...
4
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1answer
108 views

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 $\...