# Questions tagged [representation-theory]

For questions about representations or any of the tools used to classify and analyze them. A representation linearizes a group, ring, or other object by mapping it to some set of linear transformations. A common goal of representation theory is classifying all representations of some type. Representation theory is a broad field, so questions not including the word "representation" may be appropriate.

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### Alternative definition of Koszul algebra by injective resolutions?

Let $A$ be a positively graded algebra. This means that $A$ is a $k$-algebra graded non-negatively and $A_0 \cong k \times \dots \times k$ such that each degree is finite-dimensional. From here, we ...
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### Necklace Lie Bracket for Species

I am currently reading the paper "Calabi-Yau Algebras and Superpotential" by Van Den Bergh, but I find it highly confusing. In Section 10 he mention that bracket {$-,-$}$\space_{\omega_\eta}$...
1 vote
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### Are there any other examples of simple not absolutely simple lie algebra besides $so(3,1)$?

I just learned the distinction between simple and absolutely simple Lie algebras and was wondering if there are other well known examples other than the Lorentz algebra.
1 vote
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### Simple Lie algebras invariants

Do all simple Lie algebras have just one quadratic Casimir invariant as the Harish-Chandra isomorphism seems to imply or are there counter-examples?
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### Laplace-Beltrami on $SO(3)$

Denoting the double cover map $SU(2)\rightarrow SO(3)$ by $\phi$, we have an induced monomorphism $C^\infty(SO(3))\rightarrow C^\infty(SU(2))$, $f\mapsto f\circ\phi$. We know that the Lie-algebra of ...
1 vote
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### 1-dimensional representations and anti-automorphisms.

Suppose we are in the following setting. Let $A$ be an associative algebra and over a field $\mathbf{K}$, $B\subset A$ a subalgebra and $M$ be a finite dimensional left module of $A$. Let us ...
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### Definition unramified local representation

We can define unramified representation of $GL(2,\mathbb Q_p$) as a representation $(\pi , V)$ such that $V^K\neq \{0\}$ for $K$ the maximal compact subgroup i.e. $GL(2,\mathbb Z_p)$. However, I do ...
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### Are there recommended series of online videos or books that are directly related or indirectly related to the textbook? [closed]

I want to learn Group representation theory by self-study and the textbook that I will use is "The symmetric group Representations, Combinatorial, Algorithms and symmetric function" by Bruce ...
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### general structure theory of associative algebra [closed]

Recently reading literature, there is a very difficult to understand:"$F\left[ ad_{x}\right]$ is a semisimple, finite $F$ algebra and owing to general structure theory,$G$ is a completely ...
1 vote
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### Reading off module properties from the companion matrix

Let $P\in \mathbb{F}[x]$ be a monic polynomial of degree $n$ over a field $\mathbb{F}$, and $M_P$ its companion matrix. The matrix $M_P$ gives a module of $\mathbb{F}[x]$ on $\mathbb{F}^n$, by letting ...
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### Solution to system of nonlinear equations

I am trying to comple the character table of a finite group with seven conjugacy classes $c_1, \cdots, c_7$: Character table If I use the orthogonality of the column vectors of table I get a system of ...
1 vote