# Questions tagged [algebras]

For questions about algebras, their properties, and their structures. Use [tag:algebra-precalculus] or [tag:abstract-algebra] if your question is about algebra, not algebras.

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### Turning a subalgebra into an ideal by changing the algebra it is contained in

Let $B$ be an (not necessarily unital) algebra and $A \subseteq B$ a subalgebra. Is there some sort of quotient $C$ of $B$ (or any other interesting construction not necessarily found by taking ...
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### Semisimplicity implies separability for a perfect field

Let $k$ be a field, $A$ a finite-dimensional semisimple $k$-algebra. If $k$ is a perfect field (every finite field extension of $k$ is seperable), then $A$ is separable. I know a proof that uses the ...
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### Proof the quaternions are 4-dimensional?

The quaternions can be defined as $$\mathbb{R}\langle X,Y\rangle/(X^2+1,Y^2+1,XY+YX)$$ From these relations, it is relatively easy to prove that $1,X,Y,XY$ span the quaternions over $\mathbb{R}$. But ...
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### Inferring classification of Clifford algebras from classification of Clifford modules

Let $Cl_n$ be the Clifford algebra (over reals) $$Cl_n = T^{*}\mathbb{R}^n/\langle v\otimes v - q(v) \rangle.$$ There is a periodic table of $K$-representations of $Cl_n$, i.e. $\mathbb{R}$-linear ...
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### Equivalent definitions of Hopf algebras

Recently, I started to study the book Hopf algebras by Moss Sweedler, in such book, given a coalgebra $(C,\Delta,\epsilon)$ and an algebra $(A,\mu,\eta)$, the autor defines the convolution of two ...
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### Jordan-Holder theorem for group algebras

I'm currently studying the Jordan-Holder theorem for modules and representations of associative algebras over fields. I was wondering if there is a way to prove the Jordan-Holder theorem for finite ...
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### Definition of splitting field: Why do we require centrality?

Let $k$ be a field. Let $D$ be a division algebra over $k$. Call a field extension $K/k$ a splitting field for $D$ if there exists a positive integer $n$ such that $D\otimes_k K\cong M(n\times n,K).$ ...
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### Complete Subalgebras and Dense Subalgebras of Complete Boolean Algebras are Regular Subalgebras

In Thomas Jech's Set Theory book, he states that If $A$ is a complete subalgebra of a complete Boolean algebra $B$ then $A$ is a regular subalgebra of $B$. Also, if $A$ is a dense subalgebra of $B$ ...
1 vote
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### Finding the discriminant of a quaternion algebra

Consider the totally real number field $F=\mathbb{Q}(\zeta_{10}+\zeta_{10}^*)$. Consider the quaternion algebra $Q=(\frac{-1,-1}{F})$. How do I compute the discriminant of this algebra? I gave ...
1 vote
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### What is the dual Hopf algebra of T(V)?

I am reading "Geometric versus Non-Geometric Rough Paths" by Martin Hairer and David Kelly. I don't know much about Hopf algebras but $T(V)$ is the one example I'm comfortable with so far ...
1 vote
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### Is there a finite-dimensional algebra over $\mathbb C$ with no zero divisors?

Every non-trivial finite dimensional associative algebra over an algebraically closed field has zero divisors. Is there a non-trivial finite dimensional non-associative algebra over $\mathbb C$ or ...
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### When an Isomorphism between two algebras is an equivalence?

I'm currently involved in the study of algebraic structures, and there's a concept that seems to appear every so often. Given an Algebra $A=<A,F_i>$ an isomorphism $f$ is a bijective ...
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Let $R \subseteq S$ be unital rings and let $G \leq \mathrm{Aut}_R(S)$ be a finite group of automorphisms of $S$ as an $R$-algebra. We define the invariant subring: S^G = \{a \in S \mid \forall \...
### The forgetful functor $U:T-Alg\rightarrow C$ preserves limits
Suppose $F$ is an endofunctor of a complete category $C$. Let $F-Alg$ be the category that has objects the pairs $(X, \ a:FX\rightarrow X)$ where $X$ is an object in $C$, and that has morphisms \$f:(X,...