Questions tagged [associative-algebras]
For questions on algebras with an associative product.
72
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Non-rigid modules and Auslander-Reiten quiver
I have a question about proving that a module is non-rigid using Auslander-Reiten quiver. Suppose that we have some algebra $A$ and the components of its Auslander-Reiten quivers are tubes like the ...
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0
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A sufficient condition for automorphism of an exact sequence
I asked A sufficient condition for Automorphism of an exact sequence earlier on Math.StackExchange but did not get any response so am posting it here.
I am given the following commutative diagram with ...
2
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1
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A problem about extensions of division rings
For a division ring $D$ with center field $F:=Z(D)$ such that $\dim_F D = n^2$, there is a classical result saying that $D\otimes_{F}\bar{F}\cong M_n(\bar{F})$ as $\bar{F}$-algebras, where $\bar{F}$ ...
8
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Trying to understand "a refinement of the Peter–Weyl theorem" by Lusztig
"A refinement of the Peter–Weyl theorem" is the title of Chapter 29 in Lusztig's "Introduction to quantum groups" (Birkhäuser 2010, reprint of the 1994 edition). This chapter is ...
4
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Zhu's algebra for the Virasoro VOA
I am trying to understand the proof in the appendix of the following paper:
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.110.8757&rep=rep1&type=pdf
The paper discusses Zhu's ...
3
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Number of rings with additive group $(\mathbb{Z}_{16})^2$. A341547(16) in OEIS
I would like to know if somewhere the number of non-isomorphic rings with additive group $(\mathbb{Z}_{16})^2$ is mentioned. If not, is someone able to calculate it?
And (easier) the commutative case? ...
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Are there non-semisimple complex "non-unital special Frobenius algebras"?
I'm interested in "non-unital special Frobenius algebras", consisting of two linear maps (morphisms in the symmetric monoidal category of finite-dimensional complex vector spaces)
$$\mu: V\...
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A list of low-dimensional associative $\mathbb{C}$-algebras with non-trivial centers
I am looking for a list of (examples of) low(est)-dimensional non-commutative associative unital $\mathbb{C}$-algebras $A$ with non-trivial centers $Z(A)$ (hence non-semisimple). For our purposes $Z(A)...
4
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0
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divided powers of a deformation class
Let $A$ be a (unital, associative) $k$-algebra where $k$ is a field. Given a flat deformation of $A$ one gets the deformation class $h$ in the second Hochschild cohomology $HH^2(A)$. Suppose $k$ has ...
3
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Continuous differentiations of functional algebras
Let $A$ be some algebra (infinite-dimensional) of analytic functions on $\mathbb{C}^n$, and $D$ be some derivation of $A$, i.e. $D(fg)=Df \cdot g + f \cdot Dg)$ (so A may be considered as a ...
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Is there a short proof for the permutation invariance of this combinatorial map?
Consider a positive integer $n$ and integers $(c_i)_{1\le i \le 4}$, with $1 \le c_i \le n$. Conside the map:
$$f_n: (c_1,c_2,c_3,c_4) \mapsto \delta_{c_1,c_2}\delta_{c_3,c_4} - \# \{ |2n+1-2|x||, \ x ...
2
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Free resolutions of universal enveloping algebras for simple, finite dimensional Lie algebras
I'm currently studying Anick's resolution on the context of universal enveloping algebras for certain Lie algebras, namely some of the smallest cases: $A_1,A_2,A_3,B_2,G_2$, and so on.
What are some ...
4
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Literature on the polynomials and equations, in structures with zero-divisors
I need literature about zeroes of polynomials and equation resolution in associative algebraic structures with zero-divisors, but I am having difficulties to find it.
For example, there is literature ...
3
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The inner product of a Clifford Algebra
Any Clifford algebra $\operatorname{Cl}(k, p)$ carries an induced inner product, which is the "trace" on its 0-blade: $\langle AB\rangle_0$ for given elements $A, B$ of the algebra.
This inner ...
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References about transfinite socle series
I would like to know if there is any literature that discusses "transfinite socle series" of a ring module. Below is my attempt at defining the series.
Let $R$ be an associative unital ring and $...
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2
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Classification of finite-dimensional (nilpotent) associative algebras
What is known about the classification of finite-dimensional (nilpotent) associative algebras? I am assuming that algebras are over a field of characteristic zero. If it is simple, then it has to be ...
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What's an illustrative example of a tame algebra?
A finite-dimensional associative $\mathbf{k}$-algebra $\mathbf{k}Q/I$ is of tame representation type if for each dimension vector $d\geq 0$, with the exception of maybe finitely many dimension vectors ...
4
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"Non-associative" standard polynomials
I saw somewhere (I appreciate if anyone has any references to proof of this fact) that if $A$ is a finite dimensional associative algebra such that $\textrm{dim}(A)<n$, then $A$ satisfies the ...
1
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0
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Exponents in unit groups of modular group algebras
Let $p$ be a prime number, $G=:G_1$ a (non-Abelian) finite p-Group and $K$ a finite field with $\operatorname{char}(K)=p$. It is well-known that the group $G_2:=1+\operatorname{rad}(KG)$ is a p-group ...
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Number of conjugacy classes of unit groups of modular group algebras
Let $n$ be a natural number, $p$ a prime number, $G$ a finite $p$-group and $K$ a finite field with $p^n$ elements. We focus on the group $1+J(KG)$, where $J(KG)$ is the Jacobson radical of $KG$, ...
2
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1
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A weak Schur's lemma for non-semisimple finite dimensional algebras
Let $B \subseteq C$ be an inclusion of finite dimensional (associative) algebras over a field $k$. Assume that $C$ is a free $B$-module. Let $\bigoplus_i U_i$ be
a decomposition of $B$ into ...
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Infinite-dimensional representation theory of $K[x]$
Let $K$ be an algebraically closed field. The finite-dimensional representation theory of the polynomial algebra $K[x]$ is tame and completely understood, which I shall first summarise. It's ...
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Rigid $Hom$-orthogonal modules in wild hereditary algebras
Let $Q$ be a simply-laced wild quiver with at least one multiple edge, $k$ be an algebraically closed field, $1$ be the source of and $2$ be the sink of one set of such edges. Can we find rigid ...
3
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47
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$c$-matrix reduction in hereditary algebras
Let $k$ be an algebraically closed field, $Q$ be a finite connected quiver and $Q'$ a subquiver of $Q$. Let $C$ be the $c$-matrix of a chamber of the scattering diagram/semi-invariant picture of $kQ$. ...
3
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Operation of a p'-group on a set of p-power order and fix points
The question is related to Taft's Theorem about G-invariant radical complements. Let $A$ be an associative unitary finite-dimensional $K$-Algebra posessing a separable factor Algebra by ist nilradical....
2
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Maximal separable subalgebras of semisimple algebras
Is anything known about maximal separable subalgebras of semisimple algebras in finite-dimension? Are those subalgebras of unique dimension or isomorphic or conjugate?
5
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223
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Higher Braces algebra and operads
1) In [HIGHER OPERATIONS ON HOCHSCHILD COMPLEX], Gerstenhaber and Voronov showed that the Hochschild complex $C_1(\mathcal A)$ of any associative algebra (or e_1 algebra) $\mathcal A$ is naturally ...
4
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Given a representation-infinite algebra, when is every AR component infinite?
Let $A$ be a finite dimensional algebra over an algebraically closed field $K$. The Auslander-Reiten quiver $\Gamma_A$ of $A$ is a means of presenting the category of finitely generated right $A$-...
4
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Elementary proof that a central simple algebra over a field having a maximal subfield is a cyclic algebra
I'm currently reading the book "Central Simple Algebras and Galois Cohomology" written by Philippe Gille and Tamas Szamuely.
In the book, I don't understand a computational proof of the theorem that ...
1
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1
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Jordan-Hölder series of $k$-subalgebras?
I am reading Shurygin's survey "Smooth Manifolds over Local Algebras and Weil Bundles" (Journal of Math. Sciences, Vol. 108, No. 2, 2002) and it mentions the following basic fact which I don't quite ...
6
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Non-rigid indecomposable summands of simple-minded collections in bounded derived category of hereditary algebras
Let $\Lambda$ be a hereditary algebra over an algebraically closed field $k$. Let $S$ be one of the indecomposable summands of one simple-minded collection in $D^b(\Lambda)$. Is it true that $S$ is ...
3
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A differential graded Lie algebra with the Hochschild differential
Let $(V,\cdot)$ be an associative algebra and $W$ be a vector space endowed with a bimodule structure $\triangleright:V\otimes W\to W$ and $\triangleleft:W\otimes V\to W$ such that the following ...
2
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Injection of the Universal enveloping algebra
Let L1 and L2 be two Lie algebras.If U(L1)is isomorphic to U(L2)as associative algebra,then L1 is isomorphic to L2 ?
2
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Morphisms from quasi-simple regular rigid $\Lambda$-module $M$ to $\tau^m M$ when $\Lambda$ is wild hereditary
Let $\Lambda$ be a finite dimensional basic wild hereditary algebra and $M$ be an indecomposable regular quasi-simple right-$\Lambda$ that is rigid in a regular component $\mathcal{R_i}$ of the ...
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Which groups can occur as the group of units of finite-dimensional noncommutative algebras?
This is a continuation of a previous question: Connectedness of groups of units in finite-dimensional commutative algebras.
Let $k$ be an algebraically closed field of characteristic $0$. Which ...
6
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Representation-finiteness vs. $\tau$-tilting-finiteness
Setting: Throughout, $\Lambda$ is a finite dimensional associative algebra, $\operatorname{mod} \Lambda$ is the category of all finitely generated left $\Lambda$-modules, and all subcategories are ...
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Is -1 a sum of Hermitian squares in associative *-envelopes of formally real Jordan algebras?
Let $J$ be an unital Jordan algebra (over $\mathbb{R}$) - recall that this means that $J$ is an unital $\mathbb{R}$-algebra (whose product we denote by $\bullet$) satisfying $x\bullet y=y\bullet x$ ...
2
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Question about the mutation of a cluster seed associated to any word of the braid semigroup
Let $G$ be a semisimple Lie group with the set of positive simple roots $\prod$. Let $\prod^{-}$ be the set of negative simple roots and let $\mathfrak{M}$ be the semigroup freely generated by $\prod$ ...
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Projective modules over a Hochschild extension of algebras
Let $k$ be a commutative ring and let $R$ be an associative (not necessarily commutative) $k$-algebra with unit. Also, let $M$ be an $R$-bimodule. It is well known that on $S:=R+M$ we can define an ...
3
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Unique dimension of Cartan subalgebras in modular Lie algebras
If $L$ is a Lie algebra over an algebraic closed field $K$ of characteristic zero, then all Cartan subalgebras are conjugated. Hence, they have all the same dimension. If $K$ is not algebraic closed ...
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If $A$ is an integer ring such that each $P \in A_L[X]$ has a finite number of zeros in $A$, is $A$ commutative?
Let $A$ be a ring in which the product of any two nonzero elements is nonzero (we shall say that $A$ is an integral domain, even if $A$ is non commutative). It is well-known that if $A$ is commutative,...
6
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Connectedness of units in finite-dimensional commutative complex algebras
In the following, an algebra will always mean a finite-dimensional associative commutative unital algebra (over some field $k$).
Let $A$ be a $\mathbb{C}$-algebra. I am trying to understand how its ...
0
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Can we drop commutativity assumption?
Let $A$ be an associative algebra with a unit over a field $k$. fix $n > 1$. Define a $k$-algebra structure on the vector space $A^{\otimes n} = A \otimes_k \dots \otimes_k A$ (where there are $n$ ...
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degree of associative algebra
Let $A$ be a finite dimensional associative algebra with unity over a field $F$. The degree of the algebra is the degree of its generic minimum polynomial (see Nathan Jacobson, Generic norm of an ...
5
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2
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413
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Conjugation in associative algebras over finite fields
Let $A$ be a finite dimensional associative algebra (with unity) over a finite field $F$. Let $L$ be a field extension of $F$. Suppose that after extending scalars to $L$, two elements $a,b$ of $A$ ...
6
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presentations of subalgebras
Assume that I have a finitely presented algebra $A$ over the complex numbers (by which I mean that $A$ is generated over $\mathbb{C}$ by finitely many elements $x_1,...,x_n$ subject to finitely many ...
4
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Relation between Associative algebra and group algebra
Let $A$ be an associative algebra over a filed $k$.
Q) What are the condition we can impose on $A$ such that there exists a $G$ such that $A=k[G]$, the group algebra generated by $G$?
I am ...
0
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0
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300
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Differences between primitive central idempotents and primitive orthogonal idempotents
If we have a complete set of primitive orthogonal idempotents of an algebra $A$, then we can obtain simple modules, indecomposable projective modules, indecomposable injective modules of $A$.
If we ...
4
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Non-commutative normalization
Let $A$ be a (non-commutative) associative algebra with 1. Assume that $A$ contains a cental subalgebra $Z$ such that
a) $Z$ is a noetherian domain
b) $A$ is a finitely generated module over $Z$.
...
10
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Free subgroups in algebras of polynomial growth
What is known about free non-abelian subgroups in finitely generated associative algebras of polynomial growth (e.g., over finite fields, to avoid finite-dimensional free subgroups)? For example, are ...