# Questions tagged [associative-algebras]

For questions on algebras with an associative product.

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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 ...
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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 ...
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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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### 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 ...
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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 ...
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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 ...
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 $... 0answers 88 views ### Why is an operad of associative algebras Koszul? Let$Assoc$be an operad of associative algebras. What does it mean for$A$to be a Koszul operad? Is it related to standard Koszul duality for algebras? As far as I understand, if$Assoc_{\infty}$... 2answers 177 views ### Classification of finite-dimensional (nilpotent) associative algebras What is known about the classification of finite-dimensional (nilpotent) associative algebras? I am assuming that algebras are over a field of characteristic zero. If it is simple, then it has to be ... 3answers 596 views ### What's an illustrative example of a tame algebra? A finite-dimensional associative$\mathbf{k}$-algebra$\mathbf{k}Q/I$is of tame representation type if for each dimension vector$d\geq 0$, with the exception of maybe finitely many dimension vectors ... 1answer 264 views ### “Non-associative” standard polynomials I saw somewhere (I appreciate if anyone has any references to proof of this fact) that if$A$is a finite dimensional associative algebra such that$\textrm{dim}(A)<n$, then$A$satisfies the ... 0answers 18 views ### Exponents in unit groups of modular group algebras Let$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)$is a p-group ... 0answers 56 views ### Number of conjugacy classes of unit groups of modular group algebras Let$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$KG$, ... 1answer 117 views ### 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 ... 0answers 154 views ### 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 ... 0answers 34 views ### 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 ... 0answers 45 views ###$c$-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$kQ$. ... 1answer 135 views ### 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.... 0answers 52 views ### 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? 0answers 201 views ### 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 ... 1answer 89 views ### 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$A$-... 1answer 321 views ### 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 ... 1answer 74 views ### 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 ... 0answers 72 views ### 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 ... 1answer 194 views ### 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 ... 1answer 133 views ### 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 ？ 0answers 36 views ### 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 ... 0answers 192 views ### 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 ... 0answers 242 views ### 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 ... 0answers 31 views ### 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$x\bullet y=y\bullet x$... 0answers 55 views ### 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$\prod$... 0answers 66 views ### 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 ... 2answers 259 views ### 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 ... 0answers 81 views ### 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,... 2answers 322 views ### 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 ... 0answers 206 views ### 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$n$... 0answers 82 views ### 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 ... 2answers 379 views ### 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$A$... 1answer 164 views ### 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 ... 2answers 364 views ### 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 ... 0answers 277 views ### 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 ... 1answer 248 views ### 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$Z$. ... 1answer 191 views ### 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 ... 3answers 678 views ### Is the Amitsur-Levitzki identity essentially unique? Let us consider the matrix algebra.$Mat_n(\mathbb{C})$. The Amitsur-Levitzki identity states that for any matrices$X_1, X_2, ..., X_{2n} \in Mat_n(\mathbb{C})$the sum$\Sigma_{\sigma \in S_{2n}} ...
Let $A$ be a finite dimensional (unital) $K$-Algebra. By $A^{\circ}$ we denote the associated $K$-Lie-algebra of $A$ with respect to the product $a\circ b:=ab-ba$. In addition, we denote by \$rad(A^{\...