Recent zbMATH articles in MSC 16Shttps://zbmath.org/atom/cc/16S2022-09-13T20:28:31.338867ZWerkzeugA note on simple modules of quantum polynomial algebrahttps://zbmath.org/1491.160042022-09-13T20:28:31.338867Z"Mukherjee, Snehashis"https://zbmath.org/authors/?q=ai:mukherjee.snehashisSummary: The simple modules of \(\mathbb{K}_Q[x_1,\dots,x_n]\) the coordinate ring of quantum affine space, are classified in the case, where \(\mathcal{Q}=(q_{ij})\) is an \(n\times n\) matrix with \(q_{ii}=1\) and \(q_{ji}=q_{ij}^{-1}\), for all \(1\le i,j\le n\) under the assumption that the superdiagonal and the diagonals above the superdiagonal are constant and \(\mathbb{K}\) is algebraically closed.Cancellation of Morita and skew typeshttps://zbmath.org/1491.160232022-09-13T20:28:31.338867Z"Tang, Xin"https://zbmath.org/authors/?q=ai:tang.xin"Zhang, James J."https://zbmath.org/authors/?q=ai:zhang.james-j|zhang.james-yiming"Zhao, Xiangui"https://zbmath.org/authors/?q=ai:zhao.xianguiThe \textit{Zariski cancellation problem} asks whether an isomorphism \(A[x] \cong B[x]\) of \(\Bbbk\)-algebras, \(\Bbbk\) a field, implies \(A \cong B\). Though originally studied as a question in algebraic geometry and commutative algebra, there has been significant recent interest in this problems and its variants for noncommutative algebras, building off the work of \textit{J. Bell} and \textit{J. J. Zhang} [Sel. Math., New Ser. 23, No. 3, 1709--1737 (2017; Zbl 1380.16024)].
This paper explores two generalizations of the Zariski cancellation problem well-suited to the study of noncommutative algebras. The first, introduced in [\textit{D. M. Lu} et al., Can. J. Math. 72, No. 3, 708--731 (2020; Zbl 1454.16028)], is to ask whether a Morita equivalence between algebras \(A[x]\) and \(B[x]\) implies a Morita equivalence between \(A\) and \(B\). The authors of the present paper prove that an algebra with center \(\Bbbk\) is universally Morita cancellative. Furthermore, if the center \(Z\) of \(A\) is strongly retractable or \(Z/N(Z)\) is strongly retractable, where \(N(Z)\) is the nilradical of \(Z\), then \(A\) is both strongly cancellation and strongly Morita cancellative.
The second generalization replaces polynomial extensions with Ore extensions. The \textit{skew cancellation problem} asks whether an isomorphism \(A[x;\sigma,\delta] \cong B[x';\sigma',\delta']\) implies an isomorphism \(A \cong B\). \textit{J. Bergen} [Commun. Algebra 46, No. 2, 705--707 (2018; Zbl 1410.16029)] considered a version of this problem for differential operator rings. Further progress on this problem appears in \textit{J. Bell} et al. [Beitr. Algebra Geom. 62, No. 2, 295--315 (2021; Zbl 1484.16024)]. The present paper invokes several invariants to study this problem, including the divisor subalgebra and the stratiform length. In particular, suppose \(A\) is a noetherian domain that is stratiform. The authors prove that \(A\) is strongly skew cancellative in the category of noetherian stratiform domains assuming the divisor subalgebra of \(A\) generated by \(1\) is \(A\) itself.
Reviewer: Jason Gaddis (Oxford)\(\ell\)-weak identities and central polynomials for matriceshttps://zbmath.org/1491.160242022-09-13T20:28:31.338867Z"Blachar, Guy"https://zbmath.org/authors/?q=ai:blachar.guy"Matzri, Eli"https://zbmath.org/authors/?q=ai:matzri.eli"Rowen, Louis"https://zbmath.org/authors/?q=ai:rowen.louis-halle"Vishne, Uzi"https://zbmath.org/authors/?q=ai:vishne.uziThe paper deals with \(PI\)-theory. One of the most important questions is the finite basis problem. For zero characteristics (so-called Specht problem) this was done by \textit{A. R. Kemer} [Algebra Logic 27, No. 3, 167--184 (1988; Zbl 0678.16012); translation from Algebra Logika 27, No. 3, 274--294 (1988); Algebra Logika 26, No. 5, 597--641 (1987; Zbl 0646.16013)]. He also solved the problem for the local case (i.e., finitely many variables) over an infinite field [\textit{A. R. Kemer}, Math. USSR, Izv. 37, No. 1, 69--96 (1990; Zbl 0784.16016); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 54, No. 4, 726--753 (1990)]. For the local case over a finite field and noetherian associative commutative ring see [\textit{A. Y. Belov}, Izv. Math. 74, No. 1, 1--126 (2010; Zbl 1208.16022); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 74, No. 1, 3--134 (2010); \textit{A. Belov-Kanel} et al., Trans. Am. Math. Soc. 367, No. 8, 5553--5596 (2015; Zbl 1332.16015)]. For the case of positive characteristic, the first counterexamples were found in [\textit{A. Ya. Belov}, Fundam. Prikl. Mat. 5, No. 1, 47--66 (1999; Zbl 0964.16024)] and latter by \textit{A. V. Grishin} [Fundam. Prikl. Mat. 5, No. 1, 101--118 (1999; Zbl 1015.16022)] and \textit{V. V. Shchigolev} [Fundam. Prikl. Mat. 5, No. 1, 307--312 (1999; Zbl 0968.16008)].
However, a concrete basis of identities is known only in few cases, for example for \(2\times 2\) matrices over a field of characteristic zero [\textit{Ju. P. Razmyslov}, Algebra Logic 11, 108--120 (1973; Zbl 0266.20021)].
V. N. Latyshev introduced notion of \textit{weak identities} and established their relations with central polynomials. Although in many cases it is known that they have a finite basis, for concrete basis our knowledge is still poor. This idea generated series of papers including [\textit{A. Ya. Belov}, Sib. Mat. Zh. 44, No. 6, 1239--1254 (2003; Zbl 1054.16015); translation in Sib. Math. J. 44, No. 6, 969--980 (2003)]. For concrete basis of weak identities and central polynomials our knowledge is also still poor.
In order to attack the subject, the authors develop the theory of \(\ell\)-weak identities in order to provide a feasible way of studying the central polynomials of matrix algebras. They describe the weak identities of minimal degree of matrix algebras in any dimension. They provide a description of \(\ell\)-weak identities for small degrees.
This is an important step for the future theory allowing to get some concrete basis information. In this spirit, I recommend the paper [\textit{A. R. Kemer}, Izv. Vyssh. Uchebn. Zaved., Mat. 1989, No. 6(325), 71--76 (1989; Zbl 0678.16013)].
For the entire collection see [Zbl 1461.16003].
Reviewer: Alexei Kanel-Belov (Ramat-Gan)Trace identities on diagonal matrix algebrashttps://zbmath.org/1491.160252022-09-13T20:28:31.338867Z"Ioppolo, Antonio"https://zbmath.org/authors/?q=ai:ioppolo.antonio"Koshlukov, Plamen"https://zbmath.org/authors/?q=ai:koshlukov.plamen-e"La Mattina, Daniela"https://zbmath.org/authors/?q=ai:la-mattina.danielaFor \(\alpha_{i},a_{ii}\in F\) (\(i=1,\dots,n)\), define \(t_{\alpha_{1},\ldots,\alpha_{n}}(\text{diag}(a_{11},\dots,a_{nn}))=\alpha_{1}a_{11}+\cdots+\alpha_{n}a_{nn}\). Let \(D_{n}^{t_{\alpha_{1},\ldots,\alpha_{n}}}\) denote the algebra of \(n\times n\) diagonal matrices endowed with the trace \(t_{\alpha_{1},\ldots,\alpha_{n}}\). Given a trace polynomial \(f(x_{1},\ldots,x_{k},\text{Tr})\) and \(\alpha\in F\), the trace polynomial \(f^{\alpha}(x_{1},\ldots,x_{k},\text{Tr})\) is obtained in the way that for every monomial \(M_{s}(x_{1},\dots,x_{k})\) of of \(f(x_{1},\ldots,x_{k},\text{Tr})\) containing \(s\) traces, \(f^{\alpha}(x_{1},\ldots,x_{k},\text{Tr})\) contains the monomial \(\alpha^{-s}M_{s}(x_{1},\dots,x_{k})\). Also, let \(C_{n}(x_{1},\ldots,x_{k})\) be the \(k\)-th Cayley-Hamilton polynomial.
For arbitrary \(n\), there are two main results. (1) For nonzero \(\alpha\in F\), the trace \(T\)-ideal \(\text{Id}^{tr}(D_{n}^{t_{\alpha,\ldots,\alpha}})\) is generated by the polynomials \([x_{1},x_{2}]\) and \(C_{n}^{\alpha}(x_{1},...,x_{n})\). (2) If there exist \(i\) and \(j\) with distinct nonzero \(\alpha_{i}\) and \(\alpha_{j}\), then \(D_{n}^{t_{\alpha_{1},\ldots,\alpha_{n}}}\) does not satisfy any multilinear trace identity of degree \(n\) which is not a consequence of the identity \([x_{1},x_{2}]\equiv0\).
Finally, the authors study the polynomial identities satisfied by \(D_{2}\) and \(D_{3}\) endowed with all possible traces.
For the entire collection see [Zbl 1461.16003].
Reviewer: Wen-Fong Ke (Tainan)Identities in group rings, enveloping algebras and Poisson algebrashttps://zbmath.org/1491.160262022-09-13T20:28:31.338867Z"Petrogradsky, Victor"https://zbmath.org/authors/?q=ai:petrogradsky.victor-mLet \(\mathcal{V}\) be a variety of \(\Sigma\)-algebras in the sense of universal algebra. Let \(F_{\mathcal{V}}n\) be the free algebra in the variety \(\mathcal{V}\) on n generators \(x_1,\cdots,x_n\). Let \((s,t)\) be pair of members of \(F_{\mathcal{V}}n\). A \(\Sigma\)-algebra \(A\) which belongs to \(\mathcal{V}\) satisfies the identity \(s=t\) if for each \(a_1,\cdots,a_n\in A\), \(\hat{s}(a_1,\cdots,a_n)=\hat{t}(a_1,\cdots,a_n)\), where for \(u\in F_{\mathcal{V}}n\), \(\hat{u}(a_1,\cdots,a_n)\) is the image of \(u\) by the unique homomorphism \(\pi\colon F_{\mathcal{V}}n\to A\) such that \(\pi(x_i)=a_i\), \(i=1,\cdots,n\). When \(\mathcal{V}\) is the variety of all algebras over some ring \(R\), then such an identity is referred to as a polynomial identity.
The paper is a survey of known results about identities in group rings, enveloping algebras, Poisson symmetric algebras. In its second section the author recalls, for instance, the results of \textit{D. S. Passman} [J. Algebra 20, 103--117 (1972; Zbl 0226.16015)] on polynomial identities on group rings over a field. The third section concerns identities satisfied by enveloping algebras of (restricted) Lie algebras. The fourth section is about Poisson algebras which are sets carrying a commutative unital algebra structure and a Lie algebra structure which interact nicely (they satisfy the Leibniz rule). Section five is about the notion of multilinear identities of symmetric Poisson algebras (the symmetric algebra on the underlying module of Lie algebra has a structure of a Lie algebra that makes it a Poisson algebra, referred to as the symmetric Poisson algebra of the Lie algebra). The final section deals with Lie identities of symmetric Poisson algebras.
For the entire collection see [Zbl 1461.16003].
Reviewer: Laurent Poinsot (Villetaneuse)Universal enveloping algebras of generalized Poisson-Ore extensionshttps://zbmath.org/1491.160272022-09-13T20:28:31.338867Z"Shen, Yuan"https://zbmath.org/authors/?q=ai:shen.yuan|shen.yuan.1"Zheng, Xia"https://zbmath.org/authors/?q=ai:zheng.xiaSummary: The main goal of this paper is to study the Poisson universal enveloping algebra of a generalized Poisson-Ore extension. We prove such a Poisson universal enveloping algebra is a twisted tensor product.A note on twisted group rings and semilinearizationhttps://zbmath.org/1491.160282022-09-13T20:28:31.338867Z"Brazelton, Thomas"https://zbmath.org/authors/?q=ai:brazelton.thomasSummary: In this short note, we construct a right adjoint to the functor which associates to a ring \(R\) equipped with a group action its \textit{twisted group ring}. This right adjoint admits an interpretation as \textit{semilinearization}, in that it sends an \(R\)-module to the group of semilinear \(R\)-module automorphisms of the module. As an immediate corollary, we provide a novel proof of the classical observation that modules over a twisted group ring are modules over the base ring together with a semilinear action.Lie automorphisms of incidence algebrashttps://zbmath.org/1491.160292022-09-13T20:28:31.338867Z"Fornaroli, Érica Z."https://zbmath.org/authors/?q=ai:fornaroli.erica-zancanella"Khrypchenko, Mykola"https://zbmath.org/authors/?q=ai:khrypchenko.mykola-s"Santulo, Ednei A. jun."https://zbmath.org/authors/?q=ai:santulo.ednei-aparecido-junLet \(A\) be an associative algebra over a commutative ring \(R\). Then \(A\) becomes a Lie algebra under the Lie product \([x, y]=xy-yx\) for all \(x, y\in A\). A bijective \(f: A\longrightarrow A\) is called Lie automorphism if \(f\) preserves the Lie product, that is \(f([x, y])=[f(x), f(y)]\). One can understand that the Lie automorphism \(f\) of \(A\) is an automorphism in the sense of of the Lie algebra \((A, [ , ])\). Furthermore, \(f\) is said to be proper if there exist maps \(\phi\) and \(\nu\) such that \(f=\varphi+\nu\), where \(\varphi\) is an automorphism of \(A\) or the negative of an anti-automorphism of \(A\), \(\nu\) is an \(R\)-linear central-valued map on \(A\) such that \(\nu([A, A])=\{0\}\). \textit{H. Loo-Keng} [J. Chinese Math. Soc. (N.S.) 1, 110--163 (1951)] proved that each Lie automorphism of the full matrix ring \(M_n(R)\), \(n>2\), over a division ring \(R\), \(\mathrm{char}(R)\neq 2,3\), is proper. \textit{D. Ž. Đoković} [J. Algebra 170, No. 1, 101--110 (1994; Zbl 0822.17017)] showed that each Lie automorphi of the upper triangular matrix algebra \(T_n(R)\), where \(R\) is a commutative unital ring with trivial idempotents, is proper too.
The incidence algebra \(I(X,R)\) of a locally finite poset \(X\) over a commutative unital ring \(R\) is a natural generalization of \(T_n(R)\). Jordan and Lie maps on \(I(X, R)\) (and even on more general algebras) have been actively studied in more recent years. Usually, all Lie-type maps on \(I(X,R)\) are proper. However, this is no longer true for the case of Lie automorphisms of \(I(X, K)\), where \(K\) is a field and \(X\) is finite and connected, which is the main result of this paper. The authors give a full description of the Lie automorphisms of the incidence algebra \(I(X, K)\). In particular, they show that they are in general not proper.
Reviewer: Wei Feng (Beijing)On Fagundes-Mello conjecturehttps://zbmath.org/1491.160302022-09-13T20:28:31.338867Z"Luo, Yingyu"https://zbmath.org/authors/?q=ai:luo.yingyu"Wang, Yu"https://zbmath.org/authors/?q=ai:wang.yu.4|wang.yu.1|wang.yu.2|wang.yu.8|wang.yu.3|wang.yu.5|wang.yu.9|wang.yuThe famous Lvov-Kaplansky conjecture states that the image of a multilinear polynomial in non-commutative variables over a field \(K\) on the matrix algebra \(M_n(K)\), \(n\geq 2\), is a vector space. Replacing the algebra \(M_n(K)\) with the upper triangular matrix algebra \(T_n(K)\), Fagundes-de Mello conjecture [\textit{P. S. Fagundes} and \textit{T. C. de Mello}, Oper. Matrices 13, No. 1, 283--292 (2019; Zbl 1432.16023)] states that the image of a multilinear polynomial on the algebra \(T_n(K)\), \(n\geq 2\), is a vector space. In the paper under review the authors confirm the conjecture when the base field \(K\) is infinite or if it is finite but has sufficiently many elements. For the proof the authors define the \(\beta\)-index of a nonzero multilinear polynomial and show that the image of the multilinear polynomial \(p(x_1,\ldots,x_m)\) depends on its index \(\beta(p)\): \par (i) If \(\beta(p)=0\), then \(p(T_n(K))=T_n(K)\); \par (ii) If \(1\leq \beta(p)=t<n\), then \(p(T_n(K))=T_n(K)^{(t-1)}\), where \(T_n(K)^{(t-1)}=J^t(T_n(K))\) and \(J(T_n(K))\) is the Jacobson radical of \(T_n(K)\); \par (iii) If \(\beta(p)\geq n\), then \(p(T_n(K))=0\); \par The Fagundes-de Mello conjecture was solved for infinite fields independently and with other methods in [\textit{I. G. Gargate} and \textit{C. de Mello}, ``Images of multilinear polynomials on \(n\times n\) upper triangular matrices over infinite fields'', Preprint, \url{arXiv: 2106.12726}]. Comparing both proofs one can conclude that the equality \(\beta(p)=t\) is equivalent to the fact that \(p(x_1,\ldots,x_n)\) belongs to \(C^t(K\langle X\rangle)\setminus C^{t+1}(K\langle X\rangle)\), where \(C(K\langle X\rangle)=([x_1,x_2])^T\) is the T-ideal of the free associative algebra \(K\langle X\rangle\) generated by the commutator \([x_1,x_2]\).
Reviewer: Vesselin Drensky (Sofia)Graphs with disjoint cycles classification via the talented monoidhttps://zbmath.org/1491.160312022-09-13T20:28:31.338867Z"Hazrat, Roozbeh"https://zbmath.org/authors/?q=ai:hazrat.roozbeh"Sebandal, Alfilgen N."https://zbmath.org/authors/?q=ai:sebandal.alfilgen-n"Vilela, Jocelyn P."https://zbmath.org/authors/?q=ai:vilela.jocelyn-pIf \(E\) is a directed graph, the talented monoid \(T_E\) of \(E\) is a pre-ordered abelian monoid with an action of the infinite cyclic group generated by \(x.\) The Grothendieck group \(G_E\) of \(T_E\) is isomorphic to the Grothendieck group \(K^{\operatorname{gr}}_0(L_K(E))\) of the Leavitt path algebra \(L_K(E)\) of \(E\) over any field \(K\) when \(L_K(E)\) is considered as an algebra naturally graded by \(\mathbb Z\) (the action of the infinite cyclic group generated by \(x\) on \(K^{\operatorname{gr}}_0(L_K(E))\) is induced by this natural grading of \(L_K(E)\)). The Graded Classification Conjecture is stating that \(T_E\) (and, equivalently, \(G_E\)) is a complete invariant of the Leavitt path algebra \(L_K(E).\)
The work by the authors of this paper is related to this conjecture for finite graphs. The authors show that \(T_E\) of a finite graph \(E\) detects whether all cycles of \(E\) are disjoint (in the sense that every vertex is the base of at most one cycle). The cycles of \(E\) are disjoint if and only if the Gelfand-Kirillov dimension of \(L_K(E)\) is finite. If this dimension is finite, the authors compute it as the length of an order-ideal series of \(T_E.\) Using this result, they show that if two finite graphs are such that there is an order preserving \(\mathbb Z[x, x^{-1}]\)-module isomorphism of their Grothendieck groups, then their Leavitt path algebras have the same Gelfand-Kirillov dimension.
Reviewer: Lia Vas (Philadelphia)Simplicity of algebras associated to non-Hausdorff groupoidshttps://zbmath.org/1491.160322022-09-13T20:28:31.338867Z"Clark, Lisa Orloff"https://zbmath.org/authors/?q=ai:orloff-clark.lisa"Exel, Ruy"https://zbmath.org/authors/?q=ai:exel.ruy"Pardo, Enrique"https://zbmath.org/authors/?q=ai:pardo.enrique"Sims, Aidan"https://zbmath.org/authors/?q=ai:sims.aidan"Starling, Charles"https://zbmath.org/authors/?q=ai:starling.charlesSummary: We prove a uniqueness theorem and give a characterization of simplicity for Steinberg algebras associated to non-Hausdorff ample groupoids. We also prove a uniqueness theorem and give a characterization of simplicity for the \(C^*\)-algebra associated to non-Hausdorff étale groupoids. Then we show how our results apply in the setting of tight representations of inverse semigroups, groups acting on graphs, and self-similar actions. In particular, we show that the \(C^{*}\)-algebra and the complex Steinberg algebra of the self-similar action of the Grigorchuk group are simple but the Steinberg algebra with coefficients in \(\mathbb{Z}_2\) is not simple.A combinatorial approach to noninvolutive set-theoretic solutions of the Yang-Baxter equationhttps://zbmath.org/1491.160352022-09-13T20:28:31.338867Z"Gateva-Ivanova, Tatiana"https://zbmath.org/authors/?q=ai:gateva-ivanova.tatianaSummary: We study noninvolutive set-theoretic solutions \((X,r)\) of the Yang-Baxter equations in terms of the properties of the canonically associated braided monoid \(S(X,r)\), the quadratic Yang-Baxter algebra \(A= A(\mathbf{k}, X, r)\) over a field \(\mathbf{k} \), and its Koszul dual \(A^!\). More generally, we continue our systematic study of \textit{nondegenerate quadratic sets} \((X,r)\) \textit{and their associated algebraic objects}. Next we investigate the class of (noninvolutive) square-free solutions \((X,r)\). This contains the self distributive solutions (quandles). We make a detailed characterization in terms of various algebraic and combinatorial properties each of which shows the contrast between involutive and noninvolutive square-free solutions. We introduce and study a class of finite square-free braided sets \((X,r)\) of order \(n\geq 3\) which satisfy \textit{the minimality condition}, that is, \( \dim_{\mathbf{k}} A_2 =2n-1\). Examples are some simple racks of prime order \(p\). Finally, we discuss general extensions of solutions and introduce the notion of \textit{a generalized strong twisted union of braided sets}. We prove that if \((Z,r)\) is a nondegenerate \(2\)-cancellative braided set splitting as a generalized strong twisted union of \(r\)-invariant subsets \(Z = X\mathbin{\natural}^{\ast} Y\), then its braided monoid \(S_Z\) is a generalized strong twisted union \(S_Z= S_X\mathbin{\natural}^{\ast} S_Y\) of the braided monoids \(S_X\) and \(S_Y\). We propose a construction of a generalized strong twisted union \(Z = X\mathbin{\natural}^{\ast} Y\) of braided sets \((X,r_X)\) and \((Y,r_Y)\), where the map \(r\) has a high, explicitly prescribed order.Hopf algebra structure on free Rota-Baxter algebras by angularly decorated rooted treeshttps://zbmath.org/1491.170132022-09-13T20:28:31.338867Z"Zhang, Xigou"https://zbmath.org/authors/?q=ai:zhang.xigou"Xu, Anqi"https://zbmath.org/authors/?q=ai:xu.anqi"Guo, Li"https://zbmath.org/authors/?q=ai:guo.li\textit{T. Zhang}, \textit{X. Gao} and \textit{L. Guo} [J. Math. Phys. 57, No. 10, 101701, 16 p. (2016; Zbl 1351.81076)] proved that free Rota-Baxter algebras are bialgebras, with a coproduct making the Rota-Baxter operators 1-cocycles. This article gives an explicit, combinatorial description of this coproduct. For this, it uses Ebrahimi-Fard and Guo's description of free Rota-Baxter algebras in terms of angularly decorated rooted forests. The description of the coproduct uses a notion of subforest and is comparable to the Connes and Kreimer's coproduct on rooted forests given by admissible cuts. It is also proved that free Rota-Baxter algebras are Hopf algebras by an connectedness argument.
Reviewer: Loïc Foissy (Calais)Calculus of multilinear differential operators, operator \(L_\infty\)-algebras and \(IBL_\infty\)-algebrashttps://zbmath.org/1491.180232022-09-13T20:28:31.338867Z"Bashkirov, Denis"https://zbmath.org/authors/?q=ai:bashkirov.denis"Markl, Martin"https://zbmath.org/authors/?q=ai:markl.martinSummary: We propose an operadic framework suitable for describing algebraic structures with operations being multilinear differential operators of varying orders or, more generally, formal series of such operators. The framework is built upon the notion of a multifiltration of a linear operad generalizing the concept of a filtration of an associative algebra. We describe a particular way of constructing and analyzing multifiltrations based on a presentation of a linear operad in terms of generators and relations. In particular, that allows us to observe a special role played in this context by Lie, Lie-admissible and \texttt{Lie}\(_\infty \)-structures. As a main application, and the original motivation for the present work, we show how a certain generalization of the well-known big bracket construction of Lecomte-Roger and Kosmann-Schwarzbach encompassing the case of homotopy involutive Lie bialgebras can be obtained.Computing the dimension of ideals in group algebras, with an application to coding theoryhttps://zbmath.org/1491.200102022-09-13T20:28:31.338867Z"Elia, Michele"https://zbmath.org/authors/?q=ai:elia.michele"Gorla, Elisa"https://zbmath.org/authors/?q=ai:gorla.elisa(no abstract)On the primitive irreducible representations of finitely generated linear groups of finite rankhttps://zbmath.org/1491.200112022-09-13T20:28:31.338867Z"Tushev, A. V."https://zbmath.org/authors/?q=ai:tushev.anatolii-vThe \(\pi\)-semisimplicity of locally inverse semigroup algebrashttps://zbmath.org/1491.201352022-09-13T20:28:31.338867Z"Ji, Yingdan"https://zbmath.org/authors/?q=ai:ji.yingdanSummary: In this paper, we first characterize when a semigroup has completely 0-simple semigroup as its principal factors. Let \(R\) be a commutative ring with an identity, and let \(S\) be a locally inverse semigroup with the set of idempotents locally pseudofinite. Assume that the principal factors of \(S\) are all completely 0-simple. Then, we prove that the contracted semigroup algebra \(R_0[S]\) is \(\pi\)-semisimple if and only if the contracted semigroup algebras of all the principal factors of \(S\) are \(\pi\)-semisimple. Examples are provided to illustrate that the locally pseudofinite condition on the idempotent set of \(S\) cannot be removed. Notice that we extend the corresponding results on finite locally inverse semigroups.Unicity for representations of the Kauffman bracket skein algebrahttps://zbmath.org/1491.570142022-09-13T20:28:31.338867Z"Frohman, Charles"https://zbmath.org/authors/?q=ai:frohman.charles-d"Kania-Bartoszynska, Joanna"https://zbmath.org/authors/?q=ai:kania-bartoszynska.joanna"Lê, Thang"https://zbmath.org/authors/?q=ai:le-tu-quoc-thang.Summary: This paper resolves the unicity conjecture of \textit{F. Bonahon} and \textit{H. Wong} [Quantum Topol. 10, No. 2, 325--398 (2019; Zbl 1447.57017)] for the Kauffman bracket skein algebras of all oriented finite type surfaces at all roots of unity. The proof is a consequence of a general unicity theorem that says that the irreducible representations of a prime affine \(k\)-algebra over an algebraically closed field \(k\), that is finitely generated as a module over its center, are generically classified by their central characters. The center of the Kauffman bracket skein algebra of any orientable surface at any root of unity is characterized, and it is proved that the skein algebra is finitely generated as a module over its center. It is shown that for any orientable surface the center of the skein algebra at any root of unity is the coordinate ring of an affine algebraic variety.