Papers by Michael Natapov
In the last decade, group gradings and graded identities of finite dimensional central simple alg... more In the last decade, group gradings and graded identities of finite dimensional central simple algebras have been an active area of research. We refer the reader to Bahturin, et al [6] and [7]. There are two basic kinds of group grading, elementary and fine. It was proved by Bahturin and Zaicev [7] that any group grading of M n (C) is given by a certain composition of an elementary grading and a fine grading. In this paper we are concerned with fine gradings on M n (C) and their corresponding graded identities. Let R be a simple algebra, finite dimensional over its center k and G a finite group. We say that R is fine graded by G if R ∼ = ⊕ g∈G R g is a grading and dim k (R g) ≤ 1. Thus any component is either 0 or isomorphic to k as a k-vector space. It is easy to show that Supp(R), the subset of elements of G for which R g is not 0, is a subgroup of G. Moreover
We consider the algebra M k (C) of k-by-k matrices over the complex numbers and view it as a cros... more We consider the algebra M k (C) of k-by-k matrices over the complex numbers and view it as a crossed product with a group G of order k by imbedding G in the symmetric group S k via the regular representation and imbedding S k in M k (C) in the usual way. This induces a natural G−grading on M k (C) which we call a crossed product grading. This grading is the so called elementary grading defined by any k-tuple (g 1 , g 2 ,. .. , g k) of distinct elements g i ∈ G. We study the graded polynomial identities for M k (C) equipped with a crossed product grading. To each multilinear monomial in the free graded algebra we associate a directed labeled graph. This approach allows us to give new proofs of known results of Bahturin and Drensky on the generators of the T-ideal of identities and the Amitsur-Levitsky Theorem. Our most substantial new result is the determination of the asymptotic formula for the G-graded codimension of M k (C).
Let G be a finite group and let k be a field. We say that G is a projective basis of a k-algebra ... more Let G be a finite group and let k be a field. We say that G is a projective basis of a k-algebra A if it is isomorphic to a twisted group algebra kαG for some α ∈ H2(G, k×), where the action of G on k× is trivial. In a preceding paper by Aljadeff, Haile and the author it was shown that if a group G is a projective basis of a k-central division algebra, then G is nilpotent and every Sylow p-subgroup of G is on the short list of p-groups, denoted by Λ. In this paper we complete the classification of projective bases of division algebras by showing that every group on that list is a projective basis for a suitable division algebra.We also consider the question of uniqueness of a projective basis of a k-central division algebra. We show that basically all groups on the list Λ but one satisfy certain rigidity property.
We adjust sentiment analysis techniques to automatically detect customer emotion in on-line servi... more We adjust sentiment analysis techniques to automatically detect customer emotion in on-line service interactions of multiple business domains. Then we use the adjusted sentiment analysis tool to report insights about the dynamics of emotion in on-line service chats, using a large data set of Telecommunication customer service interactions. Our analyses show customer emotions starting out negative and evolving into positive as the interaction ends. Also, we identify a close relationship between customer emotion dynamicsduring the service interaction and the concepts of service failure and recovery. This connection manifests in customer service quality evaluationsafter the interaction ends. Our study shows the connection between customer emotion and service quality as service interactions unfold, and suggests the use of sentiment analysis tools for real-time monitoring and control of web-based service quality.
Let $G$ be a group of order $k$. We consider the algebra $M_k(\mathbb{C})$ of $k$ by $k$ matrices... more Let $G$ be a group of order $k$. We consider the algebra $M_k(\mathbb{C})$ of $k$ by $k$ matrices over the complex numbers and view it as a crossed product with respect to $G$ by embedding $G$ in the symmetric group $S_k$ via the regular representation and embedding $S_k$ in $M_k(\mathbb{C})$ in the usual way. This induces a natural $G$-grading on $M_k(\mathbb{C})$ which we call a crossed-product grading. We study the graded $*$-identities for $M_k(\mathbb{C})$ equipped with such a crossed-product grading and the transpose involution. To each multilinear monomial in the free graded algebra with involution we associate a directed labeled graph. Use of these graphs allows us to produce a set of generators for the $(T,*)$-ideal of identities. It also leads to new proofs of the results of Kostant and Rowen on the standard identities satisfied by skew matrices. Finally we determine an asymptotic formula for the $*$-graded codimension of $M_k(\mathbb{C})$.
On fine gradings on central simple algebras
Lecture Notes in Pure and Applied Mathematics, 2006
Journal of Algebra, 2012
We consider the algebra M k (C) of k-by-k matrices over the complex numbers and view it as a cros... more We consider the algebra M k (C) of k-by-k matrices over the complex numbers and view it as a crossed product with a group G of order k by imbedding G in the symmetric group S k via the regular representation and imbedding S k in M k (C) in the usual way. This induces a natural G−grading on M k (C) which we call a crossed product grading. This grading is the so called elementary grading defined by any k-tuple (g 1 , g 2 ,. .. , g k) of distinct elements g i ∈ G. We study the graded polynomial identities for M k (C) equipped with a crossed product grading. To each multilinear monomial in the free graded algebra we associate a directed labeled graph. This approach allows us to give new proofs of known results of Bahturin and Drensky on the generators of the T-ideal of identities and the Amitsur-Levitsky Theorem. Our most substantial new result is the determination of the asymptotic formula for the G-graded codimension of M k (C).
Transactions of the American Mathematical Society, 2010
In the last decade, group gradings and graded identities of finite dimensional central simple alg... more In the last decade, group gradings and graded identities of finite dimensional central simple algebras have been an active area of research. We refer the reader to Bahturin, et al [6] and [7]. There are two basic kinds of group grading, elementary and fine. It was proved by Bahturin and Zaicev [7] that any group grading of M n (C) is given by a certain composition of an elementary grading and a fine grading. In this paper we are concerned with fine gradings on M n (C) and their corresponding graded identities. Let R be a simple algebra, finite dimensional over its center k and G a finite group. We say that R is fine graded by G if R ∼ = ⊕ g∈G R g is a grading and dim k (R g) ≤ 1. Thus any component is either 0 or isomorphic to k as a k-vector space. It is easy to show that Supp(R), the subset of elements of G for which R g is not 0, is a subgroup of G. Moreover
Israel Journal of Mathematics, 2008
Let k be a field. For each finite group G and two-cocyle f in Z2(G,k x) (with trivial action), on... more Let k be a field. For each finite group G and two-cocyle f in Z2(G,k x) (with trivial action), one can form the twisted group algebra k.fG = (~eGkx~ where x~x~ = f(a,~r)x~ for all a,T E G. Our main result is a short list of p-groups containing all the p-groups G for which there is a field k and a cocycle such that the resulting twisted group algebra is a k-central division algebra. We also complete the proof (presented in all but one case in a previous paper by Aljadeff and Halle) that every k-central division algebra that is a twisted group algebra is isomorphic to a tensor product of cyclic algebras.
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Papers by Michael Natapov