1,720,979 research outputs found
Z(2)-Graded Gelfand-Kirillov dimension of the Grassmann algebra
No full textFundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)We consider the infinite dimensional Grassmann algebra E over a field F of characteristic 0 or p, where p > 2, and we compute its Z(2)-graded Gelfand-Kirillov (GK) dimension as a Z(2)-graded PI-algebra.243365374Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)FAPESP [2013/06752-4
Action of pontryagin dual of semilattices on graded algebras
No full textLet A be a ℂ-algebra and G be a finite abelian group. Then a G-graded algebra is merely a G-algebra and viceversa because of the fact that G and its group of characters are isomorphic. This fact is no longer true if we substitute G with infinite or non-abelian groups. In this paper we try to obtain similar results for a special class of abelian monoids, i.e., the bounded semilattices. Moreover, if S is such a monoid, we are going to investigate the role of S and its Pontryagin dual over the algebra A, in the case A is S-graded. © 2014 Copyright Taylor & Francis Group, LLC.Let A be a ℂ-algebra and G be a finite abelian group. Then a G-graded algebra is merely a G-algebra and viceversa because of the fact that G and its group of characters are isomorphic. This fact is no longer true if we substitute G with infinite or non-ab42834913506sem informaçãosem informaçãoBanakh, T., Hryniv, O., Pontryagin duality between compact and discrete abelian inverse monoids, , http://arxiv.org/pdf/1008.1175v2.pdfBlattner, R.J., Montgomery, S., A duality theorem for Hopf module algebras (1985) J. Algebra, 95, pp. 153-172Cohen, M., Montgomery, S., Group graded rings, smash products and group actions (1984) Trans. Amer. Math. Soc., 282, pp. 237-258Grosshans, F.D., Rota, G.-C., Stein, J.A., On the foundations of combinatorial theory IX: Combinatorial methods in invariant theory (1976) Stud. Appl. Math., 53, pp. 185-216La Scala, R., Levandovsky, V., Skew polynomial rings, Gröbner bases and the letterplace embedding of the free associative algebra (2012) J. Symbolic Comput., , To appearMontgomery, S., Hopf algebras and their actions on rings (1993) CBMS Regional Conference Series in Mathematics, 82. , Providence, RI: AMSMorris, S., (1977) Pontryagin duality and the structure of locally compact abelian groups, , Cambridge-New York-Melbourne: Cambridge Univ. PressNeeb, K.-H., Toric varieties and algebraic monoids (1992) Seminar Sophus Lie, 2, pp. 159-187Sweedler, M.E., (1969) Hopf Algebras, , New York, BenjaminTimmermann, T., An Invitation to Quantum Groups and Duality: From Hopf Algebras to Multiplicative Unitaries and Beyond (2008) EMS Textbooks in Mathematics. European Mathematica/Society (EMS), Zürich, p. 407. , xx+ ISBN: 978-3-03719-043-
The -graded identities of the Grassmann Algebra
Full text linksummary:Let be a finite abelian group with identity element and be an infinite dimensional -homogeneous vector space over a field of characteristic . Let be the Grassmann algebra generated by . It follows that is a -graded algebra. Let be odd, then we prove that in order to describe any ideal of -graded identities of it is sufficient to deal with -grading, where , and if . In the same spirit of the case odd, if is even it is sufficient to study only those -gradings such that , where , and all the other components are finite dimensional. We also compute graded cocharacters and codimensions of in the case and if
A note on cocharacter sequence of Jordan upper triangular matrix algebra
No full textLet UJn(F) be the Jordan algebra of n × n upper triangular matrices over a field F of characteristic zero. This paper is devoted to the study of polynomial identities satisfied by UJ2(F) and UJ3(F). In particular, the goal is twofold. On one hand, we complete the description of G-graded polynomial identities of UJ2(F), where G is a finite abelian group. On the other hand, we compute the Gelfand–Kirillov dimension of the relatively free algebra of UJ2(F) and we give a bound for the Gelfand–Kirillov dimension of the relatively free algebra of UJ3(F)
The graded Gelfand--Kirillov dimension of verbally prime algebras
No full textLet E be the infinite-dimensional Grassmann algebra over a field F of characteristic 0. In this article, we consider the verbally prime algebras M n(F), M n(E) and M a,b(E) endowed with their gradings induced by that of Vasilovsky, and we compute their graded Gelfand-Kirillov dimensions. © 2011 Copyright Taylor and Francis Group, LLC
Ordinary and ℤ<sub>2</sub>-Graded Cocharacters of<i>UT</i><sub>2</sub>(<i>E</i>)
No full textLet E be the infinite dimensional Grassmann algebra over a field F of characteristic 0. In this article we consider the algebra R of 2 × 2 matrices with entries in E and its subalgebra G, which is one of the minimal algebras of polynominal identity (PI) exponent 2. We compute firstly the Hilbert series of G and, as a consequence, we compute its cocharacter sequence. Then we find the Hilbert series of R, using the tool of proper Hilbert series, and we compute its cocharacter sequence. Finally we describe explicitely the Z2-graded cocharacters of R. © Taylor & Francis Group, LLC
The GK dimension of relatively free algebras of PI-algebras
No full textWe prove a strict relation between the Gelfand-Kirillov (GK) dimension of the relatively free (graded) algebra of a PI-algebra and its (graded) exponent. As a consequence we show a Bahturin-Zaicev type result relating the GK dimension of the relatively free algebra of a graded PI-algebra and the one of its neutral part. We also get that the growth of the relatively free graded algebra of a matrix algebra is maximal when the grading is fine. Finally we compute the graded GK dimension of the matrix algebra with any grading and the graded GK dimension of any verbally prime algebra endowed with an elementary grading223729772996CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICO - CNPQFUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULO - FAPESP303971/2015-02015/08961-5, 2018/02108-
A NOTE ON GRADED GELFAND–KIRILLOV DIMENSION OF GRADED ALGEBRAS
No full text In this paper, we consider associative P.I. algebras over a field F of characteristic 0, graded by a finite group G. More precisely, we define the G-graded Gelfand–Kirillov dimension of a G-graded P.I. algebra. We find a basis of the relatively free graded algebras of the upper triangular matrices UTn(F) and UTn(E), with entries in F and in the infinite-dimensional Grassmann algebra, respectively. As a consequence, we compute their graded Gelfand–Kirillov dimension with respect to the natural gradings defined over these algebras. We obtain similar results for the upper triangular matrix algebra UTa, b(E) = UTa+b(E)∩Ma, b(E) with respect to its natural ℤa+b × ℤ2-grading. Finally, we compute the ℤn-graded Gelfand–Kirillov dimension of Mn(F) in some particular cases and with different methods. </jats:p
Y-proper graded cocharacters and codimensions of upper triangular matrices of size 2, 3, 4
No full textLet F be a field of characteristic 0. We consider the upper triangular matrices with entries in F of size 2, 3 and 4 endowed with the grading induced by that of Vasilovsky. In this paper we give explicit computation for the multiplicities of the Y-proper graded cocharacters and codimensions of these algebras. © 2012 Elsevier Inc
The Nowicki conjecture for relatively free algebras
No full textA linear locally nilpotent derivation of the polynomial algebra K[Xm] in m variables over a field K of characteristic 0 is called a Weitzenböck derivation. It is well known from the classical theorem of Weitzenböck that the algebra of constants K[Xm]δ of a Weitzenböck derivation δ is finitely generated. Assume that δ acts on the polynomial algebra K[X2d] in 2d variables as follows: δ(x2i)=x2i−1, δ(x2i−1)=0, i=1,...,d. The Nowicki conjecture states that the algebra K[X2d]δ is generated by x1,x3....,x2d−1, and x2i−1x2j−x2ix2j−1, 1≤
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