ℤ2-Graded Gelfand–Kirillov dimension of the Grassmann algebra

2014 ◽  
Vol 24 (03) ◽  
pp. 365-374 ◽  
Author(s):  
Lucio Centrone
Keyword(s):  
Grassmann Algebra ◽  
Infinite Dimensional ◽  
Gk Dimension ◽  
Kirillov Dimension

We consider the infinite dimensional Grassmann algebra E over a field F of characteristic 0 or p, where p > 2, and we compute its ℤ2-graded Gelfand–Kirillov (GK) dimension as a ℤ2-graded PI-algebra.

A note on graded polynomial identities for tensor products by the Grassmann algebra in positive characteristic

2016 ◽  
Vol 26 (06) ◽  
pp. 1125-1140 ◽  
Author(s):  
Lucio Centrone ◽  
Viviane Ribeiro Tomaz da Silva
Keyword(s):  
Positive Characteristic ◽  
Finite Abelian Group ◽  
Grassmann Algebra ◽  
Graded Algebra ◽  
Infinite Field ◽  
Infinite Dimensional ◽  
Graded Identities ◽  
Graded Polynomial Identities ◽  
Specific Formula ◽  
Gk Dimension

Let [Formula: see text] be a finite abelian group. As a consequence of the results of Di Vincenzo and Nardozza, we have that the generators of the [Formula: see text]-ideal of [Formula: see text]-graded identities of a [Formula: see text]-graded algebra in characteristic 0 and the generators of the [Formula: see text]-ideal of [Formula: see text]-graded identities of its tensor product by the infinite-dimensional Grassmann algebra [Formula: see text] endowed with the canonical grading have pairly the same degree. In this paper, we deal with [Formula: see text]-graded identities of [Formula: see text] over an infinite field of characteristic [Formula: see text], where [Formula: see text] is [Formula: see text] endowed with a specific [Formula: see text]-grading. We find identities of degree [Formula: see text] and [Formula: see text] while the maximal degree of a generator of the [Formula: see text]-graded identities of [Formula: see text] is [Formula: see text] if [Formula: see text]. Moreover, we find a basis of the [Formula: see text]-graded identities of [Formula: see text] and also a basis of multihomogeneous polynomials for the relatively free algebra. Finally, we compute the [Formula: see text]-graded Gelfand–Kirillov (GK) dimension of [Formula: see text].

scholarly journals GK–DIMENSION OF ALGEBRAS WITH MANY GENERIC RELATIONS

2009 ◽  
Vol 51 (2) ◽  
pp. 253-256 ◽  
Author(s):  
AGATA SMOKTUNOWICZ
Keyword(s):  
Open Problems ◽  
Infinite Dimensional ◽  
Gk Dimension ◽  
Kirillov Dimension

AbstractWe prove some results on algebras, satisfying many generic relations. As an application we show that there are Golod–Shafarevich algebras which cannot be homomorphically mapped onto infinite dimensional algebras with finite Gelfand–Kirillov dimension. This answers a question of Zelmanov (Some open problems in the theory of infinite dimensional algebras, J. Korean Math. Soc. 44(5) 2007, 1185–1195).

A NOTE ON GRADED GELFAND–KIRILLOV DIMENSION OF GRADED ALGEBRAS

2011 ◽  
Vol 10 (05) ◽  
pp. 865-889 ◽  
Author(s):  
LUCIO CENTRONE
Keyword(s):  
Triangular Matrix ◽  
Grassmann Algebra ◽  
Graded Algebras ◽  
Triangular Matrices ◽  
Infinite Dimensional ◽  
Upper Triangular Matrices ◽  
Triangular Matrix Algebra ◽  
Upper Triangular Matrix ◽  
A Finite Group ◽  
Kirillov Dimension

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.

MULTIPLICITY-FREE DECOMPOSITIONS OF THE MINIMAL REPRESENTATION OF THE INDEFINITE ORTHOGONAL GROUP

2008 ◽  
Vol 19 (10) ◽  
pp. 1187-1201 ◽  
Author(s):  
MASAYASU MORIWAKI
Keyword(s):  
Unitary Representation ◽  
Orthogonal Group ◽  
Irreducible Unitary Representation ◽  
Unitary Representations ◽  
Minimal Representation ◽  
Infinite Dimensional ◽  
Irreducible Unitary Representations ◽  
Direct Product Group ◽  
Multiplicity Free ◽  
Kirillov Dimension

Kazhdan, Kostant, Binegar–Zierau and Kobayashi–Ørsted constructed a distinguished infinite-dimensional irreducible unitary representation π of the indefinite orthogonal group G = O(2p, 2q) for p, q ≥ 1 with p + q > 2, which has the smallest Gelfand–Kirillov dimension 2p + 2q - 3 among all infinite-dimensional irreducible unitary representations of G and hence is called the minimal representation. We consider, for which subgroup G′ of G, the restriction π|G′ is multiplicity-free. We prove that the restriction of π to any subgroup containing the direct product group U(p1) × U(p2) × U(q) for p1, p2 ≥ 1 with p1 + p2 = p is multiplicity-free, whereas the restriction to U(p1) × U(p2) × U(q1) × U(q2) for q1, q2 ≥ 1 with q1 + q2 = q has infinite multiplicities.

scholarly journals Branching laws for small unitary representations of GL(n, ℂ)

2014 ◽  
Vol 25 (06) ◽  
pp. 1450052
Author(s):  
Jan Möllers ◽  
Benjamin Schwarz
Keyword(s):  
Parabolic Subgroup ◽  
Principal Series ◽  
Unitary Representations ◽  
Symmetric Pair ◽  
Maximal Parabolic Subgroup ◽  
Infinite Dimensional ◽  
Branching Laws ◽  
Series Representations ◽  
Unitary Principal Series ◽  
Kirillov Dimension

The unitary principal series representations of G = GL (n, ℂ) induced from a character of the maximal parabolic subgroup P = ( GL (1, ℂ) × GL (n - 1, ℂ)) ⋉ ℂn-1 attain the minimal Gelfand–Kirillov dimension among all infinite-dimensional unitary representations of G. We find the explicit branching laws for the restriction of these representations to all reductive subgroups H of G such that (G, H) forms a symmetric pair.

scholarly journals Some projective surfaces of GK-dimension 4

2012 ◽  
Vol 148 (4) ◽  
pp. 1195-1237 ◽  
Author(s):  
D. Rogalski ◽  
Susan J. Sierra
Keyword(s):  
Global Dimension ◽  
New Techniques ◽  
Interesting Family ◽  
Gk Dimension ◽  
General Member ◽  
Kirillov Dimension ◽  
Projective Surfaces

AbstractWe construct an interesting family of connected graded domains of Gel’fand–Kirillov dimension 4, and show that the general member of this family is noetherian. The algebras we construct are Koszul and have global dimension 4. They fail to be Artin–Schelter Gorenstein, however, showing that a theorem of Zhang and Stephenson for dimension 3 algebras does not extend to dimension 4. The Auslander–Buchsbaum formula also fails to hold for these algebras. The algebras we construct are birational to ℙ2, and their existence disproves a conjecture of the first author and Stafford. The algebras can be obtained as global sections of a certain quasicoherent graded sheaf on ℙ1×ℙ1, and our key technique is to work with this sheaf. In contrast to all previously known examples of birationally commutative graded domains, the graded pieces of the sheaf fail to be ample in the sense of Van den Bergh. Our results thus require significantly new techniques.

Graph algebras and the Gelfand–Kirillov dimension

2018 ◽  
Vol 17 (05) ◽  
pp. 1850095 ◽  
Author(s):  
José M. Moreno-Fernández ◽  
Mercedes Siles Molina
Keyword(s):  
Graph Algebras ◽  
Full Generality ◽  
Path Algebras ◽  
Direct Limits ◽  
Leavitt Path Algebras ◽  
Gk Dimension ◽  
Kirillov Dimension

We study some properties of the Gelfand–Kirillov dimension in a non-necessarily unital context, in particular, its Morita invariance when the algebras have local units, and its commutativity with direct limits. We then give some applications in the context of graph algebras, which embraces, among some others, path algebras and Cohn and Leavitt path algebras. In particular, we determine the GK-dimension of these algebras in full generality, so extending the main result in A. Alahmadi, H. Alsulami, S. K. Jain and E. Zelmanov, Leavitt Path algebras of finite Gelfand–Kirillov dimension, J. Algebra Appl. 11(6) (2012) 1250225–1250231.

Graded identities for algebras with elementary gradings over an infinite field

2022 ◽  
pp. 1-21
Author(s):  
Diogo Diniz ◽  
Claudemir Fidelis ◽  
Plamen Koshlukov
Keyword(s):  
Tensor Product ◽  
Associative Algebra ◽  
Positive Characteristic ◽  
Grassmann Algebra ◽  
Alternative Proof ◽  
Infinite Field ◽  
Infinite Dimensional ◽  
Graded Identities ◽  
Graded Polynomial Identities ◽  
Field Of Positive Characteristic

Abstract Let $F$ be an infinite field of positive characteristic $p > 2$ and let $G$ be a group. In this paper, we study the graded identities satisfied by an associative algebra equipped with an elementary $G$ -grading. Let $E$ be the infinite-dimensional Grassmann algebra. For every $a$ , $b\in \mathbb {N}$ , we provide a basis for the graded polynomial identities, up to graded monomial identities, for the verbally prime algebras $M_{a,b}(E)$ , as well as their tensor products, with their elementary gradings. Moreover, we give an alternative proof of the fact that the tensor product $M_{a,b}(E)\otimes M_{r,s}(E)$ and $M_{ar+bs,as+br}(E)$ are $F$ -algebras which are not PI equivalent. Actually, we prove that the $T_{G}$ -ideal of the former algebra is contained in the $T$ -ideal of the latter. Furthermore, the inclusion is proper. Recall that it is well known that these algebras satisfy the same multilinear identities and hence in characteristic 0 they are PI equivalent.

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