scholarly journals Gelfand-Kirillov dimension of commutative subalgebras of simple infinite dimensional algebras and their quotient division algebras

2005 ◽  
Vol 2005 (582) ◽  
pp. 61-85 ◽  
Author(s):  
V. Bavula
Keyword(s):  
Division Algebras ◽  
Infinite Dimensional ◽  
Commutative Subalgebras ◽  
Kirillov Dimension

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 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).

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.

scholarly journals Nil algebras with restricted growth

2012 ◽  
Vol 55 (2) ◽  
pp. 461-475 ◽  
Author(s):  
T. H. Lenagan ◽  
Agata Smoktunowicz ◽  
Alexander A. Young
Keyword(s):  
Finitely Generated ◽  
Infinite Dimensional ◽  
Kirillov Dimension

AbstractIt is shown that over an arbitrary countable field there exists a finitely generated algebra that is nil, infinite dimensional and has Gelfand–Kirillov dimension at most 3.

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.

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