scholarly journals On the exotic Grassmannian and its nilpotent variety

2016 ◽  
Vol 20 (16) ◽  
pp. 451-481 ◽  
Author(s):  
Lucas Fresse ◽  
Kyo Nishiyama
Keyword(s):  
Nilpotent Variety

Conjugacy Classes and Nilpotent Variety of a Reductive Monoid

1998 ◽  
Vol 50 (4) ◽  
pp. 829-843 ◽  
Author(s):  
Mohan S. Putcha
Keyword(s):  
Conjugacy Classes ◽  
Strong Connection ◽  
Nilpotent Variety ◽  
Nilpotent Elements ◽  
Irreducible Components ◽  
Reductive Monoid

AbstractWe continue in this paper our study of conjugacy classes of a reductive monoid M. The main theorems establish a strong connection with the Bruhat-Renner decomposition of M. We use our results to decompose the variety Mnil of nilpotent elements of M into irreducible components. We also identify a class of nilpotent elements that we call standard and prove that the number of conjugacy classes of standard nilpotent elements is always finite.

A (locally nilpotent)-by-nilpotent variety of groups

2002 ◽  
Vol 132 (2) ◽  
pp. 193-196 ◽  
Author(s):  
PAVEL SHUMYATSKY
Keyword(s):  
Nilpotent Variety ◽  
Variety Of Groups ◽  
Locally Nilpotent ◽  
Positive Integers

Given positive integers k and n, let [Xfr ] be the class of all groups G such that γk(G) is locally nilpotent and [x1, x2, …, xk]n = 1 for any x1, x2, …, xk ∈ G. It is shown that [Xfr ] is a variety.

scholarly journals Some finitely based varieties of rings

1974 ◽  
Vol 17 (2) ◽  
pp. 246-255 ◽  
Author(s):  
Trevor Evans
Keyword(s):  
Nilpotent Group ◽  
Finitely Based ◽  
Finitely Generated ◽  
Nilpotent Variety ◽  
Moufang Loops ◽  
Associative Rings ◽  
Finitely Related

The results in this paper are consequences of an attempt many years ago to extend to loops some form of the theorem of Lyndon [12] that any nilpotent group has finitely based identities. Having failed in this, we looked for other algebras for which a similar approach might work. The algebra has to belong to a variety in which finitely generated algebras are finitely related and we must be able to bound the number of variables needed in a basis. Commutative Moufang loops, because of the extensive commutator calculus available (Bruck, [4]), provide one example (Evans, [6]). Here we give two examples from rings, namely associative rings satisfying xn = x (more generally, satisfying an identity x2 · p(x) = x) and nilpotent (non-associative) rings. We are also able to extend some results of Higman [9] on product varieties and we show that for associative rings the product of a nilpotent variety and a finitely based bariety is finitely based.

scholarly journals On almost nilpotent varieties in theclass of commutative metabelian algebras

2017 ◽  
Vol 21 (3) ◽  
pp. 21-28
Author(s):  
S.P. Mishchenko ◽  
O.V. Shulezhko
Keyword(s):  
Positive Integer ◽  
Linear Algebra ◽  
Associative Algebra ◽  
Main Field ◽  
Nilpotent Algebra ◽  
Nilpotent Variety ◽  
Zero Characteristic ◽  
Fixed Set ◽  
Engel Identity ◽  
Metabelian Algebras

A well founded way of researching the linear algebra is the study of it using the identities, consequences of which is the identity of nilpotent. We know the Nagata-Higman’s theorem that says that associative algebra with nil condition of limited index over a field of zero characteristic is nilpotent. It is well known the result of E.I.Zel’manov about nilpotent algebra with Engel identity. A set of linear algebras where a fixed set of identities takes place, following A.I. Maltsev, is called a variety. The variety is called almost nilpotent if it is not nilpotent, but each its own subvariety is nilpotent. Here in the case of the main field with zero characteristic, we proved that for any positive integer m there exist commutative metabelian almost nilpotent variety of exponent is equal to m.

scholarly journals On the number of varieties of groups

1968 ◽  
Vol 8 (3) ◽  
pp. 444-446 ◽  
Author(s):  
L. G. Kovács
Keyword(s):  
Finite Basis ◽  
Minimum Condition ◽  
Locally Finite ◽  
Converse Implication ◽  
Nilpotent Variety ◽  
Locally Nilpotent ◽  
Countable Infinity ◽  
The Continuum

There are infinitely, but at most continuously, many varieties of groups; the precise cardinal is unknown. It is easy to see that if there is no infinite properly descending chain of varieties (equivalently, if the laws of every variety have a finite basis), then the number of varieties is countable infinity; the converse implication does not seem to have been proved. This note presents an argument which implies that if the locally finite or the locally nilpotent varieties fail to satisfy the minimum condition, then there are continuously many such varieties. Alternatively, one can conclude that if a locally finite or locally nilpotent variety has a finite basis for its laws but some subvariety of has none, then there are continuously many varieties between and . This points again to the interesting question: is every locally finite [locally nilpotent] variety contained in a suitable locally finite [locally nilpotent] variety which has a finite basis for its laws? (That is, must be locally finite [locally nilpotent] for some finite n?) For, if the answer were affirmative, it would follow that the number of locally finite [locally nilpotent] varieties is either countable or the cardinal of the continuum, depending exactly on the existence of finite bases for the laws of such varieties.

A finite basis theorem for product varieties of groups

1970 ◽  
Vol 2 (1) ◽  
pp. 39-44 ◽  
Author(s):  
M.S. Brooks ◽  
L.G. Kovács ◽  
M.F. Newman
Keyword(s):  
Product Variety ◽  
Finite Basis ◽  
Sufficient Condition ◽  
Nilpotent Variety ◽  
Basis Theorem ◽  
General Sufficient Condition ◽  
Special Cases

It is shown that, if U is a subvariety of the join of a nilpotent variety and a metabelian variety and if V is a variety with a finite basis for its laws, then UV also has a finite basis for its laws. The special cases U nilpotent and U metabelian have been established by Higman (1959) and Ivanjuta (1969) respectively. The proof here, which is independent of Ivanjuta's, depends on a rather general sufficient condition for a product variety to have a finite basis for its laws.

scholarly journals Kazhdan-Lusztig Basis and a Geometric Filtration of an Affine Hecke Algebra

2006 ◽  
Vol 182 ◽  
pp. 285-311 ◽  
Author(s):  
Toshiyuki Tanisaki ◽  
Nanhua Xi
Keyword(s):  
Algebraic Group ◽  
Weyl Group ◽  
Hecke Algebra ◽  
Reductive Algebraic Group ◽  
Affine Hecke Algebra ◽  
Nilpotent Variety ◽  
Affine Weyl Group

AbstractAccording to Kazhdan-Lusztig and Ginzburg, the Hecke algebra of an affine Weyl group is identified with the equivariant K-group of Steinberg’s triple variety. The K-group is equipped with a filtration indexed by closed G-stable subvarieties of the nilpotent variety, where G is the corresponding reductive algebraic group over ℂ. In this paper we will show in the case of type A that the filtration is compatible with the Kazhdan-Lusztig basis of the Hecke algebra.

scholarly journals An almost nilpotent variety of exponent 2

2013 ◽  
Vol 199 (1) ◽  
pp. 241-257 ◽  
Author(s):  
S. Mishchenko ◽  
A. Valenti
Keyword(s):  
Nilpotent Variety
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