scholarly journals Tridiagonal pairs of type III with height one

2019 ◽  
Vol 35 ◽  
pp. 555-582 ◽  
Author(s):  
Xue Li ◽  
Bo Hou ◽  
Suogang Gao
Keyword(s):  
Vector Space ◽  
Closed Field ◽  
Positive Dimension ◽  
Algebraically Closed Field ◽  
Type Iii ◽  
Tridiagonal Pairs ◽  
Unique Integer ◽  
Tridiagonal Pair

Let K denote an algebraically closed field with characteristic 0. Let V denote a vector space over K with finite positive dimension, and let A, A∗ denote a tridiagonal pair on V  of diameter d.  Let V0, . . . , Vd  denote a standard ordering of  the eigenspaces of A on V , and let θ0, . . . , θd denote the corresponding eigenvalues of A. It is assumed that d ≥ 3.  Let ρi  denote the dimension of Vi. The sequence ρ0, ρ1, . . . , ρd is called the shape of the tridiagonal pair. It is known that ρ0 = 1 and there  exists  a  unique  integer  h (0 ≤ h ≤ d/2)  such  that  ρi−1 < ρi  for  1 ≤ i ≤ h,  ρi−1 = ρi  for  h < i ≤ d − h,  and  ρi−1 > ρi for d − h < i ≤ d. The integer h is known as the height of the tridiagonal pair. In this paper, it is showed that the shape of a tridiagonal pair of type III with height one is either 1, 2, 2, . . ., 2, 1 or 1, 3, 3, 1.  In each case, an interesting basis is found for V and the actions of A, A∗ on this basis are described.

Additive reducts of algebraically closed valued fields

2015 ◽  
Vol 27 (2) ◽  
Author(s):  
Fares Maalouf
Keyword(s):  
Vector Space ◽  
Space Structure ◽  
Closed Field ◽  
Valued Fields ◽  
Algebraically Closed Field ◽  
Open Set

AbstractIn this paper we prove a form of the Zilber's trichotomy conjecture for reducts of algebraically closed valued fields of characteristic 0 which are expansions of the valued vector space structure. We prove first that a non-modular reduct of a nontrivially valued algebraically closed field containing the valued vector space structure defines a non-semilinear curve. Then we show that the expansion of such a reduct by a non-semilinear curve defines multiplication on a nonempty open set.

scholarly journals Some Groups of Type E7

2006 ◽  
Vol 182 ◽  
pp. 259-284 ◽  
Author(s):  
T. A. Springer
Keyword(s):  
Automorphism Group ◽  
Irreducible Representation ◽  
Vector Space ◽  
Algebraic Group ◽  
Automorphism Groups ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Identity Component ◽  
Field Of Definition ◽  
Quartic Form

AbstractAn algebraic group of type E7 over an algebraically closed field has an irreducible representation in a vector space of dimension 56 and is, in fact, the identity component of the automorphism group of a quartic form on the space. This paper describes the construction of the quartic form if the characteristic is ≠ 2, 3, taking into account a field of definition F. Certain F-forms of E7 appear in the automorphism groups of quartic forms over F, as well as forms of E6. Many of the results of the paper are known, but are perhaps not easily accessible in the literature.

scholarly journals HOW TO SHARPEN A TRIDIAGONAL PAIR

2010 ◽  
Vol 09 (04) ◽  
pp. 543-552 ◽  
Author(s):  
TATSURO ITO ◽  
PAUL TERWILLIGER
Keyword(s):  
Vector Space ◽  
Primitive Idempotent ◽  
Linear Transformations ◽  
Positive Dimension ◽  
Tridiagonal Pair

Let 𝔽 denote a field and let V denote a vector space over 𝔽 with finite positive dimension. We consider a pair of linear transformations A : V → V and A* : V → V that satisfy the following conditions: (i) each of A, A* is diagonalizable; (ii) there exists an ordering [Formula: see text] of the eigenspaces of A such that A* Vi ⊆ Vi-1 + Vi + Vi+1 for 0 ≤ i ≤ d, where V-1 = 0 and Vd+1 = 0; (iii) there exists an ordering [Formula: see text] of the eigenspaces of A* such that [Formula: see text] for 0 ≤ i ≤ δ, where [Formula: see text] and [Formula: see text]; (iv) there is no subspace W of V such that AW ⊆ W, A* W ⊆ W, W ≠ 0, W ≠ V. We call such a pair a tridiagonal pair on V. It is known that d = δ, and for 0 ≤ i ≤ d the dimensions of Vi, [Formula: see text], Vd-i, [Formula: see text] coincide. Denote this common dimension by ρi and call A, A*sharp whenever ρ0 = 1. Let T denote the 𝔽-subalgebra of End 𝔽(V) generated by A, A*. We show: (i) the center Z(T) is a field whose dimension over 𝔽 is ρ0; (ii) the field Z(T) is isomorphic to each of E0TE0, EdTEd, [Formula: see text], [Formula: see text], where Ei (resp. [Formula: see text]) is the primitive idempotent of A (resp. A*) associated with Vi (resp. [Formula: see text]); (iii) with respect to the Z(T)-vector space V the pair A, A* is a sharp tridiagonal pair.

On the Tubes of Exterior Algebras

2006 ◽  
Vol 13 (01) ◽  
pp. 149-162 ◽  
Author(s):  
Jinyun Guo ◽  
Qiuxian Wu ◽  
Qianhong Wan
Keyword(s):  
Vector Space ◽  
Exterior Algebra ◽  
Closed Field ◽  
Algebraically Closed Field

In this paper, we study the Koszul modules of complexity 1 of an exterior algebra of a vector space V over an algebraically closed field. We prove that there are ℙ (V)-orthogonal families of such modules, each of which consists of Koszul modules filtered by the cyclic ones corresponding to the same point in ℙ (V). We also prove that they are invariant under syzygy, so they are all on the homogeneous tubes.

scholarly journals TWO NON-NILPOTENT LINEAR TRANSFORMATIONS THAT SATISFY THE CUBIC q-SERRE RELATIONS

2007 ◽  
Vol 06 (03) ◽  
pp. 477-503 ◽  
Author(s):  
TATSURO ITO ◽  
PAUL TERWILLIGER
Keyword(s):  
Closed Field ◽  
Linear Transformations ◽  
Algebraically Closed Field ◽  
Root Of Unity ◽  
Finite Dimensional ◽  
Tridiagonal Pairs ◽  
Nonzero Scalar

Let 𝕂 denote an algebraically closed field with characteristic 0, and let q denote a nonzero scalar in 𝕂 that is not a root of unity. Let 𝔸q denote the unital associative 𝕂-algebra defined by generators x,y and relations [Formula: see text] where [3]q = (q3 - q-3)/(q - q-1). We classify up to isomorphism the finite-dimensional irreducible 𝔸q-modules on which neither of x,y is nilpotent. We discuss how these modules are related to tridiagonal pairs.

scholarly journals The strong simple connectedness of tame algebras with separating almost cyclic coherent Auslander–Reiten components

2021 ◽  
Author(s):  
Piotr Malicki
Keyword(s):  
Closed Field ◽  
Algebraically Closed Field ◽  
Main Application ◽  
Finite Dimensional ◽  
Simple Connectedness ◽  
Tame Algebras

AbstractWe study the strong simple connectedness of finite-dimensional tame algebras over an algebraically closed field, for which the Auslander–Reiten quiver admits a separating family of almost cyclic coherent components. As the main application we describe all analytically rigid algebras in this class.

scholarly journals On Unramified Separable Abelian p-Extensions of Function Fields I

1959 ◽  
Vol 14 ◽  
pp. 223-234 ◽  
Author(s):  
Hisasi Morikawa
Keyword(s):  
Galois Group ◽  
Function Field ◽  
Function Fields ◽  
Abelian Extension ◽  
Closed Field ◽  
Jacobian Variety ◽  
Algebraically Closed Field

Let k be an algebraically closed field of characteristic p>0. Let K/k be a function field of one variable and L/K be an unramified separable abelian extension of degree pr over K. The galois automorphisms ε1, …, εpr of L/K are naturally extended to automorphisms η(ε1), … , η(εpr) of the jacobian variety JL of L/k. If we take a svstem of p-adic coordinates on JL, we get a representation {Mp(η(εv))} of the galois group G(L/K) of L/K over p-adic integers.

scholarly journals CHARACTER CLUSTERS FOR LIE ALGEBRA MODULES OVER A FIELD OF NONZERO CHARACTERISTIC

2013 ◽  
Vol 89 (2) ◽  
pp. 234-242 ◽  
Author(s):  
DONALD W. BARNES
Keyword(s):  
Lie Algebra ◽  
Saturated Formation ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Direct Sum ◽  
Finite Dimensional ◽  
Nonzero Characteristic ◽  
Composition Factors

AbstractFor a Lie algebra $L$ over an algebraically closed field $F$ of nonzero characteristic, every finite dimensional $L$-module can be decomposed into a direct sum of submodules such that all composition factors of a summand have the same character. Using the concept of a character cluster, this result is generalised to fields which are not algebraically closed. Also, it is shown that if the soluble Lie algebra $L$ is in the saturated formation $\mathfrak{F}$ and if $V, W$ are irreducible $L$-modules with the same cluster and the $p$-operation vanishes on the centre of the $p$-envelope used, then $V, W$ are either both $\mathfrak{F}$-central or both $\mathfrak{F}$-eccentric. Clusters are used to generalise the construction of induced modules.

scholarly journals Classification of one-dimensional superattracting germs in positive characteristic

2014 ◽  
Vol 35 (7) ◽  
pp. 2242-2268 ◽  
Author(s):  
MATTEO RUGGIERO
Keyword(s):  
Positive Characteristic ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
One Dimensional ◽  
Higher Dimensional ◽  
Dimensional Version

We give a classification of superattracting germs in dimension $1$ over a complete normed algebraically closed field $\mathbb{K}$ of positive characteristic up to conjugacy. In particular, we show that formal and analytic classifications coincide for these germs. We also give a higher-dimensional version of some of these results.

scholarly journals Geometric Weil representation in characteristic two

2011 ◽  
Vol 11 (2) ◽  
pp. 221-271 ◽  
Author(s):  
Alain Genestier ◽  
Sergey Lysenko
Keyword(s):  
Weil Representation ◽  
Closed Field ◽  
Triangulated Category ◽  
Algebraically Closed Field ◽  
Witt Vectors ◽  
Suitable Sense ◽  
Characteristic Two ◽  
Greenberg Realization ◽  
Geometric Analogue

AbstractLet k be an algebraically closed field of characteristic two. Let R be the ring of Witt vectors of length two over k. We construct a group stack Ĝ over k, the metaplectic extension of the Greenberg realization of $\operatorname{\mathbb{S}p}_{2n}(R)$. We also construct a geometric analogue of the Weil representation of Ĝ, this is a triangulated category on which Ĝ acts by functors. This triangulated category and the action are geometric in a suitable sense.

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