scholarly journals Linear Maps That Act Tridiagonally with Respect to Eigenbases of the Equitable Generators of Uq(sl2)

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1546
Author(s):  
Hasan Alnajjar ◽  
Brian Curtin
Keyword(s):  
Nonnegative Integer ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Root Of Unity ◽  
Finite Dimensional ◽  
Linear Maps ◽  
Nonzero Scalar

Let F denote an algebraically closed field; let q be a nonzero scalar in F such that q is not a root of unity; let d be a nonnegative integer; and let X, Y, Z be the equitable generators of Uq(sl2) over F. Let V denote a finite-dimensional irreducible Uq(sl2)-module with dimension d+1, and let R denote the set of all linear maps from V to itself that act tridiagonally on the standard ordering of the eigenbases for each of X, Y, and Z. We show that R has dimension at most seven. Indeed, we show that the actions of 1, X, Y, Z, XY, YZ, and ZX on V give a basis for R when d≥3.

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 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 Involutions on finite-dimensional algebras over real closed fields

2004 ◽  
Vol 77 (1) ◽  
pp. 123-128 ◽  
Author(s):  
W. D. Munn
Keyword(s):  
Characteristic Zero ◽  
Real Closed Field ◽  
Closed Field ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Algebraically Closed Field ◽  
Real Closed Fields ◽  
Finite Dimensional ◽  
Closed Fields ◽  
Finite Dimensional Algebras

AbstractIt is shown that the following conditions on a finite-dimensional algebra A over a real closed field or an algebraically closed field of characteristic zero are equivalent: (i) A admits a special involution, in the sense of Easdown and Munn, (ii) A admits a proper involution, (iii) A is semisimple.

Second Cohomology of the Modular Lie Superalgebra of Cartan Type K

2009 ◽  
Vol 16 (02) ◽  
pp. 309-324 ◽  
Author(s):  
Wenjuan Xie ◽  
Yongzheng Zhang
Keyword(s):  
Cohomology Group ◽  
Lie Superalgebra ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Cartan Type ◽  
Modular Lie Superalgebra ◽  
Finite Dimensional ◽  
Type K

Let 𝔽 be an algebraically closed field and char 𝔽 = p > 3. In this paper, we determine the second cohomology group of the finite-dimensional Contact superalgebra K(m,n,t).

HOPF ALGEBRAS OF DIMENSION 30

2010 ◽  
Vol 09 (01) ◽  
pp. 11-15 ◽  
Author(s):  
DAIJIRO FUKUDA
Keyword(s):  
Hopf Algebra ◽  
Group Algebra ◽  
Hopf Algebras ◽  
Characteristic Zero ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Finite Dimensional

This paper contributes to the classification of finite dimensional Hopf algebras. It is shown that every Hopf algebra of dimension 30 over an algebraically closed field of characteristic zero is semisimple and thus isomorphic to a group algebra or the dual of a group algebra.

scholarly journals Multiplicities in the Tensor Product of Finite-Dimensional Representations of Discrete Groups

1978 ◽  
Vol 21 (1) ◽  
pp. 17-19
Author(s):  
Dragomir Ž. Djoković
Keyword(s):  
Tensor Product ◽  
Hilbert Spaces ◽  
Unitary Representations ◽  
Vector Spaces ◽  
Closed Field ◽  
Discrete Groups ◽  
Algebraically Closed Field ◽  
Finite Dimensional ◽  
Irreducible Unitary Representations

Let G be a group and ρ and σ two irreducible unitary representations of G in complex Hilbert spaces and assume that dimp ρ= n < ∞. D. Poguntke [2] proved that is a sum of at most n2 irreducible subrepresentations. The case when dim a is also finite he attributed to R. Howe.We shall prove analogous results for arbitrary finite-dimensional representations, not necessarily unitary. Thus let F be an algebraically closed field of characteristic 0. We shall use the language of modules and we postulate that allour modules are finite-dimensional as F-vector spaces. The field F itself will be considered as a trivial G-module.

scholarly journals On the Cohomology and Extensions of n-ary Multiplicative Hom-Nambu-Lie Superalgebras

2020 ◽  
Vol 2020 ◽  
pp. 1-17
Author(s):  
Lijun Tian ◽  
Baoling Guan ◽  
Yao Ma
Keyword(s):  
Lie Algebras ◽  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Finite Dimensional

In this paper, we discuss the representations of n-ary multiplicative Hom-Nambu-Lie superalgebras as a generalization of the notion of representations for n-ary multiplicative Hom-Nambu-Lie algebras. We also give the cohomology of an n-ary multiplicative Hom-Nambu-Lie superalgebra and obtain a relation between extensions of an n-ary multiplicative Hom-Nambu-Lie superalgebra b by an abelian one a and Z1b,a0¯. We also introduce the notion of T∗-extensions of n-ary multiplicative Hom-Nambu-Lie superalgebras and prove that every finite-dimensional nilpotent metric n-ary multiplicative Hom-Nambu-Lie superalgebra over an algebraically closed field of characteristic not 2 in the case α is a surjection is isometric to a suitable T∗-extension.

On the Existence of the Graded Exponent for Finite Dimensional ℤp-graded Algebras

2012 ◽  
Vol 55 (2) ◽  
pp. 271-284 ◽  
Author(s):  
Onofrio M. Di Vincenzo ◽  
Vincenzo Nardozza
Keyword(s):  
Characteristic Zero ◽  
Prime Order ◽  
Closed Field ◽  
Graded Algebras ◽  
Algebraically Closed Field ◽  
Finite Dimensional

AbstractLet F be an algebraically closed field of characteristic zero, and let A be an associative unitary F-algebra graded by a group of prime order. We prove that if A is finite dimensional then the graded exponent of A exists and is an integer.

scholarly journals CONFORMAL $\mathcal{s}$$\mathcal{l}$2 ENVELOPING ALGEBRAS AS AMBISKEW POLYNOMIAL RINGS

2002 ◽  
Vol 45 (1) ◽  
pp. 91-115 ◽  
Author(s):  
Martin O’Neill
Keyword(s):  
Characteristic Zero ◽  
Subject Classification ◽  
Polynomial Rings ◽  
Irreducible Representations ◽  
Prime Ideals ◽  
Closed Field ◽  
Enveloping Algebras ◽  
Root Of Unity ◽  
Finite Dimensional ◽  
Parameter Deformation

AbstractWe study a three parameter deformation $\mathcal{U}_{abc}$ of $\mathcal{U}(\mathfrak{sl}_2)$ introduced by Le Bruyn in 1995. Working over an arbitrary algebraically closed field of characteristic zero, we determine the centres, the finite-dimensional irreducible representations, and, when the parameter $a$ is not a non-trivial root of unity, the prime ideals of those $\mathcal{U}_{abc}$, with $ac\neq0$, which are conformal as ambiskew polynomial rings.AMS 2000 Mathematics subject classification: Primary 16W35; 17B37. Secondary 16S36; 16S80

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