Finite dimensional algebras, quantum groups and finite groups of Lie type

2004 ◽  
pp. 225-244
Author(s):  
Jie Du
Keyword(s):  
Quantum Groups ◽  
Finite Groups ◽  
Finite Dimensional ◽  
Groups Of Lie Type ◽  
Finite Dimensional Algebras

Finite Dimensional Algebras and Quantum Groups

2008 ◽  
Author(s):  
Bangming Deng ◽  
Jie Du ◽  
Brian Parshall ◽  
Jianpan Wang
Keyword(s):  
Quantum Groups ◽  
Finite Dimensional ◽  
Finite Dimensional Algebras

Preservation of elementary equivalence under scalar extension

1982 ◽  
Vol 47 (4) ◽  
pp. 734-738
Author(s):  
Bruce I. Rose
Keyword(s):  
Identity Element ◽  
Positive Answer ◽  
General Question ◽  
Division Algebras ◽  
Extension Field ◽  
Elementary Equivalence ◽  
Applied Model ◽  
Finite Dimensional ◽  
Positive Statements ◽  
Finite Dimensional Algebras

In this note we show that taking a scalar extension of two elementarily equivalent finite-dimensional algebras over the same field preserves elementary equivalence. The general question of whether or not tensor product preserves elementary equivalence was originally raised in [4]. In [3] Feferman relates an example of Ersov which answers the question negatively. Eklof and Olin [7] also provide a counterexample to the general question in the context of two-sorted structures. Thus the result proved below is a partial positive answer to a general question whose status has been resolved negatively. From the viewpoint of applied model theory it seems desirable to find contexts in which positive statements of preservation can be obtained. Our result does have an application; a corollary to it increases our understanding of what it means for two division algebras to be elementarily equivalent.All algebras are finite-dimensional algebras over fields. All algebras contain an identity element, but are not necessarily associative.Recall that the center of a not necessarily associative algebra A is the set of elements which commute and “associate” with all elements of A. The notion of a scalar extension is an important one in algebra. If A is an algebra over F and G is an extension field of F, then the scalar extension of A by G is the algebra A ⊗F G.

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