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The Group-Algebraic Formalism of Quantum Probability and Its Applications in Quantum Statistical Mechanics. [PDF]
Entropy (Basel)Gu Y, Wang J.
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On Matrix Algebras Over an Algebraically Closed Field [PDF]
The Annals of Mathematics, 1942 Recently a number of writers have discussed interesting developments in the theory of not completely reducible matrix sets and non-semisimple algebras.' Here we have made use of some of these concepts and methods to study matrix algebras over an algebraically closed field.
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1995
Up to now we have not considered the possibility of multiplying two vectors to obtain another vector, though we have noted that this is possible in certain cases. For example, we can multiply elements of the vector space ℳ n×n (F) over a field F. A vector space V over a field F is an algebra over F if and only if there exists a bilinear transformation (
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Up to now we have not considered the possibility of multiplying two vectors to obtain another vector, though we have noted that this is possible in certain cases. For example, we can multiply elements of the vector space ℳ n×n (F) over a field F. A vector space V over a field F is an algebra over F if and only if there exists a bilinear transformation (
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Algebras over infinite fields, revisited
Israel Journal of Mathematics, 1996Theorem 1. If \(A\) is an algebra over an infinite field \(F\), \(\text{card }F>\dim A\) and the nonzero divisors of \(A\) are invertible, then \(A\) is algebraic over \(F\). -- This is a generalization of an earlier well known result of the first author (1956). Theorem 2.
S. A. Amitsur, Lance W. Small
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1967
In this Chapter, k will be an A-field; we use all the notations introduced for such fields in earlier Chapters, such as k A , k v , r v , etc. We shall be principally concerned with a simple algebra A over k; as stipulated in Chapter IX, it is always understood that A is central, i. e. that its center is k, and that it has a finite dimension over k; by
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In this Chapter, k will be an A-field; we use all the notations introduced for such fields in earlier Chapters, such as k A , k v , r v , etc. We shall be principally concerned with a simple algebra A over k; as stipulated in Chapter IX, it is always understood that A is central, i. e. that its center is k, and that it has a finite dimension over k; by
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ALGEBRAIC VARIETIES OVER FIELDS WITH DIFFERENTIATION [PDF]
Mathematics of the USSR-Sbornik, 1969 It is known that there do not exist algebraic homomorphisms of the multiplicative group of a field into the additive group . However, if the field has a nontrivial differentiation , then the logarithmic derivative gives a homomorphism , .Ju. I. Manin observed that for abelian varieties over a field with a nontrivial differentiation it is possible to ...
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2010
18.1. Let us first recall the notion of an algebra over a field that we introduced in §11.1. By an algebra over a field F, or simply by an F -algebra, we understand an associative ring A which is also a vector space over F such that \((ax)(by)=abxy\) for \(\,a,\,b\in F\) and \(\,x,\,y\in A.\) If A has an identity element, we denote it by \(1_A,\) or ...
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18.1. Let us first recall the notion of an algebra over a field that we introduced in §11.1. By an algebra over a field F, or simply by an F -algebra, we understand an associative ring A which is also a vector space over F such that \((ax)(by)=abxy\) for \(\,a,\,b\in F\) and \(\,x,\,y\in A.\) If A has an identity element, we denote it by \(1_A,\) or ...
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1973
This chapter is a brief introduction into the structure of algebras, mostly finite dimensional, over any field k. The main contents are the Wedderburn theorems for a finite dimensional algebras A over an algebraically closed field k. If A has no nilpotent ideals ≠ 0, then A is a finite product of total matrix algebras over k. In this case, the set d (A)
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This chapter is a brief introduction into the structure of algebras, mostly finite dimensional, over any field k. The main contents are the Wedderburn theorems for a finite dimensional algebras A over an algebraically closed field k. If A has no nilpotent ideals ≠ 0, then A is a finite product of total matrix algebras over k. In this case, the set d (A)
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Some Remarks on Algebras Over an Algebraically Closed Field
The Annals of Mathematics, 1943The theory of rings with radicals is an interesting and far reaching problem of modern algebra.' In this paper we have examined some aspects of algebras which may have radicals and whose coefficient fields are algebraically closed. Some of the methods employed clearly could be used for less restricted algebras, but a full extension of the results ...
W. M. Scott, C. Nesbitt
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