Topic
Simple algebra
About: Simple algebra is a research topic. Over the lifetime, 189 publications have been published within this topic receiving 3427 citations.
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TL;DR: In this article, the Wedderburn structure theorems for a special class of Banach algebras (called H*-algebra) were shown to be equivalent to those of the normed ring.
Abstract: Introduction. In this paper we prove analogues of two of the Wedderburn structure theorems for a special class of Banach algebras (we prefer the term "Banach algebra" to the more common term "normed ring"). Our theorems assert, first, that any algebra of the special kind we shall consider is a direct sum of simple ones (simple no closed 2-sided ideals) and, secondly, that every simple algebra of this type is a full matrix algebra-where by a full matrix algebra we mean the set of all finite or infinite (countable or uncountable) matrices for which the sum of the squares of the absolute values of the elements converges (the matrix elements being complex numbers). We use the term Banach algebra, or simply B-algebra, for a set which is an algebra over the complex numbers (though without any assumptions about a basis, finite or otherwise) whose underlying linear space has a norm with respect to which it is a Banach space, and which satisfies the condition ||xy|| < X IyI . These assumptions are the same as those of Gelfand [III] (I ) except that we assume neither commutativity nor the existence of a unit. The special algebras which we consider (we call them H*-algebras) satisfy the additional conditions: (1) the underlying Banach space is a Hilbert space (of arbitrary dimension), (2) each element x has an "adjoint" x* in a certain rather strong sense. The part played in our theory by the fact that the underlying Banach space is a Hilbert space is two-fold. It makes direct sum decompositions easier through the possibility of taking orthogonal complements, and it opens up the possibility of using the spectral theory of operators on Hilbert space. Actually we do not use the spectral theory but instead use simplified forms of some of the technique used in that theory. It is here that the adjoint elements in our algebra are essential for their existence makes it possible, through the spectral theory technique, to construct idempotents. Our consideration of these H*-algebras arose from a consideration of the L2-algebra of a compact group. Segal [VIII] has defined the group algebra of a locally compact group to be the space L1 of integrable functions (that is, complex-valued functions, integrable with respect to the Haar measure of the group) with convolution for multiplication. In the case of a compact group the space L2 (of complex-valued functions whose square is integrable with
208 citations
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TL;DR: In this article, the rational Cherednik algebra H associated to a finite Coxeter group W in a vector space h is studied, and it is shown that the algebra H is simple and Morita equivalent to D(h)#W, the cross product of W with the algebra of polynomial differential operators on h.
Abstract: We develop representation theory of the rational Cherednik algebra H associated to a finite Coxeter group W in a vector space h. It is applied to show that, for integral values of parameter `c', the algebra H is simple and Morita equivalent to D(h)#W, the cross product of W with the algebra of polynomial differential operators on h.
We further study an algebra Q of quasi-invariant polynomials on h introduced by Chalykh, Feigin, and Veselov [CV], [FV], such that C[h]^W \subset Q \subset C[h]. We prove that the algebra D(Q) of differential operators on quasi-invariants is a simple algebra, Morita equivalent to D(h). The subalgebra D(Q)^W of W-invariant operators turns out to be isomorphic to the spherical subalgebra eHe \subset H. We also show that D(Q) is generated, as an algebra, by Q and its `Fourier dual Q*, and that D(Q) is a rank one projective (Q-Q*)-module (via multiplication-action on D(Q) on opposite sides).
126 citations
TL;DR: In this paper, it was shown that all maximal orders of a normal simple algebra over the rational number field can be obtained from any one by an inner automorphism of the algebra.
Abstract: Introduction. A maximal order M of a normal division algebra D over the rational number field may be imbeddedt in a simple fashion in a maximal order of any normal simple algebra similar to D. When the normal simple algebra has degree greater than two, its class number is unity,j and it can then be shown that all maximal orders of the algebra are obtainable from any one by an inner automorphism of the algebra. Thus it is sufficient to determine a single M of each D in order to determine all maximal orders of all normal simple algebras of degree greater than two over the rational number field. This determination was made by Hull? for the case in which the degree n of D is any odd prime, using methods similar to those of Albertff for the case n=2. The methods and results of Hull are extended here to the case in which n = 7re where 7r is any odd prime, and also to the case n = 2e >2 provided that D has odd discriminant and has the real number field as splitting field. More specifically, it will be shown with the aid of the class field theory that each algebra D considered has a suitably normalized cyclic generation, and a maximal order of D will be expressed in terms of a finite number of quantities related to this generation. There are two chief points of difference between the present case and that of prime degree. The quantity ain the normalized generation (Z, S, v-) is no longer the product of the primes ramified in D, but the product of certain powers of these primes. The exponents on these powers reduce to unity in the case of prime degree. The explicit basis given for the maximal order is similar to that for prime degree
122 citations
TL;DR: This paper addresses some algorithmic problems related to computations in finite-dimensional associative algebras over finite fields by giving polynomial time algorithms to find the building blocks, the radical and the simple direct summands of the radical-free part.
120 citations
1 Jan 1986
78 citations
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Performance Metrics
| Year | Papers |
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| 2022 | 1 |
| 2021 | 5 |
| 2020 | 10 |
| 2019 | 8 |
| 2018 | 4 |
| 2017 | 6 |