scholarly journals On Modular Representation Algebras and a Class of Matrix Algebras

1982 ◽  
Vol 33 (3) ◽  
pp. 351-355 ◽  
Author(s):  
J-C. Renaud
Keyword(s):  
Tensor Product ◽  
Cyclic Group ◽  
Prime Order ◽  
Matrix Algebra ◽  
Modular Representation ◽  
Complex Field ◽  
Matrix Algebras ◽  
Direct Sum ◽  
Standard Operations

AbstractLet G be a cyclic group of prime order p and K a field of characteristic p. The set of classes of isomorphic indecomposable (K, G)-modules forms a basis over the complex field for an algebra p (Green, 1962) with addition and multiplication being derived from direct sum and tensor product operations.Algebras n with similar properties can be defined for all n ≥ 2. Each such algebra is isomorphic to a matrix algebra Mn of n × n matrices with complex entries and standard operations. The characters of elements of n are the eigenvalues of the corresponding matrices in Mn.

Intrinsic Functions on Matrices of Real Quaternions

1963 ◽  
Vol 15 ◽  
pp. 456-466 ◽  
Author(s):  
C. G. Cullen
Keyword(s):  
Division Algebra ◽  
Matrix Algebra ◽  
Simple Algebra ◽  
Real Field ◽  
Complex Field ◽  
Direct Sum ◽  
Finite Dimensional ◽  
Simple Algebras ◽  
Real Quaternions ◽  
Algebra Over A Field

It is well known that any semi-simple algebra over the real field R, or over the complex field C, is a direct sum (unique except for order) of simple algebras, and that a finite-dimensional simple algebra over a field is a total matrix algebra over a division algebra, or equivalently, a direct product of a division algebra over and a total matrix algebra over (1). The only finite division algebras over R are R, C, and , the algebra of real quaternions, while the only finite division algebra over C is C.

Group Gradings on Matrix Algebras

2002 ◽  
Vol 45 (4) ◽  
pp. 499-508 ◽  
Author(s):  
Yu. A. Bahturin ◽  
M. V. Zaicev
Keyword(s):  
Abelian Group ◽  
Tensor Product ◽  
Characteristic Zero ◽  
Matrix Algebra ◽  
Closed Field ◽  
Induction Procedure ◽  
Full Matrix ◽  
Full Matrix Algebra ◽  
Matrix Algebras ◽  
Group Gradings

AbstractLet Φ be an algebraically closed field of characteristic zero, G a finite, not necessarily abelian, group. Given a G-grading on the full matrix algebra A = Mn(Φ), we decompose A as the tensor product of graded subalgebras A = B ⊗ C, B ≅ Mp(Φ) being a graded division algebra, while the grading of C ≅ Mq(Φ) is determined by that of the vector space Φn. Now the grading of A is recovered from those of A and B using a canonical “induction” procedure.

scholarly journals The identity of certain representation algebra decompositions

1970 ◽  
Vol 3 (1) ◽  
pp. 73-74
Author(s):  
S. B. Conlon ◽  
W. D. Wallis
Keyword(s):  
Finite Group ◽  
Tensor Product ◽  
Linear Algebra ◽  
Residue Field ◽  
Complex Field ◽  
Direct Sums ◽  
Isomorphism Classes ◽  
Direct Sum ◽  
A Finite Group ◽  
Representation Algebra

Let G be a finite group and F a complete local noetherian commutative ring with residue field of characteristic p # 0. Let A(G) denote the representation algebra of G with respect to F. This is a linear algebra over the complex field whose basis elements are the isomorphism-classes of indecomposable finitely generated FG-representation modules, with addition and multiplication induced by direct sum and tensor product respectively. The two authors have separately found decompositions of A(G) as direct sums of subalgebras. In this note we show that the decompositions in one case have a common refinement given in the other's paper.

scholarly journals Primary decompositions of unital locally matrix algebras

2020 ◽  
Vol 10 (01) ◽  
pp. 2050006
Author(s):  
Oksana Bezushchak ◽  
Bogdana Oliynyk
Keyword(s):  
Tensor Product ◽  
Matrix Algebra ◽  
Primary Decomposition ◽  
Locally Finite ◽  
Matrix Algebras ◽  
Dimensional Matrix ◽  
Finite Dimensional ◽  
Number Formula

We construct a unital locally matrix algebra of uncountable dimension that (1) does not admit a primary decomposition, (2) has an infinite locally finite Steinitz number. It gives negative answers to questions from [V. M. Kurochkin, On the theory of locally simple and locally normal algebras, Mat. Sb., Nov. Ser. 22(64)(3) (1948) 443–454; O. Bezushchak and B. Oliynyk, Unital locally matrix algebras and Steinitz numbers, J. Algebra Appl. (2020), online ready]. We also show that for an arbitrary infinite Steinitz number [Formula: see text] there exists a unital locally matrix algebra [Formula: see text] having the Steinitz number [Formula: see text] and not isomorphic to a tensor product of finite-dimensional matrix algebras.

scholarly journals Recurrence relations in a modular representation algebra

1982 ◽  
Vol 26 (2) ◽  
pp. 215-219 ◽  
Author(s):  
J-C. Renaud
Keyword(s):  
Cyclic Group ◽  
Prime Order ◽  
Recurrence Relations ◽  
Binomial Coefficient ◽  
Modular Representation ◽  
Exterior Powers ◽  
Representation Algebra

In 1978 Almkvist and Fossum examined the decomposition of the exterior powers of basis modules in the modular representation algebra of a cyclic group of prime order. In particular they developed an isomorphism between these exterior powers and terms of binomial coefficient type in the algebra.We derive several recurrence relations for these terms.

scholarly journals Explicit isomorphisms of real Clifford algebras

2006 ◽  
Vol 2006 ◽  
pp. 1-13
Author(s):  
N. Değırmencı ◽  
Ş. Karapazar
Keyword(s):  
Quadratic Form ◽  
Clifford Algebra ◽  
Matrix Algebra ◽  
Clifford Algebras ◽  
Explicit Construction ◽  
The Other ◽  
Matrix Algebras ◽  
Direct Sum ◽  
Other Hand ◽  
Nondegenerate Quadratic Form

It is well known that the Clifford algebraClp,qassociated to a nondegenerate quadratic form onℝn (n=p+q)is isomorphic to a matrix algebraK(m)or direct sumK(m)⊕K(m)of matrix algebras, whereK=ℝ,ℂ,ℍ. On the other hand, there are no explicit expressions for these isomorphisms in literature. In this work, we give a method for the explicit construction of these isomorphisms.

scholarly journals The characters and structure of a class of modular representation algebras of cyclic p-groups

1978 ◽  
Vol 26 (4) ◽  
pp. 410-418 ◽  
Author(s):  
J. C. Renaud
Keyword(s):  
Local Ring ◽  
Cyclic Group ◽  
Modular Representation ◽  
Alternative Proof ◽  
Local Rings ◽  
Prime Field ◽  
Direct Sum ◽  
Primitive Idempotents ◽  
C 30 ◽  
Representation Algebra

AbstractLet p,m be the modular representation algebra of the cyclic group of order pm over the prime field Zp. The characters of p, m are derived. For p = 2, this provides an alternative proof of a result due to Carlson (1975), tha 2,m is a local ring. It is shown that for p>2, p, m is a direct sum of 2m local rings. Their dimensions and primitive idempotents are derived.Subject classification (Amer. Math. Soc. (MOS) 1970): 20 C 20, 12 C 05, 12 C 30, 33 A 65.

Some computations in the modular representation ring of a finite group

1971 ◽  
Vol 69 (1) ◽  
pp. 163-166 ◽  
Author(s):  
John Santa Pietro
Keyword(s):  
Finite Group ◽  
Tensor Product ◽  
Modular Representation ◽  
Multiplicative Structure ◽  
Characteristic P ◽  
Isomorphism Classes ◽  
Direct Sum ◽  
Representation Ring ◽  
Ring Isomorphism ◽  
A Finite Group

Let p be an odd prime and G = HB be a semi-direct product where H is a cyclic, p-Sylow subgroup and B is finite Abelian. If K is a field of characteristic p the isomorphism classes of KG-modules relative to direct sum and tensor product generate a ring a(G) called the representation ring of G over K. If K is algebraically closed it is shown in (4) that there is a ring isomorphism a(G) ≃ a(HB2)⊗a(B1) where B1 is the kernel of the action of B on H and B2 = B/B1.> 2, Aut (H) is cyclic thus HB2 is metacyclic. The study of the multiplicative structure of a(G) is thus reduced to that of the known rings a(B1) and a(HB2) (see (3)).

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