On the direct sum and tensor product of matrix games

1964 ◽  
Vol 11 (2) ◽  
pp. 205-215 ◽  
Author(s):  
Bertram Mond
Keyword(s):  
Tensor Product ◽  
Matrix Games ◽  
Direct Sum

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.

Elementary operators and subhomogeneous C*-algebras

2010 ◽  
Vol 54 (1) ◽  
pp. 99-111 ◽  
Author(s):  
Ilja Gogić
Keyword(s):  
Tensor Product ◽  
Finite Type ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
C Algebras ◽  
Direct Sum ◽  
Elementary Operators ◽  
Uniformly Bounded

AbstractLet A be a C*-algebra and let ΘA be the canonical contraction form the Haagerup tensor product of M(A) with itself to the space of completely bounded maps on A. In this paper we consider the following conditions on A: (a) A is a finitely generated module over the centre of M(A); (b) the image of ΘA is equal to the set E(A) of all elementary operators on A; and (c) the lengths of elementary operators on A are uniformly bounded. We show that A satisfies (a) if and only if it is a finite direct sum of unital homogeneous C*-algebras. We also show that if a separable A satisfies (b) or (c), then A is necessarily subhomogeneous and the C*-bundles corresponding to the homogeneous subquotients of A must be of finite type.

THE MULTIPLIER AND THE COVER OF DIRECT SUMS OF LIE ALGEBRAS

2012 ◽  
Vol 05 (02) ◽  
pp. 1250026 ◽  
Author(s):  
Ali Reza Salemkar ◽  
Behrouz Edalatzadeh
Keyword(s):  
Tensor Product ◽  
Lie Algebras ◽  
Schur Multiplier ◽  
Direct Sums ◽  
Schur Multipliers ◽  
Direct Sum ◽  
Group Case

In this paper, we prove that the Schur multiplier of the direct sum of two arbitrary Lie algebras is isomorphic to the direct sum of the Schur multipliers of the direct factors and the usual tensor product of the Lie algebras, which is similar to the work of Miller (1952) in the group case. Also, a cover for the direct sum of two Lie algebras in terms of given covers of them will be constructed.

scholarly journals Khatri-Rao Products for Operator Matrices Acting on the Direct Sum of Hilbert Spaces

2016 ◽  
Vol 2016 ◽  
pp. 1-7 ◽  
Author(s):  
Arnon Ploymukda ◽  
Pattrawut Chansangiam
Keyword(s):  
Tensor Product ◽  
Hilbert Spaces ◽  
Hadamard Product ◽  
Operator Matrices ◽  
Linear Map ◽  
Direct Sum ◽  
Positive Linear Map ◽  
Algebraic Properties ◽  
Product Of Matrices

We introduce the notion of Khatri-Rao product for operator matrices acting on the direct sum of Hilbert spaces. This notion generalizes the tensor product and Hadamard product of operators and the Khatri-Rao product of matrices. We investigate algebraic properties, positivity, and monotonicity of the Khatri-Rao product. Moreover, there is a unital positive linear map taking Tracy-Singh products to Khatri-Rao products via an isometry.

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 On the tensor product of polynomials over a ring

2001 ◽  
Vol 71 (3) ◽  
pp. 307-324 ◽  
Author(s):  
S. P. Glasby
Keyword(s):  
Tensor Product ◽  
Integral Domain ◽  
Basic Properties ◽  
Direct Sum ◽  
Direct Sum Decomposition

AbstractGiven polynomials a and b over an integral domain R, their tensor product (denoted a ⊗ b) is a polynomial over R of degree deg(a) deg(b) whose roots comprise all products αβ, where α is a root of a, and β is a root of b. This paper considers basic properties of ⊗ including how to factor a ⊗ b into irreducibles factors, and the direct sum decomposition of the ⊗-product of fields.

scholarly journals Decomposing Frobenius Heisenberg categories

2019 ◽  
Vol 19 (05) ◽  
pp. 2050094
Author(s):  
Raj Gandhi
Keyword(s):  
Tensor Product ◽  
Frobenius Algebra ◽  
Direct Sum

We give two alternate presentations of the Frobenius Heisenberg category, [Formula: see text], defined by Savage, when the Frobenius algebra [Formula: see text] decomposes as a direct sum of Frobenius subalgebras. In these alternate presentations, the morphism spaces of [Formula: see text] are given in terms of planar diagrams consisting of strands “colored” by integers [Formula: see text], where a strand of color [Formula: see text] carries tokens labelled by elements of [Formula: see text] In addition, we prove that when [Formula: see text] decomposes this way, the tensor product of Frobenius Heisenberg categories, [Formula: see text] is equivalent to a certain subcategory of the Karoubi envelope of [Formula: see text] that we call the partial Karoubi envelope of [Formula: see text].

scholarly journals NON-COCOMMUTATIVE C*-BIALGEBRA DEFINED AS THE DIRECT SUM OF FREE GROUP C*-ALGEBRAS

2015 ◽  
Vol 58 (1) ◽  
pp. 119-136
Author(s):  
KATSUNORI KAWAMURA
Keyword(s):  
Tensor Product ◽  
Free Group ◽  
Free Groups ◽  
Full Group ◽  
C Algebras ◽  
Direct Sum ◽  
Product Formulas

AbstractLeti ${\Bbb F}$n be the free group of rank n and let $\bigoplus C^{*}({\Bbb F}_{n})$ denote the direct sum of full group C*-algebras $C^{*}({\Bbb F}_{n})$ of ${\Bbb F}_{n} (1\leq n<\infty$). We introduce a new comultiplication Δϕ on $\bigoplus C^{*}({\Bbb F}_{n})$ such that $(\bigoplus C^{*}({\Bbb F}_{n}),\,\Delta_{\varphi})$ is a non-cocommutative C*-bialgebra. With respect to Δϕ, the tensor product π⊗ϕπ′ of any two representations π and π′ of free groups is defined. The operation ×ϕ is associative and non-commutative. We compute its tensor product formulas of several representations.

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