An application to representation theory: Symmetrization of inner tensor products of representations

A deep result in the theory of W*-tensor products, the Commutation theorem, states that if M and N are W*-algebras faithfully represented as von Neumann algebras on the Hilbert spaces H and K, respectively, then the commutant in L(H ⊗ K) of the W*-tensor product of M and N coincides with the W*-tensor product of M′ and N′. Although special cases of this theorem were established successively by Misonou (2) and Sakai (3), the validity of the general result remained conjectural until the advent of the Tomita-Takesaki theory of Modular Hilbert algebras (6). As formulated, the Commutation theorem is a spatial result; that is, the W*-algebras in its statement are taken to act on specific Hilbert spaces. Not surprisingly, therefore, known proofs rely heavily on techniques of representation theory.

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THE CLASSICAL LIMIT OF REPRESENTATION THEORY OF THE QUANTUM PLANE

International Journal of Mathematics ◽

10.1142/s0129167x13500316 ◽

2013 ◽

Vol 24 (04) ◽

pp. 1350031 ◽

Cited By ~ 1

Author(s):

IVAN C. H. IP

Keyword(s):

Fourier Transform ◽

Representation Theory ◽

Unitary Representation ◽

Classical Limit ◽

Mellin Transform ◽

Tensor Products ◽

Quantum Plane ◽

The Fourier Transform

We showed that there is a complete analogue of a representation of the quantum plane [Formula: see text] where |q| = 1, with the classical ax+b group. We showed that the Fourier transform of the representation of [Formula: see text] on [Formula: see text] has a limit (in the dual corepresentation) toward the Mellin transform of the unitary representation of the ax+b group, and furthermore the intertwiners of the tensor products representation has a limit toward the intertwiners of the Mellin transform of the classical ax+b representation. We also wrote explicitly the multiplicative unitary defining the quantum ax+b semigroup and showed that it defines the corepresentation that is dual to the representation of [Formula: see text] above, and also correspond precisely to the classical family of unitary representation of the ax+b group.

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Representation theory for tensor products of Banach algebras

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100058916 ◽

1981 ◽

Vol 90 (3) ◽

pp. 445-463 ◽

Cited By ~ 2

Author(s):

T. K. Carne

Keyword(s):

Representation Theory ◽

Tensor Product ◽

Banach Algebra ◽

Banach Algebras ◽

Tensor Products ◽

The Subject ◽

Algebraic Tensor Product ◽

Natural Way

The algebraic tensor product A1⊗A2 of two Banach algebras is an algebra in a natural way. There are certain norms α on this tensor product for which the multiplication is continuous so that the completion, A1αA2, is a Banach algebra. The representation theory of such tensor products is the subject of this paper. It will be shown that, under certain simple conditions, the tensor product of two semi-simple Banach algebras is semi-simple although, without these conditions, the result fails.

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A Random Walk on the Indecomposable Summands of Tensor Products of Modular Representations of SL2 $\left ({\mathbb {F}_p}\right )$

Algebras and Representation Theory ◽

10.1007/s10468-021-10034-0 ◽

2021 ◽

Author(s):

Eoghan McDowell

Keyword(s):

Random Walk ◽

Markov Chains ◽

Representation Theory ◽

Tensor Product ◽

Elementary Proof ◽

Simple Module ◽

Tensor Products ◽

Connected Components ◽

Modular Representations ◽

Stationary Distributions

AbstractIn this paper we introduce a novel family of Markov chains on the simple representations of SL2$\left ({\mathbb {F}_p}\right )$ F p in defining characteristic, defined by tensoring with a fixed simple module and choosing an indecomposable non-projective summand. We show these chains are reversible and find their connected components and their stationary distributions. We draw connections between the properties of the chain and the representation theory of SL2$\left ({\mathbb {F}_p}\right )$ F p , emphasising symmetries of the tensor product. We also provide an elementary proof of the decomposition of tensor products of simple SL2$\left ({\mathbb {F}_p}\right )$ F p -representations.

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