The relations of the non-commutative coefficient algebra of the unitary group

Glöckner, P.G.; Waldenfels, Wilhelm von

doi:10.1007/bfb0083553

Cited by 16 publications

(16 citation statements)

References 3 publications

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“…In other words, the easiness condition states that the Schur–Weyl dual of

G

comes in the ‘simplest’ possible way: from partitions. As a basic example, according to an old result of Brauer , the group

G = U_{N}

is easy, with

D = P_{2}

being the category of ‘color‐matching’ pairings. Easy as well is

U_{N}^{+}

, with

D = N C_{2} \subset P_{2}

being the category of noncrossing color‐matching pairings.…”

Section: The Limit Casesmentioning

confidence: 99%

Matrix models for noncommutative algebraic manifolds

Banica

Bichon

2017

J. London Math. Soc.

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We discuss the notion of matrix model, $\pi:C(X)\to M_K(C(T))$, for algebraic submanifolds of the free complex sphere, $X\subset S^{N-1}_{\mathbb C,+}$. When $K\in\mathbb N$ is fixed there is a universal such model, which factorizes as $\pi:C(X)\to C(X^{(K)})\subset M_K(C(T))$. We have $X^{(1)}=X_{class}$ and, under a mild assumption, inclusions $X^{(1)}\subset X^{(2)}\subset X^{(3)}\subset\ldots\subset X$. Our main results concern $X^{(2)},X^{(3)},X^{(4)},\ldots$, their relation with various half-classical versions of $X$, and lead to the construction of families of higher half-liberations of the complex spheres and of the unitary groups, all having faithful matrix models.Comment: 25 pages. Final versio

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“…In other words, the easiness condition states that the Schur–Weyl dual of

G

comes in the ‘simplest’ possible way: from partitions. As a basic example, according to an old result of Brauer , the group

G = U_{N}

is easy, with

D = P_{2}

being the category of ‘color‐matching’ pairings. Easy as well is

U_{N}^{+}

, with

D = N C_{2} \subset P_{2}

being the category of noncrossing color‐matching pairings.…”

Section: The Limit Casesmentioning

confidence: 99%

Matrix models for noncommutative algebraic manifolds

Banica

Bichon

2017

J. London Math. Soc.

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“…Itô's table its value at t = 0 gives us L(b * c). Using the initial condition, one checks that the first two terms on the right hand side of (14) give rise, under the vacuum state, to the first two terms on the right hand side of (13). We are left with the computation of the coefficient of the dt-part of dj t (b * ) · dj t (c) at t = 0.…”

Section: Proof Of Theorem 47mentioning

confidence: 99%

Haar states and Lévy processes on the unitary dual group

Cébron

Ulrich

2016

Journal of Functional Analysis

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Abstract. We study states on the universal noncommutative * -algebra generated by the coefficients of a unitary matrix, or equivalently states on the unitary dual group. Its structure of dual group in the sense of Voiculescu allows to define five natural convolutions. We prove that there exists no Haar state for those convolutions. However, we prove that there exists a weaker form of absorbing state, that we call Haar trace, for the free and the tensor convolutions. We show that the free Haar trace is the limit in distribution of the blocks of a Haar unitary matrix when the dimension tends to infinity. Finally, we study a particular class of free Lévy processes on the unitary dual group which are also the limit of the blocks of random matrices on the classical unitary group when the dimension tends to infinity.

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“…Then the family (j" t ) leR defined by our recursion formula also has them. This follows by Lemma f (JT' -JST *) daf T (6,) This shows that the sequence O'5 (6) Proof. First we show that the j st are ""-algebra homomorphisms from B into B(T).…”

Section: Theorem 4-2 For Every Seu + the Integral Equation (I) Hasmentioning

confidence: 75%

“…In [13], the following theorem is proven: This *-algebra will be denoted by K d . The name 'non-commutative coefficient algebra of the unitary group' which is sometimes used iorK d comes from an algebraic isomorphism of this *-algebra with a *-algebra of operator-valued functions, see [6]. [10].…”

Section: (A) the Non-commutative Coefficient Algebra Of The Unitary Gmentioning

confidence: 99%

Quantum stochastic differential equations on *-bialgebras

Glöckner

1991

Math. Proc. Camb. Phil. Soc.

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Many examples of quantum independent stationary increment processes are solutions of quantum stochastic differential equations. We give a common characterization of these examples by a quantum stochastic differential equation on an abstract *-bialgebra. Specializing this abstract *-bialgebra and the coefficients of the equation, we obtain the equations for the Unitary Noncommutative Stochastic processes of [12], the Quantum Wiener Process [2], the Azéma martingales [11] and for other examples. The existence and uniqueness of a solution of the general equation is shown. Assuming the boundedness of this solution, we prove that it is a continuous and stationary independent increment process.

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The relations of the non-commutative coefficient algebra of the unitary group

Cited by 16 publications

References 3 publications

Matrix models for noncommutative algebraic manifolds

Matrix models for noncommutative algebraic manifolds

Haar states and Lévy processes on the unitary dual group

Quantum stochastic differential equations on *-bialgebras

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