Extension of unitary representations of inductive limits of finite dimensional lie groups

Kosyak, A. V.

doi:10.1016/0034-4877(88)90029-8

Cited by 4 publications

(11 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We remark also [8] that GL 0 (2∞, R) = ∩ a∈A GL GL 2 (a). Denote by A the set of all weight a satisfying the mentioned condition.…”

Section: )mentioning

confidence: 89%

“…In [8] (see details in section 3.4) we have constructed for the group GL 0 (2∞, R) = lim − →n GL(2n − 1, R) a family of the Hilbert-Lie groups GL 2 (a), a ∈ A such that a) GL 0 (2∞, R) ⊂ GL 2 (a) and GL 0 (2∞, R) is dense in GL 2 (a) for all a ∈ A, b) GL 0 (2∞, R) = ∩ a∈A GL 2 (a), c) any continuous representation of the group GL 0 (2∞, R) is in fact continuous in some stronger topology, namely in a topology of a suitable Hilbert -Lie group GL 2 (a).…”

Section: 2mentioning

confidence: 99%

“…The Hilbert-Lie group GL 2 (a) we define (see [8]) by its Hilbert-Lie algebra gl 2 (a) with composition [x, y] = xy − yx…”

Section: 2mentioning

confidence: 99%

“…Theorem 6.1[8]). Every continuous unitary representation U of the group GL 0 (2∞, R) in a Hilbert space H can be extended by continuity to a unitary representation U 2 (a) : GL 2 (a) → U(H) of some Hilbert-Lie group GL 2 (a) depending on the representation.…”

mentioning

confidence: 95%

“…[8]). The Hilbert space b 2 (a) (with an operation (x, y) → xy) is a Banach algebra if and only if the weight a = (a kn ) (k,n)∈Z 2 ,k<n satisfies the conditions (3.7) a = (a kn ) k<n , a kn ≤ Ca km a mn , k < m < n, k, m, n ∈ Z.…”

mentioning

confidence: 99%

See 4 more Smart Citations

Induced representations of infinite-dimensional groups, I

Kosyak

2014

Journal of Functional Analysis

View full text Add to dashboard Cite

Abstract. The induced representation Ind G H S of a locally compact group G is the unitary representation of the group G associated with unitary representation S : H → U (V ) of a subgroup H of the group G. Our aim is to develop the concept of induced representations for infinite-dimensional groups. The induced representations for infinite-dimensional groups in not unique, as in the case of a locally compact groups. It depends on two completionsH andG of the subgroup H and the group G, on an extensionS :H → U (V ) of the representation S : H → U (V ) and on a choice of the G-quasi-invariant measure µ on an appropriate completionX =H\G of the space H\G. As the illustration we consider the "nilpotent" group B Z 0 of infinite in both directions upper triangular matrices and the induced representation corresponding to the so-called generic orbits.

show abstract

“…We remark also [8] that GL 0 (2∞, R) = ∩ a∈A GL GL 2 (a). Denote by A the set of all weight a satisfying the mentioned condition.…”

Section: )mentioning

confidence: 89%

Section: 2mentioning

confidence: 99%

“…The Hilbert-Lie group GL 2 (a) we define (see [8]) by its Hilbert-Lie algebra gl 2 (a) with composition [x, y] = xy − yx…”

Section: 2mentioning

confidence: 99%

mentioning

confidence: 95%

mentioning

confidence: 99%

See 3 more Smart Citations

Induced representations of infinite-dimensional groups, I

Kosyak

2014

Journal of Functional Analysis

View full text Add to dashboard Cite

show abstract

q-Pascal's triangle and irreducible representations of the braid group B_3 in arbitrary dimension

Albeverio,

Kosyak

2008

Preprint

View full text Add to dashboard Cite

We construct a n+1 2 + 1 parameters family of irreducible representations of the Braid group B 3 in arbitrary dimension n ∈ N, using a q−deformation of the Pascal triangle. This construction extends in particular results by S.P. Humphries [8], who constructed representations of the braid group B 3 in arbitrary dimension using the classical Pascal triangle. E. Ferrand [7] obtained an equivalent representation of B 3 by considering two special operators in the space C n [X]. Slightly more general representations were given by I. Tuba and H. Wenzl [11]. They involve [ n+1 2 ] parameters (and also use the classical Pascal triangle). The latter authors also gave the complete classification of all simple representations of B 3 for dimension n ≤ 5. Our construction generalize all mentioned results and throws a new light on some of them. We also study the irreducibility and the equivalence of the representations.In [17] we establish the connection between the constructed representation of the braid group B 3 and the highest weight modules of U (sl 2 ) and quantum group U q (sl 2 ).

show abstract

Representations of the braid group B_n and the highest weight modules of U(sl_{n-1}) and U_q(sl_{n-1})

Kosyak

2008

Preprint

View full text Add to dashboard Cite

show abstract

Extension of unitary representations of inductive limits of finite dimensional lie groups

Cited by 4 publications

References 6 publications

Induced representations of infinite-dimensional groups, I

Induced representations of infinite-dimensional groups, I

q-Pascal's triangle and irreducible representations of the braid group B_3 in arbitrary dimension

Representations of the braid group B_n and the highest weight modules of U(sl_{n-1}) and U_q(sl_{n-1})

Contact Info

Product

Resources

About