Non-commutative holomorphic functions in elements of a Lie algebra and the absolute basis problem

Dosi, Anar Adıgüzel oğlu

doi:10.1070/im2009v073n06abeh002476

Cited by 10 publications

(10 citation statements)

References 11 publications

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“…e has normal growth. A more complicated example is delivered by the following proposition proved in [10]. Proposition 6.1 Let g be a nilpotent Lie algebra of nilpotence power c, let e be a triangular basis associated with the Lie generators e 1 ;:::;e d 1 , and let…”

Section: The Growth Of a Nilpotent Lie Algebramentioning

confidence: 99%

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Local left invertibility for operator tuples and noncommutative localizations

Dosiev¹

2008

J K-Theor

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Section: The Growth Of a Nilpotent Lie Algebramentioning

confidence: 99%

“…The existence of a triangular basis whose growth is normal involves many subtle properties of the algebra O g of all entire functions in elements of a nilpotent Lie algebra, which we are summarized in the next assertion (see [5], [10]). …”

Section: The Growth Of a Nilpotent Lie Algebramentioning

confidence: 99%

Local left invertibility for operator tuples and noncommutative localizations

Dosiev¹

2008

J K-Theor

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“…Moreover, each norm · ρ is submultiplicative. We refer to [44,46,106] for explicit descriptions of Arens-Michael envelopes of some other finitely generated algebras, including quantum tori, quantum Weyl algebras, the algebra of quantum 2 × 2-matrices, and universal enveloping algebras. Further results on Arens-Michael envelopes can be found in [42,43,45,104,105].…”

Section: Preliminaries and Notationmentioning

confidence: 99%

Holomorphic functions on the quantum polydisk and on the quantum ball

Pirkovskii

2019

J. Noncommut. Geom.

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We introduce and study noncommutative (or "quantized") versions of the algebras of holomorphic functions on the polydisk and on the ball in C n . Specifically, for each q ∈ C × = C \ {0} we construct Fréchet algebras O q (D n ) and O q (B n ) such that for q = 1 they are isomorphic to the algebras of holomorphic functions on the open polydisk D n and on the open ball B n , respectively. We show that O q (D n ) and O q (B n ) are not isomorphic provided that |q| = 1 and n ≥ 2. This result can be interpreted as a q-analog of Poincaré's theorem, which asserts that D n and B n are not biholomorphically equivalent unless n = 1. In contrast, O q (D n ) and O q (B n ) are shown to be isomorphic for |q| = 1. Next we prove that O q (D n ) is isomorphic to a quotient of J. L. Taylor's "free polydisk algebra" (1972). This enables us to construct a Fréchet O(C × )-algebra O def (D n ) whose "fiber" over each q ∈ C × is isomorphic to O q (D n ). Replacing the free polydisk algebra by G. Popescu's "free ball algebra" (2006)respectively. Finally, we study relations between our deformations and formal deformations of O(D n ) and O(B n ). ]. Some parallels between Taylor's theory and the "operator algebraic" noncommutative complex analysis are discussed in [68, 94, 95]. A related approach going back to Taylor's notion of an Arens-Michael envelope [160] was developed by A. A. Dosi (Dosiev) [42-46] and the author [104-106, 109, 110]. A more algebraic view of noncommutative complex geometry is based on A. Connes' fundamental ideas [32,33]. The notion of connection introduced by Connes in [32] was used by A. Schwarz [146] to define complex structures on noncommutative tori. This line of research was further developed by M. Dieng and A. Schwarz [41] and by A. Polishchuk and A. Schwarz [112-114]. A closely related point of view was adopted by J. Rosenberg [145], M. Khalkhali, G. Landi, W. D. van Suijlekom, and A. Moatadelro [71-73], E. Beggs and S. P. Smith [13], R.Ó Buachalla [ [98][99][100]. Another approach was also initiated by Connes [33, Section VI.2], who interpreted complex structures on a compact 2-dimensional manifold M in terms of positive Hochschild cocycles on the algebra of smooth functions on M. Motivated by this, he suggested to use positivity in Hochschild cohomology as a starting point for developing noncommutative complex geometry. This point of view was developed by M. Khalkhali, G. Landi, W. D. van Suijlekom, and A. Moatadelro [loc. cit.], who found relations between complex structures on noncommutative projective spaces and twisted positive Hochschild cocycles on suitable quantized function algebras. A common feature of the above papers is that almost all concrete "noncommutative complex manifolds" that appear therein are compact. Thus section spaces of "noncommutative holomorphic bundles" over such "manifolds" are finite-dimensional, and so no functional analysis is needed for their study. We refer to [71] and [13] for a detailed discussion of this side of noncommutative complex geometry.An enormous contributio...

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“…Note that, if |q ij | = 1 for at least one pair of indices i, j, then the Arens-Michael envelope of O reg q ((C × ) n ) is trivial [38,Proposition 5.23]. We refer to [9,11,38] for explicit descriptions of Arens-Michael envelopes of some other finitely generated algebras, including quantum Weyl algebras, the algebra of quantum 2 × 2-matrices, and universal enveloping algebras. Note that, in many concrete cases, the Arens-Michael envelope of a "deformed polynomial algebra" can be interpreted as a "deformed power series algebra", similarly to (62) and (63).…”

Section: Define a Weight Function Wmentioning

confidence: 99%

“…In Subsection 7.1, we observe that the Arens-Michael envelope of any finitely generated algebra is holomorphically finitely generated. As a consequence, the algebras O q (C n ) and O q ((C × ) n ) of holomorphic functions on the quantum affine space and on the quantum torus [38,39], the algebra of entire functions on a basis of a Lie algebra [9,11], and some other algebras studied in [38] are holomorphically finitely generated. Other examples are specializations of the constructions discussed in Sections 4-6.…”

Section: Introductionmentioning

confidence: 99%

Holomorphically finitely generated algebras

Pirkovskii¹

2015

J. Noncommut. Geom.

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We introduce and study holomorphically finitely generated (HFG) Fréchet algebras, which are analytic counterparts of affine (i.e., finitely generated) C-algebras. Using a theorem of O. Forster, we prove that the category of commutative HFG algebras is anti-equivalent to the category of Stein spaces of finite embedding dimension. We also show that the class of HFG algebras is stable under some natural constructions. This enables us to give a series of concrete examples of HFG algebras, including Arens-Michael envelopes of affine algebras (such as the algebras of holomorphic functions on the quantum affine space and on the quantum torus), the algebras of holomorphic functions on the free polydisk, on the quantum polydisk, and on the quantum polyannulus.

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Non-commutative holomorphic functions in elements of a Lie algebra and the absolute basis problem

Cited by 10 publications

References 11 publications

Local left invertibility for operator tuples and noncommutative localizations

Local left invertibility for operator tuples and noncommutative localizations

Holomorphic functions on the quantum polydisk and on the quantum ball

Holomorphically finitely generated algebras

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