The Arens-Michael envelopes of the Jordanian Plane and U q (sl(2))
aa r X i v : . [ m a t h . QA ] S e p THE ARENS-MICHAEL ENVELOPES OF THE JORDANIAN PLANE AND U q ( sl (2)) DMITRII PEDCHENKO
Abstract.
The Arens-Michael functor in noncommutative geometry is an analogue of the analyti-fication functor in algebraic geometry: out of the ring of “algebraic functions” on a noncommutativespace it constructs the ring of “holomorphic functions” on it. In this paper, we explicitly compute theArens-Michael envelopes of the Jordanian plane and the quantum enveloping algebra U q ( sl (2)) of sl (2) for | q | = 1 .This is an article version of the author’s senior thesis [Ped15] at HSE University from 2015. Contents
1. Introduction 12. The Arens-Michael functor 33. Examples of the Arens-Michael envelopes 34. Theoretical constructions 55. The Arens-Michael envelope of the Jordanian plane 86. The Arens-Michael envelope of U q ( sl (2)) , | q | = 1 . 11References 151. Introduction
Noncommutative geometry appeared in the second half of the XX century as the result of rethinkingand conceptualizing classic facts about the interplay between geometry and algebra. Oftentimes allessential information about a geometric space (e.g., affine algebraic variety, smooth manifold, topologicalspace) is contained in a suitably chosen algebra of functions (polynomial, smooth, continuous) on thespace. This observation is formalized in a theorem which establishes the anti-equivalence between thecategory of spaces at hand and the respective category of algebras of functions on such spaces. Sometimesthe resulting category of commutative algebras admits an abstract description which a priori does notbind elements of the algebra to functions on any space. To illustrate this point, consider Hilbert’sNullstellensatz which implies that for an algebraically closed field the category of affine algebraic varietiesis anti-equivalent to the category of commutative finitely generated reduced algebras. The Gelfand-Naimark Theorem implies that the category of compact Hausdorff topological spaces is anti-equivalentto the category of commutative C ∗ -algebras.The basic idea of noncommutative geometry is to view an arbitrary (noncommutative) algebra asan “algebra of functions on a noncommutave space”. This idea is based on an observation that manyimportant geometric concepts and constructions stated in algebraic terms remain meaningful for non-commutative algebras providing us with the tools and intuition for studying these algebras. Date : September 2020.2010
Mathematics Subject Classification.
Primary: 46H05 , 46H20. Secondary: 46H25.
Key words and phrases.
Arens-Michael functor, Arens-Michael envelope, Ore extension, noncommutative geometry. “Noncommutative geometry” includes several mathematical disciplines which have different researchobjects but are unified by that aforementioned idea. Noncommutative measure theory studies von Neu-mann algebras (note that commutative von Neumann algebras are exactly the algebras of essentiallybounded measurable functions on measure spaces); noncommutative topology studies C ∗ -algebras (be-cause of the Gelfand-Naimark theorem mentioned above); noncommutative affine algebraic geometrystudies finitely generated algebras; noncommutative differential geometry studies dense subalgebras of C ∗ -algebras equipped with a special “differential” structure.Notice that an important discipline is missing from this list - noncommutative complex analytic ge-ometry which in the commutative case bridges differential and algebraic geometries. One of the mainreasons of why this theory is underdeveloped is that it is unclear which algebras should be considered thenoncommutative generalizations of the algebras of holomorphic functions on complex analytic spaces.The ideal starting point for the development of any kind of noncommutate geometry is having a category A of associative algebras such that the full subcategory of A consisting of commutative algebras isanti-equivalent to a certain category C of “spaces”. In such a situation we may think of algebras be-longing to A as the noncommutative analogues of the spaces belonging to C . Therefore, in the caseof noncommutative complex analytc geometry we would like to start with some category consisting ofalgebras such that the commutative ones are exactly the algebras of holomorphic functions on complexanalytic spaces. As Pirkovskii states in [Pir08a], apparently in full generality such a class of algebras hasnot yet been introduced.However, we can simplify our problem as follows. Any affine algebraic variety over C is certainlya complex analytic space. So we can narrow down the category C of complex analytic spaces to thecategory of affine algebraic varieties over C viewed as complex analytic spaces. It turns out that in thiscase we have a construction that for each finitely generated C -algebra A (deemed as the “algebra ofregular algebraic functions on a noncommutative affine scheme of finite type”) assigns a new algebra b A (deemed as the “algebra of holomorphic functions on that scheme”) such that if our algebra was acommutative algebra of regular algebraic functions A = O alg ( X ) on an affine scheme X of finite type,then we get the algebra of holomorphic functions on X : b A = O hol ( X ) .The resulting algebra b A is known as the Arens-Michael envelope of algebra A . Therefore, we viewthe Arens-Michael envelopes of finitely generated algebras as the algebras of holomorphic functions onnoncommutative affine schemes of finite type. So far, the Arens-Michael envelopes are explicitly knownonly for a handful of noncommutative algebras (see §3 and [Pir08b, §5]), and it is an important for thedevelopment of the theory and its scope of applicability to grow the body of examples.In this paper, we add two algebras to the list of examples: we explicitly compute the Arens-Michaelenvelopes of the Jordanian plane and the quantum enveloping algebra U q ( sl (2)) for | q | = 1 following therecently developed techniques in this area.Finally, let us mention that this paper is an improved version of the author’s senior thesis [Ped15] atthe National Research University - Higher School of Economics from 2015. Organization of the paper.
In §2, we review the basic definitions pertaining to the Arens-Michaelfunctor. In §3, we recall some known examples of algebras for which the Arens-Michal envelope isexplicitly known. In §4, we review the theoretical constructions necessary for our computations following[Pir08b].Finally, we present our key results in §5 and §6. In §5, we explicitly compute the Arens-Michaelenvelope of the Jordanian plane. In §6, we explicitly compute the Arens-Michael envelope of the quantumenveloping algebra U q ( sl (2)) for | q | = 1 . Acknowledgments.
We would like to thank Alexei Pirkovskii for inspiring academic advising and helpfuldiscussions.
HE ARENS-MICHAEL ENVELOPES OF THE JORDANIAN PLANE AND U q ( sl (2)) The Arens-Michael functor
In what follows all vector spaces and algebras are taken over the field of complex numbers C ; all algebrasare assumed to be associative and unital. The seminorm k · k on an algebra A is called submultiplicative if k ab k ≤ k a kk b k for all a, b ∈ A . A complete topological algebra with a topology generated by a familyof submultiplicative seminorms is called an Arens-Michael algebra . We start with our main definitions.
Definition 2.1.
Let A be a topological algebra. The pair ( b A, ι A ) , consisting of an Arens-Michael algebra b A and a continuous homomorphism ι A : A → b A , is called the Arens-Michael envelope of algebra A iffor an arbitrary Arens-Michael algebra B and an arbitrary continuous homomorphism ϕ : A → B , thereexists a unique continuous homomorphism b ϕ : b A → B such that the following diagram commutes: b A BA b ϕϕι A Arens-Michael envelopes were introduced by Taylor [Tay72], but here we are using the terminology ofHelemskii [Hel89] and Pirkovskii [Pir08b, §3].It is clear from the definition that the Arens-Michael envelope is unique up to a unique isomorphism oftopological algebras over A . Moreover, it always exists [Tay72], and it can be obtained as the completionof A with respect to all continuous submultiplicative seminorms on A . Note that the topology inducedby submultiplicative seminorms might be non-Hausdorff so that before taking the completion we shouldtake the quotient by the closure of { } . Therefore, the canonical homomorphism ι A : A → b A mighthave a nontrivial kernel. Definition 2.2.
Let A be an algebra without a topology. The Arens-Michael envelope of A is the Arens-Michael envelope of A endowed with the strongest locally convex topology in the sense of Definition2.1.Finally, let us note that the association A b A extends to algebra homomorphisms A → B sothat we obtain a functor from the category of algebras to the category of Arens-Michael algebras (theArens-Michael functor) .3. Examples of the Arens-Michael envelopes
Next we discuss some known examples of the Arens-Michael envelopes.The next example and proposition justify our assertions from the introduction about the Arens-Michaelfunctor being a noncommutative analogue of the analytification functor in algebraic geometry.
Example 3.1.
As was noted by Taylor [Tay72], the Arens-Michael envelope of the polynomial algebra C [ x , ..., x n ] = O alg ( C n ) is the algebra of holomorphic functions O hol ( C n ) with compact-open topology.Pirkovskii generalized this to statement to affine algebraic varieties. Proposition 3.2 ([Pir08b, Example 3.6]) . Let X be an affine algebraic variety over C and let A = O alg ( X ) be the algebra of regular algebraic functions on X . The Arens-Michael envelope of A is the algebra O hol ( X an ) of holomophic functions on X when we view X as a complex analytic space,with compact-open topology. The same is true for the affine schemes of finite type over C . From this proposition we see that the geometric analytification functor associating to an affine algebraicscheme X a complex analytic space X an corresponds to the algebraic or functional-analytic Arens-Michaelfunctor when we instead work with functions on those spaces. As we noted in the introduction, the finitely D. PEDCHENKO generated noncommutative algebras are the natural candidates for the noncommutative affine schemesof finite type, so we view the Arens-Michael envelopes of the finitely generated noncommutative algebrasas the algebras of holomorphic functions on the noncommutative affine schemes of finite type.Here is the “most noncommutative” example:
Example 3.3 (The free algebra) . Let F n = C h x , ..., x n i be a free algebra with n generators. For each k -tuple α = ( α , ..., α k ) of integers, ≤ α i ≤ n , set x α = x α · · · x α k and | α | = k . Then each elementof F n is written as a noncommutative polynomial P | α |≤ N c α x α . Denote the set of all α as W n .Taylor [Tay72] showed that c F n = { a = X α ∈ W n c α x α : k a k ρ = X α ∈ W n | c α | ρ α < ∞ for any ρ > } . The topology on c F n is defined by the family of seminorms {k · k ρ : ρ ∈ R > } . Example 3.4 (The quantum plane) . Fix a complex number q ∈ C \ { } . The quantum plane is analgebra (denoted O algq ( C ) ) with two generators x, y , subject to a relation xy = qyx . The monomials x i y j ( i, j ≥ form a basis of O algq ( C ) so that this algebra can be viewed as an algebra of polynomialswith a “twisted” multiplication.Denote the Arens-Michael envelope of O algq ( C ) by O holq ( C ) , and view it as an algebra of holomorphicfunctions on the quantum plane. The next result is due to Pirkovskii. Proposition 3.5 ([Pir08b, Corollary 5.14]) . Let q ∈ C \ { } . (1) If | q | ≥ , then O holq ( C ) = { a = ∞ X i,j =0 c ij x i y j : k a k ρ = ∞ X i,j =0 | c ij | ρ i + j < ∞ for any ρ > } . (2) If | q | ≤ , then O holq ( C ) = { a = ∞ X i,j =0 c ij x i y j : k a k ρ = ∞ X i,j =0 | c ij || q | ij ρ i + j < ∞ for any ρ > } . In both cases the topology on O holq ( C ) is generated by the family of seminorms {k · k ρ : ρ ∈ R > } and the multiplication is defined by the relation xy = qyx . Example 3.6.
Consider the universal enveloping algebra of the Lie algebra g with basis { x, y } and thecommuting relation [ x, y ] = y . Due to the Poincare - Birkhoff - Witt theorem, the universal envelopingalgebra U ( g ) can be viewed as a polynomial algebra with a “twisted” multiplication. As shown in [Pir08b,Example 5.1] b U ( g ) = { a = ∞ X i,j =0 c ij x i y j : ∞ X i =0 | c ij | ρ i < ∞ , ∀ j ∈ Z + , ∀ ρ > } . The topology of b U ( g ) is generated by the family of seminorms {k · k n,ρ : k ∞ X i,j =0 c ij x i y j k n,ρ = n X j =0 ∞ X i =0 | c ij | ρ i < ∞ , n ∈ Z + , ρ ∈ R > } . In the previous examples, the Arens-Michael envelopes of polynomial algebras with a “twisted” multi-plications happened to be algebras of “noncommutative power series”, as one would expect by comparingto the commutative case. Interestingly, the next example (due to Taylor [Tay72]) shows that this is notalways the case.
HE ARENS-MICHAEL ENVELOPES OF THE JORDANIAN PLANE AND U q ( sl (2)) Example 3.7 (The universal enveloping algebra of a semisimple Lie algebra) . Suppose g is a semisimpleLie algebra. Every finite-dimensional irreducible representation π λ of algebra g extends to a homomor-phism π λ : U ( g ) → M d λ ( C ) ( d λ = dim π λ ) . If we denote the set of equivalence classes of irreducible finite-dimensional representations of g by b g , weget a homomorphism Y λ ∈ b g π λ : U ( g ) → Y λ ∈ b g M d λ ( C ) . The algebra Q λ ∈ b g M d λ ( C ) with the product topology and the homomorphism Q λ ∈ b g π λ form the Arens-Michael envelope of U ( g ) .This example is a bit discouraging since contrary to the above examples, this time the Arens-Michaelenvelope looks completely different from the initial algebra (for example, U ( g ) is an integral domain but Q λ ∈ b g M d λ ( C ) is not). Nonetheless, the canonical homomorphism A → b A is injective (as it was in all otherexamples so far).The next example shows the worst possible situation. Example 3.8 (Weyl algebra) . Weyl algebra A is an algebra with two generators x, ∂ with the commutingrelation [ ∂, x ] = 1 . It is well-known that in a non-zero normed algebra there are no elements with thiscommuting relation. Therefore b A = 0 and the canonical homomorphism is not injective.It is interesting to note that if we quantize Weyl algebra by taking the commuting relation to be ∂x − qx∂ = 1 ( q = 0 , , the resulting Arens-Michael envelope would again be the algebra of “noncommutative” polynomials (see[Pir08b, Corollary 5.19]).For more examples, see [Pir08a, Pir08b].4. Theoretical constructions
In this section, we collect the theoretical facts necessary for our computations following [Pir03] and[Pir08b]. We will be referring to a complete, Hausdorff, locally convex topological algebra with jointlycontinuous multiplication as a b ⊗ - algebra .4.1. The Arens-Michael envelopes and tensor product.
First, we recall how to describe the topologyon the projective tensor product of two b ⊗ -modules. Proposition 4.1 ([Pir08b, Proposition 2.3 (vi]) . Suppose A is a b ⊗ -algebra, X is a right A - b ⊗ -module, Y is left A - b ⊗ -module. Furthermore, suppose that both X and Y have countable or finite dimension andthe topology on X and Y is the strongest locally convex topology. Then the algebraic tensor product X ⊗ A Y with the strongest locally convex topology coincides with the projective tensor product X b ⊗ A Y . The next proposition shows that the Arens-Michael envelope of the projective tensor product of two b ⊗ -algebras can be computed as the projective tensor product of their Arens-Michael envelopes. Proposition 4.2 ([Pir03, Proposition 6.4]) . Let
A, B be b ⊗ -algebras. Then there exists a topologicalalgebra isomorphism ( A b ⊗ B ) b ∼ = b A b ⊗ b B. In other words, the operations of taking the Arens-Michael envelope and taking the projective tensorproduct can be interchanged.
D. PEDCHENKO
The Arens-Michael envelopes and Ore extensions.
The computation of the Arens-Michaelenvelopes of many polynomial algebras (including Examples 3.4 and 3.6) is greatly facilitated by atheoretical construction known as the
Ore extension .4.2.1.
Algebraic Ore extensions.
First, we consider a purely algebraic construction.
Definition 4.3.
Let R be an associative C -algebra (without a topology) and α : R → R an algebraendomorphism. A C -linear map δ : R → R is called α - differentiation if δ ( ab ) = δ ( a ) b + α ( a ) δ ( b ) for any a, b ∈ R . The Ore extension R [ z ; α, δ ] is a noncommutative algebra obtained by endowing theleft R -module of polynomials n P i =0 r i z i with a “twisted” multiplication with a relation zr = α ( r ) z + δ ( r ) for r ∈ R . Note that the natural inclusions R ֒ → R [ z ; α, δ ] and C [ z ] ֒ → R [ z ; α, δ ] become algebrahomomorphisms.Let us also recall a useful formula describing multiplication in R [ z ; α, δ ] . For any k, n ∈ Z > with k ≤ n , let S n,k : R → R denote an operator defined as the sum of all (cid:0) nk (cid:1) different compositions of k differentiations δ and n − k homomorpisms α . Then for any r ∈ R we have the following formula forhow to commute z n and r (see [Pir08b, §4.1]:(4.3.1) z n r = n X k =0 S n,k ( r ) z n − k . Turning back to our examples, we see that the quantum plane from Example 3.4 is the Ore extension C [ x ][ y ; α, , where α ( x ) = q − x and the commuting relation becomes yx = q − xy . The universalenveloping algebra U ( g ) from Example 3.6 is the Ore extension C [ y ][ x ; id, y ddy ] , and the commutingrelation becomes xy = yx + y .4.2.2. Analytic Ore extensions.
Next we consider a locally convex counterpart of the algebra R [ z ; α, δ ] -an analytical Ore extension O ( C , R ; α, δ ) - and state the theorem telling us the conditions under whichthe algebra O ( C , R ; α, δ ) (or some variant of it) becomes the Arens-Michael envelope of R [ z ; α, δ ] . Belowwe explain the key steps in the construction of O ( C , R ; α, δ ) . This theoretical framework is explained indetail in [Pir08b].First, we recall the following two technical definitions. Definition 4.4 ([Pir08b, Definition 4.1]) . Let E be a vector space and let T be a family of linearoperators on E . A seminorm on E is T - stable if for any T ∈ T there exists C > such that k T v k ≤ C k v k for every v ∈ E . Definition 4.5 ([Pir08b, Definition 4.2]) . Let E be a locally convex topological space. A family T oflinear operators on E is called localizable if the topology on E can be defined by a family of T -stableseminorms. A single operator T is called localizable if the singleton family T = { T } is localizable.Let now R be a b ⊗ -algebra equipped with a localizable endomorphism α : R → R and a localizabledifferentiation δ : R → R . The next two lemmas will show that we can equip the space O ( C , R ) of R -valued entire functions with a “twisted” multiplication which coincides with the multiplication on theOre extension R [ z ; α, δ ] when we restrict to the polynomial subspace in O ( C , R ) . Recall that O ( C , R ) is isomorphic to the projective tensor product R b ⊗O ( C ) both as a locally convex topological space and HE ARENS-MICHAEL ENVELOPES OF THE JORDANIAN PLANE AND U q ( sl (2)) as a left R - b ⊗ -module. Explicitly, for any family of seminorms {k · k λ } λ ∈ Λ defining the topology on R ,the space O ( C , R ) is described as convergent Taylor series { f ( z ) = X n c n z n : c n ∈ R, k f k λ,ρ < ∞ for any λ ∈ Λ , ρ > } , where k f k λ,ρ = P ∞ n =0 k c n k λ ρ n . The topology on O ( C , R ) is defined by the family of seminorms {k · k λ,ρ : λ ∈ Λ , ρ ∈ R > } . Consider the inclusion of locally convex topological vector spaces R [ z ; α, δ ] ֒ → O ( C , R ) , where R [ z ; α, δ ] is equipped with the “twisted multiplication” from Definition 4.3 and the induced topologyfrom O ( C , R ) . The next lemma shows that we can extend (4.3.1) from the dense subspace R [ z ; α, δ ] tothe whole space O ( C , R ) . Lemma 4.6 ([Pir08b, Lemma 4.2]) . Suppose R is a b ⊗ -algebra, α : R → R - a localizable endomorphismand δ : R → R - a localizable differentiation. Then there exists a unique continuous linear map τ : O ( C ) b ⊗ R → R b ⊗O ( C ) , such that τ ( z n ⊗ r ) = n X k =0 S n,k ( r ) ⊗ z n − k for all r ∈ R and n ∈ Z ≥ . Now set A = O ( C , R ) ∼ = R b ⊗O ( C ) . Define the multiplication map m A : A b ⊗ A → A as the composi-tion R b ⊗O ( C ) b ⊗ R b ⊗O ( C ) R ⊗ τ ⊗ O ( C ) −−−−−−−−→ R b ⊗ R b ⊗O ( C ) b ⊗O ( C ) m R b ⊗ m O ( C ) −−−−−−−→ R b ⊗O ( C ) . Proposition 4.7 ([Pir08b, Proposition 4.3]) . The map m A : A b ⊗ A → A turns A = O ( C , R ) into a b ⊗ -algebra, such that the inclusion map i : R [ z ; α, δ ] ֒ → O ( C , R ) is an algebra homomorphism. The last proposition allows us to give the following definition.
Definition 4.8 ([Pir08b, Definition 4.3]) . The algebra A = R b ⊗O ( C ) with the above multiplication mapwill be denoted O ( C , R ; α, δ ) and called the analytical Ore extension of algebra R .Note that O ( C , R ; α, δ ) contains R as a closed subalgebra and is therefore a R - b ⊗ -algebra.Next, we strengthen the above result in the case when R is moreover an Arens-Michael algebra. First,we have the following refinement of Definition 4.5. Definition 4.9 ([Pir08b, Definition 4.4]) . Let R be an Arens-Michael algebra. A family T of linearoperators on R is called m - localizable if the topology on R can be defined by a family of T -stablesubmultiplicative seminorms. A single operator T is called m - localizable if the singleton family T = { T } is m -localizable.The next proposition shows that if R is an Arens-Michael algebra and operators α and δ form an m -localizable family, then the analytic Ore extension O ( C , R ; α, δ ) is itself an Arens-Michael algebra. Proposition 4.10 ([Pir08b, Proposition 4.5]) . Let R be an Arens-Michael algebra, α : R → R - analgebra endomorphism, δ : R → R - an α -differentiation. Suppose that the set { α, δ } is m -localizable.Then O ( C , R ; α, δ ) is an Arens-Michael algebra. D. PEDCHENKO
Now suppose R is an algebra (without a topology), α = id : R → R is the identity map and δ : R → R is a differentiation. Denote by R δ the Arens-Michael algebra, obtained as the completion of R by thesystem of all δ -stable submiltiplicative seminorms. Let j be the canonical homomorphism j : R → R δ .Clearly δ defines a unique m -localizable differentiation b δ of R δ with the property b δ ◦ j = j ◦ δ . Thereforewe get homomorphisms R [ z ; id, δ ] → R δ [ z ; id, b δ ] ֒ → O ( C , R δ ; id, b δ ) , where the first one coincides with j on R and maps z to z , and the second one is a canonical inclusion.Let ι R [ z ; id,δ ] be the composition homomorphism. The next result describes the Arens-Michael envelopeof R [ z ; id, δ ] as the analytic Ore extension O ( C , R δ ; id, b δ ) . Theorem 4.11 ([Pir08b, Theorem 5.1]) . The pair ( O ( C , R δ ; id, b δ ) , ι R [ z ; id,δ ] ) is the Arens-Michael en-velope of the algebra R [ z ; id, δ ] . The situation when α = id is harder to handle. Let R be an algebra, let α : R → R be itsendomorphism and let X be an R -bimodule. We will denote by α X the R -bimodule obtained byendowing the underlying abelian group of X with a new R -multiplication rule • : r • x = α ( r ) x, x • r = xr, r ∈ R, x ∈ X. Let again R be an algebra without a topology, α : R → R be an endomorpism of R , and δ : R → R be an α -differentiation. By applying the Arens-Michael functor to α we obtain an endomorphism b α : b R → b R of the Arens-Michael envelope of R satisfying b α ◦ ι R = ι R ◦ α. Since we can view ι R : R → b R as a morphism α R → b α b R of R -bimodules, we get that the composition R δ −→ R ι R −→ b α b R is a differentiation. By applying the universal property of the Arens-Michael envelopes (or rather itsversion for R -modules, see [Pir08b, Definition 3.2]), we obtain a unique b α -differentiation b δ : b R → b α b R satisfying b δ ◦ ι R = ι R ◦ δ. We have the following general result describing the Arens-Michael envelope of the algebraic Oreextension as an analytic Ore extension when certain technical conditions are met.
Theorem 4.12 ([Pir08b, Theorem 5.17]) . In the above setup, if the family { b α, b δ } is m -localizable, thenthere exists a unique R -homomorphism ι R [ z ; α,δ ] : R [ z ; α, δ ] → O ( C , b R ; b α, b δ ) such that z z. The algebra O ( C , b R ; b α, b δ ) with the homomorphism ι R [ z ; α,δ ] is the Arens-Michael envelope of R [ z ; α, δ ] . The Arens-Michael envelope of the Jordanian plane
Now we turn to the main results of this paper. We first compute the Arens-Michael envelope of theJordanian plane.
Definition 5.1.
The Jordanian plane over K is the K -algebra Λ ( K ) given by generators x and y anda commuting relation yx = xy + y , i.e. Λ ( K ) = K h x, y i / ( yx − xy − y ) . HE ARENS-MICHAEL ENVELOPES OF THE JORDANIAN PLANE AND U q ( sl (2)) We would be interested in the case when K = C and we would denote Λ ( C ) =: Λ .Following a simple induction argument it is easy to check that the monomials { x i y j | i, j ∈ Z + } span Λ . It is shown in [Shi18] that they are also linearly independent and, therefore, form the basis of Λ .As a result, we can again view Λ as a polynomial algebra with a “twisted” multiplication.Comparing Definition 5.1 to Definition 4.3 we see that the Jordanian plane is the Ore extension C [ y ][ x ; id, − y ddy ] . Therefore, in order to apply Theorem 4.11 we need to describe the system of allsubmiltiplicative δ -stable seminorms on C [ y ] , where δ = − y ddy .Write an element a ∈ C [ y ] as a polynomial n P i =0 a i y i . We first have the following result. Proposition 5.2.
The family {k · k ρ : k a k ρ = n X i =0 | a i | i − ρ i < ∞ , ρ ∈ R > } , ( −
1! := 1 ,
0! := 1) is equivalent to the family of all submultiplicative δ -stable seminorms on C [ y ] with δ = − y ddy .Proof. We split the proof of the proposition into steps for the reader’s convenience.
Step 1.
First note that k · k ρ is indeed a seminorm. Moreover, it is submultiplicative: k y k y l k ρ = k y k + l k ρ = ρ k + l ( k + l − ≤ ρ k ( k − ρ l ( l − k y k k ρ k y l k ρ , ∀ k, l ≥ . Now for a, b ∈ C [ y ] we have: k ab k ρ = k n X i =0 a i y i · m X j =0 b j y j k ρ = k n,m X i,j =0 a i b j y i y j k ρ ≤ n,m X i,j =0 | a i || b j |k y i y j k ρ ≤ n,m X i,j =0 | a i || b j |k y i k ρ k y j k ρ = n X i =0 | a i |k y i k ρ · m X j =0 | b j |k y j k ρ = k n X i =0 a i y i k ρ · k m X j =0 b j y j k ρ = k a k ρ k b k ρ . Step 2.
Next, we show that k · k ρ is δ -stable. Note that δ ( y i ) = − y ddy ( y i ) = − iy i +1 , ∀ i ≥ and δ ( y ) = δ (1) = − y ddy (1) = 0 . We have k δ ( a ) k ρ = k δ ( n X i =0 a i y i ) k ρ = k n X i =0 a i δ ( y i ) k ρ = k n X i =1 a i ( − iy i +1 ) k ρ = k n X i =1 a i iy i +1 k ρ = k n +1 X j =2 a j − ( j − y j k ρ = n +1 X j =2 | a j − | j − j − ρ j = ρ n +1 X j =2 | a j − | j − ρ j − ≤ ρ n X i =1 | a i | i − ρ i ≤ ρ n X i =0 | a i | i − ρ i = ρ k a k ρ . Step 3.
Finally, we show that any submultiplicative δ -stable seminorm k · k is dominated by k · k ρ forsome ρ > .Note that by induction we get δ j ( y ) = ( − j j ! y j +1 , j ≥ . From δ -stability we get k δ j ( a ) k ≤ C k δ j − ( a ) k ≤ ... ≤ C j k a k . Now setting a = y we have k δ j ( y ) k = j ! k y j +1 k ≤ C j k y k , so that k y j +1 k ≤ C j k y k j ! , j ≥ . Note that if we pick C ≥ and set ρ := C max {k y k , } , we have C j k y k ≤ C j max {k y k , } ≤ C j (max {k y k , } ) j = ρ j , and therefore k y j +1 k ≤ C j k y k j ! ≤ ρ j j ! ≤ ρ j +1 j ! = k y j +1 k ρ , j ≥ . For j = 0 we have k y k = k y k ρ ρ ≤ D k y k ρ , where D = max { k y k ρ , } .Finally, k a k = k n X i =0 a i y i k ≤ n X i =0 | a i |k y i k ≤ | a | + | a | D k y k ρ + n X i =2 | a i |k y i k ρ ≤ D n X i =0 | a i |k y i k ρ = D k a k ρ . (cid:3) Next, we pass to a simpler family of seminorms.
Lemma 5.3.
The family of seminorms on C [ y ] P = {k · k ρ : k a k ρ = n X i =0 | a i | i − ρ i < ∞ , ρ ∈ R > , } , ( −
1! := 1 ,
0! := 1) is equivalent to the family Q = {k · k q : k a k q = n X i =0 | a i | i ! q i < ∞ , q ∈ R > } , (0! := 1) , where a = n P i =0 a i y i ∈ C [ y ] . Proof.
First, we observe n X i =0 | a i | i ! q i ≤ n X i =0 | a i | i − q i , and, therefore, Q ≺ P .Since for i ≥ i ≤ i ⇔ i − ≤ i i ! , HE ARENS-MICHAEL ENVELOPES OF THE JORDANIAN PLANE AND U q ( sl (2)) we have n X i =0 | a i | i − ρ i ≤ n X i =0 | a i | i i ! ρ i = n X i =0 | a i | i ! q i , q = 2 ρ, and P ≺ Q . (cid:3) Finally, we can apply Theorem 4.11 to get the description of the Arens-Michael envelope of theJordanian plane Λ . Theorem 5.4.
The Arens-Michael envelope of the Jordanian plane Λ is b Λ := { a = ∞ X i,j =0 a ij x i y j : k a k ρ < ∞ for any ρ > } , where k a k ρ = ∞ X i,j =0 | a ij | j ! ρ i + j . The topology on b Λ is generated by the system {k · k ρ : ρ ∈ R > } , and multiplication is characterizedby the relation yx = xy + y .Proof. By Theorem 4.11, the Arens-Michael envelope of Λ = C [ y ][ x ; id, − y ddy ] is given by the analyticOre extension O ( C , C [ y ] δ ; id, b δ ) , where C [ y ] δ is the completion of C [ y ] with respect to the family Q ofseminorms by Lemma 5.3.By the discussion preceding Theorem 4.11, the space O ( C , C [ y ] δ ; id, b δ ) can be described as the set { a = ∞ X i =0 ∞ X j =0 a ij x i y j : k a k q,ρ < ∞ for any q, ρ > } , where k a k q,ρ = ∞ X i =0 k ∞ X j =0 a ij y j k q ρ i = ∞ X i =0 ∞ X j =0 | a ij | j ! q j ρ i , with topology given by the family of seminorms {k · k q,ρ : q, ρ ∈ R > } . It is easy to check that thisdescription is equivalent to the description in the statement of the theorem. (cid:3)
We see that again the Arens-Michael envelope of a polynomial algebra with a “twisted” multiplicationis a power series algebra with the same multiplication rule.6.
The Arens-Michael envelope of U q ( sl (2)) , | q | = 1 . In the section, we turn to the second main result of this paper.The quantum enveloping algebra U q ( sl (2)) of the Lie algebra sl (2) plays one of the most importantroles in noncommutative geometry. It can be defined in two slightly different ways. Definition 6.1.
For q ∈ C \ { , − } , consider an algebra U q ( sl (2)) on four generators E, F, K, K − subject to the following relations:(1) KK − = K − K = 1 ,(2) KEK − = q E, KF K − = q − F ,(3) [ E, F ] = K − K − q − q − . Definition 6.2.
For q ∈ C \ { , − } , consider an algebra U ′ q ( sl (2)) on five generators E, F, K, K − , L with relations: (1) KK − = K − K = 1 ,(2) KEK − = q E, KF K − = q − F ,(3) [ E, F ] = L, ( q − q − ) L = K − K − ,(4) [ L, E ] = q ( EK + K − E ) , [ L, F ] = − q − ( F K + K − F ) .Clearly, U ′ q ( sl (2)) is isomorphic to U q ( sl (2)) via a map that sends L ∈ U ′ q ( sl (2)) to [ E, F ] ∈ U q ( sl (2)) and leaves other generators intact. Note that the second definition allows us to consider the limiting case q = 1 where our quantum enveloping algebra almost becomes the usual enveloping algebra U ( sl (2)) . Infact, we have the following isomorphisms (see [Kas12]):(6.2.1) U ′ ( sl (2)) ∼ = U ( sl (2))[ K ]( K − ∼ = U ( sl (2)) ⊗ C [ K ]( K − . For more information on U q ( sl (2)) , see [Kas12].We obtain the following description of the Arens-Michael envelope of U q ( sl (2)) for | q | = 1 . Theorem 6.3.
Let U q ( sl (2)) be the quantum enveloping algebra of sl (2) , and let \ U q ( sl (2)) be itsArens-Michael envelope. (1) If q = 1 , then \ U ( sl (2) ∼ = \ U ( sl (2) b ⊗ C [ K ]( K − . (2) | q | = 1 , q = 1 , − , then \ U q ( sl (2)) = { c = X i ∈ Z ,n,m ≥ c i,n,m K i F n E m : k c k ρ < ∞ for any ρ > } , where k c k ρ = X i ∈ Z ,n,m ≥ | c i,n,m | ρ i + n + m The topology on \ U q ( sl (2)) is generated by the system {k · k ρ : ρ ∈ R > } . Proof.
We split the proof of the theorem into two cases according to how the theorem is stated.
Proof of (1).
Definition 6.2 allows us to consider the limiting case q = 1 where our quantum envelopingalgebra almost becomes the usual enveloping algebra U ( sl (2)) as we mentioned in (6.2.1).The representation of U ′ ( sl (2)) as the tensor product of two algebras for which the Arens-Michaelenvelope is already known allows us to quickly compute the Arens-Michael envelope of U ′ ( sl (2)) usingthe results of §4.1. Specifically, endow U ′ ( sl (2)) with the strongest locally convex topology τ str . ByProposition 4.1, we have the following isomorphism of b ⊗ -algebras ( U ′ ( sl (2)) , τ str ) ∼ = ( U ( sl (2)) , τ str ) b ⊗ (cid:18) C [ K ] K − , τ str (cid:19) . Next, we apply Proposition 4.2 (along with Definition 2.2) to the above projective tensor product to get \ U ′ ( sl (2)) ∼ = \ U ( sl (2)) b ⊗ (cid:18) C [ K ] K − (cid:19) b . Since C [ K ]( K − is a finite-dimensional vector space, its Arens-Michael envelope coincides with C [ K ]( K − .Therefore, we finally get \ U ′ ( sl (2) ∼ = \ U ( sl (2) b ⊗ C [ K ]( K − . The Arens-Michael envelope \ U ( sl (2) was shown to be the direct product of matrix algebras in Example3.7. HE ARENS-MICHAEL ENVELOPES OF THE JORDANIAN PLANE AND U q ( sl (2)) Proof of (2).
For q = 1 , − , note that the algebra U q ( sl (2)) is an iterated Ore extension (see §4.2).Namely, start with A = C [ K, K − ] along with a C -linear algebra homomorphism α : A → A defined by α ( K ) = q K. Next, consider the Ore extension A = A [ F, α , δ = 0] equipped with a C -linear algebra homomorphism α and a C -linear differentiation δ defined by α ( F j K l ) = q − l F j K l , δ ( K ) = 0 , δ ( F j K l ) = j − X i =0 F j − δ q − i K ( F ) K l , where δ K ( F ) = K − K q − q − is a Laurent polynomial in K . Finally, consider the Ore extension A = A [ E, α , δ ] . One easily checks by comparing the above sequence of Ore extensions to Definition 6.1 that A ∼ = U q ( sl (2)) . Therefore, we can apply the results of §4.2 to each consecutive Ore extension to calculate the Arens-Michael envelope of A .The Arens-Michael envelope of the algebra A = C [ K, K − ] is very simple: c A = { a = X i ∈ Z a i K i : k a k ρ = X i ∈ Z | a i | ρ i < ∞ for all ρ > } , where k a k ρ = P i ∈ Z | a i | ρ i , and the topology is generated by the family {k · k ρ : ρ ∈ R > } .After extending α : A → A to c α : c A → c A , we check that c α is indeed m -localizable (use | q | = 1 ): k c α ( X i ∈ Z a i K i ) k ρ = k X i ∈ Z a i q i K i k ρ = X i ∈ Z | a i || q i | ρ i = X i ∈ Z | a i | ρ i = k X i ∈ Z a i K i k ρ . Applying Theorem 4.12, we obtain c A = O ( C , c A ; c α , { b = X i ∈ Z ,n ≥ b i,n K i F n : k b k ρ < ∞ for any ρ > } , where k b k ρ = P i ∈ Z ,n ≥ | b i,n | ρ i + n , and the topology is generated by the family {k · k ρ : ρ ∈ R > } .Finally, the operators α and δ simply extend to c A by their action on the generators K and F . Wecheck that { c α , b δ } is an m -localizable family. In the calculations below, we make extensive use of therelations between the generators K i F n = q − in F n K l . c α is m -localizable: k c α ( X i ∈ Z ,n ≥ b i,n K i F n ) k ρ = k c α ( X i ∈ Z ,n ≥ b i,n q − in F n K i ) k ρ = k X i ∈ Z ,n ≥ b i,n q − in q − i F n K i k ρ = X i ∈ Z ,n ≥ | b i,n || q − in || q − i | ρ i + n = X i ∈ Z ,n ≥ | b i,n || q − in | ρ i + n = k X i ∈ Z ,n ≥ b i,n q − in F n K i k ρ = k X i ∈ Z ,n ≥ b i,n K i F n k ρ . Before proving that b δ is m -localizable, let us perform one auxiliary computation: δ ( K i F n ) = δ ( q − in F n K i ) = q − in n − X j =0 F n − δ q − j K ( F ) K i == q − in F n − K i q − q − h ( K − K − ) + ( q − K − q K ) + ... + ( q − n − K − q n − K ) i = q − in F n − K i q − q − h K (1 + q − + ... + q − n − ) − K − (1 + q + ... + q n − i = q − in F n − K i q − q − (cid:20) K (cid:18) − q − n − q − (cid:19) − K − (cid:18) − q n − q (cid:19)(cid:21) . Now we can show that b δ is m -localizable: k b δ ( X i ∈ Z ,n ≥ b i,n K i F n ) k ρ = k X i ∈ Z ,n ≥ b i,n δ ( K i F n ) k ρ = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X i ∈ Z ,n ≥ b i,n q − in F n − K i q − q − (cid:20) K ( 1 − q − n − q − ) − K − ( 1 − q n − q ) (cid:21)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ρ = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X i ∈ Z ,n ≥ b i,n q − in (cid:20) F n − K i +1 − q − n ( q − q − )(1 − q − ) − F n − K i − − q n ( q − q − )(1 − q ) (cid:21)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ρ = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X i ∈ Z ,n ≥ (cid:20) b i − ,n +1 q − i − n +1) − q − n +1) ( q − q − )(1 − q − ) − b i +1 ,n +1 q − i +1)( n +1) − q n +1) ( q − q − )(1 − q ) (cid:21) F n K i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ρ = X i ∈ Z ,n ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:20) b i − ,n +1 q − i − n +1) − q − n +1) ( q − q − )(1 − q − ) − b i +1 ,n +1 q − i +1)( n +1) − q n +1) ( q − q − )(1 − q ) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) ρ i + n ≤ X i ∈ Z ,n ≥ (cid:20)(cid:12)(cid:12)(cid:12)(cid:12) b i − ,n +1 q − i − n +1) − q − n +1) ( q − q − )(1 − q − ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) b i +1 ,n +1 q − i +1)( n +1) − q n +1) ( q − q − )(1 − q ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) ρ i + n ≤ X i ∈ Z ,n ≥ (cid:20) | b i − ,n +1 | (cid:12)(cid:12)(cid:12) q − i − n +1) (cid:12)(cid:12)(cid:12) | ( q − q − )(1 − q − ) | + | b i +1 ,n +1 | (cid:12)(cid:12)(cid:12) q − i +1)( n +1) (cid:12)(cid:12)(cid:12) | ( q − q − )(1 − q ) | (cid:21) ρ i + n ≤ | ( q − q − )(1 − q − ) | X i ∈ Z ,n ≥ | b i,n | ρ i + n + 2 | ( q − q − )(1 − q − ) | X i ∈ Z ,n ≥ | b i +1 ,n +1 | ρ i + n +2 ρ ≤ C X i ∈ Z ,n ≥ | b i,n | ρ i + n + Cρ X i ∈ Z ,n ≥ | b i,n | ρ i + n ≤ ( C + Cρ ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X i ∈ Z ,n ≥ b i,n K i F n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ρ . Applying Theorem 4.12, we conclude that the Arens-Michael envelope of U q ( sl (2)) for | q | = 1 , q = 1 , − is \ U q ( sl (2)) = c A = O ( C , c A ; c α , b δ )= { c = X i ∈ Z ,n,m ≥ c i,n,m K i F n E m : k c k ρ < ∞ for any ρ > } , where k c k ρ = P i ∈ Z ,n,m ≥ | c i,n,m | ρ i + n + m , and the topology is generated by the family {k · k ρ : ρ ∈ R > } . (cid:3) Remark 6.4.
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Department of Mathematics, The Pennsylvania State University, University Park, PA 16802
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