Relationships between quantized algebras and their semiclassical limits

A Poisson C-algebra R appears in classical mechanical system and its quantized algebra appearing in quantum mechanical system is a C[[~]]-algebra Q = R[[~]] with star product ∗ such that for any a, b ∈ R ⊆ Q, a ∗ b = ab+B1(a, b)~ +B2(a, b)~ + . . . subject to {a, b} = ~−1(a ∗ b− b ∗ a)|~=0, · · · (∗∗) where Bi : R× R −→ R are bilinear products. The given Poisson algebra R is recovered from its quantized algebra Q by R = Q/~Q with Poisson bracket (∗∗), which is called its semiclassical limit. But it seems that the star product in Q is complicate and that Q is difficult to understand at an algebraic point of view since it is too big. For instance, if λ is a nonzero element of C then ~−λ is a unit in Q and thus a so-called deformation of R, Q/(~− λ)Q, is trivial. Hence it seems that we need an appropriate F-subalgebra A of Q such that A contains all generators of Q, ~ ∈ A and A is understandable at an algebraic point of view, where F is a subring of C[[~]]. Here we discuss how to find nontrivial deformations from quantized algebras and how similar quantized algebras are to their semiclassical limits. Results are illustrated by examples.

where B i : R × R −→ R are bilinear products.The given Poisson algebra R is recovered from its quantized algebra Q by R = Q/ Q with Poisson bracket ( * * ), which is called its semiclassical limit.But it seems that the star product in Q is complicate and that Q is difficult to understand at an algebraic point of view since it is too big.For instance, if λ is a nonzero element of C then − λ is a unit in Q and thus a so-called deformation of R, Q/( − λ)Q, is trivial.Hence it seems that we need an appropriate F-subalgebra A of Q such that A contains all generators of Q, ∈ A and A is understandable at an algebraic point of view, where F is a subring of C[[ ]].
Here we discuss how to find nontrivial deformations from quantized algebras and the natural map in [6] from a class of infinite deformations onto its semiclassical limit.The results are illustrated by examples.
1 Motivation and Quantization

Star product
A commutative C-algebra R is said to be a Poisson algebra if there exists a bilinear product Let us see the following example.

Poisson Weyl algebra
The Poisson Weyl algebra is the for all f, g ∈ R, namely {y, x} = 1.Define a multiplication on the set R[[ ]] of formal power series over R = C[x, y] by -algebra and is a nonzero, nonunit, non-zero-divisor and central element such that In particular, {y, x} = 1.For λ ∈ C, let Q λ be the set of formal elements f | =λ for all f ∈ Q.For the case λ = 0, observe that Q 0 = R.For 0 λ ∈ C and f = 1 + x + 2 + 3 + . . .∈ Q, we do not know mathematical meaning of the formal form In particular, we should observe that The Weyl algebra W is the C-algebra generated by x, y subject to the relation Let A be a C[ ]-algebra of Q generated by x and y.Then A is a C[ ]-algebra generated by x, y subject to the relation yx − xy = .Note that the element ∈ A satisfies the following: • is a nonzero, nonunit, non-zero-divisor and central element.
• A/ A is commutative.
Hence the bilinear product {−, −} on A/ A R defined by is well defined and thus A/ A is a Poisson algebra isomorphic to the Poisson Weyl algebra and for each 0 λ ∈ C, A/( − λ)A is the C-algebra generated by x, y subject to the relation yx = xy + λ, which is isomorphic to the Weyl algebra W. In particular, makes sense mathematically and is isomorphic to A/( − λ)A as a C-algebra.

Semiclassical limit
Let A be a C-algebra.An element ∈ A is said to be a regular element if it is a nonzero, nonunit, non-zero-divisor and central element such that A/ A is commutative.
Let be a regular element.Then 0 A is a proper ideal such that the factor Let R and Q be as in §1.1.Then is a regular element of Q and the semiclassical limit Q/ Q is isomorphic to R as a Poisson algebra.In §1.2, is a regular element of A and the semiclassical limit A/ A is isomorphic to the Poisson Weyl algebra C[x, y] as a Poisson algebra.

Deformations
Retain the notations in §2.1.Suppose that there is an element 0 λ ∈ C such that − λ is a nonunit in A. Then ( − λ)A is a proper ideal of A and thus the factor A λ := A/( − λ)A is a nontrivial C-algebra such that its multiplication is induced by that of A. The factor A λ is called a deformation of A. For instance, the algebra A λ in §1.2 is a deformation of the Poisson Weyl algebra.
An algorithm to obtain a deformation is given as follows.Let F be a subring of C[[ ]] containing C[ ] and let A be an F-algebra generated by x 1 , . . ., x n subject to relations f 1 , . . ., f r .For λ ∈ C, assume that f i | =λ makes sense mathematically for each i = 1, . . ., r.The Calgebra generated by x 1 , . . ., x n subject to the relations f 1 | =λ , . . ., f r | =λ , still denoted by A λ , is deeply related to A. If g| =λ makes sense mathematically for all g ∈ A then the evaluation map is a C-algebra epimorphism and thus A λ A/ ker ϕ λ and the multiplication of A λ is induced by that of A. If is a regular element then A λ is a deformation of A. An example of this algorithm is given in §3.

Weyl algebra
As shown in §1.
Then is a regular element in A, since is regular in Q, and A/ ker φ C[x, y] as a C-algebra.For each 0 λ ∈ C, let A λ be the C-algebra generated by cos λ, sin λ, x, y subject to the relation yx = (cos λ)xy + sin λ.

Then the evaluation map
is an algebra ephimorphism and thus A/ ker ϕ λ A λ .In particular, the C-algebra A π for the case λ = π is generated by x, y subject to the relation yx + xy = 0, which can be considered as a deformation of the Poisson Weyl algebra.Then for each 0, ±1 q ∈ C, the deformation A q = A/( − q)A is the C-algebra generated by x, y subject to the relation yx = qxy and thus A q is the so-called coordinate ring O q (C 2 ) of affine 2-space.

A natural map from a class of infinite deformations onto its semiclassical limit
We assume the following conditions (i)-(vii) in [6, Notation 1.1].
(i) Assume that K is an infinite subset of the set C \ {0}.
(ii) Assume that F is a subring of the ring of regular functions on (iii) Let F x 1 , . . ., x n be the free F-algebra on the set {x 1 , . . ., x n }.A finite product x of x i 's (repetitions allowed) is called a monomial.For each i = 1, . . ., r, let f i be an F-linear combination of monomials Set A = F x 1 , . . ., x n /I, where I is the ideal of F x 1 , . . ., x n generated by f 1 , . . ., f r .That is, A is the F-algebra generated by x 1 , . . ., x n subject to the relations f 1 , . . ., f r .
(iv) Assume that is a regular element.Hence there exists the semiclassical limit A/ A. Denote by ϕ 0 the canonical projection (v) For each λ ∈ K, let A λ be the C-algebra generated by x 1 , . . ., x n subject to the relations

Note that a i
x (λ) is a well-defined element of C by (4) and that the evaluation map is an epimorphism of C-algebras.In other words, A λ is a deformation of the Poisson algebra A/ A. (vi) Let λ∈K A λ be the product of infinite deformations A λ and let ϕ be the homomorphism of C-algebras defined by Note that ϕ( ) is an invertible element of λ∈K A λ since 0 K. (vii) Assume that there exists an F-basis {ξ i | i ∈ I} of A such that {ϕ 0 (ξ i )|i ∈ I} and {ϕ λ (ξ i )|i ∈ I} are C-bases of A/ A and A λ , respectively, for each λ ∈ K. Hence every element a ∈ A is expressed uniquely by Note that a i (λ) and a i (0) are well-defined elements of C by (4).
(iv) Let A q be the C-algebra obtained by replacing in A by q and let be the map defined by :

Theorem [Oh, 2017]
Here we write a natural map from a class of infinite deformations onto its semiclassical limit in [6, §1].Set A = −1 (ϕ(A)) ⊂ A q , the inverse image of ϕ(A) by the map .Since ϕ is a monomorphism of C-algebras by [6, Lemma 1.2], there exists the composition which is an epimorphism of C-algebras such that γ( q) = 0 and γ(x i ) = ϕ 0 (x i ) for i = 1, . . ., r.
Here we write applications of the map γ in (6).
• One should observe [6,Example 1.6] in which the map γ in ( 6) induces a homeomorphism between the primitive spectrum of the coordinate ring O q (C 2 ) of quantized affine 2-space and the Poisson primitive spectrum of its semiclassical limit O(C 2 ).
• One should observe [4,Theorem 4.2] in which the map γ in (6) induces a monomorphism from the group of automorphisms of Weyl algebra into the group of Poisson automorphisms of Poisson Weyl algebra.

2 . 4
2, let A be the C[ ]-subalgebra of Q = R[[ ]] generated by x and y.Then ∈ A is a regular element and its semiclassical limit A = A/ A is the Poisson Weyl algebra R = C[x, y] with the Poisson bracket {y, x} = 1.For each 0 λ ∈ C, deformation A λ = A/( − λ)A is the C-algebra generated by x and y subject to the relation yx − xy = λ.Note that every deformation A λ is isomorphic to the Weyl algebra W by [4, Proposition 3.4].ITM Web of Conferences 20, 01008 (2018) https://doi.org/10.1051/itmconf/20182001008ICM 2018 Another deformation of Poisson Weyl algebra Here we recall [5, Example 3.8].Let Q be a C[[ ]]-algebra generated by x, y subject to the relation yx = (cos )xy + sin .Denote by A the C[ ]-subalgebra of Q generated by cos , sin and x, y.Consider the composition of two natural maps A/ ker φ by (3).Then the Poisson bracket in A/ ker φ is {y, x} = −(sin | =0 )xy + cos | =0 = 1 and thus A/ ker φ C[x, y], the Poisson Weyl algebra as a Poisson C-algebra.
and let A be the F-algebra generated by x, y subject to the relation yx = xy.Note that − 1 is a regular element in A and the semiclassical limit A = A/( − 1) is the Poisson algebra C[x, y] with Poisson bracket {y, x} = xy.