arXiv:math-ph/0007031v1 25 Jul 2000 ON CROSSED PRODUCT OF ALGEBRAS∗ Andrzej Borowiec and Wladyslaw Marcinek Institute of Theoretical Physics, University of Wroclaw Plac Maxa Borna 9, 50-204 Wroclaw POLAND Abstract The concept of a crossed tensor product of algebras is studied from a few points of views. Some related constructions are considered. Crossed enveloping algebras and their representations are discussed. Applications to the noncommutative geometry and particle systems with generalized statistics are indicated. PACS. 02. 40. +m - Differential geometry in theoretical physics. PACS. 03. 65. Fd - Algebraic methods in quantum theory ∗ to be published in J. Math. Phys. v41 no10 (2000) I INTRODUCTION I 2 Introduction The notion of a crossed product of Hopf algebras is well–known [1]. It is also known that there is an algebra analogue for such product called a crossed (braided) product of algebras. It has been used in several constructions in the area of the nonocommutative geometry, quantum groups and braided categories. For example this product in braided categories has been studied intensively by Majid [2, 3, 4, 5]. It is interesting that there is a more general notion of a crossed product of algebras without the notion of the braided categories. Namely, if A and B are two unital and associative algebras over a field k, then such product is formed by a bigger algebra W. This algebra contains algebras A and B as subalgebras in such a way that W is generated as an algebra just by A and B. This product is called in general a crossed (or a twisted) tensor product and it has been recently studied on an abstract algebraic level by Van Daele and Van Keer [6], by Čap, Schichl and Vanžura [7]. An application of such product in the area of C ∗ –algebras has been considered by Woronowicz [8]. A crossed tensor product has been also used by Zakrzewski [9] in the study of quantum Lorentz and Poincaré groups. An interesting approach for the study of noncommutative de Rham complexes has been developed by Manin [10]. It is based on the notion of the so–called skew product of algebras. Similar concept corresponding to the algebra of differential forms on a full matrix bialgebra has been developed by Sudbery [11]. According to his construction such algebra is a skew product of an algebra of functions and an algebra of differential forms with constant coefficients. It is obvious that such skew product provide an example of a crossed tensor product. Related subject has been also considered by Wambst [12]. One can see that in general the algebra W of differential operators acting on an algebra A can be described as a crossed product of the algebra and the algebra of vector fields corresponding for an arbitrary noncommutative differential calculus [13, 14, 15, 16]. In the present paper we are going to study the concept of a crossed tensor product of algebras from a few different points of views: module theory, Hopf algebras, free product of algebras and some constructions related to the noncommutative geometry. Our considerations are motivated by the application to the construction of the so–called crossed enveloping algebras and their representations. Note that Wick algebras considered previously in the study of deformed commutation relations [17] are particular examples of crossed enveloping algebras. Such algebras has been also important in the study of systems with generalized quantum statistics [18, 19, 20]. The paper is organized as follows. We recall the definition of the crossed product in the Section 2. The corresponding module structures are considered in this section. The relation between this product and the smash product or the semi–simple product of Hopf algebras is given. The connection with the free product of algebras is studied in Section 3. The construction of twisted product for free algebras is described in details in the Section 4. Ideals in the twisted products and corresponding quotient constructions are studied in the Section 5. Consistency conditions for such constructions are described as consequences of axioms for the twisted product. Some examples are given. In the Section 6 representations of the twisted tensor product are considered. Crossed enveloping algebras are described as a twisted tensor product of a pair of conjugated algebras. Representations of crossed enveloping algebras are also considered. As an example the Fock space representation for a system with generalized statistics is given. 3 II PRELIMINARIES II Preliminaries In this note k is a field of complex (or real) numbers . All objects considered here are first of all k-vector spaces. All maps are assumed to be k-linear maps. The tensor product ⊗ means ⊗k . In what follows algebra means associative unital k-algebra and homomorphisms are assumed to be unital. If A is an algebra then Aop denotes algebra with the opposite multiplication: a ·op a′ = a′ a . For comultiplication ∆ we shall use a P P ′ ′′ shorthand Sweedler (sigma) notation ∆(a) = i ai ⊗ ai ≡ a(1) ⊗ a(2) (with i omitted). Likewise, throughout the paper we will use the Sweedler type notation for a twisting map (see below) τ : B ⊗ A −→ A ⊗ B :i.e. we will write τ (b ⊗ a) ≡ a(1) ⊗ b(2) ; again the summation is assumed here but not written explicitly. Let A and B be two unital and associative algebras over k. The multiplication in these algebras is denoted by mA and mB , respectively. Let us recall briefly the concept of a crossed product of algebras [7]. Definition: An associative algebra W equipped with two injective algebra homomorphisms iA : A ֒→ W and iB : B ֒→ W such that the canonical linear map Φ : A ⊗k B −→ W defined by Φ(a ⊗ b) := mW ◦ (iA ⊗ iB )(a ⊗ b) (1) is a linear isomorphism is said to be a crossed (twisted) product of A and B. The above definition means that the crossed product of algebras A and B is a bigger algebra W which contain these two algebras as subalgebras in such a way that W is generated by A and B. In particular as a linear space the algebra W is isomorphic to A ⊗ B. The definition is given up to the isomorphism of algebras. As an example, we can consider the standard tensor product of algebras with multiplication given by the formula (a ⊗ b)(a′ ⊗ b′ ) := aa′ ⊗ bb′ for a ⊗ b, a′ ⊗ b′ ∈ A ⊗ B. For our purposes here, we shall denote by A ⊗c B an algebra being the standard tensor product of two algebras A and B. One will see in the moment that this example does not exhaust all possible cases. First, let us study module structures on the above crossed product of algebras. If A and B are algebras, then an arbitrary linear space M is said to be a (A, B)–bimodule if M is left A–module and right B–module and both structures commute, i. e. (a.m).b = a.(m.b), a ∈ A, b ∈ B, m ∈ M. (2) In other words, (A, B)–bimodules are left modules over A⊗c Bop . We shall also identify A– bimodules with (A, A)–bimodules. Observe that the tensor product A⊗B has a canonical left A–module and right B–module structure defined by a.(a′ ⊗ b) := aa′ ⊗ b, (a ⊗ b).b′ := a ⊗ bb′ , (3) respectively. This means that A ⊗ B inherits a (A, B)–bimodule structure in natural way. The (B, A)–bimodule structure on A ⊗ B is a problem. We use the concept of module cross [16] for the study of this problem. Let τ : B ⊗ A −→ A ⊗ B be a linear mapping, then we define the left, right and two-sided universal lift of τ as mappings (cf. [16]) u τ : A ⊗ B ⊗ A −→ A ⊗ B, τ (a ⊗ b ⊗ a′ ) := a′ .τ (b ⊗ a), τ u : B ⊗ A ⊗ B −→ A ⊗ B, τ u (b′ ⊗ a ⊗ b) := τ (b ⊗ a).b′ , u u τ : B ⊗ A ⊗ B ⊗ A −→ A ⊗ B, u τ u (b′ ⊗ a ⊗ b ⊗ a′ ) := a′ .τ (b ⊗ a).b′ , u respectively. (4) 4 II PRELIMINARIES Lemma: A mapping τ : B ⊗ A −→ A ⊗ B define on A ⊗ B a structure of: (i) a right A–module, (ii) left B–module, (iii) (A, B)–bimodule, if and only if the corresponding universal lift u τ , τ u or u τ u is the algebra homomorphism u (i) τ ∈ alg(Aop , Endk (A ⊗ B)), (ii) τ u ∈ alg(B, Endk (A ⊗ B)), (iii) u τ u ∈ alg(B ⊗c Aop , Endk (A ⊗ B)), (5) respectively. Proof: Let M be a left B-module. For each b ∈ B define Lb ∈ Endk M by Lb (m) := b.m. Then L : B −→ Endk M is an algebra homomorphism. Similarly, right A-module structures on M are in one-to-one correspondence with left Aop structures on M. Now (iii) is obvious, since any (B, A)-bimodule structure is in fact, due to commutativity (2), a left B ⊗c Aop structure. ✷ Definition: A (A, B)–bimodule W which is at the same time a (B, A)–bimodule, Abimodule and B–bimodule is said to be a crossed (A, B)–bimodule. Theorem: There is one to one correspondence between crossed (A, B)–bimodule structure on A ⊗ B and linear mappings τ : B ⊗ A −→ A ⊗ B satisfying the following relations and τ (1 ⊗ a) = a ⊗ 1 τ ◦ (mB ⊗ idA ) = (idA ⊗ mB ) ◦ (τ ⊗ idB ) ◦ (idB ⊗ τ ) (6) τ (b ⊗ 1) = 1 ⊗ b τ ◦ (idB ⊗ mA ) = (mA ⊗ idB ) ◦ (idA ⊗ τ ) ◦ (τ ⊗ idA ), (7) Proof: The left A–module and a right B–module acting on A ⊗ B is given by formulae (3). We define a right A–module and a left B–module action on A ⊗ B by formulae (a ⊗ b).a′ := a′ τ (b ⊗ a) = u τ (a ⊗ b ⊗ a′ ), b′ .(a ⊗ b) := τ (b ⊗ a)b′ = τ u (b′ ⊗ a ⊗ b), respectively. (8) ✷ Definition: A k-linear mapping τ : B ⊗A −→ A ⊗B satisfying the condition (6) is called a left B-module cross. Similarly, if the relation (7) is satisfied, then τ is called a right A-module cross. The map τ is said to be an algebra cross if it is both a left B- and right A-module cross. It is obvious that the standard twist (switch) τ : B⊗A −→ A⊗B defined by τ (b⊗a) := a⊗b satisfies all conditions for the cross. It give rise to the standard tensor product of algebras A⊗c B. The second example is a graded algebra version of the previous one. It is provided by the following graded twist τ (b ⊗ a) := (−1)mn a ⊗ b, (9) where a ∈ A, b ∈ B are homogeneous elements of graded algebras A and B of grade m and n, respectively. Let τ : B ⊗ A −→ A ⊗ B be an algebra cross, then according to the last theorem there exists a structure of a crossed (A, B)–bimodule on A ⊗ B. This structure will be denoted by A >✁τ B. 5 II PRELIMINARIES Lemma: Let W be a crossed (A, B)–bimodule. Assume that the algebra A as k-submodule universally generates W as a left A–module and similarly B generates W as a right B– module. Then there exist the unique algebra cross τ : B ⊗ A −→ A ⊗ B such that W = A >✁τ B. Proof: We denote by Φ : A ⊗τ B −→ W the mapping which is a left A–module and a right B–module isomorphism. We define τ (b ⊗ a) := Φ−1 (ba). ✷ Theorem: There is one to one correspondence between algebra cross τ : B ⊗A −→ A ⊗B and crossed product W of algebras A and B. Proof: Let us assume that the algebra W is universally generated crossed product of algebras A and B. We define a linear mapping τW : B ⊗ A −→ A ⊗ B by the following formula τW (b ⊗ a) := [mW ◦ (iA ⊗ iB )]−1 (ba). (10) It can be deduced that the above mapping is an algebra cross. Moreover, the map mW ◦ (iA ⊗iB ) is an algebra isomorphism of A >✁τ B onto W. Conversely, let τ : B⊗A −→ A⊗B be an algebra cross, then the tensor product A ⊗ B of algebras A and B equipped with the multiplication mτ : (A ⊗ B) ⊗ (A ⊗ B) −→ A ⊗ B defined by the formula mτ := (mA ⊗ mB ) ◦ (idA ⊗ τ ⊗ idB ) (11) is associative [6, 7]. In this case both relations (7b) and (6b) can be written equivalently by τ ◦ (mB ⊗ mA ) = mτ ◦ (τ ⊗ τ ) ◦ (idB ⊗ τ ⊗ idA ). (12) It is easy to see that A ⊗ B equipped with the above multiplication is a crossed product of algebras A and B. The inclusion iA and iB are defined as follows iA (a) := a ⊗ 1, iB (b) := 1 ⊗ b. (13) ✷ Observe that the multiplication (11) in the crossed product A >✁τ B of algebras A and B can be given in the following form (a′ ⊗ b)(a ⊗ b′ ) = a′ a(1) ⊗ b(2) b′ , (14) where, as already mentioned, the Sweedler type notation for the cross τ has been used, i.e. τ (b ⊗ a) := a(1) ⊗ b(2) . (15) In particular, we have the relations (a ⊗ 1)(1 ⊗ b) = a ⊗ b, (1 ⊗ b)(a ⊗ 1) = a(1) ⊗ b(2) . (16) It is interesting that elements of the algebra A >✁τ B can be ordered in such a way that all elements of the algebra B are to the right and elements of the algebra A are to the left. Such ordering is said to be Wick ordering. As we already mentioned before, the crossed product related to the standard twist is known as the tensor product of algebras. If we use the graded twist, then we obtain the 6 II PRELIMINARIES graded tensor product of graded algebras. In the general case, however, an algebra cross is not so simple. The construction of all possible crossed products for a given pair of algebras A, B is a problem. It is difficult to describe a general method for an arbitrary pair of algebras. Hence we restrict our attention to some particular classes of algebras. Let B be a bialgebra. This means that there is an algebra homomorphism ∆ : B −→ B⊗B and the counit ε. We have here the following well–known conditions: the coassociativity (id ⊗ ∆) ◦ ∆ = (∆ ⊗ id) ◦ ∆, (17) and two relations for the counit (ε ⊗ id) ◦ ∆ = 1 ⊗ id, (id ⊗ ε) ◦ ∆ = id ⊗ 1. (18) An algebra A is said to be a left B–module algebra if there is an action ✄ : B −→ A such that b ✄ (aa′ ) = (b(1) ✄ a)(b(2) ✄ a′ ), (19) 1 ✄ a = a. We have the following: Lemma: If A is a left B–module algebra, then there is an algebra cross τ : B⊗A −→ A⊗B defined by the relation τ (b ⊗ a) = (b(1) ✄ a) ⊗ b(2) (20) for a ∈ A and b ∈ B. Proof: We shall show that two condition (7, 6) for the cross (20) are satisfied. For the left hand side of the first relation (7) we calculate [τ ◦ (idB ⊗ mA )] (b ⊗ a ⊗ a′ ) ′ (2) = [b(1) h ✄ m(a ⊗ a ) ⊗ b ] i = m (b(1) ✄ a) ⊗ (b(2) ✄ a′ ) ⊗ b(3) , where (id ⊗ ∆) ◦ ∆(b) = b(1) ⊗ b(2) ⊗ b(3) and the coassociativity condition have been used. For right hand side we obtain [(mA ⊗ idB )h ◦ (idA ⊗ τ ) ◦ (τ ⊗ idAi)] (b ⊗ a ⊗ a′ ) = m (b(1) ✄ a) ⊗ (b(2) ✄ a′ ) ⊗ b(3) . The second relation (6) can be calculated in a similar way. ✷ If τ is the cross defined by the formula (20), then the multiplication in the corresponding crossed product A >✁τ B is given by (a ⊗ b)(a′ ⊗ b′ ) = a(b(1) ✄ a′ ) ⊗ b(2) b′ . (21) We can see, in this case, that the crossed product A >✁τ B is exactly the so–called smash product A = //B, see Ref.[22, 1]. If in addition A and B are endowed with a Hopf algebra structure, then the corresponding crossed product is the semi–simple product of Hopf algebras introduced by Molnar [21]. Let A and B be a dual pair of Hopf algebras [25]. This means that we have a bilinear pairing < ., . >: B ⊗ A −→ k such that < ∆(b), a ⊗ a′ >=< b, aa′ >, < bb′ , a >=< b ⊗ b′ , ∆(a) > . (22) 7 III FREE PRODUCT OF ALGEBRAS Observe that there is a left action of the algebra B on A b ✄ a =< b, a(2) > a(1) , (23) One can prove that the algebra A is a (left) B–module algebra and the mapping τ : B ⊗ A −→ A ⊗ B defined by τ (b ⊗ a) ≡ a(1) ⊗ b(2) := b(1) ✄ a ⊗ b(2) =< b(1) , a(2) > a(1) ⊗ b(2) (24) is a cross. This is interesting point that the corresponding crossed product A >✁τ B contains all information about noncommutative differential operators [24] on A. It means that we can forget the Hopf algebra structures in A and B and restrict our attention to the algebra structure only. In this case we obtain the so–called crossed product of algebras [7]. III Free product of algebras Let A and B be two unital associative algebras over a field k. Then there is an algebra of polynomials containing elements of these two algebras. This algebra is said to be a free product of algebras A and B [23]. Namely, we have here the Definition: An (algebraic) free product of algebras A and B is the algebra A ∗ B formed by all formal finite sums of monomials of the form a1 ∗ b1 ∗ a2 ∗ . . . or b1 ∗ a1 ∗ b2 ∗ . . ., where ai ∈ A, bi ∈ B, i = 1, 2, . . . are non-scalar elements. In other words A ∗ B is the algebra generated by two algebras A and B with no relations except for the identification of unit element, i.e. 1A = 1B = 1. One can see that this free product of algebras is commutative and associative A ∗ B = B ∗ A, (A ∗ B) ∗ C = A ∗ (B ∗ C). (25) Moreover, if A1 is a subalgebra of A and B1 is a subalgebra of B, then A1 ∗ B1 is a subalgebra of A ∗ B. In particular, the algebras A and B are subalgebras of A ∗ B. It is known that the product A ∗ B possesses the following universal property: Lemma: For every pair of algebra maps u : A −→ C and v : B −→ C there exist one and only one algebra map w such that u = w ◦ jA and v = w ◦ jB or in other words the following diagram A u jA ↓ ց A ∗ B −→ C jB ↑ B ր w (26) v commutes. Here, jA (resp. jB ) denotes the natural inclusion of A (resp. B) into A ∗ B. Note that w is onto if and only if C is generated by images u(A) and v(B). Proof: The proof can be immediately seen if one defines w(. . . b ∗ a . . .) = . . . v(b)u(a) . . . III FREE PRODUCT OF ALGEBRAS 8 ✷ Let us consider a simple example of an algebraic free product. Example: Let U and W be two k-vector space. Then the tensor algebra over the direct sum U ⊕ W is a free product of tensors algebras T U and T W , i.e. we have the relation T (U ⊕ W ) = T U ∗ T W. Let us consider the free product of maps. Definition: Let f : A −→ B and g : C −→ D be two algebra maps. Then a mapping f ∗ g : A ∗ B −→ C ∗ D defined by f ∗g (. . . b ∗ a . . .) = . . . g(b) ∗ f (a) . . . (27) for a ∈ A and b ∈ B, is called a free product of f and g. We have the following simple lemma. Lemma: The map f ∗ g is injective (resp. surjective) if and only if the maps f and g are injective (resp. surjective). ✷ Now we are going to study ideals in the free product of algebras. It is interesting that in a free product A ∗ B of algebras A and B may exists an ideal J such that the quotient (A ∗ B)/J can be also expressed as a free product of certain algebras. Lemma: Let IA and IB be ideals in algebras A and B, respectively. Then J(IA , IB ) := IA ∗ B + A ∗ IB (28) forms an ideal in the free product A ∗ B such that A ∗ B/J(IA , IB ) = (A/IA) ∗ (B/IB ) . (29) Proof: Let πA : A −→ A/IA and πB : B −→ B/IB be canonical projections, i.e. IA := kerπA and IB := kerπB . It is obvious that the free product πA ∗ πB : A ∗ B −→ (A/IA ) ∗ (B/IB ) is surjective and (A/IA) ∗ (B/IB ) = (A ∗ B)/ker(πA ∗ πB ). One can see that ker(πA ∗ πB ) = ker(πA ) ∗ B + A ∗ ker(πB ) = J(IA , IB ). ✷ The ideal J(IA , IB ) from the above lemma is called a free ideal in A ∗ B generated by IA and IB . If A and B are two k-algebras and τ : B ⊗A −→ A⊗B is a cross, then the crossed product A >✁τ B of these algebras can be given by their free product modulo certain ideal. More precisely, we have here the following Lemma: For the crossed product A >✁τ B of algebras A and B we have the formula A >✁τ B = (A ∗ B)/Iτ , (30) where Iτ is an ideal generated by the relation Iτ = gen{b ∗ a − a(1) ∗ b(2) } (31) for a ∈ A, b ∈ B and τ (b ⊗ a) := a(1) ⊗ b(2) . Proof: We use the universality of the free product. If C ≡ A >✁τ B, then u ≡ iA , v ≡ iB , and there exist unique morphism w ≡ iA ∗ iB such that iA = w ◦ jA and iB = w ◦ jB . Observe that w is a morphism from A∗B to A >✁τ B, and his kernel is equal to the ideal Iτ . ✷ 9 IV CROSSED PRODUCT OF FREE ALGEBRAS IV Crossed product of free algebras Let A and B are graded algebras. This means that we have the following decompositions A= ∞ M Ak , B= k=0 ∞ M Bk , (32) k=0 where A0 ∼ = k. Let τ : B ⊗ A −→ A ⊗ B be an arbitrary algebra cross. Then the = B0 ∼ algebra cross τ can be given by the relation τ= ∞ M τk,l , (33) k,l=0 where τk,l is the restriction of the algebra cross τ to the space Bk ⊗ Al , (k, l = 1, 2, . . .). In such a way the algebra cross τ can be reduced by a set of mappings {τi,j : Bi ⊗ Aj −→ A ⊗ B}. Observe that we always have τ0,0 ≡ id, τk,0(Bk ⊗ 1) := 1 ⊗ Bk , τ0,m (1 ⊗ Am ) := Am ⊗ 1. (34) We have here the following problem: Find the conditions under which all τk,l for k, l > 1 can be constructed starting from τ1,1 . The mapping τ1,1 is given as an initial data for such construction. We restrict our attention to certain particular cases. Let us consider the crossed product of free algebras in details. Let A be a free algebra generated by x1 , . . . , xm and let B be a free algebra generated by y 1, . . . , y n . We identify these free algebras A, B with tensor algebras T E and T F , respectively, where E is a linear span of generators x1 , . . . , xm of A and F is a linear span of y 1, . . . , y n . This means that A1 ≡ E, Ak ≡ E ⊗k , and similarly for B. Note that E and F are said to be generating spaces for algebras A and B, respectively. Let us consider the structure of the crossed product of free algebras T E and T F in more details. Remark: Let τ1,1 : F ⊗ E −→ E ⊗ F be a linear mapping, then there is a unique algebra cross τ : T F ⊗ T E −→ T E ⊗ T F such that τ |F ⊗E = τ1,1 . Indeed, if τ1,1 : F ⊗ E −→ E ⊗ F is a linear mapping, then we can introduce a set of mappings {τi,j : F ⊗i ⊗E ⊗j −→ T E ⊗T F } as follows: Obviously τ0,0 ≡ id, τk,0 (F ⊗k ⊗1) := 1 ⊗ F ⊗k , and τ0,m (1 ⊗ E ⊗m ) := E ⊗m ⊗ 1. Then the algebra cross τ can be defined by the relations (6) and (33). For example τ2,1 = (τ1,1 ⊗ id) ◦ (id ⊗ τ1,1 ). (35) τ1,2 = (id ⊗ τ1,1 ) ◦ (τ1,1 ⊗ id). (36) Similarly We can calculate τk,l for arbitrary k, l in a similar way. Let us consider this case in more details. Definition: Let A and B be two graded algebras. An algebra cross τ is said to be homogeneous if the image of τk,l lies in Al ⊗ Bk for all k, l = 1, 2, . . .. It is obvious that he homogeneous cross can be determined uniquely by a set of linear mappings τk,l : Bk ⊗ Al −→ Al ⊗ Bk such that τk,l+m ◦ (idA ⊗ mA ) = (mA ⊗ idB ) ◦ (idA ⊗ τk,m ) ◦ (τk,l ⊗ idA ), τk+l,m ◦ (mB ⊗ idA ) = (idA ⊗ mB ) ◦ (τk,m ⊗ idB ) ◦ (idB ⊗ τl,m ). (37) 10 IV CROSSED PRODUCT OF FREE ALGEBRAS for arbitrary integers k, l, m > 0. Consider two free algebras A := T E and B := T F with their natural gradings. Choose a basis x1 , . . . , xm in E and a basis y 1 , . . . , y n in F . Now, the linear operator τ1,1 ≡ τ̂ : F ⊗ E −→ E ⊗ F can be expressed by τ̂ (y i ⊗ xj ) = τ̂klij xk ⊗ y l , (38) its matrix elements τ̂klij . Let us calculate all components τk,l : F ⊗k ⊗E ⊗l −→ E ⊗l ⊗F ⊗k for this cross. Obviously for k = 1 and arbitrary l > 1 we obtain the map τ1,l : F ⊗ E ⊗l −→ E ⊗l ⊗ F , where (l) (1) (39) τ1,l := τ̂l ◦ . . . ◦ τ̂l , (i) (i) (i+1) (i) and τ̂l : El −→ El , El := E ⊗ . . . ⊗ E ⊗ F ⊗ E ⊗ . . . ⊗ E (l + 1-factors, F on the i-th place, 1 ≤ i ≤ l) is given by the relation (i) τ̂l := idE ⊗ . . . ⊗ τ̂ ⊗ . . . ⊗ idE , {z | } l times where τ̂ is on the i-th place. One verifies that (i) (j) τ̂l ◦ τ̂l (j) = τ̂l (i) ◦ τ̂l if |i − j| ≥ 2. For arbitrary k ≥ 1 and l ≥ 1 we obtain the map τk,l : F ⊗k ⊗ E ⊗l −→ E ⊗l ⊗ F ⊗k , where τk,l := (τ1,l )(1) ◦ . . . ◦ (τ1,l )(k) , (40) (i) where (τ1,l )(i) is defined in similar way like τ̂l . In this way we obtain the result: Lemma: Let T E and T F be free algebras and τ̂ be a linear operator defined on generators of these algebras by the relation (38), then there is a homogeneous algebra cross τ : T F ⊗ T E −→ T E ⊗ T F which is given by the relations (39) and (40). Proof: We must prove that for the map τ defined by relations (39) and (40) the identities (37) hold true. Observe that we have mA (a ⊗ a′ ) ≡ a ⊗ a′ for a ∈ E ⊗l , a′ ∈ E ⊗m and similarly for mB , i.e. mA and mB act as identity operators in this case. Therefore, the relations (37) can be rewritten in a simpler form τk,l+m = (idT E ⊗ τk,m ) ◦ (τk,l ⊗ idT E ), τk+l,m = (τk,m ⊗ idT F ) ◦ (idT F ⊗ τl,m ). (41) After substituting the definition (40) of the maps τk,l into (41) and some calculations we obtain our result. ✷ We have here the following Theorem: Let τ : T F ⊗ T E −→ T E ⊗ T F be an arbitrary cross. Then for the corresponding crossed product we have the following relation T E >✁τ T F = T (E ⊕ F )/Iτ , (42) Iτ := gen{b ⊗ a − a(1) ⊗ b(2) } (43) where 11 V IDEALS IN CROSSED PRODUCT is an ideal in T (E ⊕ F ). If the cross τ is homogeneous, then Iτ := gen{y i ⊗ xj − τ̂klij xl ⊗ y k }. (44) Proof: According to the last Lemma of the previous Section for the crossed product we have the relation T E >✁τ T F = (T E ∗ T F )/Iτ = T (E ⊕ F )/Iτ . ✷ Now it is obvious that in the study of noncommutative de Rham complexes and noncommutative calculi with partial derivatives there are several examples of algebras which can be described as algebra crossed product [10, 12, 15, 28, 29, 30]. If the operator τ̂ is given by the diagonal matrix τ̂klij := tij δli δkj , tij ∈ k \ {0}, then we obtain a simple example of cross for free algebras, namely the so called color cross τ (y i ⊗ xj ) = tij xj ⊗ y i , (45) If we assume that k ≡ C I and tij ≡ q, q ∈ C I \ {0}, then we obtain the q-cross. V Ideals in crossed product Let us assume that A >✁τ B and A′ >✁τ ′ B are crossed product of algebras A, B and A′, B′ with respect to a cross τ and τ ′ , respectively. It is natural to define a morphism of such two crossed products of algebras as a map which transform the first crossed product in the second one. Definition: An algebra morphism h : A >✁τ B −→ A′ >✁τ ′ B′ is said to be a crossed product algebra morphism if there exist two algebra morphism: hA : A −→ A′ and hB : B −→ B′ such that h = hA ⊗ hB . The above definition means that the crossed product algebra morphism h : A >✁τ B −→ A′ >✁τ ′ B is described as a pair of algebra homomorphisms hA : A −→ A′ and hB : B −→ B′ . Observe that in the opposite case when we have an arbitrary pair of algebra homomorphism, then their tensor product is not a crossed product algebra morphism, however, there is the following lemma: Lemma: Let hA : A −→ A′ and hB : B −→ B′ be two algebra morphism. Then h = hA ⊗ hB is a crossed product algebra morphism if and only if we have the relation (hA ⊗ hB ) ◦ τ = τ ′ ◦ (hB ⊗ hA ), (46) or in other words the following diagram commutes τ B ⊗ A −→ A ⊗ B ↓hA ⊗hB hB ⊗hA ↓ ′ τ B′ ⊗ A′ −→ A′ ⊗ B′ (47) ✷ We introduce the notion of ideals in crossed product of algebras. Let A >✁τ B be a crossed product of algebras A and B with respect to a cross τ , then we have the following: 12 V IDEALS IN CROSSED PRODUCT Definition: A two–sided ideal J in A >✁τ B is said to be a crossed ideal in A >✁τ B if the quotient map π : A >✁τ B −→ (A >✁τ B)/J is a morphism of crossed products of algebras. The above definition means that the factor algebra (A >✁τ B)/J, where J is a crossed ideal must be a crossed product of certain algebras A′ , B′ with respect to a certain new cross τ ′ . Thus we must have the relation (A >✁τ B)/J ∼ = A′ >✁τ ′ B′ . (48) Let us consider this problem in more details. If π : A >✁τ B −→ (A >✁τ B)/J is a surjective morphism of a crossed product of algebras, then there is a pair of surjective algebra homomorphisms πA : A −→ A′ and πB : B −→ B′ . Observe that these mappings are in fact quotient ones. This means that A′ ≡ A/IA , and B′ ≡ B/IB where IA (the kernel of πA ) is a two-sided ideal in A and IB is a two-sided one in B. One can see that there is a cross τ ′ : B/IB ⊗ A/IA −→ A/IA ⊗ B/IB such that the following diagram is commutative τ B⊗A −→ A⊗B ↓ ↓ (49) πB ⊗πA πA ⊗πB τ′ B/IB ⊗ A/IA −→ A/IA ⊗ B/IB . In this way we obtain the following: Lemma: If J is a crossed ideal in the crossed product A >✁τ B, then there is a pair of ideals (IA , IB ) in algebras A and B, respectively and the cross τ ′ : B/IB ⊗ A/IA −→ A/IA ⊗ B/IB such that we have the relation (48). ✷ It is interesting to investigate the opposite statement. For a given ideals (IA , IB ) in A and B, respectively find a corresponding ideal in A >✁τ B. First, we consider a particular case when one of the ideal in the above pair is trivial. Definition: A two–sided ideal IA in the algebra A such that IA ⊗ B is a crossed ideal in the algebra W ≡ A >✁τ B is said to be a left τ -ideal in A. Observe that we have the following criterion (cf. Proposition 3.2.4 in [16]) Lemma: An ideal IA in A is a left τ -ideal in A >✁τ B if and only if τ (B ⊗ IA ) ⊂ IA ⊗ B. (50) Proof: How can be easily seen, the condition (50) is equivalent to the fact that J := IA ⊗B is a two-sided ideal in A >✁τ B. Therefore, one has to prove that (50) implies that J is a crossed ideal as well. Indeed: the vector space quotient (A ⊗ B)/J is isomorphic to A/IA ⊗ B. Since J is an ideal, the projection map πA ⊗ idB : A >✁τ B −→ A/IA ⊗ B, where πA (a) = [a] and [a] ∈ A/IA denotes the equivalence class of a ∈ A, is an algebra map. In particular, πA ⊗ idB ((1 ⊗ b)(a ⊗ 1)) = ([1] ⊗ b) ⊗ ([a] ⊗ 1) = [a(1) ] ⊗ b(2) . This means that τ ′ ([a] ⊗ b) := [a(1) ] ⊗ b(2) is a new twist converting A/IA ⊗ B into a crossed product algebra. Moreover, the following diagram τ B⊗A −→ A⊗B ↓ ↓ . idB ⊗πA πA ⊗idB τ′ B ⊗ A/IA −→ A/IA ⊗ B (51) 13 V IDEALS IN CROSSED PRODUCT must commutes. The formula (50) gives the condition for the commutativity of the above diagram. ✷ It follows immediately from the proof of the previous lemma that we have: Lemma: If IA is a left τ -ideal in the algebra A, then there is a new cross τ ′ : B ⊗ A/IA −→ A/IA ⊗ B such that the quotient algebra (A >✁τ B)/(IA ⊗ B) is isomorphic to the crossed product ✷ A/IA >✁τ ′ B. This lemma means that for a left τ –ideal IA in A we have the relation (A >✁τ B)/(IA ⊗ B) ∼ = A/IA >✁τ ′ B. (52) This algebra is said to be a left factor of the crossed product A >✁τ B. We can define a right τ -ideal IB in B in a similar way. It is easy to see that for this ideal we have similar results as for the left τ -ideal. In this way we obtain a right factor of the crossed product A >✁τ B as the following quotient (A >✁τ B)/(A ⊗ IB ) ∼ = A >✁τ ′ B/IB . (53) A two-sided ideal JIA ,IB := IA ⊗ B + A ⊗ IB in A >✁τ B, where IA is a left τ -ideal in A and IB is a right τ -ideal in B, is said to be a crossed ideal generated by IA , IB . Theorem: If JIA ,IB is a crossed ideal in the algebra A >✁τ B generated by τ –ideals IA , IB , then there is a new cross τ ′ : B/IB ⊗A/IA −→ A/IA ⊗B/IB such that the quotient algebra (A >✁τ B)/JIA ,IB is isomorphic to the crossed product of A/IA and B/IB . ✷ The quotient algebra (A >✁τ B)/JIA ,IB is said to be a factor of the crossed product A >✁τ B with respect to τ –ideals (IA , IB ). Definition: Let A >✁τ B be a crossed product of algebras A and B with respect to a given cross τ : B ⊗ A −→ A ⊗ B. If there exist a pair of algebras Ã, B̃ and a cross τ̃ : B̃ ⊗ à −→ à ⊗ B̃ such that the product A >✁τ B is an image of à ⊗τ̃ B̃ under certain surjective morphism h = (hA , hB ) of crossed products, i.e. the following diagram τ̃ B̃ ⊗ à −→ à ⊗ B̃ ↓hA ⊗hB hB ⊗hA ↓ τ B ⊗ A −→ A ⊗ B (54) is commutative, then à ⊗τ̃ B̃ is said to be a cover crossed product for A >✁τ B. Lemma: Assume that A and B are algebras with presentation A := T E/IA and B := T F/IB . If τ̃ : T F ⊗ T E −→ T E ⊗ T F is a cross, then the corresponding crossed product T E >✁τ̃ T F is cover for a product A >✁τ B with certain cross τ : B ⊗ A −→ A ⊗ B if and only if the ideal IA is a left τ̃ -ideal in T E and IB is a right τ̃ -ideal in T F . ✷ Lemma: Let A >✁τ B be a crossed product of algebras A and B with presentation A := T E/IA and B := T F/IB , respectively. If the crossed product T E >✁τ̃ T F is a cover for the product A >✁τ B, then we have the relation A >✁τ B ≡ T (E ⊕ F )/I, (55) 14 V IDEALS IN CROSSED PRODUCT where I is the ideal in the tensor algebra T (E ⊕ F ) of the form I ≡ I1 + I2 + Iτ̃ , (56) I1 := hIA iT (E⊕F ) is an ideal in T (E ⊕ F ) generated by the τ̃ –ideal IA , similarly I2 := hIB iT (E⊕F ), and Iτ̃ is an ideal in T (E ⊕ F ) defined by the relation Iτ̃ := hv ⊗ u − τ̃ (v ⊗ u)iT (E⊕F ), for every u ∈ E and v ∈ F . (57) ✷ Lemma: Assume that E, F are two linear spaces and R : E ⊗E −→ E ⊗E, S : F ⊗F −→ F ⊗ F are two linear operators. Let A and B be two quadratic algebras generated by E and F . It means that we have the quotients A := T E/IR , B := T F/IS , (58) where ideals are given by the quadratic relations IR = hid − RiT E , IS = hid − SiT F . Assume further, that a homogeneous cross τ̃ : T F ⊗ T E −→ T E ⊗ T F is induced by a linear operator C : F ⊗ E −→ E ⊗ F . Then there is a cross τ : B ⊗ A −→ A ⊗ B and the corresponding crossed product A >✁τ B if and only if we have the following relations (id ⊗ C) ◦ (C ⊗ id) ◦ (id − (id ⊗ R)) = (id − (R ⊗ id)) ◦ (id ⊗ C) ◦ (C ⊗ id), (C ⊗ id) ◦ (id ⊗ C) ◦ (id − (S ⊗ id)) = (id − (id ⊗ S)) ◦ (C ⊗ id) ◦ (id ⊗ C). Moreover T (E ⊕ F )/I ∼ = T E/IR >✁τ T F/IS , (59) (60) where I is an ideal of the form I ≡ I1 + I2 + IC , (61) I1 := hIR iT (E⊕F ) , I2 := hIS iT (E⊕F ) and IC is an ideal given by the relation IC := hv ⊗ u − C(v ⊗ u)iT (E⊕F ), (62) for every u ∈ E and v ∈ F . Proof: One checks that (59) are equivalent to the τ̃ -ideal conditions (50) for IR and IS respectively. ✷ Lemma: Let R, C and S be three linear operators like in the previous lemma. If (R ⊗ id)(id ⊗ C)(C ⊗ id) = (id ⊗ C)(C ⊗ id)(id ⊗ R), (id ⊗ S)(C ⊗ id)(id ⊗ C) = (C ⊗ id)(id ⊗ C)(S ⊗ id), (63) then the conditions (59) are satisfied. ✷ Let us consider an example of an algebra crossed product. It is well–known that the VI CROSSED ENVELOPINGS AND REPRESENTATIONS 15 noncommutative differential calculi with partial derivatives on the quantum plane is determined by an R–matrix satisfying the following braid relation R(1) R(2) R(1) = R(2) R(1) R(2) , (64) (R − q)(R + q −1 ) = 0, (65) and the Hecke condition where q ∈ C I \ {0}. Here R1 means R ⊗ idE and analogously R2 := idE ⊗ R. The noncommutative coordinate algebra and the corresponding partial derivatives algebra can be expressed as the following quotients A := T E/IA and B := T E ′ /IB , where ideals IA and IB are generated by the quadratic relations IA := hid − q −1 RiT E , IB := hid − q −1 Rt iT E ′ , Rt is the transpose of R and E ′ is the algebraic dual of E. It follows from the previous lemma and Wess–Zumino consistency conditions [31], that there is a cross τR and the corresponding crossed product W(R) := A >✁τR B. In this case we must replace R by q −1 R, S := q −1 Rt , and C := qR. The Hecke relation solves a linear Wess-Zumino consistency condition (see also [14, 15, 16] in this context). Observe that W(R) is just the quantum Weyl algebra considered previously by Giaquinto and Zhang [32]. As we already mentioned, (A, B)–modules can be identify with left A ⊗c Bop –modules. Thus left A >✁τ Bop –modules supply a generalization for the (A, B)–bimodule structure (2). Below, by using the Fock space representation methods (the quantization), we are going to describe some class of A >✁τ Aop –modules. VI Crossed envelopings and representations The notion of ∗–algebras is well–known. An associative algebra A is said to be a ∗–algebra if we have the following relations for the ∗–operation (ab)∗ = b∗ a∗ , a∗∗ = a, (αa)∗ = αa∗ , (66) where a, b, a∗ , b∗ ∈ A, α ∈ C, I α is a complex conjugated to α. In this section we assume that k ≡ C I is the field of complex numbers. It is obvious that not every algebra is a ∗–algebra. Observe that if A is a ∗–algebra, then the ∗–operation can be described in two equivalent ways: as an involutive anti-isomorphism of A or an involutive isomorphism op between A and A . If A is not a ∗–algebra, then it is interesting to describe all possible ∗–algebra extensions of it. We introduce here the concept of conjugated algebras and crossed enveloping algebras for this goal. Definition: If A be an arbitrary associative algebra, then an algebra B is said to be conjugated to A if there is an antilinear anti-isomorphism (in the complex case) (−)⋆ : A −→ B such that (ab)⋆ = b⋆ a⋆ , (αa)⋆ = αa⋆ , (67) where a, b ∈ A and a⋆ , b⋆ are their images under the isomorphism (−)⋆ . The inverse isomorphism B −→ A will be denoted by the same symbol, i.e. (a⋆ )⋆ = a. (68) 16 VI CROSSED ENVELOPINGS AND REPRESENTATIONS If A is an algebra, then the conjugated algebra will be denoted by A⋆ . It follows immediately from the definition that for a given algebra A the conjugate algebra A⋆ always exists. The algebra A⋆ as a vector space is isomorphic to the complex conjugate space A, op and as an algebra – to the opposite one Aop , i. e. A ≡ A , (in the real case it coincides with the opposite algebra Aop ). Consider a crossed product Wτ (A) := A >✁τ A⋆ of an algebra with its conjugate. We can try to define the natural ∗-operation in Wτ (A) by the relation (a ⊗ b⋆ )∗ := b ⊗ a⋆ (69) for a, b ∈ A. Then the following holds Lemma: The algebra Wτ (A) is a ∗-algebra if and only if (τ (b⋆ ⊗ a))∗ = τ (a⋆ ⊗ b) (70) for any a, b ∈ A. Proof: One needs the property [(1 ⊗ b⋆ )(a ⊗ 1)]∗ = (1 ⊗ a⋆ )(b ⊗ 1) or b(2) ⊗ a⋆(1) = b(1) ⊗ a⋆(2) which is equivalent to the the relation (70). ✷ An algebra cross τ : A⋆ ⊗ A −→ A ⊗ A⋆ satisfying the relation (70) is called a ⋆–cross. Definition: If τ : A⋆ ⊗ A −→ A ⊗ A⋆ is a ⋆– cross, then the crossed product Wτ (A) := A >✁τ A⋆ . (71) is called a crossed enveloping algebra of A with respect to τ . Note that the switch σ(b⋆ ⊗ a) := a ⊗ b⋆ satisfies (70). It implies that crossed enveloping algebra generalizes the concept of enveloping algebras for associative algebra [35]. From now on, we assume that every algebra cross considered below is a ⋆–cross. We shall also identify our two ”star”–operations and use the symbol ” ⋆ ” for both of them. Let us consider the crossed enveloping algebra Wτ (A), where A ≡ T E is a free algebra and the generating space E is a (finite or infinite dimensional) complex Hilbert space equipped with an orthonormal basis {xi : i = 1, · · · , N}, and τ is an arbitrary cross. Note that similar algebras have been studied previously by a few authors [33, 34]. Observe that the conjugated algebra A⋆ can be identified with the tensor algebra T E ⋆ , where E ⋆ is the complex conjugation space. The pairing (.|.) : E ⋆ ⊗ E −→ C I and the corresponding scalar product is given by gE (x⋆i ⊗ xj ) ≡ (x⋆i |xj ) = hxi |xj i := δ ij . (72) Let τ̂ : E ⋆ ⊗ E −→ E ⊗ E ⋆ be a linear and Hermitian operator with matrix elements τ̂ (x⋆i ⊗ xj ) = Σ τ̂klij xk ⊗ x⋆l , (73) W(τ̂ ) = T (E ⊕ E ⋆ )/Iτ̂ , (74) then the quotient where the ideal Iτ̂ is given by the relation Iτ̂ := gen{x⋆i ⊗ xj − Σ τ̂klij xk ⊗ x⋆l − (x⋆i |xj )} (75) VI CROSSED ENVELOPINGS AND REPRESENTATIONS 17 is said to be Hermitian Wick algebra [17]. Theorem.( Jørgensen, Schmitt and Werner [17]) The Hermitian Wick algebra W(τ̂ ) is isomorphic to the crossed enveloping algebra Wτ (T E) of T E with respect to the (nonhomogeneous) cross generated by τ̂ + gE . Proof: It has been shownn in [17] that the Wick ordered monomials form a basis in W(τ̂ ). In our language it means that W(τ̂ ) as a vector space is isomorphic to T E ⊗ T E ∗ . Moreover, T E and T E ∗ are subalgebras in W(τ̂ ). This implies that W(τ̂ ) is a crossed product, i.e. W(τ̂ ) ∼ = T E >✁τ T E ∗ for a certain cross τ . Since τ̂ is Hermitian operator and h | i is Hermitian scalar product then Iτ̂ is ⋆–ideal (i.e. Iτ̂∗ ⊂ Iτ̂ ). As a consequence, the cross τ is ⋆–cross. ✷ Let A be an algebra with the presentation A := T E/IA , then for the algebra A⋆ we have the presentation A⋆ := T E ⋆ /IA⋆ . Observe that if IA is a left τ –ideal then IA⋆ is automatically a right τ –ideal (remember that τ is a ⋆–cross). Then there is a cross τ ′ : A⋆ ⊗ A −→ A ⊗ A⋆ and the corresponding crossed enveloping algebra Wτ ′ (A) = A > ✁τ ′ A⋆ . Let H be a k-vector space. We denote by L(H) the algebra of linear operators acting on H. Let A and B be two arbitrary k-algebras and τ : B ⊗ A −→ A ⊗ B be a cross. Theorem: Let πA and πB be representations of the algebras A and B in L(H), respectively. If the condition πB (b)πA (a) = πA (a(1) )πB (b(2) ) (76) holds for all a ∈ A, b ∈ B , then there exist unique representation π of the crossed product A >✁τ B in L(H) such that π|A = πA and π|B = πB . Proof: The representation π : A >✁τ B −→ L(H) is defined by π(a ⊗ b) := πA (a)πB (b). (77) ✷ This theorem allows us to introduce the following definition: Definition: The representation π from the above theorem is said to be a crossed product of representations πA and πB and it is denoted by πA >✁τ πB . It is not difficult to prove the converse: Theorem: If π is a representation of the crossed product A >✁τ B of algebras A and B in H, then there exist representations πA and πB of A and B, respectively, such that π = πA >✁τ πB . (78) Proof: Representations πA and πB are defined by the formulae πA (a) := π(a ⊗ 1), πB (b) := π(1 ⊗ b). (79) ✷ VI CROSSED ENVELOPINGS AND REPRESENTATIONS 18 Let us consider representations of crossed enveloping algebras. For a given representation π : A −→ L(H) of A in a Hilbert space H one can define a conjugate representation π+ (a⋆ ) := π(a)+ of the algebra A⋆ , where + stands for the Hermitian conjugation in L(H). Thus we have: Theorem: Let W ≡ A >✁τ A⋆ be a crossed enveloping algebra. If π : A −→ L(H) is a representation in a Hilbert space H, such that π(b)+ π(a) = π(a(1) )π(b(2) )+ (80) then there is a unique Hermitian or ⋆–representation πW : W −→ L(H) such that πW = π >✁τ π+ (81) Conversely, any Hermitian representation of πW in a Hilbert space has the form (81). Proof: It is a direct consequence of two proceeding Theorems. One also easily verifies the Hermiticity condition: πW (w ⋆) = πW (w)+ for w ∈ W. ✷ As an example, we outline the Fock space representation construction of a cross enveloping algebra Wτ (A) ≡ A >✁τ A⋆ . For this purpose we assume that A is a pre-Hilbert space with an unitary scalar product h | i. Its completion will be denoted by H. In this case we have at our disposal a canonical representation (the quantization) Π acting on the algebra A by means of the left (or right) multiplications in A. For x, f ∈ A it writes Π(x)f := xf (82) The operators Π(x) are, in general, unbounded operators in H. Thus, hΠ+ (x⋆ )f |gi ≡ hΠ(x)+ f |gi = hf |Π(x)gi. (83) Note that the relations (80) are said to be a commutation relation if they are satisfied for a given (Hermitian) scalar product h | i on A. A proper definition of the action of the operators Π(x)+ on the whole algebra A can be a problem. In a case it can be solved, it leads to the canonical (Fock type) Hermitian representation ΠW ≡ Π >✁τ Π+ of the cross enveloping algebra Wτ (A) on A. If the algebra A has generators {xi }i=1,...,N , we use the notation Π(xi ) ≡ a+ i , Π+ (xi⋆ ) ≡ ai , (84) It is customary to call them creation and annihilation operators. Thus the commutation relations play a role of the compatibility conditions relating a+ i , τ and h | i, since ai are + + Hermitian conjugate to ai . For non-free algebra ai -s have to satisfy a set of generating relations of the algebra A. These give rise to the supplementary commutation relations. For the ground state |0i ≡ 1 ∈ A and annihilation operators we usually assume h0|0i = 0, ai |0i = |0i . (85) If the action Π admits non-degenerate, positive definite (pre-) Hermitian scalar product such that the annihilation operators are well defined and the creation and annihilation operators satisfy the commutation relations (80) together with (85), then we say that we have the well–defined Fock representation for a crossed enveloping algebra. Of course, the canonical commutation relations (CCR) and the canonical anti-commutation relations (CAR) provide the most familiar examples of this type. Some other examples can be found in [17] and references therein. 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