The Lusztig automorphism of the $q$-Onsager algebra
Paul Terwilliger
TL;DR
This paper provides a new explicit expression for the Lusztig automorphism of the $q$-Onsager algebra, proves its existence without computer assistance, and explores its implications for related algebraic structures and modules.
Contribution
It introduces explicit formulas for the Lusztig automorphism, proves its existence independently, and extends the automorphism to related current algebras and modules.
Findings
Explicit formulas for $L$ and $L^{-1}$ as quantum adjoint sums
Proof of the existence of $L$ without computational methods
Description of the automorphism's effect on finite-dimensional modules
Abstract
Pascal Baseilhac and Stefan Kolb recently introduced the Lusztig automorphism of the -Onsager algebra . In this paper, we express each of as a formal sum involving some quantum adjoints. In addition, (i) we give a computer-free proof that exists; (ii) we establish the higher order -Dolan/Grady relations previously conjectured by Baseilhac and Thao Vu; (iii) we obtain a Lusztig automorphism for the current algebra associated with ; (iv) we describe what happens when a finite-dimensional irreducible -module is twisted via .
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The Lusztig automorphism
of the
-Onsager algebra
Paul Terwilliger
Abstract
Pascal Baseilhac and Stefan Kolb recently introduced the Lusztig automorphism of the -Onsager algebra . In this paper, we express each of as a formal sum involving some quantum adjoints. In addition, (i) we give a computer-free proof that exists; (ii) we establish the higher order -Dolan/Grady relations previously conjectured by Baseilhac and Thao Vu; (iii) we obtain a Lusztig automorphism for the current algebra associated with ; (iv) we describe what happens when a finite-dimensional irreducible -module is twisted via .
Keywords. -Onsager algebra, tridiagonal pair. 2010 Mathematics Subject Classification. Primary: 33D80. Secondary 17B40.
1 Introduction
Throughout this paper denotes a field. Fix that is not a root of unity. Recall the notation
[TABLE]
We will be discussing algebras. An algebra is meant to be associative and have a . A subalgebra has the same as the parent algebra.
Definition 1.1**.**
(See [2, Section 2], [23, Definition 3.9].)* *Let denote the -algebra with generators and relations
[TABLE]
We call the -Onsager algebra. The relations (1), (2) are called the -Dolan/Grady relations.
We now give some background on ; for more information see [24]. There is a family of algebras called tridiagonal algebras [23, Definition 3.9] that arise in the study of ( and )-polynomial association schemes [21, Lemma 5.4] and tridiagonal pairs [16, Theorem 10.1], [23, Theorem 3.10]. The algebra is the “most general” example of a tridiagonal algebra [17, p. 70]. Applications of to tridiagonal pairs can be found in [4, 15, 16, 17, 18, 22, 23, 27]. The algebra has applications to quantum integrable models [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], reflection equation algebras [12], and coideal subalgebras [14, 19, 20]. There is an algebra homomorphism from into the algebra [26, Proposition 5.6], and the universal Askey-Wilson algebra [25, Sections 9,10].
In [11] Pascal Baseilhac and Stefan Kolb found an automorphism of that acts as follows:
[TABLE]
They called the Lusztig automorphism of . In our view is a profound discovery, and worthy of much further study. In this paper, we express each of , as a formal sum that involves some quantum adjoints of . In addition, (i) we obtain a computer-free proof that exists; (ii) we establish the higher order -Dolan/Grady relations previously conjectured by Baseilhac and Thao Vu [13]; (iii) we obtain a Lusztig automorphism for the current algebra [12, Definition 3.1] associated with ; (iv) we describe what happens when a finite-dimensional irreducible -module is twisted via .
2 Statement of the main result
We will state our main result after a few comments. Recall the natural numbers and integers . Let denote an -algebra. For , the corresponding adjoint map is , . For define the quantum adjoint map , . We have . Note that , commute for .
We now state our main result.
Theorem 2.1**.**
The Lusztig automorphism of satisfies
[TABLE]
Moreover for all , in the above sums the large parenthetical expression vanishes at for all but finitely .
We mention two consequences of Theorem 2.1.
Corollary 2.2**.**
The automorphism fixes every element of that commutes with .
Corollary 2.3**.**
Pick such that
[TABLE]
Then sends
[TABLE]
and sends
[TABLE]
We will obtain Theorem 2.1 as a consequence of a more general result, which we now summarize. Let denote an -algebra and let . Consider the formal sums
[TABLE]
An element is called -standard whenever the large parenthetical expression in (7), (8) vanishes at for all but finitely many . The algebra is called -standard whenever each element of is -standard. Assume that is -standard. We will show that and act on as an automorphism, and these automorphisms are inverses. Also, we will show that the algebra is -standard and , .
3 Some identities for the quantum adjoint
As we work towards Theorem 2.1, our first goal is to establish some identities for the quantum adjoint, that apply to any -algebra. Let denote an -algebra, and fix . Recall the sums , from (7), (8). We will be discussing the terms in these sums. To simplify this discussion we introduce a “balanced” version of , called . Let denote the set of positive integers.
Definition 3.1**.**
Define
[TABLE]
and
[TABLE]
Further define
[TABLE]
We interpret .
Definition 3.2**.**
Define and
[TABLE]
Further define and
[TABLE]
Lemma 3.3**.**
In the above notation the sums (7), (8) become
[TABLE]
Our next goal is to prove Proposition 3.10 below. To this end we give some identities that hold in .
Lemma 3.4**.**
For ,
[TABLE]
Proof.
Routine. ∎
Lemma 3.5**.**
For ,
[TABLE]
Proof.
Use Definition 3.2 and Lemma 3.4. ∎
Lemma 3.6**.**
For ,
[TABLE]
Proof.
Use Definition 3.1. ∎
Lemma 3.7**.**
For ,
[TABLE]
Proof.
Use Definition 3.2 and Lemma 3.6. ∎
Lemma 3.8**.**
For ,
[TABLE]
Proof.
Routine using Definition 3.1. ∎
Lemma 3.9**.**
For ,
[TABLE]
Proof.
Use Definition 3.2 and Lemma 3.8. ∎
Proposition 3.10**.**
For ,
[TABLE]
Proof.
The proof is by induction on . Let denote the left-hand side minus the right-hand side. We show that . One routinely obtains , so assume . To show that , it suffices to show that . In the expression , eliminate the terms , , using Lemmas 3.5, 3.7, 3.9. After a routine simplification we obtain , so . ∎
Our next goal is to prove Proposition 3.19 below. To this end we give some more identities that hold in .
Lemma 3.11**.**
For distinct and ,
[TABLE]
Proof.
Routine. ∎
Lemma 3.12**.**
For and and ,
[TABLE]
Proof.
Use Definition 3.2 and Lemma 3.11. ∎
Lemma 3.13**.**
For and and ,
[TABLE]
Proof.
Use Definitions 3.1, 3.2 and Lemma 3.11. ∎
Lemma 3.14**.**
For and ,
[TABLE]
Proof.
Routine. ∎
The next four lemmas are routinely obtained using Lemmas 3.11–3.14.
Lemma 3.15**.**
For and and ,
[TABLE]
Lemma 3.16**.**
For and and ,
[TABLE]
Lemma 3.17**.**
For and and and ,
[TABLE]
Lemma 3.18**.**
For and and and ,
[TABLE]
Proposition 3.19**.**
For and ,
[TABLE]
Proof.
The proof is by induction on . Let (resp. ) denote the first (resp. second) displayed equation in the proposition statement. The equation holds since the identity map. The equation holds by Lemma 3.14 at , , . To get from and , apply to each side of , and evaluate the result using Lemmas 3.15–3.18. One obtains after a brief calculation. For the rest of this proof, assume that . To get from and , apply to each side of , and evaluate the result using Lemmas 3.15–3.18. One obtains after a brief calculation. The result follows from these comments. ∎
The following result is a variation on Proposition 3.19.
Proposition 3.20**.**
For and ,
[TABLE]
Proof.
In Proposition 3.19, replace by and evaluate the result using Definitions 3.1, 3.2. ∎
Strictly speaking we do not need the following result; we mention it for the sake of completeness.
Proposition 3.21**.**
For and ,
[TABLE]
Proof.
Using Definition 3.2 and Lemma 3.11 we find that for ,
[TABLE]
In the relations from the proposition statement, eliminate the terms , using (10), (11) and compare the results with the second equation in Proposition 3.19. ∎
Proposition 3.22**.**
Given and such that
[TABLE]
Then
[TABLE]
Proof.
Use the second equation in Proposition 3.19 or 3.20. Alternatively use either equation in Proposition 3.21. ∎
4 The subalgebra
We continue to work with the element of the -algebra .
Definition 4.1**.**
For let denote the set of elements in at which vanishes. Note that is a subspace of the -vector space .
Example 4.2**.**
The subspace consists of the elements in that commute with .
Example 4.3**.**
The subspace consists of the elements in such that
[TABLE]
Lemma 4.4**.**
We have for .
Proof.
Lemma 4.5**.**
Pick . Then for the maps vanish on . Moreover on ,
[TABLE]
Proof.
By (9) the map is a factor of . By Definition 3.2, the map is a factor of and . Consequently and vanish on . The equations (12) are from Lemma 3.3. ∎
By Lemma 4.5, and are well defined -linear maps on for all .
Lemma 4.6**.**
For the subspace is invariant under and .
Proof.
The map commutes with and for . ∎
Lemma 4.7**.**
For the maps and are inverses.
Proof.
By Proposition 3.10. ∎
Example 4.8**.**
The maps and fix everything in .
Proof.
On we have and . ∎
Example 4.9**.**
Pick . Then sends
[TABLE]
and sends
[TABLE]
Proof.
On we have and . ∎
Lemma 4.10**.**
We have for .
Proof.
By Proposition 3.22. ∎
Definition 4.11**.**
Define .
Lemma 4.12**.**
The set is a subalgebra of that contains .
Proof.
By Definition 4.1 and Lemma 4.4, is a subspace of the -vector space . By Example 4.2, the subspace contains 1 and . By Lemma 4.10, the subspace is closed under multiplication. The result follows. ∎
By Definition 4.11 along with Lemma 4.6 and the comment above it, we obtain -linear maps and .
Proposition 4.13**.**
The maps and act on the algebra as an automorphism, and these automorphisms are inverses.
Proof.
To get the first assertion use Propositions 3.19, 3.20. The last assertion follows from Lemma 4.7. ∎
5 -Standard algebras and their Lusztig automorphism
We continue to work with the element of the -algebra .
Definition 5.1**.**
An element is called -standard whenever there exists a positive integer such that . Note that consists of the -standard elements of .
Definition 5.2**.**
The algebra is called -standard whenever each element of is -standard.
Lemma 5.3**.**
The following (i)–(iii) are equivalent:
- (i)
* is -standard;* 2. (ii)
; 3. (iii)
* has a generating set whose elements are -standard.*
Proof.
Clear.
By Lemma 4.12 is a subalgebra of . By Definition 5.1 contains each -standard element of . The result follows. ∎
Theorem 5.4**.**
Assume that is -standard. Then and act on as an automorphism, and these automorphisms are inverses.
Proof.
Apply Proposition 4.13 to the algebra . ∎
Recall the -Onsager algebra and its generators , .
Proposition 5.5**.**
For the following (i)–(iv) hold:
- (i)
* and ;* 2. (ii)
the algebra is -standard; 3. (iii)
* sends*
[TABLE]
and sends
[TABLE] 4. (iv)
* and .*
Proof.
(i) We have by Example 4.2, and by Example 4.3.
(ii) The generators , are -standard by (i) above. Now is -standard by Lemma 5.3.
(iii) By (i) above and Examples 4.8, 4.9.
(iv) Compare (3), (4) with (iii) above. ∎
Theorem 2.1 follows from Theorem 5.4 and Proposition 5.5. Combining Theorem 5.4 and Proposition 5.5(i)–(iii), we get a computer-free proof that there exists an automorphism of that satisfies (3), (4).
We return our attention to the algebra , and the element . The following definition is motivated by Proposition 5.5.
Definition 5.6**.**
Assume that is -standard, and consider its automorphism from Theorem 5.4. We call the Lusztig automorphism of that corresponds to .
6 The higher order -Dolan/Grady relations
In this section we establish the higher order -Dolan/Grady relations conjectured by Baseilhac and Vu [13]. Let denote an -algebra and fix .
Theorem 6.1**.**
Given such that
[TABLE]
Then
[TABLE]
We are using the notation (9).
Proof.
By Example 4.3 we have . By Lemma 4.10 we have . The result follows by Definition 4.1. ∎
7 The current algebra and its Lusztig automorphism
In [12] Baseilhac and K. Shigechi introduce the current algebra for , and they discuss how is related to . This relationship is discussed further in [7], where it is conjectured that is a homomorphic image of [7, Conjecture 2]. The algebra is defined by generators and relations [12, Definition 3.1]. The generators are denoted , , , , where . In [7, Lemma 2.1], Baseilhac and S. Belliard display some central elements for . In [7, Corollary 3.1], it is shown that is generated by these central elements together with and . The elements are known to satisfy the -Dolan/Grady relations (1), (2) [7, eqn. (3.7)]. In this section we show that is -standard, and describe how the corresponding Lusztig automorphism acts on the elements mentioned above. We now recall the definition of .
Definition 7.1**.**
(See [12, Definition 3.1].) Let denote the -algebra with generators , , , and the following relations:
[TABLE]
In the above equations and . We are using the notation and .
For the algebra , consider the element and the corresponding subspaces , from Examples 4.2, 4.3.
Lemma 7.2**.**
*For the algebra the following (i)–(v) hold for :
(i) ; (ii) ; (iii) ; (iv) ; (v) .*
Proof.
(i) The elements commute by (16). The result follows in view of Example 4.2.
(ii) We show that
[TABLE]
By (13),
[TABLE]
Using linear algebra and (14),
[TABLE]
Using in order (25), (26), (14), (16) we obtain
[TABLE]
We have shown (24). The result follows in view of Example 4.3.
(iii) We show that
[TABLE]
Using in order (14), (16), (13) we obtain
[TABLE]
Now in (27), the left-hand side minus the right-hand side is equal to
[TABLE]
and this is zero by (14). We have shown (27). The result follows in view of Example 4.3.
(iv) We show that
[TABLE]
Using in order (14), (16), (13) we obtain
[TABLE]
Now in (28), the left-hand side minus the right-hand side is equal to
[TABLE]
and this is zero by (14). We have shown (28). The result follows in view of Example 4.3.
(v) The element is central, so it commutes with . The result follows in view of Example 4.2. ∎
Proposition 7.3**.**
The algebra is -standard.
Proof.
Consider the generators of from Definition 7.1. By Lemma 7.2 these generators are -standard. Now is -standard by Lemma 5.3. ∎
Since the algebra is -standard, we may speak of the corresponding Lusztig automorphism of , from Theorem 5.4 and Definition 5.6.
Proposition 7.4**.**
For the automorphism sends
[TABLE]
Moreover sends
[TABLE]
Proof.
By Theorem 5.4, Lemma 7.2, and Examples 4.8, 4.9. ∎
8 Finite-dimensional -modules
Recall the generators , for the -Onsager algebr . Throughout this section denotes a finite-dimensional irreducible -module on which and are diagonalizable. To avoid trivialities, we always assume that has dimension at least 2. We describe what happens when is twisted via the Lusztig automorphism of . By [23, Theorem 3.10] the elements , act on as a tridiagonal pair. The tridiagonal pair concept is defined in [16, Definition 1.1], and described further in [15, 17, 18, 24]. In what follows, we freely invoke the notation and theory of tridiagonal pairs. Fix a standard ordering of the eigenvalues of on , and a standard ordering of the eigenvalues of on . By construction are mutually distinct and contained in . Similarly are mutually distinct and contained in . Note that ; otherwise and which contradicts the irreducibility of . For let (resp. ) denote the projection onto the eigenspace of (resp. ) for (resp. ). By linear algebra,
[TABLE]
By [16, Lemma 2.4] the following hold for :
[TABLE]
The following result can be found in [16, Theorem 11.2]; we give a short proof for the sake of completeness.
Lemma 8.1**.**
(See [16, Theorem 11.2].)* There exist nonzero such that*
[TABLE]
for .
Proof.
We verify the equation on the left in (30). For we multiply each side of (1) on the left by and on the right by . Simplify the result to get
[TABLE]
Now assuming and using ,
[TABLE]
Let denote the right-hand side of (32). For ,
[TABLE]
By the above recurrence there exist such that
[TABLE]
Since we have the equation . Evaluate this equation using (33) to obtain . This yields the equation on the left in (30). The equation on the right in (30) is similarly obtained. ∎
Definition 8.2**.**
Define
[TABLE]
The following calculation will be useful.
Lemma 8.3**.**
For such that ,
[TABLE]
and
[TABLE]
Proof.
Each side of (34) is equal to (if ), (if ), and (if ). Each side of (35) is equal to (if ), (if ), and (if ). ∎
Definition 8.4**.**
Define
[TABLE]
Lemma 8.5**.**
The map is invertible, and
[TABLE]
Proof.
Since and for . ∎
Theorem 8.6**.**
For the following holds on :
[TABLE]
Proof.
It suffices to show and . Certainly , since and commutes with . We now verify . Since it suffices to show for . Let be given. Using the definition (3) of , one finds that is equal to times the scalar on the right in (34). Using Definition 8.4 one finds that is equal to times the scalar on the left in (34). For the moment assume . Then by Lemma 8.3. Next assume . Then since . We have shown . ∎
In Lemma 8.3 we related the parameters and . We now give a more general result along this line.
Theorem 8.7**.**
For we have
[TABLE]
and
[TABLE]
Moreover, in the above sums the large parenthetical expression is zero for .
Proof.
We verify the first displayed equation. Since the -module is irreducible, there exists such that . For this , equation (38) holds on . In equation (38), multiply each side on the left by and on the right by . Evaluate the results using (5), (36), (37) together with . This yields the first displayed equation after a brief calculation. The second displayed equation is similarly obtained using . The last assertion of the theorem statement can be checked directly using (30). ∎
Note 8.8**.**
It is natural to ask how we discovered Theorem 2.1. The answer is that we first discovered Theorem 8.7, and then considered the implications for .
9 Acknowledgment
The author thanks Pascal Baseilhac and Stefan Kolb for sharing their preprint [11] prior to publication.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
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