1. Introduction
The term “quantum groups” was popularized by V. G. Drinfel’d in his address to the International Congress of Mathematicians (ICM) in Berkeley (1986). However, quantum groups are actually not groups; they are nontrivial deformations of the universal enveloping algebras of semisimple Lie algebras, also called quantized enveloping algebras. These algebras were introduced independently by Drinfel’d [1] (in his definition, these algebras were infinitesimal, i.e., they were Hopf algebras over the field of formal power series) and Jimbo [2] (in his definition, these algebras were Hopf algebras over the field of rational functions in one variable) in 1985 in their study of exactly solvable models in the statistical mechanics. Quantum groups play an important role in the study of Lie groups, Lie algebras, algebraic groups, Hopf algebras, etc.; they are also closely linked with conformal field theory, quiver theory and knot theory.
The positive part of a quantum group has a kind of important basis, i.e., canonical basis introduced by Lusztig [3] , which plays an important role in the theory of quantum groups and their representations. However, it is difficult to determine the elements in canonical basis, which is interested in seeking the simplest elements in canonical basis, i.e., monomial basis elements. Some efforts have been done for monomial basis elements in quantum group of type An. Lusztig firstly introduced algebraic definition of canonical basis of quantum groups for the simply laced case (i.e., An, Dn, En), and gave explicitly the longest monomials for type A1, A2, which were all of canonical basis elements (see [3] ). Then, Lusztig [4] associated a quadratic form to every monomial. He showed that, given certain linear conditions, the monomial was tight, i.e., it belonged to canonical basis (respectively, semitight, i.e., it was a linear combination of elements in canonical basis with constant coefficients in
) provided that the quadratic form satisfied a certain positivity condition (respectively, nonnegativity condition). He showed that the positivity condition (for tightness) always held in type A3 and computed 8 longest tight monomials of type A3. He also asked when we had (semi)tightness in type An. Based on Lusztig’s work, Xi [5] found explicitly all 14 canonical basis elements of type A3 (consisting of 8 longest monomials and 6 polynomials with one-dimensional support). For type A4, Hu, Ye and Yue [6] determined all 62 longest monomials in canonical basis, Hu and Ye [7] gave all 144 polynomials with one-dimensional support in canonical basis, and Li and Hu [8] got 112 polynomials with two-dimensional support in canonical basis. For type An (n ≥ 5), Marsh [9] carried out thorough investigation. He presented a semitight longest monomial for type A5. However, he proved that a class of special longest monomials did not satisfy sufficient condition of tightness or semitightness for type An (n ≥ 6) (although it might turn out that the corresponding monomials were still tight). Reineke [10] associated a new quadratic form to every monomial, and gave a sufficient and necessary condition for the monomial to be tight for the simply laced case in terms of the quadratic form. By use of this criterion, Wang [11] listed all tight monomials for type A3, in which 8 longest tight monomials were the same as Lusztig and Xi’s results.
Based on Reineke’s criterion and some other results, all tight monomials for type A5 with t ≤ 6 are determined in this paper.
2. Preliminaries
Let
be a Cartan matrix of finite type,
be a diagonal matrix with integer en-
tries making the matrix DC symmetric. Let
be the complex semisimple Lie algebra associated with C, and let
(here v is an indeterminate) be the corresponding quantized enveloping algebra, whose positive part U+ is the
-subalgebra of U generated by
, subject to the relations
,
where
. Let
, U+ be the ![](//html.scirp.org/file/1-2230079x16.png)
-subalgebra of U+ generated by
. Corresponding to every reduced expression i of the longest element of the Weyl group of
, one constructs a PBW basis Bi of U+. Lusztig proved that the
-lattice
spanned by Bi is independent of the choice of i, write
; and the image of Bi in the
-module
is a basis B of
independent of i. Let
be the image of
under the bar map of U+ de- fined by
and
. Canonical basis B is the preimage of B under
-module isomorphism
.
A monomial in U+ is an element of the form
(*)
where
. When
is the longest element of Weyl group, the monomial (*) is called the longest monomial. We say that (*) is tight if it belongs to B; we say that (*) is semitight if it is a linear combination of elements in B with constant coefficients.
Let
be a finite quiver with vertex set Q0 and arrow set Q1. Write
as
, where hρ
and tρ denote the head and the tail of ρ respectively. An automorphism σ of Q is a permutation on the vertices of
Q and on the arrows of Q such that
and
for any
. Denote the quiver with
automorphism σ as
. Attach to the pair
a valued quiver
as follows. Its vertex set
and arrow set
are simply the sets of σ-orbits in Q0 and Q1, respectively. The valuation of ![]()
is given by
,
;
,
. The
Euler form of
is defined to be the bilinear form
given by
,
where
, so
is the symmetric Euler form. The valued quiver
defines a Cartan matrix
, where
![]()
Let t be a non-negative integer. Let
and
. We write
.
Define
![]()
where
![]()
Obviously,
.
The following results are very useful in the determination of tight monomials.
Theorem 2.1 [4] (Lusztig, 1993). Let U be the quantum group of type
,
as before. If the following quadratic form takes only values < 0 on
, then monomial
is tight.
![]()
Theorem 2.2 [10] (Reineke, 2001). Let U be the quantum group of type An, Dn, En,
as before, the monomial
is tight if and only if the following quadratic form takes only values < 0 on ![]()
![]()
If
are mutually different, then
, by Theorem 2.2, we have the following Corollaries.
Corollary 2.3. When
are mutually different, monomial
is tight.
Corollary 2.4. If
is tight, then for any mutually different ![]()
and any mutually different
, and
,
![]()
is also tight.
Theorem 2.5 [12] (Deng-Du, 2010). Let
and
. If
is tight, then
(a) For
, monomial
is also tight;
(b) For
,
.
Theorem 2.6 [4] (Lusztig, 1993). Let
be the non-trivial automorphism of U+ induced by Dynkin diagram
automorphism of
, and
be the unique
-algebra isomorphism such that
.
If
is tight, then
and
are all tight.
3. Main Results
Let
. For convenience, we abbreviate a monomial ![]()
as a word
(1 as 0), an inequality
as
. For example,
a monomial
is abbreviated to
, a monomial
to 1234, etc.
By Theorem 2.5(b), we only consider those words
with
in determining tight monomials, in this case, we call
the word with t-value,
the monomial with t-value. If
for some
, we identify the word
with the word
. Let us present the so called word-procedure for making the words with
- value from the words with t-value. Let
be a word with t-value, we firstly add a number
different from i1 (or it) in the front (or behind) of i1 (or it), secondly delete the words with t-value, lastly apply the automorphism
and isomorphism
. After the above procedure put into practice for all the words with t-value, we get all words with
-value by deleting repeated words. For example, by applying the above word-procedure to the word 13 with 2-value, we get the words with 3-value as follows: 132, 134, 135, 143, 213, 235, 325, 354, 435.
Theorem 3.1. Let Mt be the set of all tight monomials with t-value in quantum group for type A5, we have the following results.
(1) t = 0,
, tight monomial has only one;
(2) t = 1, if
, then
, tight monomials have 5 families;
(3) t = 2, if
, then
, tight monomials have 14 families;
(4) t = 3, if
, where
,
,
then
, tight monomials have 33 families;
(5) t = 4, if
where
![]()
![]()
![]()
then
, tight monomials have 67 families;
(6) t = 5, if
where
![]()
![]()
![]()
![]()
![]()
![]()
![]()
![]()
then
, tight monomials have 125 families;
(7) If t = 6,
where
![]()
![]()
![]()
![]()
![]()
![]()
![]()
![]()
![]()
![]()
![]()
![]()
![]()
![]()
![]()
![]()
![]()
then
, tight monomials have 222 families;
4. Proof of Theorem 3.1
Consider the quiver
of type A5, where
,
. Let
id be the identity automorphism of Q, then valued quiver of
is
. The valuation is given by
. Euler form
on
is
,
Symmetric Euler form
on
is
,
where
.
By simple computation, we have
,
, and
.
Let us prove Theorem 3.1.
Case 1.
. By Corollary 2.3, monomials with
are all tight.
Case 2. t = 3. Applying the word-procedure on S2, we get 33 words with 3-value. By considering
and
, we get
. By Corollary 2.3, monomials in
are all tight. For
, it suffices to consider
. For any
, we have
, where
![]()
and
![]()
Obviously,
if and only if
. So monomial
is tight by Theorem 2.2.
Case 3. t = 4. Applying the word-procedure on
, we get 75 words with 4-value. By considering
and
, we get
. When
, for any
, we have
,
where
![]()
and
![]()
Obviously,
if and only if
, this is a contradiction. Applying
, one gets that the monomials corresponding to
![]()
are all not tight for any
.
Monomials in
are all tight by Corollary 2.3. By
and Corollary 2.4, monomials in
are all tight. For
, it suffices to consider
. For any
, we have
,
where
![]()
and
![]()
if and only if
. So
is tight by Theorem 2.2.
Case 4. t = 5. Applying the word-procedure on
, and deleting words including subwords 1212, 2121, 2323, 3232, 3434, 4343, 4545 and 5454 (considering Theorem 2.5(a)), we get 125 words with 5-value. By considering
and
, we get
. By Corollary 2.3, monomials in
are all tight. Monomials in
are all tight by
and Corollary 2.4. Monomials in
are all tight by
and Corollary 2.4.
For
, it suffices to consider
. For any
, we have
,
where
![]()
and
![]()
if and only if
. So
![]()
is tight by Theorem 2.2.
For
, it suffices to consider
. For any
, we have
,
where
![]()
and
![]()
if and only if
. So
![]()
is tight by Theorem 2.2.
For
, it suffices to consider
. For any
, we have
,
where
![]()
and
![]()
if and only if
. So
![]()
is tight by Theorem 2.2.
For
, it suffices to consider
. For any
, we have
,
where
![]()
and
![]()
if and only if
. So
![]()
is tight by Theorem 2.2.
Case 5. t = 6. Applying the word-procedure on S5, and deleting words including subwords 1212, 2121, 2323, 3232, 3434, 4343, 4545 and 5454(considering Theorem 2.5(a)), we get 228 words with 6-value. By considering Φ and Ψ, we get
. When
, for any
, we have
,
where
![]()
and
![]()
if and only if
. This is a contradiction. Applying
, one gets that the monomials corresponding to
![]()
are all not tight for any
.
By Corollary 2.4, we have
,
,
,
,
,
,
, and
.
For
, it suffices to consider
. For any
, we have
![]()
where
![]()
and
![]()
if and only if
. So
![]()
is tight by Theorem 2.2.
For
, it suffices to consider
. For any
, we have
![]()
where
![]()
and
![]()
if and only if
. So
![]()
is tight by Theorem 2.2.
For
, it suffices to consider
. For any
, we have
![]()
where
![]()
and
![]()
if and only if
. So
![]()
is tight by Theorem 2.2.
For
, it suffices to consider
. For any
, we have
![]()
where
![]()
and
![]()
if and only if
. So
![]()
is tight by Theorem 2.2.
Funding
This paper is supported by the NSF of China (No. 11471333) and Basic and advanced technology research project of Henan Province (142300410449).
NOTES
*Corresponding author.