Yangians¶

AUTHORS:

• Travis Scrimshaw (2013-10-08): Initial version
class sage.algebras.yangian.GeneratorIndexingSet(index_set, level=None)

Helper class for the indexing set of the generators.

an_element()

Initialize self.

cardinality()

Return the cardinality of self.

Base class for graded algebras associated to a Yangian.

The associated graded algebra corresponding to a Yangian $$\operatorname{gr} Y(\mathfrak{gl}_n)$$ with the filtration of $$\deg t_{ij}^{(r)} = r - 1$$.

Using this filtration for the Yangian, the associated graded algebra is isomorphic to $$U(\mathfrak{gl}_n[z])$$, the universal enveloping algebra of the loop algebra of $$\mathfrak{gl}_n$$.

INPUT:

• Y – a Yangian with the loop filtration
antipode_on_basis(m)

Return the antipode on a basis element indexed by m.

EXAMPLES:

sage: grY = Yangian(QQ, 4).graded_algebra()
-tbar(2)[1,1]

sage: x = grY.an_element(); x
tbar(1)[1,1]*tbar(1)[1,2]^2*tbar(1)[1,3]^3*tbar(3)[1,1]
-tbar(1)[1,1]*tbar(1)[1,2]^2*tbar(1)[1,3]^3*tbar(3)[1,1]
- 2*tbar(1)[1,1]*tbar(1)[1,2]*tbar(1)[1,3]^3*tbar(3)[1,2]
- 3*tbar(1)[1,1]*tbar(1)[1,2]^2*tbar(1)[1,3]^2*tbar(3)[1,3]
+ 5*tbar(1)[1,2]^2*tbar(1)[1,3]^3*tbar(3)[1,1]
+ 10*tbar(1)[1,2]*tbar(1)[1,3]^3*tbar(3)[1,2]
+ 15*tbar(1)[1,2]^2*tbar(1)[1,3]^2*tbar(3)[1,3]
coproduct_on_basis(m)

Return the coproduct on the basis element indexed by m.

EXAMPLES:

sage: grY = Yangian(QQ, 4).graded_algebra()
1 # tbar(2)[1,1] + tbar(2)[1,1] # 1
sage: grY.gen(2,3,1).coproduct()
1 # tbar(2)[3,1] + tbar(2)[3,1] # 1
counit_on_basis(m)

Return the antipode on the basis element indexed by m.

EXAMPLES:

sage: grY = Yangian(QQ, 4).graded_algebra()
0
sage: grY.gen(0,0,0).counit()
1

The associated graded algebra corresponding to a Yangian $$\operatorname{gr} Y(\mathfrak{gl}_n)$$ with the natural filtration of $$\deg t_{ij}^{(r)} = r$$.

INPUT:

• Y – a Yangian with the natural filtration
product_on_basis(x, y)

Return the product on basis elements given by the indices x and y.

EXAMPLES:

sage: grY = Yangian(QQ, 4, filtration='natural').graded_algebra()
sage: x = grY.gen(12, 2, 1) * grY.gen(2, 1, 1) # indirect doctest
sage: x
tbar(2)[1,1]*tbar(12)[2,1]
sage: x == grY.gen(2, 1, 1) * grY.gen(12, 2, 1)
True
class sage.algebras.yangian.Yangian(base_ring, n, variable_name, filtration)

The Yangian $$Y(\mathfrak{gl}_n)$$.

Let $$A$$ be a commutative ring with unity. The Yangian $$Y(\mathfrak{gl}_n)$$, associated with the Lie algebra $$\mathfrak{gl}_n$$ for $$n \geq 1$$, is defined to be the unital associative algebra generated by $$\{t_{ij}^{(r)} \mid 1 \leq i,j \leq n , r \geq 1\}$$ subject to the relations

$[t_{ij}^{(M+1)}, t_{k\ell}^{(L)}] - [t_{ij}^{(M)}, t_{k\ell}^{(L+1)}] = t_{kj}^{(M)} t_{i\ell}^{(L)} - t_{kj}^{(L)} t_{i\ell}^{(M)},$

where $$L,M \geq 0$$ and $$t_{ij}^{(0)} = \delta_{ij} \cdot 1$$. This system of quadratic relations is equivalent to the system of commutation relations

$[t_{ij}^{(r)}, t_{k\ell}^{(s)}] = \sum_{p=0}^{\min\{r,s\}-1} \bigl(t_{kj}^{(p)} t_{i\ell}^{(r+s-1-p)} - t_{kj}^{(r+s-1-p)} t_{i\ell}^{(p)} \bigr),$

where $$1 \leq i,j,k,\ell \leq n$$ and $$r,s \geq 1$$.

Let $$u$$ be a formal variable and, for $$1 \leq i,j \leq n$$, define

$t_{ij}(u) = \delta_{ij} + \sum_{r=1}^\infty t_{ij}^{(r)} u^{-r} \in Y(\mathfrak{gl}_n)[\![u^{-1}]\!].$

Thus, we can write the defining relations as

\begin{aligned} (u - v)[t_{ij}(u), t_{k\ell}(v)] & = t_{kj}(u) t_{i\ell}(v) - t_{kj}(v) t_{i\ell}(u). \end{aligned}

These series can be combined into a single matrix:

$T(u) := \sum_{i,j=1}^n t_{ij}(u) \otimes E_{ij} \in Y(\mathfrak{gl}_n) [\![u^{-1}]\!] \otimes \operatorname{End}(\CC^n),$

where $$E_{ij}$$ is the matrix with a $$1$$ in the $$(i,j)$$ position and zeros elsewhere.

For $$m \geq 2$$, define formal variables $$u_1, \ldots, u_m$$. For any $$1 \leq k \leq m$$, set

$T_k(u_k) := \sum_{i,j=1}^n t_{ij}(u_k) \otimes (E_{ij})_k \in Y(\mathfrak{gl}_n)[\![u_1^{-1},\dots,u_m^{-1}]\!] \otimes \operatorname{End}(\CC^n)^{\otimes m},$

where $$(E_{ij})_k = 1^{\otimes (k-1)} \otimes E_{ij} \otimes 1^{\otimes (m-k)}$$. If we consider $$m = 2$$, we can then also write the defining relations as

$R(u - v) T_1(u) T_2(v) = T_2(v) T_1(u) R(u - v),$

where $$R(u) = 1 - Pu^{-1}$$ and $$P$$ is the permutation operator that swaps the two factors. Moreover, we can write the Hopf algebra structure as

$\Delta \colon T(u) \mapsto T_{}(u) T_{}(u), \qquad S \colon T(u) \mapsto T^{-1}(u), \qquad \epsilon \colon T(u) \mapsto 1,$

where $$T_{[a]} = \sum_{i,j=1}^n (1^{\otimes a-1} \otimes t_{ij}(u) \otimes 1^{2-a}) \otimes (E_{ij})_1$$.

We can also impose two filtrations on $$Y(\mathfrak{gl}_n)$$: the natural filtration $$\deg t_{ij}^{(r)} = r$$ and the loop filtration $$\deg t_{ij}^{(r)} = r - 1$$. The natural filtration has a graded homomorphism with $$U(\mathfrak{gl}_n)$$ by $$t_{ij}^{(r)} \mapsto (E^r)_{ij}$$ and an associated graded algebra being polynomial algebra. Moreover, this shows a PBW theorem for the Yangian, that for any fixed order, we can write elements as unique linear combinations of ordered monomials using $$t_{ij}^{(r)}$$. For the loop filtration, the associated graded algebra is isomorphic (as Hopf algebras) to $$U(\mathfrak{gl}_n[z])$$ given by $$\overline{t}_{ij}^{(r)} \mapsto E_{ij} x^{r-1}$$, where $$\overline{t}_{ij}^{(r)}$$ is the image of $$t_{ij}^{(r)}$$ in the $$(r - 1)$$-th component of $$\operatorname{gr}Y(\mathfrak{gl}_n)$$.

INPUT:

• base_ring – the base ring
• n – the size $$n$$
• level – (optional) the level of the Yangian
• variable_name – (default: 't') the name of the variable
• filtration – (default: 'loop') the filtration and can be one of the following:
• 'natural' – the filtration is given by $$\deg t_{ij}^{(r)} = r$$
• 'loop' – the filtration is given by $$\deg t_{ij}^{(r)} = r - 1$$

Todo

Implement the antipode.

EXAMPLES:

sage: Y = Yangian(QQ, 4)
sage: t = Y.algebra_generators()
sage: t[6,2,1] * t[2,3,2]
-t(1)[2,2]*t(6)[3,1] + t(1)[3,1]*t(6)[2,2]
+ t(2)[3,2]*t(6)[2,1] - t(7)[3,1]
sage: t[6,2,1] * t[3,1,4]
t(1)[1,1]*t(7)[2,4] + t(1)[1,4]*t(6)[2,1] - t(1)[2,1]*t(6)[1,4]
- t(1)[2,4]*t(7)[1,1] + t(2)[1,1]*t(6)[2,4] - t(2)[2,4]*t(6)[1,1]
+ t(3)[1,4]*t(6)[2,1] + t(6)[2,4] + t(8)[2,4]

We check that the natural filtration has a homomorphism to $$U(\mathfrak{gl}_n)$$ as algebras:

sage: Y = Yangian(QQ, 4, filtration='natural')
sage: t = Y.algebra_generators()
sage: gl4 = lie_algebras.gl(QQ, 4)
sage: Ugl4 = gl4.pbw_basis()
sage: E = matrix(Ugl4, 4, 4, Ugl4.gens())
sage: Esq = E^2
sage: t[2,1,3] * t[1,2,1]
t(1)[2,1]*t(2)[1,3] - t(2)[2,3]
sage: Esq[0,2] * E[1,0] == E[1,0] * Esq[0,2] - Esq[1,2]
True

sage: Em = [E^k for k in range(1,5)]
sage: S = list(t.some_elements())[:30:3]
sage: def convert(x):
....:     return sum(c * prod(Em[t-1][t-1,t-1] ** e
....:                         for t,e in m._sorted_items())
....:                for m,c in x)
sage: for x in S:
....:     for y in S:
....:         ret = x * y
....:         rhs = convert(x) * convert(y)
....:         assert rhs == convert(ret)
....:         assert ret.maximal_degree() == rhs.maximal_degree()

REFERENCES:

algebra_generators()

Return the algebra generators of self.

EXAMPLES:

sage: Y = Yangian(QQ, 4)
sage: Y.algebra_generators()
Lazy family (generator(i))_{i in Cartesian product of
Positive integers, (1, 2, 3, 4), (1, 2, 3, 4)}
coproduct_on_basis(m)

Return the coproduct on the basis element indexed by m.

The coproduct $$\Delta\colon Y(\mathfrak{gl}_n) \longrightarrow Y(\mathfrak{gl}_n) \otimes Y(\mathfrak{gl}_n)$$ is defined by

$\Delta(t_{ij}(u)) = \sum_{a=1}^n t_{ia}(u) \otimes t_{aj}(u).$

EXAMPLES:

sage: Y = Yangian(QQ, 4)
sage: Y.gen(2,1,1).coproduct() # indirect doctest
1 # t(2)[1,1] + t(1)[1,1] # t(1)[1,1] + t(1)[1,2] # t(1)[2,1]
+ t(1)[1,3] # t(1)[3,1] + t(1)[1,4] # t(1)[4,1] + t(2)[1,1] # 1
sage: Y.gen(2,3,1).coproduct()
1 # t(2)[3,1] + t(1)[3,1] # t(1)[1,1] + t(1)[3,2] # t(1)[2,1]
+ t(1)[3,3] # t(1)[3,1] + t(1)[3,4] # t(1)[4,1] + t(2)[3,1] # 1
sage: Y.gen(2,2,3).coproduct()
1 # t(2)[2,3] + t(1)[2,1] # t(1)[1,3] + t(1)[2,2] # t(1)[2,3]
+ t(1)[2,3] # t(1)[3,3] + t(1)[2,4] # t(1)[4,3] + t(2)[2,3] # 1
counit_on_basis(m)

Return the counit on the basis element indexed by m.

EXAMPLES:

sage: Y = Yangian(QQ, 4)
sage: Y.gen(2,3,1).counit() # indirect doctest
0
sage: Y.gen(0,0,0).counit()
1
degree_on_basis(m)

Return the degree of the monomial index by m.

The degree of $$t_{ij}^{(r)}$$ is equal to $$r - 1$$ if filtration = 'loop' and is equal to $$r$$ if filtration = 'natural'.

EXAMPLES:

sage: Y = Yangian(QQ, 4)
1
sage: x = Y.gen(5,2,3)^4
16
sage: elt = Y.gen(10,3,1) * Y.gen(2,1,1) * Y.gen(1,2,4); elt
t(1)[1,1]*t(1)[2,4]*t(10)[3,1] - t(1)[2,4]*t(1)[3,1]*t(10)[1,1]
+ t(1)[2,4]*t(2)[1,1]*t(10)[3,1] + t(1)[2,4]*t(10)[3,1]
+ t(1)[2,4]*t(11)[3,1]
sage: for s in sorted(elt.support(), key=str): s, Y.degree_on_basis(s)
(t(1, 1, 1)*t(1, 2, 4)*t(10, 3, 1), 9)
(t(1, 2, 4)*t(1, 3, 1)*t(10, 1, 1), 9)
(t(1, 2, 4)*t(10, 3, 1), 9)
(t(1, 2, 4)*t(11, 3, 1), 10)
(t(1, 2, 4)*t(2, 1, 1)*t(10, 3, 1), 10)

sage: Y = Yangian(QQ, 4, filtration='natural')
2
sage: x = Y.gen(5,2,3)^4
20
sage: elt = Y.gen(10,3,1) * Y.gen(2,1,1) * Y.gen(1,2,4)
sage: for s in sorted(elt.support(), key=str): s, Y.degree_on_basis(s)
(t(1, 1, 1)*t(1, 2, 4)*t(10, 3, 1), 12)
(t(1, 2, 4)*t(1, 3, 1)*t(10, 1, 1), 12)
(t(1, 2, 4)*t(10, 3, 1), 11)
(t(1, 2, 4)*t(11, 3, 1), 12)
(t(1, 2, 4)*t(2, 1, 1)*t(10, 3, 1), 13)
dimension()

Return the dimension of self, which is $$\infty$$.

EXAMPLES:

sage: Y = Yangian(QQ, 4)
sage: Y.dimension()
+Infinity
gen(r, i=None, j=None)

Return the generator $$t^{(r)}_{ij}$$ of self.

EXAMPLES:

sage: Y = Yangian(QQ, 4)
sage: Y.gen(2, 1, 3)
t(2)[1,3]
sage: Y.gen(12, 2, 1)
t(12)[2,1]
sage: Y.gen(0, 1, 1)
1
sage: Y.gen(0, 1, 3)
0

Return the associated graded algebra of self.

EXAMPLES:

Graded Algebra of Yangian of gl(4) in the loop filtration over Rational Field
sage: Yangian(QQ, 4, filtration='natural').graded_algebra()
Graded Algebra of Yangian of gl(4) in the natural filtration over Rational Field
one_basis()

Return the basis index of the element $$1$$.

EXAMPLES:

sage: Y = Yangian(QQ, 4)
sage: Y.one_basis()
1
product_on_basis(x, y)

Return the product of two monomials given by x and y.

EXAMPLES:

sage: Y = Yangian(QQ, 4)
sage: Y.gen(12, 2, 1) * Y.gen(2, 1, 1) # indirect doctest
t(1)[1,1]*t(12)[2,1] - t(1)[2,1]*t(12)[1,1]
+ t(2)[1,1]*t(12)[2,1] + t(12)[2,1] + t(13)[2,1]
product_on_gens(a, b)

Return the product on two generators indexed by a and b.

We assume $$(r, i, j) \geq (s, k, \ell)$$, and we start with the basic relation:

$[t_{ij}^{(r)}, t_{k\ell}^{(s)}] - [t_{ij}^{(r-1)}, t_{k\ell}^{(s+1)}] = t_{kj}^{(r-1)} t_{i\ell}^{(s)} - t_{kj}^{(s)} t_{i\ell}^{(r-1)}.$

Solving for the first term and using induction we get:

$[t_{ij}^{(r)}, t_{k\ell}^{(s)}] = \sum_{a=1}^s \left( t_{kj}^{(a-1)} t_{i\ell}^{(r+s-a)} - t_{kj}^{(r+s-a)} t_{i\ell}^{(a-1)} \right).$

Next applying induction on this we get

$t_{ij}^{(r)} t_{k\ell}^{(s)} = t_{k\ell}^{(s)} t_{ij}^{(r)} + \sum C_{abcd}^{m\ell} t_{ab}^{(m)} t_{cd}^{(\ell)}$

where $$m + \ell < r + s$$ and $$t_{ab}^{(m)} < t_{cd}^{(\ell)}$$.

EXAMPLES:

sage: Y = Yangian(QQ, 4)
sage: Y.product_on_gens((2,1,1), (12,2,1))
t(2)[1,1]*t(12)[2,1]
sage: Y.gen(2, 1, 1) * Y.gen(12, 2, 1)
t(2)[1,1]*t(12)[2,1]
sage: Y.product_on_gens((12,2,1), (2,1,1))
t(1)[1,1]*t(12)[2,1] - t(1)[2,1]*t(12)[1,1]
+ t(2)[1,1]*t(12)[2,1] + t(12)[2,1] + t(13)[2,1]
sage: Y.gen(12, 2, 1) * Y.gen(2, 1, 1)
t(1)[1,1]*t(12)[2,1] - t(1)[2,1]*t(12)[1,1]
+ t(2)[1,1]*t(12)[2,1] + t(12)[2,1] + t(13)[2,1]
class sage.algebras.yangian.YangianLevel(base_ring, n, level, variable_name, filtration)

The Yangian $$Y_{\ell}(\mathfrak{gl_n})$$ of level $$\ell$$.

The Yangian of level $$\ell$$ is the quotient of the Yangian $$Y(\mathfrak{gl}_n)$$ by the two-sided ideal generated by $$t_{ij}^{(r)}$$ for all $$r > p$$ and all $$i,j \in \{1, \ldots, n\}$$.

EXAMPLES:

sage: Y = Yangian(QQ, 4, 3)
sage: elt = Y.gen(3,2,1) * Y.gen(1,1,3)
sage: elt * Y.gen(1, 1, 2)
t(1)[1,2]*t(1)[1,3]*t(3)[2,1] + t(1)[1,2]*t(3)[2,3]
- t(1)[1,3]*t(3)[1,1] + t(1)[1,3]*t(3)[2,2] - t(3)[1,3]
defining_polynomial(i, j, u=None)

Return the defining polynomial of i and j.

The defining polynomial is given by:

$T_{ij}(u) = \delta_{ij} u^{\ell} + \sum_{k=1}^{\ell} t_{ij}^{(k)} u^{\ell-k}.$

EXAMPLES:

sage: Y = Yangian(QQ, 3, 5)
sage: Y.defining_polynomial(3, 2)
t(1)[3,2]*u^4 + t(2)[3,2]*u^3 + t(3)[3,2]*u^2 + t(4)[3,2]*u + t(5)[3,2]
sage: Y.defining_polynomial(1, 1)
u^5 + t(1)[1,1]*u^4 + t(2)[1,1]*u^3 + t(3)[1,1]*u^2 + t(4)[1,1]*u + t(5)[1,1]
gen(r, i=None, j=None)

Return the generator $$t^{(r)}_{ij}$$ of self.

EXAMPLES:

sage: Y = Yangian(QQ, 4, 3)
sage: Y.gen(2, 1, 3)
t(2)[1,3]
sage: Y.gen(12, 2, 1)
0
sage: Y.gen(0, 1, 1)
1
sage: Y.gen(0, 1, 3)
0
gens()

Return the generators of self.

EXAMPLES:

sage: Y = Yangian(QQ, 2, 2)
sage: Y.gens()
(t(1)[1,1], t(2)[1,1], t(1)[1,2], t(2)[1,2], t(1)[2,1],
t(2)[2,1], t(1)[2,2], t(2)[2,2])
level()

Return the level of self.

EXAMPLES:

sage: Y = Yangian(QQ, 3, 5)
sage: Y.level()
5
product_on_gens(a, b)

Return the product on two generators indexed by a and b.

EXAMPLES:

sage: Y = Yangian(QQ, 4, 3)
sage: Y.gen(1,2,2) * Y.gen(2,1,3) # indirect doctest
t(1)[2,2]*t(2)[1,3]
sage: Y.gen(1,2,1) * Y.gen(2,1,3) # indirect doctest
t(1)[2,1]*t(2)[1,3]
sage: Y.gen(3,2,1) * Y.gen(1,1,3) # indirect doctest
t(1)[1,3]*t(3)[2,1] + t(3)[2,3]
quantum_determinant(u=None)

Return the quantum determinant of self.

The quantum determinant is defined by:

$\operatorname{qdet}(u) = \sum_{\sigma \in S_n} (-1)^{\sigma} \prod_{k=1}^n T_{\sigma(k),k}(u - k + 1).$

EXAMPLES:

sage: Y = Yangian(QQ, 2, 2)
sage: Y.quantum_determinant()
u^4 + (-2 + t(1)[1,1] + t(1)[2,2])*u^3
+ (1 - t(1)[1,1] + t(1)[1,1]*t(1)[2,2] - t(1)[1,2]*t(1)[2,1]
- 2*t(1)[2,2] + t(2)[1,1] + t(2)[2,2])*u^2
+ (-t(1)[1,1]*t(1)[2,2] + t(1)[1,1]*t(2)[2,2]
+ t(1)[1,2]*t(1)[2,1] - t(1)[1,2]*t(2)[2,1]
- t(1)[2,1]*t(2)[1,2] + t(1)[2,2] + t(1)[2,2]*t(2)[1,1]
- t(2)[1,1] - t(2)[2,2])*u
- t(1)[1,1]*t(2)[2,2] + t(1)[1,2]*t(2)[2,1] + t(2)[1,1]*t(2)[2,2]
- t(2)[1,2]*t(2)[2,1] + t(2)[2,2]