Yangians

AUTHORS:

  • Travis Scrimshaw (2013-10-08): Initial version
sage.algebras.yangian.GeneratorIndexingSet

Helper class for the indexing set of the generators.

sage.algebras.yangian.GradedYangianBase

Base class for graded algebras associated to a Yangian.

sage.algebras.yangian.GradedYangianLoop

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
sage.algebras.yangian.GradedYangianNatural

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
sage.algebras.yangian.Yangian

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

\[\Delta \colon T(u) \mapsto T_{[1]}(u) T_{[2]}(u), \qquad S \colon T(u) \mapsto T^{-1}(u), \qquad \epsilon \colon T(u) \mapsto 1,\]

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[0]-1][t[1]-1,t[2]-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:

sage.algebras.yangian.YangianLevel

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]