Onsager Algebra

AUTHORS:

  • Travis Scrimshaw (2017-07): Initial version
sage.algebras.lie_algebras.onsager.OnsagerAlgebra

The Onsager (Lie) algebra.

The Onsager (Lie) algebra \(\mathcal{O}\) is a Lie algebra with generators \(A_0, A_1\) that satisfy

\[[A_0, [A_0, [A_0, A_1]]] = -4 [A_0, A_1], \qquad [A_1, [A_1, [A_1, A_0]]] = -4 [A_1, A_0].\]

Note

We are using a rescaled version of the usual defining generators.

There exist a basis \(\{A_m, G_n \mid m \in \ZZ, n \in \ZZ_{>0}\}\) for \(\mathcal{O}\) with structure coefficients

\[[A_m, A_{m'}] = G_{m-m'}, \qquad [G_n, G_{n'}] = 0, \qquad [G_n, A_m] = 2A_{m-n} - 2A_{m+n},\]

where \(m > m'\).

The Onsager algebra is isomorphic to the subalgebra of the affine Lie algebra \(\widehat{\mathfrak{sl}}_2 = \mathfrak{sl}_2 \otimes \CC[t,t^{-1}] \oplus \CC K \oplus \CC d\) that is invariant under the Chevalley involution. In particular, we have

\[A_i \mapsto f \otimes t^i - e \otimes t^{-i}, \qquad G_i \mapsto h \otimes t^{-i} - h \otimes t^i.\]

where \(e,f,h\) are the Chevalley generators of \(\mathfrak{sl}_2\).

EXAMPLES:

We construct the Onsager algebra and do some basic computations:

sage: O = lie_algebras.OnsagerAlgebra(QQ)
sage: O.inject_variables()
Defining A0, A1

We verify the defining relations:

sage: O([A0, [A0, [A0, A1]]]) == -4 * O([A0, A1])
True
sage: O([A1, [A1, [A1, A0]]]) == -4 * O([A1, A0])
True

We check the embedding into \(\widehat{\mathfrak{sl}}_2\):

sage: L = LieAlgebra(QQ, cartan_type=['A',1,1])
sage: B = L.basis()
sage: al = RootSystem(['A',1]).root_lattice().simple_root(1)
sage: ac = al.associated_coroot()
sage: def emb_A(i): return B[-al,i] - B[al,-i]
sage: def emb_G(i): return B[ac,i] - B[ac,-i]
sage: a0 = emb_A(0)
sage: a1 = emb_A(1)
sage: L([a0, [a0, [a0, a1]]]) == -4 * L([a0, a1])
True
sage: L([a1, [a1, [a1, a0]]]) == -4 * L([a1, a0])
True

sage: all(emb_G(n).bracket(emb_A(m)) == 2*emb_A(m-n) - 2*emb_A(m+n)
....:     for m in range(-10, 10) for n in range(1,10))
True
sage: all(emb_A(m).bracket(emb_A(mp)) == emb_G(m-mp)
....:     for m in range(-10,10) for mp in range(m-10, m))
True

REFERENCES:

sage.algebras.lie_algebras.onsager.QuantumOnsagerAlgebra

The quantum Onsager algebra.

The quantum Onsager algebra, or \(q\)-Onsager algebra, is a quantum group analog of the Onsager algebra. It is the left (or right) coideal subalgebra of the quantum group \(U_q(\widehat{\mathfrak{sl}}_2)\) and is the simplest example of a quantum symmetric pair coideal subalgebra of affine type.

The \(q\)-Onsager algebra depends on a parameter \(c\) such that \(c(1) = 1\). The \(q\)-Onsager algebra with parameter \(c\) is denoted \(U_q(\mathcal{O}_R)_c\), where \(R\) is the base ring of the defining Onsager algebra.

EXAMPLES:

We create the \(q\)-Onsager algebra and its generators:

sage: O = lie_algebras.OnsagerAlgebra(QQ)
sage: Q = O.quantum_group()
sage: G = Q.algebra_generators()

The generators are given as pairs, where \(G[0,n]\) is the generator \(B_{n\delta+\alpha_1}\) and \(G[1,n]\) is the generator \(B_{n\delta}\). We use the convention that \(n\delta + \alpha_1 \equiv (-n-1)\delta + \alpha_0\).

sage: G[0,5]
B[5d+a1]
sage: G[0,-5]
B[4d+a0]
sage: G[1,5]
B[5d]
sage: (G[0,5] + G[0,-3]) * (G[1,2] - G[0,3])
B[2d+a0]*B[2d] - B[2d+a0]*B[3d+a1]
 + ((-q^4+1)/q^2)*B[1d]*B[6d+a1]
 + ((q^4-1)/q^2)*B[1d]*B[4d+a1] + B[2d]*B[5d+a1]
 - B[5d+a1]*B[3d+a1] + ((q^2+1)/q^2)*B[7d+a1]
 + ((q^6+q^4-q^2-1)/q^2)*B[5d+a1] + (-q^4-q^2)*B[3d+a1]
sage: (G[0,5] + G[0,-3] + G[1,4]) * (G[0,2] - G[1,3])
-B[2d+a0]*B[3d] + B[2d+a0]*B[2d+a1]
 + ((q^4-1)/q^4)*B[1d]*B[7d+a1]
 + ((q^8-2*q^4+1)/q^4)*B[1d]*B[5d+a1]
 + (-q^4+1)*B[1d]*B[3d+a1] + ((q^4-1)/q^2)*B[2d]*B[6d+a1]
 + ((-q^4+1)/q^2)*B[2d]*B[4d+a1] - B[3d]*B[4d]
 - B[3d]*B[5d+a1] + B[4d]*B[2d+a1] + B[5d+a1]*B[2d+a1]
 + ((-q^2-1)/q^4)*B[8d+a1] + ((-q^6-q^4+q^2+1)/q^4)*B[6d+a1]
 + (-q^6-q^4+q^2+1)*B[4d+a1] + (q^6+q^4)*B[2d+a1]

We check the \(q\)-Dolan-Grady relations:

sage: def q_dolan_grady(a, b, q):
....:     x = q*a*b - ~q*b*a
....:     y = ~q*a*x - q*x*a
....:     return a*y - y*a
sage: A0, A1 = G[0,-1], G[0,0]
sage: q = Q.q()
sage: q_dolan_grady(A1, A0, q) == (q^4 + 2*q^2 + 1) * (A0*A1 - A1*A0)
True
sage: q_dolan_grady(A0, A1, q) == (q^4 + 2*q^2 + 1) * (A1*A0 - A0*A1)
True

REFERENCES: