sagemathgh-39168: Addition of Kähler algebras as a category

    
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This PR adds Kähler algebras as a category `KahlerAlgebras` as defined
in Theorem 2.28 of [this paper](https://arxiv.org/abs/2309.14312). It
changes the category of Chow rings to `KahlerAlgebras() &
CommutativeRings().Quotients()`. A method, `lefschetz_element()` is also
implemented for Chow rings which returns a Lefschetz element as provided
in Corollary 2.30 of the paper above, which is used for the Hodge-
Riemann relations computation.

@tscrim

### 📝 Checklist

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- [x] I have linked a relevant issue or discussion.
- [x] I have created tests covering the changes.
- [x] I have updated the documentation and checked the documentation
preview.

### ⌛ Dependencies

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URL: sagemath#39168
Reported by: 25shriya
Reviewer(s): 25shriya, Travis Scrimshaw

src/doc/en/reference/categories/index.rst

@@ -137,6 +137,7 @@ Individual Categories
    sage/categories/integral_domains
    sage/categories/j_trivial_semigroups
    sage/categories/kac_moody_algebras
+   sage/categories/kahler_algebras
    sage/categories/lambda_bracket_algebras
    sage/categories/lambda_bracket_algebras_with_basis
    sage/categories/lattice_posets

src/sage/categories/category_types.py

@@ -202,12 +202,13 @@ def _test_category_over_bases(self, **options):
         """
         tester = self._tester(**options)
         from sage.categories.category_singleton import Category_singleton
-
+        from sage.categories.category_with_axiom import CategoryWithAxiom_over_base_ring
         from .bimodules import Bimodules
         from .schemes import Schemes
         for cat in self.super_categories():
             tester.assertTrue(isinstance(cat, (Category_singleton, Category_over_base,
-                                            Bimodules, Schemes)),
+                                               CategoryWithAxiom_over_base_ring,
+                                               Bimodules, Schemes)),
                            "The super categories of a category over base should"
                            " be a category over base (or the related Bimodules)"
                            " or a singleton category")

src/sage/categories/graded_algebras_with_basis.py

@@ -13,6 +13,7 @@
 from sage.categories.graded_modules import GradedModulesCategory
 from sage.categories.signed_tensor import SignedTensorProductsCategory, tensor_signed
 from sage.misc.cachefunc import cached_method
+from sage.categories.category_with_axiom import CategoryWithAxiom_over_base_ring
 
 
 class GradedAlgebrasWithBasis(GradedModulesCategory):
@@ -154,6 +155,24 @@ def formal_series_ring(self):
     class ElementMethods:
         pass
 
+    class FiniteDimensional(CategoryWithAxiom_over_base_ring):
+        class ParentMethods:
+            @cached_method
+            def top_degree(self):
+                r"""
+                Return the top degree of the finite dimensional graded algebra.
+
+                EXAMPLES::
+
+                    sage: ch = matroids.Uniform(4,6).chow_ring(QQ, False)
+                    sage: ch.top_degree()
+                    3
+                    sage: ch = matroids.Wheel(3).chow_ring(QQ, True, 'atom-free')
+                    sage: ch.top_degree()
+                    3
+                """
+                return max(b.degree() for b in self.basis())
+
     class SignedTensorProducts(SignedTensorProductsCategory):
         """
         The category of algebras with basis constructed by signed tensor

src/sage/categories/kahler_algebras.py

@@ -0,0 +1,200 @@
+r"""
+Kähler Algebras
+
+AUTHORS:
+
+- Shriya M
+"""
+# ****************************************************************************
+#       Copyright (C) 2024 Shriya M <25shriya at gmail.com>
+#
+# This program is free software: you can redistribute it and/or modify
+# it under the terms of the GNU General Public License as published by
+# the Free Software Foundation, either version 2 of the License, or
+# (at your option) any later version.
+#                  https://www.gnu.org/licenses/
+# ****************************************************************************
+
+from sage.categories.category_types import Category_over_base_ring
+from sage.categories.graded_algebras_with_basis import GradedAlgebrasWithBasis
+from sage.categories.finite_dimensional_algebras_with_basis import FiniteDimensionalAlgebrasWithBasis
+from sage.categories.filtered_modules_with_basis import FilteredModulesWithBasis
+from sage.misc.abstract_method import abstract_method
+from sage.quadratic_forms.quadratic_form import QuadraticForm
+from sage.misc.cachefunc import cached_method
+
+
+class KahlerAlgebras(Category_over_base_ring):
+    r"""
+    The category of graded algebras satisfying the Kähler package.
+    A finite-dimensional graded algebra `\bigoplus_{k=1}^{r}A^k` satisfies
+    the *Kähler package* if the following properties hold:
+
+    - Poincaré duality: There exists a perfect `\ZZ`-bilinear pairing
+      given by
+
+      .. MATH::
+
+          A^k \times A^{r-k} \longrightarrow \ZZ \\
+            (a,b) \mapsto \deg(a \cdot b).
+
+    - Hard-Lefschetz Theorem: The graded algebra contains *Lefschetz elements*
+      `\omega \in A^{1}_{\RR}` such that multiplication by `\omega` is
+      an injection from `A^k_{\RR} \longrightarrow A^{k+1}_{\RR}`
+      for all `k < \frac{r}{2}`.
+
+    - Hodge-Riemann-Minikowski Relations: Every Lefchetz element `\omega`,
+      define quadratic forms on `A^{k}_{\RR}` given by
+
+      .. MATH::
+
+          a \mapsto (-1)^k \deg(a \cdot \omega^{r-2k} \cdot a)
+
+      This quadratic form becomes positive definite upon restriction to the
+      kernel of the following map
+
+      .. MATH::
+
+          A^k_\RR \longrightarrow A^{r-k+1}_\RR \\
+                a \mapsto a \cdot \omega^{r-2k+1}.
+
+    REFERENCES:
+
+    - [ANR2023]_
+
+    TESTS::
+
+        sage: from sage.categories.kahler_algebras import KahlerAlgebras
+
+        sage: C = KahlerAlgebras(QQ)
+        sage: TestSuite(C).run()
+    """
+    def super_categories(self):
+        r"""
+        Return the super categories of ``self``.
+
+        EXAMPLES::
+
+            sage: from sage.categories.kahler_algebras import KahlerAlgebras
+
+            sage: C = KahlerAlgebras(QQ); C
+            Category of kahler algebras over Rational Field
+            sage: sorted(C.super_categories(), key=str)
+            [Category of finite dimensional graded algebras with basis over
+            Rational Field]
+        """
+        return [GradedAlgebrasWithBasis(self.base_ring()).FiniteDimensional()]
+
+    class ParentMethods:
+        @abstract_method
+        def poincare_pairing(self, a, b):
+            r"""
+            Return the Poincaré pairing of two elements of the Kähler algebra.
+
+            EXAMPLES::
+
+                sage: ch = matroids.catalog.Fano().chow_ring(QQ, True, 'fy')
+                sage: Ba, Bb, Bc, Bd, Be, Bf, Bg, Babf, Bace, Badg, Bbcd, Bbeg, Bcfg, Bdef, Babcdefg = ch.gens()[8:]
+                sage: u = ch(-Babf^2 + Bcfg^2 - 8/7*Bc*Babcdefg + 1/2*Bd*Babcdefg - Bf*Babcdefg - Bg*Babcdefg); u
+                -Babf^2 + Bcfg^2 - 8/7*Bc*Babcdefg + 1/2*Bd*Babcdefg - Bf*Babcdefg - Bg*Babcdefg
+                sage: v = ch(Bg - 2/37*Babf + Badg + Bbeg + Bcfg + Babcdefg); v
+                Bg - 2/37*Babf + Badg + Bbeg + Bcfg + Babcdefg
+                sage: ch.poincare_pairing(v, u)
+                3
+            """
+
+        @abstract_method
+        def lefschetz_element(self):
+            r"""
+            Return one Lefschetz element of the given Kähler algebra.
+
+            EXAMPLES::
+
+                sage: U46 = matroids.Uniform(4, 6)
+                sage: C = U46.chow_ring(QQ, False)
+                sage: w = C.lefschetz_element(); w
+                -2*A01 - 2*A02 - 2*A03 - 2*A04 - 2*A05 - 2*A12 - 2*A13 - 2*A14
+                - 2*A15 - 2*A23 - 2*A24 - 2*A25 - 2*A34 - 2*A35 - 2*A45 - 6*A012
+                - 6*A013 - 6*A014 - 6*A015 - 6*A023 - 6*A024 - 6*A025 - 6*A034
+                - 6*A035 - 6*A045 - 6*A123 - 6*A124 - 6*A125 - 6*A134 - 6*A135
+                - 6*A145 - 6*A234 - 6*A235 - 6*A245 - 6*A345 - 30*A012345
+                sage: basis_deg = {}
+                sage: for b in C.basis():
+                ....:     deg = b.homogeneous_degree()
+                ....:     if deg not in basis_deg:
+                ....:         basis_deg[deg] = []
+                ....:     basis_deg[deg].append(b)
+                sage: m = max(basis_deg); m
+                3
+                sage: len(basis_deg[1]) == len(basis_deg[2])
+                True
+                sage: matrix([(w*b).to_vector() for b in basis_deg[1]]).rank()
+                36
+                sage: len(basis_deg[2])
+                36
+            """
+
+        def hodge_riemann_relations(self, k):
+            r"""
+            Return the quadratic form for the corresponding ``k``
+            (`< \frac{r}{2}`) for the Kähler algebra, where `r` is the top degree.
+
+            EXAMPLES::
+
+                sage: ch = matroids.Uniform(4, 6).chow_ring(QQ, False)
+                sage: ch.hodge_riemann_relations(1)
+                Quadratic form in 36 variables over Rational Field with coefficients:
+                [ 3 -1 -1 3 -1 -1 -1 -1 3 -1 -1 -1 -1 -1 -1 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 3 -1 -1 -1 -1 -1 3 ]
+                [ * 3 -1 3 -1 -1 -1 -1 -1 3 -1 -1 -1 -1 -1 -1 -1 -1 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 3 -1 3 ]
+                [ * * 3 3 -1 -1 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 3 3 ]
+                [ * * * 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 ]
+                [ * * * * 3 -1 3 -1 3 -1 -1 -1 -1 -1 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 3 -1 -1 -1 -1 -1 -1 3 ]
+                [ * * * * * 3 3 -1 -1 3 -1 -1 -1 -1 -1 -1 -1 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 3 -1 -1 3 ]
+                [ * * * * * * 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 ]
+                [ * * * * * * * 3 3 3 -1 -1 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 3 ]
+                [ * * * * * * * * 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 ]
+                [ * * * * * * * * * 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 ]
+                [ * * * * * * * * * * 3 -1 3 -1 -1 3 -1 -1 3 -1 -1 -1 -1 -1 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 3 ]
+                [ * * * * * * * * * * * 3 3 -1 3 -1 -1 3 -1 -1 -1 -1 -1 -1 -1 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 3 ]
+                [ * * * * * * * * * * * * 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 ]
+                [ * * * * * * * * * * * * * 3 3 3 -1 -1 -1 3 -1 -1 -1 -1 -1 -1 -1 3 -1 -1 -1 -1 -1 -1 -1 3 ]
+                [ * * * * * * * * * * * * * * 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 ]
+                [ * * * * * * * * * * * * * * * 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 ]
+                [ * * * * * * * * * * * * * * * * 3 3 3 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 3 -1 -1 -1 3 ]
+                [ * * * * * * * * * * * * * * * * * 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 ]
+                [ * * * * * * * * * * * * * * * * * * 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 ]
+                [ * * * * * * * * * * * * * * * * * * * 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 ]
+                [ * * * * * * * * * * * * * * * * * * * * 3 -1 3 -1 3 -1 -1 -1 -1 3 -1 -1 -1 3 -1 3 ]
+                [ * * * * * * * * * * * * * * * * * * * * * 3 3 -1 -1 3 -1 -1 3 -1 -1 -1 3 -1 -1 3 ]
+                [ * * * * * * * * * * * * * * * * * * * * * * 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 ]
+                [ * * * * * * * * * * * * * * * * * * * * * * * 3 3 3 -1 3 -1 -1 -1 3 -1 -1 -1 3 ]
+                [ * * * * * * * * * * * * * * * * * * * * * * * * 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 ]
+                [ * * * * * * * * * * * * * * * * * * * * * * * * * 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 ]
+                [ * * * * * * * * * * * * * * * * * * * * * * * * * * 3 3 3 3 -1 -1 -1 -1 3 3 ]
+                [ * * * * * * * * * * * * * * * * * * * * * * * * * * * 3 -1 -1 -1 -1 -1 -1 -1 -1 ]
+                [ * * * * * * * * * * * * * * * * * * * * * * * * * * * * 3 -1 -1 -1 -1 -1 -1 -1 ]
+                [ * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 3 -1 -1 -1 -1 -1 -1 ]
+                [ * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 3 3 3 3 3 3 ]
+                [ * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 3 -1 -1 -1 -1 ]
+                [ * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 3 -1 -1 -1 ]
+                [ * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 3 -1 -1 ]
+                [ * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 3 -1 ]
+                [ * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 3 ]
+                sage: ch.hodge_riemann_relations(3)
+                Traceback (most recent call last):
+                ...
+                ValueError: k must be less than r/2 < 2
+            """
+            r = self.top_degree()
+            if k > (r/2):
+                raise ValueError("k must be less than r/2 < 2")
+            basis_k = []
+            lefschetz_el = self.lefschetz_element()
+            for b in self.basis():
+                if b.homogeneous_degree() == k:
+                    basis_k.append(b)
+            coeff = []
+            for i,el in enumerate(basis_k):
+                for j in range(i, len(basis_k)):
+                    coeff.append((el * (lefschetz_el ** (r-(2*k)) * basis_k[j])).degree())
+            return QuadraticForm(self.base_ring(), len(basis_k), coeff)

src/sage/categories/meson.build

@@ -120,6 +120,7 @@ py.install_sources(
   'isomorphic_objects.py',
   'j_trivial_semigroups.py',
   'kac_moody_algebras.py',
+  'kahler_algebras.py',
   'l_trivial_semigroups.py',
   'lambda_bracket_algebras.py',
   'lambda_bracket_algebras_with_basis.py',