diff --git a/gtsam/base/Lie.h b/gtsam/base/Lie.h
index ce6541d21..827141e58 100644
--- a/gtsam/base/Lie.h
+++ b/gtsam/base/Lie.h
@@ -171,21 +171,21 @@ namespace internal {
 /// Assumes existence of: identity, dimension, localCoordinates, retract,
 /// and additionally Logmap, Expmap, compose, between, and inverse
 template<class Class>
-struct LieGroupTraits: GetDimensionImpl<Class, Class::dimension> {
-  typedef lie_group_tag structure_category;
+struct LieGroupTraits: public GetDimensionImpl<Class, Class::dimension> {
+  using structure_category = lie_group_tag;
 
   /// @name Group
   /// @{
-  typedef multiplicative_group_tag group_flavor;
+  using group_flavor = multiplicative_group_tag;
   static Class Identity() { return Class::Identity();}
   /// @}
 
   /// @name Manifold
   /// @{
-  typedef Class ManifoldType;
+  using ManifoldType = Class;
   inline constexpr static auto dimension = Class::dimension;
-  typedef Eigen::Matrix<double, dimension, 1> TangentVector;
-  typedef OptionalJacobian<dimension, dimension> ChartJacobian;
+  using TangentVector = Eigen::Matrix<double, dimension, 1>;
+  using ChartJacobian = OptionalJacobian<dimension, dimension>;
 
   static TangentVector Local(const Class& origin, const Class& other,
       ChartJacobian Horigin = {}, ChartJacobian Hother = {}) {
@@ -225,10 +225,28 @@ struct LieGroupTraits: GetDimensionImpl<Class, Class::dimension> {
   /// @}
 };
 
+
 /// Both LieGroupTraits and Testable
 template<class Class> struct LieGroup: LieGroupTraits<Class>, Testable<Class> {};
 
-} // \ namepsace internal
+/// Adds LieAlgebra, Hat, and Vie to LieGroupTraits
+template<class Class> struct MatrixLieGroupTraits: LieGroupTraits<Class> {
+  using LieAlgebra = typename Class::LieAlgebra;
+  using TangentVector = typename LieGroupTraits<Class>::TangentVector;
+
+  static LieAlgebra Hat(const TangentVector& v) {
+    return Class::Hat(v);
+  }
+
+  static TangentVector Vee(const LieAlgebra& X) {
+    return Class::Vee(X);
+  }
+};
+
+/// Both LieGroupTraits and Testable
+template<class Class> struct MatrixLieGroup: MatrixLieGroupTraits<Class>, Testable<Class> {};
+
+} // \ namespace internal
 
 /**
  * These core global functions can be specialized by new Lie types
@@ -269,7 +287,7 @@ public:
         (std::is_base_of<lie_group_tag, structure_category_tag>::value),
         "This type's trait does not assert it is a Lie group (or derived)");
 
-    // group opertations with Jacobians
+    // group operations with Jacobians
     g = traits<T>::Compose(g, h, Hg, Hh);
     g = traits<T>::Between(g, h, Hg, Hh);
     g = traits<T>::Inverse(g, Hg);
@@ -286,6 +304,25 @@ private:
   ChartJacobian Hg, Hh;
 };
 
+/**
+ * Matrix Lie Group Concept
+ */
+template<typename T>
+class IsMatrixLieGroup: public IsLieGroup<T> {
+public:
+typedef typename traits<T>::LieAlgebra LieAlgebra;
+typedef typename traits<T>::TangentVector TangentVector;
+
+  BOOST_CONCEPT_USAGE(IsMatrixLieGroup) {
+    // hat and vee
+    X = traits<T>::Hat(xi);
+    xi = traits<T>::Vee(X);
+  }
+private:
+  LieAlgebra X;
+  TangentVector xi;
+};
+
 /**
  *  Three term approximation of the Baker-Campbell-Hausdorff formula
  *  In non-commutative Lie groups, when composing exp(Z) = exp(X)exp(Y)
@@ -301,22 +338,21 @@ T BCH(const T& X, const T& Y) {
   return T(X + Y + _2 * X_Y + _12 * bracket(X - Y, X_Y) - _24 * bracket(Y, bracket(X, X_Y)));
 }
 
-/**
- * Declaration of wedge (see Murray94book) used to convert
- * from n exponential coordinates to n*n element of the Lie algebra
- */
+#ifdef GTSAM_ALLOW_DEPRECATED_SINCE_V43
+/// @deprecated: use T::Hat
 template <class T> Matrix wedge(const Vector& x);
+#endif
 
 /**
  * Exponential map given exponential coordinates
- * class T needs a wedge<> function and a constructor from Matrix
+ * class T needs a constructor from Matrix.
  * @param x exponential coordinates, vector of size n
  * @ return a T
  */
 template <class T>
-T expm(const Vector& x, int K=7) {
-  Matrix xhat = wedge<T>(x);
-  return T(expm(xhat,K));
+T expm(const Vector& x, int K = 7) {
+  auto xhat = T::Hat(x);
+  return T(expm(xhat, K));
 }
 
 /**
diff --git a/gtsam/base/VectorSpace.h b/gtsam/base/VectorSpace.h
index c5c4ad622..21c8710f1 100644
--- a/gtsam/base/VectorSpace.h
+++ b/gtsam/base/VectorSpace.h
@@ -51,6 +51,8 @@ struct VectorSpaceImpl {
   /// @name Lie Group
   /// @{
 
+  typedef Eigen::Matrix<double, N, 1> LieAlgebra;
+
   static TangentVector Logmap(const Class& m, ChartJacobian Hm = {}) {
     if (Hm) *Hm = Jacobian::Identity();
     return m.vector();
@@ -80,6 +82,13 @@ struct VectorSpaceImpl {
     return -v;
   }
 
+  static LieAlgebra Hat(const TangentVector& v) {
+    return v;
+  }
+
+  static TangentVector Vee(const LieAlgebra& X) {
+    return X;
+  }
   /// @}
 };
 
@@ -411,6 +420,8 @@ struct DynamicTraits {
 
   /// @name Lie Group
   /// @{
+  using LieAlgebra = Dynamic;
+    
   static TangentVector Logmap(const Dynamic& m, ChartJacobian H = {}) {
     if (H) *H = Eye(m);
     TangentVector result(GetDimension(m));
@@ -441,6 +452,15 @@ struct DynamicTraits {
     if (H2) *H2 = Eye(v1);
     return v2 - v1;
   }
+  
+  static LieAlgebra Hat(const TangentVector& v) {
+    return v;
+  }
+  
+  static TangentVector Vee(const LieAlgebra& X) {
+    return X;
+  }
+  
   /// @}
 
 };
diff --git a/gtsam/geometry/Pose2.cpp b/gtsam/geometry/Pose2.cpp
index 640481406..b5c426597 100644
--- a/gtsam/geometry/Pose2.cpp
+++ b/gtsam/geometry/Pose2.cpp
@@ -203,6 +203,20 @@ Pose2 Pose2::inverse() const {
   return Pose2(r_.inverse(), r_.unrotate(Point2(-t_.x(), -t_.y())));
 }
 
+/* ************************************************************************* */
+Matrix3 Pose2::Hat(const Pose2::TangentVector& xi) {
+  Matrix3 X;
+  X << 0., -xi.z(), xi.x(),
+    xi.z(), 0., xi.y(),
+    0., 0., 0.;
+  return X;
+}
+
+/* ************************************************************************* */
+Pose2::TangentVector Pose2::Vee(const Matrix3& X) {
+  return TangentVector(X(0, 2), X(1, 2), X(1,0));
+}
+
 /* ************************************************************************* */
 // see doc/math.lyx, SE(2) section
 Point2 Pose2::transformTo(const Point2& point,
diff --git a/gtsam/geometry/Pose2.h b/gtsam/geometry/Pose2.h
index da7bc922e..93bc6751b 100644
--- a/gtsam/geometry/Pose2.h
+++ b/gtsam/geometry/Pose2.h
@@ -40,9 +40,12 @@ class GTSAM_EXPORT Pose2: public LieGroup<Pose2, 3> {
 
 public:
 
-  /** Pose Concept requirements */
-  typedef Rot2 Rotation;
-  typedef Point2 Translation;
+  /// Pose Concept requirements
+  using Rotation = Rot2;
+  using Translation = Point2;
+
+  /// LieGroup Concept requirements
+  using LieAlgebra = Matrix3;
 
 private:
 
@@ -183,21 +186,6 @@ public:
   static Matrix3 adjointMap_(const Vector3 &xi) { return adjointMap(xi);}
   static Vector3 adjoint_(const Vector3 &xi, const Vector3 &y) { return adjoint(xi, y);}
 
-  /**
-   * wedge for SE(2):
-   * @param xi 3-dim twist (v,omega) where
-   *  omega is angular velocity
-   *  v (vx,vy) = 2D velocity
-   * @return xihat, 3*3 element of Lie algebra that can be exponentiated
-   */
-  static inline Matrix3 wedge(double vx, double vy, double w) {
-    Matrix3 m;
-    m << 0.,-w,  vx,
-         w,  0., vy,
-         0., 0.,  0.;
-    return m;
-  }
-
   /// Derivative of Expmap
   static Matrix3 ExpmapDerivative(const Vector3& v);
 
@@ -212,6 +200,12 @@ public:
 
   using LieGroup<Pose2, 3>::inverse; // version with derivative
 
+  /// Hat maps from tangent vector to Lie algebra
+  static Matrix3 Hat(const Vector3& xi);
+
+  /// Vee maps from Lie algebra to tangent vector
+  static Vector3 Vee(const Matrix3& X);
+
   /// @}
   /// @name Group Action on Point2
   /// @{
@@ -339,6 +333,16 @@ public:
   friend std::ostream &operator<<(std::ostream &os, const Pose2& p);
 
   /// @}
+  /// @name deprecated
+  /// @{
+
+#ifdef GTSAM_ALLOW_DEPRECATED_SINCE_V43
+  /// @deprecated: use Hat
+  static inline Matrix3 wedge(double vx, double vy, double w) {
+    return Hat(TangentVector(vx, vy, w));
+  }
+#endif
+  /// @}
 
  private:
 
@@ -357,22 +361,24 @@ public:
   GTSAM_MAKE_ALIGNED_OPERATOR_NEW
 }; // Pose2
 
-/** specialization for pose2 wedge function (generic template in Lie.h) */
+#ifdef GTSAM_ALLOW_DEPRECATED_SINCE_V43
+/// @deprecated: use T::Hat
 template <>
 inline Matrix wedge<Pose2>(const Vector& xi) {
   // NOTE(chris): Need eval() as workaround for Apple clang + avx2.
-  return Matrix(Pose2::wedge(xi(0),xi(1),xi(2))).eval();
+  return Matrix(Pose2::Hat(xi)).eval();
 }
+#endif
 
 // Convenience typedef
 using Pose2Pair = std::pair<Pose2, Pose2>;
 using Pose2Pairs = std::vector<Pose2Pair>;
 
 template <>
-struct traits<Pose2> : public internal::LieGroup<Pose2> {};
+struct traits<Pose2> : public internal::MatrixLieGroup<Pose2> {};
 
 template <>
-struct traits<const Pose2> : public internal::LieGroup<Pose2> {};
+struct traits<const Pose2> : public internal::MatrixLieGroup<Pose2> {};
 
 // bearing and range traits, used in RangeFactor
 template <typename T>
diff --git a/gtsam/geometry/Pose3.cpp b/gtsam/geometry/Pose3.cpp
index a92bb84ef..1a17e61df 100644
--- a/gtsam/geometry/Pose3.cpp
+++ b/gtsam/geometry/Pose3.cpp
@@ -160,6 +160,21 @@ Vector6 Pose3::adjointTranspose(const Vector6& xi, const Vector6& y,
   return adT_xi * y;
 }
 
+/* ************************************************************************* */
+Matrix4 Pose3::Hat(const Vector6& xi) {
+  Matrix4 X;
+  const double wx = xi(0), wy = xi(1), wz = xi(2), vx = xi(3), vy = xi(4), vz = xi(5);
+  X << 0., -wz, wy, vx, wz, 0., -wx, vy, -wy, wx, 0., vz, 0., 0., 0., 0.;
+  return X;
+}
+
+/* ************************************************************************* */
+Vector6 Pose3::Vee(const Matrix4& Xi) {
+  Vector6 xi;
+  xi << Xi(2, 1), Xi(0, 2), Xi(1, 0), Xi(0, 3), Xi(1, 3), Xi(2, 3);
+  return xi;
+}
+
 /* ************************************************************************* */
 void Pose3::print(const std::string& s) const {
   std::cout << (s.empty() ? s : s + " ") << *this << std::endl;
diff --git a/gtsam/geometry/Pose3.h b/gtsam/geometry/Pose3.h
index dd2030567..154e24a36 100644
--- a/gtsam/geometry/Pose3.h
+++ b/gtsam/geometry/Pose3.h
@@ -141,6 +141,8 @@ public:
   /// @name Lie Group
   /// @{
 
+  using LieAlgebra = Matrix4;
+
   /// Exponential map at identity - create a rotation from canonical coordinates \f$ [R_x,R_y,R_z,T_x,T_y,T_z] \f$
   static Pose3 Expmap(const Vector6& xi, OptionalJacobian<6, 6> Hxi = {});
 
@@ -230,16 +232,16 @@ public:
   using LieGroup<Pose3, 6>::inverse; // version with derivative
 
   /**
-   * wedge for Pose3:
+   * Hat for Pose3:
    * @param xi 6-dim twist (omega,v) where
    *  omega = (wx,wy,wz) 3D angular velocity
    *  v (vx,vy,vz) = 3D velocity
    * @return xihat, 4*4 element of Lie algebra that can be exponentiated
    */
-  static Matrix wedge(double wx, double wy, double wz, double vx, double vy,
-      double vz) {
-    return (Matrix(4, 4) << 0., -wz, wy, vx, wz, 0., -wx, vy, -wy, wx, 0., vz, 0., 0., 0., 0.).finished();
-  }
+  static Matrix4 Hat(const Vector6& xi);
+
+  /// Vee maps from Lie algebra to tangent vector
+  static Vector6 Vee(const Matrix4& X);
 
   /// @}
   /// @name Group Action on Point3
@@ -394,6 +396,19 @@ public:
   GTSAM_EXPORT
   friend std::ostream &operator<<(std::ostream &os, const Pose3& p);
 
+  /// @}
+  /// @name deprecated
+  /// @{
+
+#ifdef GTSAM_ALLOW_DEPRECATED_SINCE_V43
+  /// @deprecated: use Hat
+  static inline LieAlgebra wedge(double wx, double wy, double wz, double vx, double vy,
+    double vz) {
+    return Hat(TangentVector(wx, wy, wz, vx, vy, vz));
+  }
+#endif
+  /// @}
+
  private:
 #if GTSAM_ENABLE_BOOST_SERIALIZATION
   /** Serialization function */
@@ -414,17 +429,14 @@ public:
 };
 // Pose3 class
 
-/**
- * wedge for Pose3:
- * @param xi 6-dim twist (omega,v) where
- *  omega = 3D angular velocity
- *  v = 3D velocity
- * @return xihat, 4*4 element of Lie algebra that can be exponentiated
- */
+#ifdef GTSAM_ALLOW_DEPRECATED_SINCE_V43
+/// @deprecated: use T::Hat
 template<>
 inline Matrix wedge<Pose3>(const Vector& xi) {
-  return Pose3::wedge(xi(0), xi(1), xi(2), xi(3), xi(4), xi(5));
+  // NOTE(chris): Need eval() as workaround for Apple clang + avx2.
+  return Matrix(Pose3::Hat(xi)).eval();
 }
+#endif
 
 // Convenience typedef
 using Pose3Pair = std::pair<Pose3, Pose3>;
@@ -434,10 +446,10 @@ using Pose3Pairs = std::vector<std::pair<Pose3, Pose3> >;
 typedef std::vector<Pose3> Pose3Vector;
 
 template <>
-struct traits<Pose3> : public internal::LieGroup<Pose3> {};
+struct traits<Pose3> : public internal::MatrixLieGroup<Pose3> {};
 
 template <>
-struct traits<const Pose3> : public internal::LieGroup<Pose3> {};
+struct traits<const Pose3> : public internal::MatrixLieGroup<Pose3> {};
 
 // bearing and range traits, used in RangeFactor
 template <>
diff --git a/gtsam/geometry/Quaternion.h b/gtsam/geometry/Quaternion.h
index 36f891505..f0deba5a9 100644
--- a/gtsam/geometry/Quaternion.h
+++ b/gtsam/geometry/Quaternion.h
@@ -138,6 +138,16 @@ struct traits<QUATERNION_TYPE> {
     return omega;
   }
 
+  using LieAlgebra = Matrix3;
+
+  static Matrix3 Hat(const Vector3& v) {
+    return SO3::Hat(v);
+  }
+  
+  static Vector3 Vee(const Matrix3& X) {
+    return SO3::Vee(X);
+  }
+
   /// @}
   /// @name Manifold traits
   /// @{
diff --git a/gtsam/geometry/Rot2.cpp b/gtsam/geometry/Rot2.cpp
index 9bf631e50..f1eca66d8 100644
--- a/gtsam/geometry/Rot2.cpp
+++ b/gtsam/geometry/Rot2.cpp
@@ -81,6 +81,21 @@ Vector1 Rot2::Logmap(const Rot2& r, OptionalJacobian<1, 1> H) {
   v << r.theta();
   return v;
 }
+/* ************************************************************************* */
+Matrix2 Rot2::Hat(const Vector1& xi) {
+  Matrix2 X;
+  X << 0., -xi.x(),
+    xi.x(), 0.;
+  return X;
+}
+
+/* ************************************************************************* */
+Vector1 Rot2::Vee(const Matrix2& X) {
+  TangentVector v;
+  v << X(1, 0);
+  return v;
+}
+
 /* ************************************************************************* */
 Matrix2 Rot2::matrix() const {
   Matrix2 rvalue_;
diff --git a/gtsam/geometry/Rot2.h b/gtsam/geometry/Rot2.h
index faa0e79a2..85b41e72c 100644
--- a/gtsam/geometry/Rot2.h
+++ b/gtsam/geometry/Rot2.h
@@ -33,7 +33,6 @@ namespace gtsam {
    * \nosubgrouping
    */
   class GTSAM_EXPORT Rot2 : public LieGroup<Rot2, 1> {
-
     /** we store cos(theta) and sin(theta) */
     double c_, s_;
 
@@ -125,6 +124,8 @@ namespace gtsam {
     /// @name Lie Group
     /// @{
 
+    using LieAlgebra = Matrix2;
+
     /// Exponential map at identity - create a rotation from canonical coordinates
     static Rot2 Expmap(const Vector1& v, ChartJacobian H = {});
 
@@ -156,6 +157,12 @@ namespace gtsam {
 
     using LieGroup<Rot2, 1>::inverse; // version with derivative
 
+    /// Hat maps from tangent vector to Lie algebra
+    static Matrix2 Hat(const Vector1& xi);
+
+    /// Vee maps from Lie algebra to tangent vector
+    static Vector1 Vee(const Matrix2& X);
+
     /// @}
     /// @name Group Action on Point2
     /// @{
@@ -216,7 +223,9 @@ namespace gtsam {
     /** Find closest valid rotation matrix, given a 2x2 matrix */
     static Rot2 ClosestTo(const Matrix2& M);
 
-  private:
+    /// @}
+
+    private:
 #if GTSAM_ENABLE_BOOST_SERIALIZATION
     /** Serialization function */
     friend class boost::serialization::access;
@@ -230,9 +239,9 @@ namespace gtsam {
   };
 
   template<>
-  struct traits<Rot2> : public internal::LieGroup<Rot2> {};
+  struct traits<Rot2> : public internal::MatrixLieGroup<Rot2> {};
 
   template<>
-  struct traits<const Rot2> : public internal::LieGroup<Rot2> {};
+  struct traits<const Rot2> : public internal::MatrixLieGroup<Rot2> {};
 
 } // gtsam
diff --git a/gtsam/geometry/Rot3.h b/gtsam/geometry/Rot3.h
index 71f4cbacd..5768399f8 100644
--- a/gtsam/geometry/Rot3.h
+++ b/gtsam/geometry/Rot3.h
@@ -371,6 +371,8 @@ class GTSAM_EXPORT Rot3 : public LieGroup<Rot3, 3> {
     /// @name Lie Group
     /// @{
 
+    using LieAlgebra = Matrix3;
+
     /**
      * Exponential map at identity - create a rotation from canonical coordinates
      * \f$ [R_x,R_y,R_z] \f$ using Rodrigues' formula
@@ -407,6 +409,12 @@ class GTSAM_EXPORT Rot3 : public LieGroup<Rot3, 3> {
 
     using LieGroup<Rot3, 3>::inverse; // version with derivative
 
+    /// Hat maps from tangent vector to Lie algebra
+    static inline Matrix3 Hat(const Vector3& xi) { return SO3::Hat(xi); }
+
+    /// Vee maps from Lie algebra to tangent vector
+    static inline Vector3 Vee(const Matrix3& X) { return SO3::Vee(X); }
+
     /// @}
     /// @name Group Action on Point3
     /// @{
@@ -581,10 +589,10 @@ class GTSAM_EXPORT Rot3 : public LieGroup<Rot3, 3> {
       const Matrix3& A, OptionalJacobian<3, 9> H = {});
 
   template<>
-  struct traits<Rot3> : public internal::LieGroup<Rot3> {};
+  struct traits<Rot3> : public internal::MatrixLieGroup<Rot3> {};
 
   template<>
-  struct traits<const Rot3> : public internal::LieGroup<Rot3> {};
+  struct traits<const Rot3> : public internal::MatrixLieGroup<Rot3> {};
   
 }  // namespace gtsam
 
diff --git a/gtsam/geometry/SO3.h b/gtsam/geometry/SO3.h
index 1bdcd82c7..ee51f4d83 100644
--- a/gtsam/geometry/SO3.h
+++ b/gtsam/geometry/SO3.h
@@ -224,13 +224,13 @@ struct GTSAM_EXPORT DexpFunctor : public ExpmapFunctor {
 }  //  namespace so3
 
 /*
- * Define the traits. internal::LieGroup provides both Lie group and Testable
+ * Define the traits. internal::MatrixLieGroup provides both Lie group and Testable
  */
 
 template <>
-struct traits<SO3> : public internal::LieGroup<SO3> {};
+struct traits<SO3> : public internal::MatrixLieGroup<SO3> {};
 
 template <>
-struct traits<const SO3> : public internal::LieGroup<SO3> {};
+struct traits<const SO3> : public internal::MatrixLieGroup<SO3> {};
 
 }  // end namespace gtsam
diff --git a/gtsam/geometry/SO4.h b/gtsam/geometry/SO4.h
index 43b74f7aa..6a37f84bb 100644
--- a/gtsam/geometry/SO4.h
+++ b/gtsam/geometry/SO4.h
@@ -106,13 +106,13 @@ void serialize(Archive &ar, SO4 &Q, const unsigned int /*version*/) {
 #endif
 
 /*
- * Define the traits. internal::LieGroup provides both Lie group and Testable
+ * Define the traits. internal::MatrixLieGroup provides both Lie group and Testable
  */
 
 template <>
-struct traits<SO4> : public internal::LieGroup<SO4> {};
+struct traits<SO4> : public internal::MatrixLieGroup<SO4> {};
 
 template <>
-struct traits<const SO4> : public internal::LieGroup<SO4> {};
+struct traits<const SO4> : public internal::MatrixLieGroup<SO4> {};
 
 }  // end namespace gtsam
diff --git a/gtsam/geometry/SOn.h b/gtsam/geometry/SOn.h
index e8e1f641d..b24070c91 100644
--- a/gtsam/geometry/SOn.h
+++ b/gtsam/geometry/SOn.h
@@ -59,6 +59,9 @@ class SO : public LieGroup<SO<N>, internal::DimensionSO(N)> {
   using VectorN2 = Eigen::Matrix<double, internal::NSquaredSO(N), 1>;
   using MatrixDD = Eigen::Matrix<double, dimension, dimension>;
 
+  /// LieGroup Concept requirements
+  using LieAlgebra = MatrixNN;
+
   GTSAM_MAKE_ALIGNED_OPERATOR_NEW_IF(true)
 
  protected:
@@ -393,14 +396,14 @@ void serialize(
 #endif
 
 /*
- * Define the traits. internal::LieGroup provides both Lie group and Testable
+ * Define the traits. internal::MatrixLieGroup provides both Lie group and Testable
  */
 
 template <int N>
-struct traits<SO<N>> : public internal::LieGroup<SO<N>> {};
+struct traits<SO<N>> : public internal::MatrixLieGroup<SO<N>> {};
 
 template <int N>
-struct traits<const SO<N>> : public internal::LieGroup<SO<N>> {};
+struct traits<const SO<N>> : public internal::MatrixLieGroup<SO<N>> {};
 
 }  // namespace gtsam
 
diff --git a/gtsam/geometry/Similarity2.cpp b/gtsam/geometry/Similarity2.cpp
index c78604953..a48a6b6ef 100644
--- a/gtsam/geometry/Similarity2.cpp
+++ b/gtsam/geometry/Similarity2.cpp
@@ -222,6 +222,24 @@ Matrix4 Similarity2::AdjointMap() const {
   throw std::runtime_error("Similarity2::AdjointMap not implemented");
 }
 
+Matrix3 Similarity2::Hat(const Vector4 &xi) {
+  const auto w = xi[2];
+  const auto u = xi.head<2>();
+  const double lambda = xi[3];
+  Matrix3 W;
+  W << 0, -w, u[0],
+       w,  0, u[1],
+       0,  0, -lambda;
+  return W;
+}
+
+Vector4 Similarity2::Vee(const Matrix3 &Xi) {
+  Vector4 xi;
+  xi[2] = Xi(1, 0);
+  xi.head<2>() = Xi.topRightCorner<2, 1>();
+  xi[3] = -Xi(2, 2);
+  return xi;
+}
 std::ostream& operator<<(std::ostream& os, const Similarity2& p) {
   os << "[" << p.rotation().theta() << " " << p.translation().transpose() << " "
      << p.scale() << "]\';";
diff --git a/gtsam/geometry/Similarity2.h b/gtsam/geometry/Similarity2.h
index 6bec716ea..cc4ffc792 100644
--- a/gtsam/geometry/Similarity2.h
+++ b/gtsam/geometry/Similarity2.h
@@ -139,6 +139,8 @@ class GTSAM_EXPORT Similarity2 : public LieGroup<Similarity2, 4> {
   /// @name Lie Group
   /// @{
 
+  using LieAlgebra = Matrix3;
+
   /**
    * Log map at the identity
    * \f$ [t_x, t_y, \delta, \lambda] \f$
@@ -167,6 +169,12 @@ class GTSAM_EXPORT Similarity2 : public LieGroup<Similarity2, 4> {
 
   using LieGroup<Similarity2, 4>::inverse;
 
+  /// Hat maps from tangent vector to Lie algebra
+  static Matrix3 Hat(const Vector4& xi);
+
+  /// Vee maps from Lie algebra to tangent vector
+  static Vector4 Vee(const Matrix3& X);
+
   /// @}
   /// @name Standard interface
   /// @{
@@ -193,9 +201,9 @@ class GTSAM_EXPORT Similarity2 : public LieGroup<Similarity2, 4> {
 };
 
 template <>
-struct traits<Similarity2> : public internal::LieGroup<Similarity2> {};
+struct traits<Similarity2> : public internal::MatrixLieGroup<Similarity2> {};
 
 template <>
-struct traits<const Similarity2> : public internal::LieGroup<Similarity2> {};
+struct traits<const Similarity2> : public internal::MatrixLieGroup<Similarity2> {};
 
 }  // namespace gtsam
diff --git a/gtsam/geometry/Similarity3.cpp b/gtsam/geometry/Similarity3.cpp
index e569755ef..f0f97ef33 100644
--- a/gtsam/geometry/Similarity3.cpp
+++ b/gtsam/geometry/Similarity3.cpp
@@ -192,7 +192,7 @@ Similarity3 Similarity3::Align(const Pose3Pairs &abPosePairs) {
   return internal::alignGivenR(abPointPairs, aRb_estimate);
 }
 
-Matrix4 Similarity3::wedge(const Vector7 &xi) {
+Matrix4 Similarity3::Hat(const Vector7 &xi) {
   // http://www.ethaneade.org/latex2html/lie/node29.html
   const auto w = xi.head<3>();
   const auto u = xi.segment<3>(3);
@@ -202,6 +202,14 @@ Matrix4 Similarity3::wedge(const Vector7 &xi) {
   return W;
 }
 
+Vector7 Similarity3::Vee(const Matrix4 &Xi) {
+  Vector7 xi;
+  xi.head<3>() = Rot3::Vee(Xi.topLeftCorner<3, 3>());
+  xi.segment<3>(3) = Xi.topRightCorner<3, 1>();
+  xi[6] = -Xi(3, 3);
+  return xi;
+}
+
 Matrix7 Similarity3::AdjointMap() const {
   // http://www.ethaneade.org/latex2html/lie/node30.html
   const Matrix3 R = R_.matrix();
diff --git a/gtsam/geometry/Similarity3.h b/gtsam/geometry/Similarity3.h
index 8f7f30be8..6eb1d5584 100644
--- a/gtsam/geometry/Similarity3.h
+++ b/gtsam/geometry/Similarity3.h
@@ -139,6 +139,8 @@ class GTSAM_EXPORT Similarity3 : public LieGroup<Similarity3, 7> {
   /// @name Lie Group
   /// @{
 
+  using LieAlgebra = Matrix4;
+
   /** Log map at the identity
    * \f$ [R_x,R_y,R_z, t_x, t_y, t_z, \lambda] \f$
    */
@@ -164,17 +166,19 @@ class GTSAM_EXPORT Similarity3 : public LieGroup<Similarity3, 7> {
 
   using LieGroup<Similarity3, 7>::inverse;
 
-  /**
-   * wedge for Similarity3:
-   * @param xi 7-dim twist (w,u,lambda) where
-   * @return 4*4 element of Lie algebra that can be exponentiated
-   * TODO(frank): rename to Hat, make part of traits
-   */
-  static Matrix4 wedge(const Vector7& xi);
-
   /// Project from one tangent space to another
   Matrix7 AdjointMap() const;
 
+  /**
+   * Hat for Similarity3:
+   * @param xi 7-dim twist (w,u,lambda) where
+   * @return 4*4 element of Lie algebra that can be exponentiated
+   */
+  static Matrix4 Hat(const Vector7& xi);
+
+  /// Vee maps from Lie algebra to tangent vector
+  static Vector7 Vee(const Matrix4& X);
+
   /// @}
   /// @name Standard interface
   /// @{
@@ -197,6 +201,17 @@ class GTSAM_EXPORT Similarity3 : public LieGroup<Similarity3, 7> {
   /// Dimensionality of tangent space = 7 DOF
   inline size_t dim() const { return 7; }
 
+  /// @}
+  /// @name Deprecated
+  /// @{
+
+#ifdef GTSAM_ALLOW_DEPRECATED_SINCE_V43
+  /// @deprecated: use Similarity3::Hat
+    static Matrix4 wedge(const Vector7& xi) {
+      return Similarity3::Hat(xi);
+    }
+#endif
+  
   /// @}
   /// @name Helper functions
   /// @{
@@ -220,15 +235,17 @@ class GTSAM_EXPORT Similarity3 : public LieGroup<Similarity3, 7> {
   /// @}
 };
 
+#ifdef GTSAM_ALLOW_DEPRECATED_SINCE_V43
+/// @deprecated: use Similarity3::Hat
 template <>
 inline Matrix wedge<Similarity3>(const Vector& xi) {
-  return Similarity3::wedge(xi);
+  return Similarity3::Hat(xi);
 }
+#endif
+template <>
+struct traits<Similarity3> : public internal::MatrixLieGroup<Similarity3> {};
 
 template <>
-struct traits<Similarity3> : public internal::LieGroup<Similarity3> {};
-
-template <>
-struct traits<const Similarity3> : public internal::LieGroup<Similarity3> {};
+struct traits<const Similarity3> : public internal::MatrixLieGroup<Similarity3> {};
 
 }  // namespace gtsam
diff --git a/gtsam/geometry/geometry.i b/gtsam/geometry/geometry.i
index a8af78f2f..2430105e9 100644
--- a/gtsam/geometry/geometry.i
+++ b/gtsam/geometry/geometry.i
@@ -174,7 +174,10 @@ class Rot2 {
   // Lie Group
   static gtsam::Rot2 Expmap(gtsam::Vector v);
   static gtsam::Vector Logmap(const gtsam::Rot2& p);
+  gtsam::Rot2 expmap(gtsam::Vector v);
   gtsam::Vector logmap(const gtsam::Rot2& p);
+  static gtsam::Matrix Hat(const gtsam::Vector& xi);
+  static gtsam::Vector Vee(const gtsam::Matrix& xi);
 
   // Group Action on Point2
   gtsam::Point2 rotate(const gtsam::Point2& point) const;
@@ -216,6 +219,14 @@ class SO3 {
   // Operator Overloads
   gtsam::SO3 operator*(const gtsam::SO3& R) const;
 
+  // Lie Group
+  static gtsam::SO3 Expmap(gtsam::Vector v);
+  static gtsam::Vector Logmap(const gtsam::SO3& p);
+  gtsam::SO3 expmap(gtsam::Vector v);
+  gtsam::Vector logmap(const gtsam::SO3& p);
+  static gtsam::Matrix Hat(const gtsam::Vector& xi);
+  static gtsam::Vector Vee(const gtsam::Matrix& xi);
+
   // Manifold
   gtsam::SO3 retract(gtsam::Vector v) const;
   gtsam::Vector localCoordinates(const gtsam::SO3& R) const;
@@ -246,6 +257,14 @@ class SO4 {
   // Operator Overloads
   gtsam::SO4 operator*(const gtsam::SO4& Q) const;
 
+  // Lie Group
+  static gtsam::SO4 Expmap(gtsam::Vector v);
+  static gtsam::Vector Logmap(const gtsam::SO4& p);
+  gtsam::SO4 expmap(gtsam::Vector v);
+  gtsam::Vector logmap(const gtsam::SO4& p);
+  static gtsam::Matrix Hat(const gtsam::Vector& xi);
+  static gtsam::Vector Vee(const gtsam::Matrix& xi);
+
   // Manifold
   gtsam::SO4 retract(gtsam::Vector v) const;
   gtsam::Vector localCoordinates(const gtsam::SO4& Q) const;
@@ -276,6 +295,14 @@ class SOn {
   // Operator Overloads
   gtsam::SOn operator*(const gtsam::SOn& Q) const;
 
+  // Lie Group
+  static gtsam::SOn Expmap(gtsam::Vector v);
+  static gtsam::Vector Logmap(const gtsam::SOn& p);
+  gtsam::SOn expmap(gtsam::Vector v);
+  gtsam::Vector logmap(const gtsam::SOn& p);
+  static gtsam::Matrix Hat(const gtsam::Vector& xi);
+  static gtsam::Vector Vee(const gtsam::Matrix& xi);
+
   // Manifold
   gtsam::SOn retract(gtsam::Vector v) const;
   gtsam::Vector localCoordinates(const gtsam::SOn& Q) const;
@@ -346,6 +373,14 @@ class Rot3 {
   gtsam::Rot3 retract(gtsam::Vector v) const;
   gtsam::Vector localCoordinates(const gtsam::Rot3& p) const;
 
+  // Lie group
+  static gtsam::Rot3 Expmap(gtsam::Vector v);
+  static gtsam::Vector Logmap(const gtsam::Rot3& p);
+  gtsam::Rot3 expmap(const gtsam::Vector& v);
+  gtsam::Vector logmap(const gtsam::Rot3& p);
+  static gtsam::Matrix Hat(const gtsam::Vector& xi);
+  static gtsam::Vector Vee(const gtsam::Matrix& xi);
+
   // Group Action on Point3
   gtsam::Point3 rotate(const gtsam::Point3& p) const;
   gtsam::Point3 unrotate(const gtsam::Point3& p) const;
@@ -358,10 +393,6 @@ class Rot3 {
   gtsam::Unit3 unrotate(const gtsam::Unit3& p) const;
   
   // Standard Interface
-  static gtsam::Rot3 Expmap(gtsam::Vector v);
-  static gtsam::Vector Logmap(const gtsam::Rot3& p);
-  gtsam::Rot3 expmap(const gtsam::Vector& v);
-  gtsam::Vector logmap(const gtsam::Rot3& p);
   gtsam::Matrix matrix() const;
   gtsam::Matrix transpose() const;
   gtsam::Point3 column(size_t index) const;
@@ -417,8 +448,12 @@ class Pose2 {
   // Lie Group
   static gtsam::Pose2 Expmap(gtsam::Vector v);
   static gtsam::Vector Logmap(const gtsam::Pose2& p);
+  static gtsam::Pose2 Expmap(gtsam::Vector v, Eigen::Ref<Eigen::MatrixXd> H);
+  static gtsam::Vector Logmap(const gtsam::Pose2& p, Eigen::Ref<Eigen::MatrixXd> H);
+  gtsam::Pose2 expmap(gtsam::Vector v);
+  gtsam::Pose2 expmap(gtsam::Vector v, Eigen::Ref<Eigen::MatrixXd> H1, Eigen::Ref<Eigen::MatrixXd> H2);
   gtsam::Vector logmap(const gtsam::Pose2& p);
-  gtsam::Vector logmap(const gtsam::Pose2& p, Eigen::Ref<Eigen::MatrixXd> H);
+  gtsam::Vector logmap(const gtsam::Pose2& p, Eigen::Ref<Eigen::MatrixXd> H1, Eigen::Ref<Eigen::MatrixXd> H2);
   static gtsam::Matrix ExpmapDerivative(gtsam::Vector v);
   static gtsam::Matrix LogmapDerivative(const gtsam::Pose2& v);
   gtsam::Matrix AdjointMap() const;
@@ -426,7 +461,8 @@ class Pose2 {
   static gtsam::Matrix adjointMap_(gtsam::Vector v);
   static gtsam::Vector adjoint_(gtsam::Vector xi, gtsam::Vector y);
   static gtsam::Vector adjointTranspose(gtsam::Vector xi, gtsam::Vector y);
-  static gtsam::Matrix wedge(double vx, double vy, double w);
+  static gtsam::Matrix Hat(const gtsam::Vector& xi);
+  static gtsam::Vector Vee(const gtsam::Matrix& xi);
 
   // Group Actions on Point2
   gtsam::Point2 transformFrom(const gtsam::Point2& p) const;
@@ -496,11 +532,13 @@ class Pose3 {
 
   // Lie Group
   static gtsam::Pose3 Expmap(gtsam::Vector v);
-  static gtsam::Pose3 Expmap(gtsam::Vector v, Eigen::Ref<Eigen::MatrixXd> Hxi);
-  static gtsam::Vector Logmap(const gtsam::Pose3& pose);
-  static gtsam::Vector Logmap(const gtsam::Pose3& pose, Eigen::Ref<Eigen::MatrixXd> Hpose);
+  static gtsam::Vector Logmap(const gtsam::Pose3& p);
+  static gtsam::Pose3 Expmap(gtsam::Vector v, Eigen::Ref<Eigen::MatrixXd> H);
+  static gtsam::Vector Logmap(const gtsam::Pose3& p, Eigen::Ref<Eigen::MatrixXd> H);
   gtsam::Pose3 expmap(gtsam::Vector v);
-  gtsam::Vector logmap(const gtsam::Pose3& pose);
+  gtsam::Pose3 expmap(gtsam::Vector v, Eigen::Ref<Eigen::MatrixXd> H1, Eigen::Ref<Eigen::MatrixXd> H2);
+  gtsam::Vector logmap(const gtsam::Pose3& p);
+  gtsam::Vector logmap(const gtsam::Pose3& p, Eigen::Ref<Eigen::MatrixXd> H1, Eigen::Ref<Eigen::MatrixXd> H2);
   gtsam::Matrix AdjointMap() const;
   gtsam::Vector Adjoint(gtsam::Vector xi) const;
   gtsam::Vector Adjoint(gtsam::Vector xi, Eigen::Ref<Eigen::MatrixXd> H_this,
@@ -515,8 +553,8 @@ class Pose3 {
   static gtsam::Vector adjointTranspose(gtsam::Vector xi, gtsam::Vector y);
   static gtsam::Matrix ExpmapDerivative(gtsam::Vector xi);
   static gtsam::Matrix LogmapDerivative(const gtsam::Pose3& xi);
-  static gtsam::Matrix wedge(double wx, double wy, double wz, double vx, double vy,
-                      double vz);
+  static gtsam::Matrix Hat(const gtsam::Vector& xi);
+  static gtsam::Vector Vee(const gtsam::Matrix& xi);
 
   // Group Action on Point3
   gtsam::Point3 transformFrom(const gtsam::Point3& point) const;
@@ -1147,6 +1185,14 @@ class Similarity2 {
   static gtsam::Similarity2 Align(const gtsam::Point2Pairs& abPointPairs);
   static gtsam::Similarity2 Align(const gtsam::Pose2Pairs& abPosePairs);
 
+  // Lie group
+  static gtsam::Similarity2 Expmap(gtsam::Vector v);
+  static gtsam::Vector Logmap(const gtsam::Similarity2& p);
+  gtsam::Similarity2 expmap(const gtsam::Vector& v);
+  gtsam::Vector logmap(const gtsam::Similarity2& p);
+  static gtsam::Matrix Hat(const gtsam::Vector& xi);
+  static gtsam::Vector Vee(const gtsam::Matrix& xi);
+
   // Standard Interface
   bool equals(const gtsam::Similarity2& sim, double tol) const;
   void print(const std::string& s = "") const;
@@ -1171,6 +1217,14 @@ class Similarity3 {
   static gtsam::Similarity3 Align(const gtsam::Point3Pairs& abPointPairs);
   static gtsam::Similarity3 Align(const gtsam::Pose3Pairs& abPosePairs);
 
+  // Lie group
+  static gtsam::Similarity3 Expmap(gtsam::Vector v);
+  static gtsam::Vector Logmap(const gtsam::Similarity3& p);
+  gtsam::Similarity3 expmap(const gtsam::Vector& v);
+  gtsam::Vector logmap(const gtsam::Similarity3& p);
+  static gtsam::Matrix Hat(const gtsam::Vector& xi);
+  static gtsam::Vector Vee(const gtsam::Matrix& xi);
+
   // Standard Interface
   bool equals(const gtsam::Similarity3& sim, double tol) const;
   void print(const std::string& s = "") const;
diff --git a/gtsam/geometry/tests/testPose2.cpp b/gtsam/geometry/tests/testPose2.cpp
index 9e4efe799..9bbc8f5ca 100644
--- a/gtsam/geometry/tests/testPose2.cpp
+++ b/gtsam/geometry/tests/testPose2.cpp
@@ -37,7 +37,7 @@ GTSAM_CONCEPT_LIE_INST(Pose2)
 TEST(Pose2 , Concept) {
   GTSAM_CONCEPT_ASSERT(IsGroup<Pose2 >);
   GTSAM_CONCEPT_ASSERT(IsManifold<Pose2 >);
-  GTSAM_CONCEPT_ASSERT(IsLieGroup<Pose2 >);
+  GTSAM_CONCEPT_ASSERT(IsMatrixLieGroup<Pose2 >);
 }
 
 /* ************************************************************************* */
@@ -141,6 +141,27 @@ TEST(Pose2, expmap0d) {
   EXPECT(assert_equal(expected, actual, 1e-5));
 }
 
+/* ************************************************************************* */
+TEST(Pose2, HatAndVee) {
+  // Create a few test vectors
+  Vector3 v1(1, 2, 3);
+  Vector3 v2(0.1, -0.5, 1.0);
+  Vector3 v3(0.0, 0.0, 0.0);
+
+  // Test that Vee(Hat(v)) == v for various inputs
+  EXPECT(assert_equal(v1, Pose2::Vee(Pose2::Hat(v1))));
+  EXPECT(assert_equal(v2, Pose2::Vee(Pose2::Hat(v2))));
+  EXPECT(assert_equal(v3, Pose2::Vee(Pose2::Hat(v3))));
+
+  // Check the structure of the Lie Algebra element
+  Matrix3 expected;
+  expected << 0, -3, 1,
+    3, 0, 2,
+    0, 0, 0;
+
+  EXPECT(assert_equal(expected, Pose2::Hat(v1)));
+}
+
 /* ************************************************************************* */
 // test case for screw motion in the plane
 namespace screwPose2 {
@@ -186,17 +207,16 @@ TEST(Pose2, Adjoint_full) {
 }
 
 /* ************************************************************************* */
-// assert that T*wedge(xi)*T^-1 is equal to wedge(Ad_T(xi))
+// assert that T*Hat(xi)*T^-1 is equal to Hat(Ad_T(xi))
 TEST(Pose2, Adjoint_hat) {
   Pose2 T(1, 2, 3);
-  auto hat = [](const Vector& xi) { return ::wedge<Pose2>(xi); };
-  Matrix3 expected = T.matrix() * hat(screwPose2::xi) * T.matrix().inverse();
-  Matrix3 xiprime = hat(T.Adjoint(screwPose2::xi));
+  Matrix3 expected = T.matrix() * Pose2::Hat(screwPose2::xi) * T.matrix().inverse();
+  Matrix3 xiprime = Pose2::Hat(T.Adjoint(screwPose2::xi));
   EXPECT(assert_equal(expected, xiprime, 1e-6));
 
   Vector3 xi2(4, 5, 6);
-  Matrix3 expected2 = T.matrix() * hat(xi2) * T.matrix().inverse();
-  Matrix3 xiprime2 = hat(T.Adjoint(xi2));
+  Matrix3 expected2 = T.matrix() * Pose2::Hat(xi2) * T.matrix().inverse();
+  Matrix3 xiprime2 = Pose2::Hat(T.Adjoint(xi2));
   EXPECT(assert_equal(expected2, xiprime2, 1e-6));
 }
 
diff --git a/gtsam/geometry/tests/testPose3.cpp b/gtsam/geometry/tests/testPose3.cpp
index f3ad34b98..d14af7ac0 100644
--- a/gtsam/geometry/tests/testPose3.cpp
+++ b/gtsam/geometry/tests/testPose3.cpp
@@ -41,6 +41,13 @@ static const Pose3 T2(Rot3::Rodrigues(0.3,0.2,0.1),P2);
 static const Pose3 T3(Rot3::Rodrigues(-90, 0, 0), Point3(1, 2, 3));
 static const double tol=1e-5;
 
+//******************************************************************************
+TEST(Pose3 , Concept) {
+  GTSAM_CONCEPT_ASSERT(IsGroup<Pose3 >);
+  GTSAM_CONCEPT_ASSERT(IsManifold<Pose3 >);
+  GTSAM_CONCEPT_ASSERT(IsMatrixLieGroup<Pose3 >);
+}
+
 /* ************************************************************************* */
 TEST( Pose3, equals)
 {
@@ -221,20 +228,41 @@ TEST(Pose3, AdjointTranspose)
 }
 
 /* ************************************************************************* */
-// assert that T*wedge(xi)*T^-1 is equal to wedge(Ad_T(xi))
+TEST(Pose3, HatAndVee) {
+  // Create a few test vectors
+  Vector6 v1(1, 2, 3, 4, 5, 6);
+  Vector6 v2(0.1, -0.5, 1.0, -1.0, 0.5, 2.0);
+  Vector6 v3(0.0, 0.0, 0.0, 0.0, 0.0, 0.0);
+
+  // Test that Vee(Hat(v)) == v for various inputs
+  EXPECT(assert_equal(v1, Pose3::Vee(Pose3::Hat(v1))));
+  EXPECT(assert_equal(v2, Pose3::Vee(Pose3::Hat(v2))));
+  EXPECT(assert_equal(v3, Pose3::Vee(Pose3::Hat(v3))));
+
+  // Check the structure of the Lie Algebra element
+  Matrix4 expected;
+  expected << 0, -3, 2, 4,
+    3, 0, -1, 5,
+    -2, 1, 0, 6,
+    0, 0, 0, 0;
+
+  EXPECT(assert_equal(expected, Pose3::Hat(v1)));
+}
+
+/* ************************************************************************* */
+// assert that T*Hat(xi)*T^-1 is equal to Hat(Ad_T(xi))
 TEST(Pose3, Adjoint_hat)
 {
-  auto hat = [](const Vector& xi) { return ::wedge<Pose3>(xi); };
-  Matrix4 expected = T.matrix() * hat(screwPose3::xi) * T.matrix().inverse();
-  Matrix4 xiprime = hat(T.Adjoint(screwPose3::xi));
+  Matrix4 expected = T.matrix() * Pose3::Hat(screwPose3::xi) * T.matrix().inverse();
+  Matrix4 xiprime = Pose3::Hat(T.Adjoint(screwPose3::xi));
   EXPECT(assert_equal(expected, xiprime, 1e-6));
 
-  Matrix4 expected2 = T2.matrix() * hat(screwPose3::xi) * T2.matrix().inverse();
-  Matrix4 xiprime2 = hat(T2.Adjoint(screwPose3::xi));
+  Matrix4 expected2 = T2.matrix() * Pose3::Hat(screwPose3::xi) * T2.matrix().inverse();
+  Matrix4 xiprime2 = Pose3::Hat(T2.Adjoint(screwPose3::xi));
   EXPECT(assert_equal(expected2, xiprime2, 1e-6));
 
-  Matrix4 expected3 = T3.matrix() * hat(screwPose3::xi) * T3.matrix().inverse();
-  Matrix4 xiprime3 = hat(T3.Adjoint(screwPose3::xi));
+  Matrix4 expected3 = T3.matrix() * Pose3::Hat(screwPose3::xi) * T3.matrix().inverse();
+  Matrix4 xiprime3 = Pose3::Hat(T3.Adjoint(screwPose3::xi));
   EXPECT(assert_equal(expected3, xiprime3, 1e-6));
 }
 
diff --git a/gtsam/geometry/tests/testQuaternion.cpp b/gtsam/geometry/tests/testQuaternion.cpp
index c99c67e0f..40781b832 100644
--- a/gtsam/geometry/tests/testQuaternion.cpp
+++ b/gtsam/geometry/tests/testQuaternion.cpp
@@ -31,7 +31,7 @@ typedef traits<Q>::ChartJacobian QuaternionJacobian;
 TEST(Quaternion , Concept) {
   GTSAM_CONCEPT_ASSERT(IsGroup<Quaternion >);
   GTSAM_CONCEPT_ASSERT(IsManifold<Quaternion >);
-  GTSAM_CONCEPT_ASSERT(IsLieGroup<Quaternion >);
+  GTSAM_CONCEPT_ASSERT(IsMatrixLieGroup<Quaternion >);
 }
 
 //******************************************************************************
diff --git a/gtsam/geometry/tests/testRot2.cpp b/gtsam/geometry/tests/testRot2.cpp
index 5a087edcd..790ff7a3e 100644
--- a/gtsam/geometry/tests/testRot2.cpp
+++ b/gtsam/geometry/tests/testRot2.cpp
@@ -103,6 +103,25 @@ TEST(Rot2, logmap)
   CHECK(assert_equal(expected, actual));
 }
 
+/* ************************************************************************* */
+TEST(Rot2, HatAndVee) {
+  // Create a few test vectors
+  Vector1 v1 = (Vector1() << 1).finished();
+  Vector1 v2 = (Vector1() << 0.1).finished();
+  Vector1 v3 = (Vector1() << 0.0).finished();
+
+  // Test that Vee(Hat(v)) == v for various inputs
+  EXPECT(assert_equal(v1, Rot2::Vee(Rot2::Hat(v1))));
+  EXPECT(assert_equal(v2, Rot2::Vee(Rot2::Hat(v2))));
+  EXPECT(assert_equal(v3, Rot2::Vee(Rot2::Hat(v3))));
+
+  // Check the structure of the Lie Algebra element
+  Matrix2 expected;
+  expected << 0., -1., 1., 0.;
+
+  EXPECT(assert_equal(expected, Rot2::Hat(v1)));
+}
+
 /* ************************************************************************* */
 // rotate and derivatives
 inline Point2 rotate_(const Rot2 & R, const Point2& p) {return R.rotate(p);}
diff --git a/gtsam/geometry/tests/testRot3.cpp b/gtsam/geometry/tests/testRot3.cpp
index ce056edb6..65b97fc09 100644
--- a/gtsam/geometry/tests/testRot3.cpp
+++ b/gtsam/geometry/tests/testRot3.cpp
@@ -40,7 +40,7 @@ static double error = 1e-9, epsilon = 0.001;
 TEST(Rot3 , Concept) {
   GTSAM_CONCEPT_ASSERT(IsGroup<Rot3 >);
   GTSAM_CONCEPT_ASSERT(IsManifold<Rot3 >);
-  GTSAM_CONCEPT_ASSERT(IsLieGroup<Rot3 >);
+  GTSAM_CONCEPT_ASSERT(IsMatrixLieGroup<Rot3 >);
 }
 
 /* ************************************************************************* */
@@ -324,6 +324,34 @@ TEST(Rot3, manifold_expmap)
   CHECK(assert_equal(R5,R3*R2));
 }
 
+/* ************************************************************************* */
+TEST(Rot3, HatAndVee) {
+  // Create a few test vectors
+  Vector3 v1(1, 2, 3);
+  Vector3 v2(0.1, -0.5, 1.0);
+  Vector3 v3(0.0, 0.0, 0.0);
+
+  // Test that Vee(Hat(v)) == v for various inputs
+  EXPECT(assert_equal(v1, Rot3::Vee(Rot3::Hat(v1))));
+  EXPECT(assert_equal(v2, Rot3::Vee(Rot3::Hat(v2))));
+  EXPECT(assert_equal(v3, Rot3::Vee(Rot3::Hat(v3))));
+
+  // Check the structure of the Lie Algebra element
+  Matrix3 expected;
+  expected << 0, -3, 2,
+    3, 0, -1,
+    -2, 1, 0;
+
+  EXPECT(assert_equal(expected, Rot3::Hat(v1)));
+}
+
+/* ************************************************************************* */
+// Checks correct exponential map (Expmap) with brute force matrix exponential
+TEST(Rot3, BruteForceExpmap) {
+  const Vector3 xi(0.1, 0.2, 0.3);
+  EXPECT(assert_equal(Rot3::Expmap(xi), expm<Rot3>(xi), 1e-6));
+}
+
 /* ************************************************************************* */
 class AngularVelocity : public Vector3 {
  public:
diff --git a/gtsam/geometry/tests/testSO3.cpp b/gtsam/geometry/tests/testSO3.cpp
index 17b27daea..3f06624e1 100644
--- a/gtsam/geometry/tests/testSO3.cpp
+++ b/gtsam/geometry/tests/testSO3.cpp
@@ -38,7 +38,7 @@ TEST(SO3, Identity) {
 TEST(SO3, Concept) {
   GTSAM_CONCEPT_ASSERT(IsGroup<SO3>);
   GTSAM_CONCEPT_ASSERT(IsManifold<SO3>);
-  GTSAM_CONCEPT_ASSERT(IsLieGroup<SO3>);
+  GTSAM_CONCEPT_ASSERT(IsMatrixLieGroup<SO3>);
 }
 
 //******************************************************************************
diff --git a/gtsam/geometry/tests/testSO4.cpp b/gtsam/geometry/tests/testSO4.cpp
index 7a3198901..b084ace35 100644
--- a/gtsam/geometry/tests/testSO4.cpp
+++ b/gtsam/geometry/tests/testSO4.cpp
@@ -44,7 +44,7 @@ TEST(SO4, Identity) {
 TEST(SO4, Concept) {
   GTSAM_CONCEPT_ASSERT(IsGroup<SO4>);
   GTSAM_CONCEPT_ASSERT(IsManifold<SO4>);
-  GTSAM_CONCEPT_ASSERT(IsLieGroup<SO4>);
+  GTSAM_CONCEPT_ASSERT(IsMatrixLieGroup<SO4>);
 }
 
 //******************************************************************************
diff --git a/gtsam/geometry/tests/testSOn.cpp b/gtsam/geometry/tests/testSOn.cpp
index 741ddd3d0..b9609116b 100644
--- a/gtsam/geometry/tests/testSOn.cpp
+++ b/gtsam/geometry/tests/testSOn.cpp
@@ -86,7 +86,7 @@ TEST(SOn, SO5) {
 TEST(SOn, Concept) {
   GTSAM_CONCEPT_ASSERT(IsGroup<SOn>);
   GTSAM_CONCEPT_ASSERT(IsManifold<SOn>);
-  GTSAM_CONCEPT_ASSERT(IsLieGroup<SOn>);
+  GTSAM_CONCEPT_ASSERT(IsMatrixLieGroup<SOn>);
 }
 
 //******************************************************************************
diff --git a/gtsam/geometry/tests/testSimilarity2.cpp b/gtsam/geometry/tests/testSimilarity2.cpp
index 9a64ba1c9..78b25b50f 100644
--- a/gtsam/geometry/tests/testSimilarity2.cpp
+++ b/gtsam/geometry/tests/testSimilarity2.cpp
@@ -37,7 +37,7 @@ static const double s = 4;
 TEST(Similarity2, Concepts) {
   GTSAM_CONCEPT_ASSERT(IsGroup<Similarity2>);
   GTSAM_CONCEPT_ASSERT(IsManifold<Similarity2>);
-  GTSAM_CONCEPT_ASSERT(IsLieGroup<Similarity2>);
+  GTSAM_CONCEPT_ASSERT(IsMatrixLieGroup<Similarity2>);
 }
 
 //******************************************************************************
@@ -56,6 +56,34 @@ TEST(Similarity2, Getters) {
   EXPECT_DOUBLES_EQUAL(1.0, sim2_default.scale(), 1e-9);
 }
 
+/* ************************************************************************* */
+TEST(Similarity2, HatAndVee) {
+  // Create a few test vectors
+  Vector4 v1(1, 2, 3, 4);
+  Vector4 v2(0.1, -0.5, 1.0, -1.0);
+  Vector4 v3(0.0, 0.0, 0.0, 0.0);
+
+  // Test that Vee(Hat(v)) == v for various inputs
+  EXPECT(assert_equal(v1, Similarity2::Vee(Similarity2::Hat(v1))));
+  EXPECT(assert_equal(v2, Similarity2::Vee(Similarity2::Hat(v2))));
+  EXPECT(assert_equal(v3, Similarity2::Vee(Similarity2::Hat(v3))));
+
+  // Check the structure of the Lie Algebra element
+  Matrix3 expected;
+  expected << 0, -3, 1,
+              3, 0, 2,
+              0, 0, -4;
+
+  EXPECT(assert_equal(expected, Similarity2::Hat(v1)));
+}
+
+/* ************************************************************************* */
+// Checks correct exponential map (Expmap) with brute force matrix exponential
+TEST_DISABLED(Similarity2, BruteForceExpmap) {
+  const Vector4 xi(0.1, 0.2, 0.3, 0.4);
+  EXPECT(assert_equal(Similarity2::Expmap(xi), expm<Similarity2>(xi), 1e-4));
+}
+
 //******************************************************************************
 int main() {
   TestResult tr;
diff --git a/gtsam/geometry/tests/testSimilarity3.cpp b/gtsam/geometry/tests/testSimilarity3.cpp
index e4f68f22f..ce0814f7c 100644
--- a/gtsam/geometry/tests/testSimilarity3.cpp
+++ b/gtsam/geometry/tests/testSimilarity3.cpp
@@ -56,7 +56,7 @@ const double degree = M_PI / 180;
 TEST(Similarity3, Concepts) {
   GTSAM_CONCEPT_ASSERT(IsGroup<Similarity3 >);
   GTSAM_CONCEPT_ASSERT(IsManifold<Similarity3 >);
-  GTSAM_CONCEPT_ASSERT(IsLieGroup<Similarity3 >);
+  GTSAM_CONCEPT_ASSERT(IsMatrixLieGroup<Similarity3 >);
 }
 
 //******************************************************************************
@@ -80,14 +80,43 @@ TEST(Similarity3, Getters) {
   EXPECT_DOUBLES_EQUAL(7.0, sim3.scale(), 1e-9);
 }
 
+/* ************************************************************************* */
+TEST(Similarity3, HatAndVee) {
+  // Create a few test vectors
+  Vector7 v1(1, 2, 3, 4, 5, 6, 7);
+  Vector7 v2(0.1, -0.5, 1.0, -1.0, 0.5, 2.0, -0.3);
+  Vector7 v3(0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0);
+
+  // Test that Vee(Hat(v)) == v for various inputs
+  EXPECT(assert_equal(v1, Similarity3::Vee(Similarity3::Hat(v1))));
+  EXPECT(assert_equal(v2, Similarity3::Vee(Similarity3::Hat(v2))));
+  EXPECT(assert_equal(v3, Similarity3::Vee(Similarity3::Hat(v3))));
+
+  // Check the structure of the Lie Algebra element
+  Matrix4 expected;
+  expected << 0, -3, 2, 4,
+              3, 0, -1, 5,
+              -2, 1, 0, 6,
+              0, 0, 0, -7;
+
+  EXPECT(assert_equal(expected, Similarity3::Hat(v1)));
+}
+
+/* ************************************************************************* */
+// Checks correct exponential map (Expmap) with brute force matrix exponential
+TEST(Similarity3, BruteForceExpmap) {
+  const Vector7 xi(0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7);
+  EXPECT(assert_equal(Similarity3::Expmap(xi), expm<Similarity3>(xi), 1e-4));
+}
+
 //******************************************************************************
 TEST(Similarity3, AdjointMap) {
   const Matrix4 T = T2.matrix();
   // Check Ad with actual definition
   Vector7 delta;
   delta << 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7;
-  Matrix4 W = Similarity3::wedge(delta);
-  Matrix4 TW = Similarity3::wedge(T2.AdjointMap() * delta);
+  Matrix4 W = Similarity3::Hat(delta);
+  Matrix4 TW = Similarity3::Hat(T2.AdjointMap() * delta);
   EXPECT(assert_equal(TW, Matrix4(T * W * T.inverse()), 1e-9));
 }
 
@@ -209,10 +238,6 @@ TEST(Similarity3, ExpLogMap) {
   Similarity3 expZero = Similarity3::Expmap(zeros);
   Similarity3 ident = Similarity3::Identity();
   EXPECT(assert_equal(expZero, ident));
-
-  // Compare to matrix exponential, using expm in Lie.h
-  EXPECT(
-      assert_equal(expm<Similarity3>(delta), Similarity3::Expmap(delta), 1e-3));
 }
 
 //******************************************************************************
diff --git a/gtsam/navigation/NavState.cpp b/gtsam/navigation/NavState.cpp
index 116efe90a..50907db25 100644
--- a/gtsam/navigation/NavState.cpp
+++ b/gtsam/navigation/NavState.cpp
@@ -270,6 +270,29 @@ Matrix9 NavState::LogmapDerivative(const NavState& state) {
   return J;
 }
 
+//------------------------------------------------------------------------------
+Matrix5 NavState::Hat(const Vector9& xi) {
+  Matrix5 X;
+  const double wx = xi(0), wy = xi(1), wz = xi(2);
+  const double px = xi(3), py = xi(4), pz = xi(5);
+  const double vx = xi(6), vy = xi(7), vz = xi(8);
+  X << 0., -wz, wy, px, vx,
+    wz, 0., -wx, py, vy,
+    -wy, wx, 0., pz, vz,
+    0., 0., 0., 0., 0.,
+    0., 0., 0., 0., 0.;
+  return X;
+}
+
+//------------------------------------------------------------------------------
+Vector9 NavState::Vee(const Matrix5& Xi) {
+  Vector9 xi;
+  xi << Xi(2, 1), Xi(0, 2), Xi(1, 0),
+    Xi(0, 3), Xi(1, 3), Xi(2, 3),
+    Xi(0, 4), Xi(1, 4), Xi(2, 4);
+  return xi;
+}
+
 //------------------------------------------------------------------------------
 NavState NavState::ChartAtOrigin::Retract(const Vector9& xi,
                                           ChartJacobian Hxi) {
diff --git a/gtsam/navigation/NavState.h b/gtsam/navigation/NavState.h
index e246cbe27..6c93a6b94 100644
--- a/gtsam/navigation/NavState.h
+++ b/gtsam/navigation/NavState.h
@@ -56,18 +56,27 @@ public:
   NavState() :
       t_(0, 0, 0), v_(Vector3::Zero()) {
   }
+
   /// Construct from attitude, position, velocity
   NavState(const Rot3& R, const Point3& t, const Velocity3& v) :
       R_(R), t_(t), v_(v) {
   }
+
   /// Construct from pose and velocity
   NavState(const Pose3& pose, const Velocity3& v) :
       R_(pose.rotation()), t_(pose.translation()), v_(v) {
   }
+
   /// Construct from SO(3) and R^6
   NavState(const Matrix3& R, const Vector6& tv) :
       R_(R), t_(tv.head<3>()), v_(tv.tail<3>()) {
   }
+
+  /// Construct from Matrix5
+  NavState(const Matrix5& T) :
+    R_(T.block<3, 3>(0, 0)), t_(T.block<3, 1>(0, 3)), v_(T.block<3, 1>(0, 4)) {
+  }
+
   /// Named constructor with derivatives
   static NavState Create(const Rot3& R, const Point3& t, const Velocity3& v,
                          OptionalJacobian<9, 3> H1 = {},
@@ -154,10 +163,6 @@ public:
   /// Syntactic sugar
   const Rot3& rotation() const { return attitude(); };
 
-  /// @}
-  /// @name Lie Group
-  /// @{
-
   // Tangent space sugar.
   // TODO(frank): move to private navstate namespace in cpp
   static Eigen::Block<Vector9, 3, 1> dR(Vector9& v) {
@@ -189,6 +194,12 @@ public:
       OptionalJacobian<9, 9> H1 = {}, OptionalJacobian<9, 9> H2 =
           {}) const;
 
+  /// @}
+  /// @name Lie Group
+  /// @{
+
+  using LieAlgebra = Matrix5;
+
   /**
    * Exponential map at identity - create a NavState from canonical coordinates
    * \f$ [R_x,R_y,R_z,T_x,T_y,T_z,V_x,V_y,V_z] \f$
@@ -242,6 +253,12 @@ public:
     static Vector9 Local(const NavState& state, ChartJacobian Hstate = {});
   };
 
+  /// Hat maps from tangent vector to Lie algebra
+  static Matrix5 Hat(const Vector9& xi);
+
+  /// Vee maps from Lie algebra to tangent vector
+  static Vector9 Vee(const Matrix5& X);
+
   /// @}
   /// @name Dynamics
   /// @{
@@ -283,9 +300,9 @@ private:
 
 // Specialize NavState traits to use a Retract/Local that agrees with IMUFactors
 template <>
-struct traits<NavState> : public internal::LieGroup<NavState> {};
+struct traits<NavState> : public internal::MatrixLieGroup<NavState> {};
 
 template <>
-struct traits<const NavState> : public internal::LieGroup<NavState> {};
+struct traits<const NavState> : public internal::MatrixLieGroup<NavState> {};
 
 } // namespace gtsam
diff --git a/gtsam/navigation/navigation.i b/gtsam/navigation/navigation.i
index ceeab3b35..99567509a 100644
--- a/gtsam/navigation/navigation.i
+++ b/gtsam/navigation/navigation.i
@@ -54,9 +54,24 @@ class NavState {
   gtsam::Vector velocity() const;
   gtsam::Pose3 pose() const;
 
+  // Manifold
   gtsam::NavState retract(const gtsam::Vector& x) const;
   gtsam::Vector localCoordinates(const gtsam::NavState& g) const;
 
+  // Lie Group
+  static gtsam::NavState Expmap(gtsam::Vector v);
+  static gtsam::Vector Logmap(const gtsam::NavState& p);
+  static gtsam::NavState Expmap(gtsam::Vector v, Eigen::Ref<Eigen::MatrixXd> H);
+  static gtsam::Vector Logmap(const gtsam::NavState& p, Eigen::Ref<Eigen::MatrixXd> H);
+  gtsam::NavState expmap(gtsam::Vector v);
+  gtsam::NavState expmap(gtsam::Vector v, Eigen::Ref<Eigen::MatrixXd> H1, Eigen::Ref<Eigen::MatrixXd> H2);
+  gtsam::Vector logmap(const gtsam::NavState& p);
+  gtsam::Vector logmap(const gtsam::NavState& p, Eigen::Ref<Eigen::MatrixXd> H1, Eigen::Ref<Eigen::MatrixXd> H2);
+  gtsam::Matrix AdjointMap() const;
+  gtsam::Vector Adjoint(gtsam::Vector xi) const;
+  static gtsam::Matrix Hat(const gtsam::Vector& xi);
+  static gtsam::Vector Vee(const gtsam::Matrix& xi);
+
   // enabling serialization functionality
   void serialize() const;
 };
diff --git a/gtsam/navigation/tests/testNavState.cpp b/gtsam/navigation/tests/testNavState.cpp
index 427531415..e36b654b0 100644
--- a/gtsam/navigation/tests/testNavState.cpp
+++ b/gtsam/navigation/tests/testNavState.cpp
@@ -52,6 +52,13 @@ static const NavState T2(Rot3::Rodrigues(0.3, 0.2, 0.1), P2, V2);
 static const NavState T3(Rot3::Rodrigues(-90, 0, 0), Point3(5, 6, 7),
                          Point3(1, 2, 3));
 
+//******************************************************************************
+TEST(NavState, Concept) {
+  GTSAM_CONCEPT_ASSERT(IsGroup<NavState >);
+  GTSAM_CONCEPT_ASSERT(IsManifold<NavState >);
+  GTSAM_CONCEPT_ASSERT(IsMatrixLieGroup<NavState >);
+}
+
 /* ************************************************************************* */
 TEST(NavState, Constructor) {
   std::function<NavState(const Rot3&, const Point3&, const Vector3&)> create =
@@ -533,6 +540,63 @@ TEST(NavState, Adjoint_compose_full) {
   EXPECT(assert_equal(expected, actual, 1e-6));
 }
 
+/* ************************************************************************* */
+TEST(NavState, HatAndVee) {
+  // Create a few test vectors
+  Vector9 v1(1, 2, 3, 4, 5, 6, 7, 8, 9);
+  Vector9 v2(0.1, -0.5, 1.0, -1.0, 0.5, 2.0, 0.3, -0.2, 0.8);
+  Vector9 v3 = Vector9::Zero();
+
+  // Test that Vee(Hat(v)) == v for various inputs
+  EXPECT(assert_equal(v1, NavState::Vee(NavState::Hat(v1))));
+  EXPECT(assert_equal(v2, NavState::Vee(NavState::Hat(v2))));
+  EXPECT(assert_equal(v3, NavState::Vee(NavState::Hat(v3))));
+
+  // Check the structure of the Lie Algebra element
+  Matrix5 expected;
+  expected << 0, -3, 2, 4, 7,
+    3, 0, -1, 5, 8,
+    -2, 1, 0, 6, 9,
+    0, 0, 0, 0, 0,
+    0, 0, 0, 0, 0;
+
+  EXPECT(assert_equal(expected, NavState::Hat(v1)));
+}
+
+/* ************************************************************************* */
+// Checks correct exponential map (Expmap) with brute force matrix exponential
+TEST(NavState, BruteForceExpmap1) {
+  const Vector9 xi(0, 0, 0, 14, 24, 34, 15, 25, 35);
+  EXPECT(assert_equal(NavState::Expmap(xi), expm<NavState>(xi), 1e-6));
+}
+
+TEST(NavState, BruteForceExpmap2) {
+  const Vector9 xi(0.1, 0.2, 0.3, 0, 0, 0, 0, 0, 0);
+  EXPECT(assert_equal(NavState::Expmap(xi), expm<NavState>(xi), 1e-6));
+}
+
+TEST(NavState, BruteForceExpmap3) {
+  const Vector9 xi(0.1, 0.2, 0.3, 4, 5, 6, 7, 8, 9);
+  EXPECT(assert_equal(NavState::Expmap(xi), expm<NavState>(xi), 1e-6));
+}
+
+/* ************************************************************************* */
+// assert that T*Hat(xi)*T^-1 is equal to Hat(Ad_T(xi))
+TEST(NavState, Adjoint_hat)
+{
+  Matrix5 expected = T.matrix() * NavState::Hat(screwNavState::xi) * T.matrix().inverse();
+  Matrix5 xiprime = NavState::Hat(T.Adjoint(screwNavState::xi));
+  EXPECT(assert_equal(expected, xiprime, 1e-6));
+
+  Matrix5 expected2 = T2.matrix() * NavState::Hat(screwNavState::xi) * T2.matrix().inverse();
+  Matrix5 xiprime2 = NavState::Hat(T2.Adjoint(screwNavState::xi));
+  EXPECT(assert_equal(expected2, xiprime2, 1e-6));
+
+  Matrix5 expected3 = T3.matrix() * NavState::Hat(screwNavState::xi) * T3.matrix().inverse();
+  Matrix5 xiprime3 = NavState::Hat(T3.Adjoint(screwNavState::xi));
+  EXPECT(assert_equal(expected3, xiprime3, 1e-6));
+}
+
 /* ************************************************************************* */
 TEST(NavState, Retract_LocalCoordinates) {
   Vector9 d;
diff --git a/python/gtsam/tests/test_DiscreteBayesTree.py b/python/gtsam/tests/test_DiscreteBayesTree.py
index e8943fc80..b75067d3a 100644
--- a/python/gtsam/tests/test_DiscreteBayesTree.py
+++ b/python/gtsam/tests/test_DiscreteBayesTree.py
@@ -98,6 +98,7 @@ class TestDiscreteBayesNet(GtsamTestCase):
         self.assertFalse(bayesTree.empty())
         self.assertEqual(12, bayesTree.size())
 
+    @unittest.skip("Too Slow")
     def test_discrete_bayes_tree_lookup(self):
         """Check that we can have a multi-frontal lookup table."""
         # Make a small planning-like graph: 3 states, 2 actions